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STABILITY OF LARGE-AMPLITUDE SHOCK WAVESOF COMPRESSIBLE NAVIER–STOKES EQUATIONS

Kevin Zumbrun

Mathematics Department, Indiana UniversityBloomington, IN 47405-4301, USA

email: [email protected]

September 5, 2003; Revised: February 11, 2004.

Abstract. We summarize recent progress on one- and multi-dimensional stabil-ity of viscous shock wave solutions of compressible Navier–Stokes equations and re-lated symmetrizable hyperbolic–parabolic systems, with an emphasis on the large-amplitude regime where transition from stability to instability may be expected tooccur. The main result is the establishment of rigorous necessary and su�cient con-ditions for linearized and nonlinear planar viscous stability, agreeing in one dimensionand separated in multi-dimensions by a codimension one set, that both extend andsharpen the formal conditions of structural and dynamical stability found in classicalphysical literature. The su�cient condition in multi-dimensions is new, and repre-sents the main mathematical contribution of this article. The su�cient condition forstability is always satisfied for su�ciently small-amplitude shocks, while the necessarycondition is known to fail under certain circumstances for su�ciently large-amplitudeshocks; both are readily evaluable numerically. The precise conditions under and thenature in which transition from stability to instability occurs are outstanding openquestions in the theory.

Preface.

Five years ago, we and coathors introduced in the pair of papers [GZ] and [ZH]a new, “dynamical systems” approach to stability of viscous shock waves based onEvans function and inverse Laplace transform techniques, suitable for the treatmentof large-amplitude and or strongly nonlinear waves such as arise in the regimewhere physical transition from stability to instability may be expected to occur.Previous results for systems (see, e.g., [MN.1, KMN, Go.1–2, L.1–3, SzX, LZ.1–2]and references therein) had concerned only su�cient conditions for stability, andhad been confined almost exclusively to the small-amplitude case.1 The philosophyof [GZ, ZH] was, rather, to determine useful necessary and su�cient conditions forstability in terms of the Evans function, which could then be investigated either

Prepared for Handbook of Fluid Mechanics, Volume IV. This work was supported in part bythe National Science Foundation under Grants No. DMS-0070765, and DMS-0300487.

1The single exception is the partial, zero-mass stability analysis of [MN.1].

1

2 KEVIN ZUMBRUN

numerically [Br.1–2, BrZ, KL] or analytically [MN.1, KMN, Go.1–2, Z.5, HuZ,Hu.1–3, Li, FreS.2, PZ] in cases of interest.2

This approach has proven to be extremely fruitful. In particular, the method hasbeen generalized in [ZS, Z.3, HoZ.1–2] and [God.1, Z.3–4, MaZ.1–5], respectively, tomulti-dimensions and to real viscosity and relaxation systems, two improvementsof great importance in physical applications. Extensions in other directions includegeneralizations to sonic shocks [H.4–5, HZ.3], boundary layers [GG, R], discrete andsemidiscrete models [Se.1, God.2, B-G.3, BHR], combustion [Ly, LyZ.1–2, JLy.1–2],and periodic traveling-wave solutions in phase-transitional models [OZ.1–2]. Mostrecently, the qualitative connection to inviscid theory pointed out in [ZS] has beenmade explicit in [MeZ.1, GMWZ.1–4] by the introduction of Kreiss symmetrizerand pseudodi↵erential techniques, leading to results on existence and stability ofcurved shock and boundary layers in the inviscid limit.

In this article, we attempt to summarize the developments of the past five yearsas they pertain to the specific context of compressible Navier–Stokes equations andrelated hyperbolic–parabolic systems. As the theory continues in a state of rapiddevelopment, this survey is to some extent a snapshot of a field in motion. For thisreason, we have chosen to emphasize those aspects that appear to be complete, inparticular, to expose the basic planar theory and technical tools on which currentdevelopments are based. It is our hope and expectation that developments overthe next five years will substantially change the complexion of the field, from thetechnical study of stability conditions as carried out here to their exploitation inunderstanding phenomena of importance in physical applications.

As the list of references makes clear, the development of this program has beenthe work of several di↵erent groups and individuals, each contributing their uniquepoint of view. We gratefully acknowledge the contribution and companionship ofour colleagues and co-authors in this venture. Special thanks go to former graduatestudents Peter Howard, Len Brin, Myungyun Oh, Je↵rey Humpherys, and GregoryLyng (IU Bloomington, advisor K. Zumbrun); Pauline Godillon, Pierre Huot, andFrederic Rousset (ENS Lyons, respective advisors D. Serre, S. Benzoni–Gavage, andD. Serre); and Ramon Plaza (NYU, advisor J. Goodman); and former postdoctoralresearchers Corrado Mascia and Kristian Jenssen (IU Bloomington), whose interestand innovation have changed the Evans function approach to conservation laws froma handful of papers into an emerging subfield.

Special thanks also to Rob Gardner, Todd Kapitula, and Chris Jones, whoseinitial contributions were instrumental to the development of the Evans functionapproach; to Denis Serre, Sylvie Benzoni–Gavage, and Heinrich Freistuhler, whosevision and innovation immediately widened the scope of investigations, and whoseideas have contributed still more than is indicated by the record of their joint andindividually published articles; to Olivier Gues, Mark Williams, and Guy Metivier,

2A similar philosophy may be found in the partial, zero-mass stability analysis of [KK], whichwas carried out simultaneously to and independently of [GZ, ZH]; that work, however, concernedonly su�cient stability conditions.

STABILITY OF LARGE-AMPLITUDE SHOCK WAVES 3

whose ideas have not only driven the newest developments in the field, but also pro-foundly a↵ected our view of the old; and to Arthur Azevedo, Constantin Dafermos,Jonathan Goodman, David Ho↵, Tai-Ping Liu, Dan Marchesin, Brad Plohr, KeithPromislow, Je↵rey Rauch, Joel Smoller, Anders Szepessy, Bjorn Sandstede, BlakeTemple, and Yanni Zeng for their generously o↵ered ideas and continued supportand encouragement outside the bounds of collaboration. Thanks to Denis Serre andthe anonymous referee for their careful reading of the manuscript and many helpfulcorrections. Finally, thanks to my colleagues in Bloomington for their friendship,aid, and stimulating mathematical company, and to my family for their patience,love, and support.

K. Zumbrun, May 20, 2003.

CONTENTS

Main Sections:

1. Introduction1.1. Equations and assumptions1.2. Structure and classification of profiles1.3. Classical (inviscid) stability analysis

1.3.1. Structural stability1.3.2. Dynamical stability

1.4. Viscous stability analysis1.4.1. Connection with classical theory1.4.2. One-dimensional stability1.4.3. Multi-dimensional stability

2. Preliminaries2.1. The spectral resolution formulae2.2. Evans function framework

2.2.1. The gap/conjugation lemma2.2.2. The tracking/reduction lemma2.2.3. Formal block-diagonalization2.2.4. Splitting of a block-Jordan block2.2.5. Approximation of stable/unstable manifolds

2.3. Hyperbolic–parabolic smoothing2.4. Construction of the resolvent

2.4.1. Domain of consistent splitting2.4.2. Basic construction2.4.3. Generalized spectral decomposition

3. The Evans function, and its low-frequency limit3.1. The Evans function3.2. The low-frequency limit

4 KEVIN ZUMBRUN

4. One-dimensional stability: the stability index4.1. Necessary conditions: the stability index4.2. Su�cient conditions

4.2.1. Linearized estimates4.2.2. Linearized stability4.2.3. Auxiliary energy estimate4.2.4. Nonlinear stability

4.3. Proof of the linearized bounds4.3.1. Low-frequency bounds on the resolvent kernel4.3.2. High-frequency bounds on the resolvent kernel4.3.3. Extended spectral theory4.3.4. Bounds on the Green distribution4.3.5. Inner layer dynamics

5. Multi-dimensional stability: the refined Lopatinski condition5.1. Necessary conditions5.2. Su�cient conditions

5.2.1. Linearized estimates5.2.2. Auxiliary energy estimate5.2.3. Nonlinear stability

5.3. Proof of the linearized estimates5.3.1. Low-frequency bounds on the resolvent kernel5.3.2. High-frequency bounds5.3.3. Bounds on the solution operator

Appendices:

A. Auxiliary results/calculationsA.1. Applications to example systemsA.2. Structure of viscous profilesA.3. Asymptotic ODE estimatesA.4. Expansion of the one-dimensional Fourier symbol

B. A weak version of Metivier’s Theorem

C. Evaluation of the Lopatinski determinant for gas dynamics3

1. INTRODUCTION.

Fluid- and gas-dynamical flow, though familiar in everyday experience, are suf-ficiently complex as to defy simple description. Reacting flow, plastic–elastic flowin solids, and magnetohydrodynamic flow are still more complicated, and more-over are removed from direct observation in our daily lives. Historically, therefore,much of the progress in understanding these phenomena has come from the studyof simple flows occurring in special situations, and their stability and bifurcation.

3Contributed by K. Jenssen and G. Lyng.


For example, one may neglect compressibility of the conducting medium for lowMach number flow, or viscosity for high Reynolds number, or both, in each caseobtaining a reduction in the complexity of the governing equations. Alternatively,one may consider solutions with special symmetry, reducing the spatial dimensionof solutions: for example laminar (shear) flows exhibiting local symmetry parallelto the direction of flow, or compressive (shock) flows exhibiting local symmetrynormal to the flow.

Once existence of such solutions has been ascertained, there remains the inter-esting question of stability of the flow/validity of simplifying assumptions; moreprecisely, one would like to know the parameter range under which such solutionsmay actually be found in nature and to understand the corresponding bifurcationat the point where stability is lost. These questions comprise the classical subjectof hydrodynamic stability; see e.g. [Ke.1–2, Ra.1–6, T, Ch, DR, Lin]. With theadvent of powerful and a↵ordable processors, this subject has lost its central placeto computational fluid dynamics, which in e↵ect brings the most remote param-eter regimes into the realm of our direct experience. However, we would arguethat this development makes all the more interesting the rigorous understanding ofstability/validity. For, numerical approximation adds another layer of e↵ects be-tween model and experiment that must be carefully separated from actual behavior,and this is an important practical role for rigorous analysis. Likewise, numericalexperiments suggest new phenomena for which analysis can yield qualitative un-derstanding.

Stability and the related issue of validity of formal or numerical approximationsare di�cult questions, and unfortunately analysis has often lagged behind the needsof practical application. However, this area is currently in a period of rapid devel-opment, to the extent that practical applications now seem not only realistic butimminent. In this article, we give an account of recent developments in the studyof stability and behavior of compressive “shock-type” fronts, with an eye towardboth mathematical completeness and eventual physical applications. The lattercriterion means that we must consider stability in the regime where transition fromstability to instability is likely to occur, and this means in most cases that we mustconsider simplified flow within the context of the full model, without simplifyingidealizations, i.e., as occurring in “real media” and usually in the large-amplituderegime. This requirement leads to substantial technical di�culties, and much ofthe mathematical interest of the analysis.

Consider a general system of viscous conservation laws

(1.1)Ut +

X

j

F j(U)xj

= ⌫X

j,k

(Bjk(U)Uxk

)xj

,

x 2 Rd;U, F j 2 Rn; Bjk 2 Rn⇥n,

modeling flow in a compressible medium, where ⌫ is a constant measuring transporte↵ects (e.g. viscosity or heat conduction).

6 KEVIN ZUMBRUN

An important class of solutions are planar viscous shock waves

(1.2) U(x, t) = U⇣x

1

� st

⌫

⌘, lim

x1!±1 U(x1

) = U±,

satisfying the traveling-wave ordinary di↵erential equation (ODE)

(1.3) B11(U)U 0 = F 1(U)� F 1(U�)� s(U � U�),

which reduce in the vanishing-viscosity limit ⌫ ! 0 to “ideal shocks,” or planardiscontinuities between constant states U± of the corresponding “inviscid” system

(1.4) Ut +X

j

F j(U)xj

= 0.

The shape function U is often called a “viscous shock profile”. Planar shock wavesare archetypal of more general, curved shock solutions, which have locally planarstructure (see [M.1–4], [GW, GMWZ.2–3] for discussions of existence of curvedshocks in the respective contexts (1.1), (1.4)). The equations of reacting flow, con-taining additional reaction terms Q(u) on the righthand sides of (1.1) and (1.4),likewise feature planar solutions (1.2), known variously as strong or weak detona-tions, or strong or weak deflagrations, depending on their specific physical roles;see, e.g., [FD, CF, GS, Ly] for details.

The theories of stability of shock and detonation waves are historically inter-twined. For physical reasons, the more complicated detonation case seems to havedriven the study of stability, at least initially. For, experiments early on indicatedseveral possible types of instabilities that could occur, in particular: galloping lon-gitudinal instabilities in which the steadily traveling planar front is replaced bya time-periodic planar solution moving with time-periodic speed; cellular instabil-ities, in which it is replaced by a nonplanar traveling wave with approximatelyperiodic transverse spatial structure and di↵erent speed; and spinning instabilities,in which it is replaced by a nonplanar, steadily traveling and rotating wave; see, e.g.,[FD, Ly, LyZ.1–2] for further discussion. The latter two types are the ones mostfrequently observed; note that these are transverse instabilities, and thus multi-dimensional in character. The questions of main physical interest seem to be: 1.the origins/mechanism for instability; 2. location of the transition from stabilityto instability; 3. understanding/prediction of the associated bifurcation to morecomplicated solutions; and 4. the e↵ect of front geometry (curvature) on stability.

Shock waves, on the other hand, were apparently thought for a long time tobe universally stable. Indeed, the possibility of instability was first predicted ana-lytically, through the normal modes analyses of [Kr, D, Er.1–2], in a regime thatwas at the time not accessible to experiment; see [BE] for an account of these andother interesting details of the early theory. On the other hand, in the inviscid case⌫ = 0, the stability condition can be explicitly calculated for shock waves, whereas


for detonation waves it cannot. For this and other reasons, the shock case is muchbetter understood; in particular, there exists a rather complete analytical multi-dimensional stability theory for inviscid shock waves, see [M.1–4, Me.1–5], whereasthe corresponding (ZND) theory for detonation waves is mainly numerical, withsensitive numerical issues limiting conclusions; see [LS] for a fairly recent survey.

As we shall discuss further below, the inviscid shock theory gives partial answersto question 1, above, but does not rule out possible additional mechanisms con-nected with viscosity or other transport e↵ects. Likewise, it gives only a partialanswer to question 2, reducing the location of the transition point to an open re-gion in parameter space, and so question 3 cannot even be addressed. Question4 inherently belongs to the viscous theory, as there is no e↵ect of shock curvaturewithout finite shock width. In this article, we present a mathematical frameworkfor the study of viscous shock stability, based on the Evans function of [E.1–4,AGJ, PW], that has emerged through the series of papers [GZ, ZH, BSZ.1, ZS,Z.1–3, MaZ.1–5] and references therein, and which features necessary and su�cientconditions for planar viscous stability analogous to but somewhat sharper than theweak and strong (or “uniform”) Lopatinski conditions of the inviscid theory [Kr,M.1–4, Me.5]. These are su�cient in principle to resolve questions 1–2, and givealso a useful starting point for the study of question 3. Question 4 remains for themoment open, and is an important direction for further investigation.

Aside from their intrinsic interest, these results are significant for their bearing onthe originally motivating problem of detonation and on related problems associatedwith nonclassical shocks in phase-transitional or magnetohydrodynamical flows, forwhich significant stability questions remain even in “typical” regimes occurringfrequently in applications. Though we shall not discuss it here, the technical toolswe develop in this article can be applied also in these more exotic cases; see [ZH,Z.5, Z.3, HZ.2, BMSZ, FreZ.2, LyZ.2] for discussions in the strictly parabolic case,and [Ly, LyZ.1] in the case of “real”, or physical viscosity.

1.1. Equations and Assumptions.

We begin by identifying an abstract class of equations, generalizing the Kawashimaclass of [Kaw], that isolates those qualities relevant to the investigation of shockstability in compressible flow. The Navier–Stokes equations of compressible gas-or magnetohydrodynamics (MHD) in dimension d, may be expressed in terms ofconserved quantities U in the standard form (1.1), with

(1.5) U =✓

uI

uII

◆, Bjk =

✓0 0

bjk1

bjk2

◆,

uI 2 Rn�r, uII 2 Rr, and

(1.6) Re �X

⇠j⇠kbjk2

� ✓|⇠|2,

✓ > 0, for all ⇠ 2 Rd; for details, see Appendix A.1. Since in this work we areinterested in time-asymptotic stability for a fixed, finite viscosity rather than the

8 KEVIN ZUMBRUN

vanishing-viscosity limit, we set ⌫ = 1 from here forward, and suppress the param-eter ⌫.

Near states of stable thermodynamic equilibrium, where there exists an associ-ated convex entropy ⌘, they can be written, alternatively, in terms of the “entropyvariable” W := d⌘(U), in symmetric hyperbolic–parabolic form

(1.7) U(W )t +X

j

F j(W )xj

=X

j,k

(BjkWxk

)xj

, W =✓

wI

wII

◆,

where

(1.8) dF j = dF j(@U/@W ), Bjk = Bjk(@U/@W ) =✓

0 00 bjk

◆

are symmetric, and (@U/@W ) andP⇠j⇠k bjk are (uniformly) symmetric positive

definite,

(1.9)X

⇠j⇠k bjk � ✓|⇠|2, ✓ > 0,

for all ⇠ 2 Rd; see [Kaw], and references therein, or Appendix A.1. This fundamentalobservation generalizes the corresponding observation of Godunov [G] (see also[Fri,M,Bo]) in the inviscid setting Bjk ⌘ 0.

In either setting, symmetric form (1.7)–(1.9) has the important property that itsstructure guarantees local existence under small perturbations of equilibrium, asmay be established by standard energy estimates [Fri,Kaw]. The viscous equations,moreover, satisfy the further condition of genuine coupling

(1.10) No eigenvector ofX

j

⇠jdF j lies in the kernel ofX

⇠j⇠kBjk,

for all nonzero ⇠ 2 Rd (equivalently,P

j ⇠jdF j ,P⇠j⇠kBjk), which implies that the

system is “dissipative” in a certain sense; in particular, it implies, together withform (1.7), global existence and decay under small perturbations of equilibrium, viaa series of clever energy estimates revealing “hyperbolic compensation” for absentparabolic terms [Kaw]. We refer to equations of form (1.7) satisfying (1.8)–(1.10)as “Kawashima class”.

Here, we generalize these ideas to the situation of a viscous shock solution (1.2) of(1.1) consisting of a smooth shock profile connecting two di↵erent thermodynami-cally stable equilibria, and small perturbations thereof. Specifically, we assume that,by some invertible change of coordinates U ! W (U), possibly but not necessarilyconnected with a global convex entropy, followed if necessary by multiplication onthe left by a nonsingular matrix function S(W ), equations (1.1) may be written inthe quasilinear, partially symmetric hyperbolic-parabolic form

(1.11) A0Wt +X

j

AjWxj

=X

j,k

(BjkWxk

)xj

+ G, W =✓

wI

wII

◆,


wI 2 Rn�r, wII 2 Rr, x 2 Rd, t 2 R, where, defining W± := W (U±):

(A1) Aj(W±), Aj⇤ := Aj

11

, A0 are symmetric, A0 > 0.

(A2) No eigenvector ofP⇠jdF j(U±) lies in the kernel of

P⇠j⇠kBjk(U±), for

all nonzero ⇠ 2 Rd. (Equivalently, no eigenvector ofP⇠jAj(A0)�1(W±) lies in the

kernel ofP⇠j⇠kBjk(W±).)

(A3) Bjk =✓

0 00 bjk

◆, G =

✓0g

◆, with Re

P⇠j⇠k bjk(W ) � ✓|⇠|2 for some

✓ > 0, for all W and all ⇠ 2 Rd, and g(Wx,Wx) = O(|Wx|2).

Here, the coe�cients of (1.11) may be expressed in terms of the original equation(1.1), the coordinate change U ! W (U), and the approximate symmetrizer S(W ),as

(1.12)A0 := S(W )(@U/@W ), Aj := S(W )dF j(@U/@W ),

Bjk := S(W )Bjk(@U/@W ), G = �X

jk

(dSWxj

)Bjk(@U/@W )Wxk

.

Note that, in accord with the general philosophy of [ZH, Z.3] in the strictly par-abolic case, the conditions of dissipative symmetric hyperbolic–parabolicity con-nected with stability of equilibrium states are required only at the endstates ofthe profile, the conditions along the profile– symmetrizable hyperbolicity, dF j

11

, inthe first equation and parabolicity,

P⇠j⇠k bjk > 0, in the second equation– being

concerned with local well-posedness rather than time-asymptotic stability of anyintermediate state. This allows for interesting applications to phase-transitional orvan der Waals gas dynamics, for which hyperbolicity of the associated ideal system(1.4) may be lost along the profile; see Appendix A.1.

Remark 1.1. The di↵erential symbol A⇠⇤ =

Pj ⇠jA

j⇤ defined by (A1) governs

convection in “hyperbolic” modes, as described in [MaZ.3]; for general equations ofform (1.1), this must be replaced by the pseudodi↵erential symbol A⇠

⇤ := dF ⇠11

�dF ⇠

12

(b⇠⇠2

)�1b⇠⇠1

, where dF ⇠ :=P

j dF j⇠j and b⇠⇠ :=P

j bjk⇠j⇠k. That is, the form(1.11) imposes interesting structure also at the linearized symbolic level.

Remark 1.2. Assumptions (A1)–(A3) were introduced in [MaZ.4] under therestriction G ⌘ 0, or equivalently S ⌘ I. This is su�cient to treat isentropic vander Waals gas dynamics, as discussed in the example at the end of the introductionof [MaZ.4]; however, it is not clear whether there exists such a coordinate changefor the full equations of gas dynamics with van der Waals equation of state. Theimproved version G 6⌘ 0, introduced in [GMWZ.4], is much easier to check, and inparticular holds for the full van der Waals gas equations under the single (clearlynecessary) condition Te > 0, where T = T (⇢, e) is temperature, and ⇢ and e aredensity and internal energy; see Appendix A.1. Still more general versions of (1.11)

10 KEVIN ZUMBRUN

(in particular, including zero-order terms with vanishing 1�1 block) might be acco-modated in our analysis; however, so far there seems to be no need in applications.Of course, equations in the Kawashima class, in particular systems possessing aglobal entropy, are included under (A1)–(A3) as a special case.

Along with the above structural assumptions, we make the technical hypotheses:

(H0) F j , Bjk, W , S 2 Cs, with s � 2 in our analysis of necessary conditionsfor linearized stability, s � 5 in our analysis of su�cient conditions for linearizedstability,4 and s � s(d) := [d/2] + 5 in our analysis of nonlinear stability.

(H1) The eigenvalues of A1

⇤ are (i) distinct from the shock speed s; (ii) of commonsign relative to s; and (iii) of constant multiplicity with respect to U .5

(H2) The eigenvalues of dF 1(U±) are distinct from s.(H3) Local to U(·), solutions of (1.2)–(1.3) form a smooth manifold {U�(·)},

� 2 U ⇢ R`.Conditions (H0)–(H2) may be checked algebraically, and are generically satisfied

for profiles of the equations of gas dynamics and MHD; see Appendix A.1.6 Weremark that (H1)(i) arises naturally as the condition that the traveling-wave ODEbe of nondegenerate type, while (H2) is the condition for normal hyperbolicity ofU± as rest points of that ODE; see Section 1.2. The “weak transversality” condition(H3) is more di�cult to verify; however, it is automatically satisfied for extreme Laxshocks such as arise in gas dynamics; see Remark 2, Section 1.2. More generally, itis implied by but does not imply transversality of the traveling-wave connection Uas an orbit of (1.3).

In our analysis of su�cient conditions for stability, we make two further hypothe-ses at the level of the inviscid equations (1.4), analogous to but somewhat strongerthan the block structure condition of the inviscid stability theory [Kr, M.1–4, Me.5].The first is, simply:

(H4) The eigenvalues ofP⇠jdF j(U±) are of constant multiplicity with respect

to ⇠ 2 Rd \ {0}.This condition was shown in [Me.4] to imply block structure, and is satisfied forall physical examples for which the block structure condition has currently beenverified. In particular, it holds always for gas dynamics, but fails for MHD; see,respectively, Appendix C of this article and [Je, JeT].

The second concerns the structure of the glancing set of the symbol

(1.13) P±(⇠, ⌧) := i⌧ +X

i⇠jdF j(U±)

4As mentioned in [MaZ.3], we suspect that C3 should su�ce, with further work.5In our analysis of necessary conditions for stability, we require only (H1)(i). Condition

(H1)(iii) can be dropped, at the expense of some detail in the linear estimates [Z.4].6In particular, Aj

⇤ = ujIn�r

, where uj is the jth component of fluid velocity, and so (H1) istrivially satisfied. That is, hyperbolic modes experience only passive, scalar convection for theseequations.


of the inviscid equations (1.4) at x1

= ±1. By symmetrizability assumption (A1),together with hypothesis (H4), the characteristic equation det P±(⇠, ⌧) = 0 has nroots

(1.14) ⌧ = �ar(⇠), r = 1, . . . , n,

locally analytic and homogeneous degree one in ⇠, where ar are the eigenvalues ofP⇠jdF j(U±).Definition 1.3. Setting ⇠ = (⇠

1

, . . . , ⇠d) =: (⇠1

, ⇠), we define the glancing set ofP± as the set of frequencies (⇠, ⌧) for which ⌧ = ar(⇠1, ⇠) and (@ar/@⇠1) = 0 forsome real ⇠

1

and 1 r n.

Remark 1.4. The word “glancing” is used in Definition 1.3 because null bichar-acteristics of P± through points (⇠, ⌧) with (@ar/@⇠1) = 0 are parallel to x

1

= 0.

The significance of the glancing set for our analysis is that it corresponds to theset of frequencies ⇠, ⌧ for which the (rotated) inverse functions i⇠

1

= µr(⇠, ⌧) associ-ated with ⌧ = ar(⇠1, ⇠), corresponding to rates of spatial decay in x

1

for the inviscidresolvent equation, are pure imaginary and have branch singularities of degree sr

equal to the multiplicity of root ⇠1

. As pointed out by Kreiss [Kr], these are thefrequencies for which resolvent estimates become delicate, and structural assump-tions become important (e.g., strict hyperbolicity as in [Kr], or, more generally,block structure as in [M.1–4, Me.5]).

Our final technical hypothesis is, then:

(H5) The glancing set of P± is the union of k (possibly intersecting) smoothcurves ⌧ = ⌘q(⇠), 0 k n, defined as the locii on which ⌧ = �aq(⇠1, ⇠) and(@aq/@⇠1) = 0 for some real ⇠

1

, on which root ⇠1

has constant multiplicity sq,defined as the order of the first nonvanishing partial derivative (@saq/@⇠s

1

) withrespect to ⇠

1

, i.e., the associated inverse function ⇠q1

(⇠, ⌧) has constant degree ofsingularity sq.

Condition (H5) is automatic in dimensions d = 1, 2 and in any dimension forrotationally invariant problems. In one dimension, the glancing set is empty. Inthe two-dimensional case, the homogeneity of ar and its derivatives implies that theray through (⇠, ⌧) is the graph of ⌧(⇠) and that (H5) holds there. By the ImplicitFunction Theorem, (H5) holds also in the case that all branch singularities areof square root type, degree sq = 2, with ⌘q defined implicitly by the requirement(@aq/@⇠1) = 0 (indeed, ⌘q is in this case analytic). In particular, it holds in the casethat all eigenvalues ar(⇠) are either linear or else strictly convex/concave in ⇠

1

for⇠ 6= 0.7 Thus, (H5) is satisfied in all dimensions for the equations of gas dynamics,for which the characteristic eigenvalues ar are linear combinations of (⇠, ⌘) and |⇠, ⌘|(see Appendix C), hence clearly linear or else strictly concave/convex for ⇠ 6= 0.

7More generally, just those involved in glancing. Likewise, constant multiplicity (H4) is onlyneeded in our analysis for eigenvalues involved in glancing.

12 KEVIN ZUMBRUN

Similarly, it may be calculated (see [MeZ.3]) that all characteristic eigenvalues ofthe equations of MHD are linear or else strictly concave/convex for ⇠ 6= 0, with theexceptions of the two “slow” magnetoacoustic characteristics, which together mayproduce a singularity of degree at most two. Thus, again (H5) is always satisfied.

Remark 1.5. Assuming symmetrizability, (A1), condition (H5) like (H4) impliesblock structure; see the direct matrix perturbation calculations of Section 5, [Z.3],or Section 5 of this article. As was the block structure condition in the inviscidcase, conditions (H4)–(H5) are used in the viscous case to make convenient certainestimates on the resolvent, and should be viewed as “first-order nondegeneracyconditions”. There is ample motivation in the example of MHD to carry out arefined analysis at the next level of degeneracy, with these conditions relaxed: inparticular, the constant multiplicity requirement (H4), which presently restricts thestability analysis for MHD to the one-dimensional case. See [MeZ.3] for some workin this direction.

1.2. Structure and classification of profiles.

We next state some general results from [MaZ.3, Z.3] regarding structure andclassification of profiles, analogous to those proved in [MP] in the strictly paraboliccase. Proofs may be found in Appendix A.2; see also [MaZ.3, Z.3, Ly, LyZ.1].

Observe first, given det b11

2

6= 0, (A2), that (H1)(i) is equivalent, by determinant

identity det✓

dF 1

11

� s dF 1

12

b11

1

b11

2

◆= det (dF 1

11

� s � b11

1

(b11

2

)�1dF 1

12

)det b11

2

, to the

condition

(1.15) det✓

dF 1

11

� s dF 1

12

b11

1

b11

2

◆6= 0.

Writing the traveling-wave ODE (1.3) as

(1.16) f I(U)� suI ⌘ f I(U�)� suI�,

(1.17) b11

1

(uI)0 + b11

2

(uII)0 = f II(U)� f II(U�)� s(uII � uII� ),

where F 1 =:✓

f I

f II

◆, we find that (1.15) is in turn the condition that (1.17) describe

a nondegenerate ODE on the r-dimensional manifold described by (1.16): well-defined by the Implicit Function Theorem plus full rank of (dF 1

11

� s, dF 1

12

).

Lemma 1.6[Z.3, MaZ.3]. Given (H1)–(H3), the endstates U± are hyperbolic restpoints of the ODE determined by (1.17) on the r-dimensional manifold (1.16), i.e.,the coe�cients of the linearized equations about U±, written in local coordinates,have no center subspace. In particular, under regularity (H0), standing-wave solu-tions (1.2) satisfy

(1.18) |(d/dx1

)k(U � U±)| Ce�✓|x1|, k = 0, . . . , 4,


as x1

! ±1.

Lemma 1.6 verifies the hypotheses of the gap lemma, [GZ,ZH]8, on which weshall rely to obtain the basic ODE estimates underlying our analysis in the low-frequency/large-time regime.

Let i+

denote the dimension of the stable subspace of dF 1(U+

), i� denote thedimension of the unstable subspace of dF 1(U�), and i := i

+

+ i�. Indices i± countthe number of incoming characteristics from the right/left of the shock, while icounts the total number of incoming characteristics toward the shock. Then, thehyperbolic classification of U(·), i.e., the classification of the associated hyperbolicshock (U�, U

+

), is:

8><

>:

Lax type if i = n + 1,

Undercompressive (u.c.) if i n,

Overcompressive (o.c.) if i � n + 2.

In case all characteristics are incoming on one side, i.e. i+

= n or i� = n, a shockis called extreme.

As in the strictly parabolic case, there is a close connection between the hy-perbolic type of a shock and the nature of the corresponding connecting profile.Considering the standing-wave ODE as an r-dimensional ODE on manifold (1.16),let us denote by 1 d± r the dimensions of stable manifold at U

+

and unstablemanifold at U�, and d := d

+

+ d�. Then, we have:

Lemma 1.7 [MaZ.3, LyZ.1]. Given (H1)–(H3), there hold relations

(1.19)n� i

+

= r � d+

+ dim U(A⇤+),n� i� = r � d� + dim S(A⇤�),

where A⇤ := dF 1

11

�dF 1

12

(b11

2

)�1b11

1

, and U(M) and S(M) denote unstable and stablesubspaces of a matrix M . In particular, existence of a connecting profile impliesn� i = r � d.

The final assertion implies that the “viscous” and “hyperbolic” types of shockconnections agree, i.e. Lax, under-, and overcompressive designations imply cor-responding information about connections. The more detailed information (1.19)implies that “extreme” shocks have “extreme” connections:

Corollary 1.8 [MaZ.3, LyZ.1]. For (right) extreme shocks, i+

= n, there holdsalso d

+

= r, i.e. the connection is also extreme, and also dim U(A⇤) ⌘ 0.

Proof. Immediate from (1.19), using d+

r and dim U(A⇤+) � 0. ⇤

8See also the version established in [KS], independently of and simultaneously to that of [GZ].

14 KEVIN ZUMBRUN

A complete description of the connection, of course, requires the further index` defined in (H4) as the dimension ` of the connecting manifold between (u±, v±)in the traveling-wave ODE. Generically, one expects that ` should be equal to thesurplus d� r = i� n. In case the connection is “dimensionally transverse” in thissense, i.e.:

(1.20) ` =⇢

1 undercompressive or Lax case,i� n overcompressive case,

we call the shock “pure” type, and classify it according to its hyperbolic type;otherwise, we call it “mixed” under/overcompressive type. Throughout this article,we shall assume that (1.20) holds, so that all viscous profiles are of pure, hyperbolictype. Indeed, we shall restrict attention mainly to the classical case of pure, Lax-typeprofiles such as occur in the standard gas-dynamical case, confining our discussionof other cases to brief remarks. For more detailed discussion of the nonclassical,over- and undercompressive cases, we refer the reader to [Z.3] and especially [ZS].

Remarks 1.9. 1. Transverse Lax and overcompressive connections of (1.3) per-sist under change of parameters (U�, s), while transverse undercompressive con-nections are of codimension q := n + 1� i (the “degree of undercompressivity”) inparameter space (U�, s).

2. For extreme indices d+

= r or d� = r, profiles if they exist are alwaystransverse, and likewise for indices that are “minimal” in the sense that d

+

= ` ord� = `. In particular, extreme Lax or overcompressive connections (if they exist) arealways transverse, hence satisfy (H3). Undercompressive shocks are never extreme.

3. The relation 0 dim U(A+

)�dim U(A⇤+) = (r�d+

) r, may be viewed asa sort of subcharacteristic relation, by analogy with the relaxation case. Indeed, asdescribed in [LyZ.1], it may be used to obtain the full subcharacteristic condition

aj < a⇤j < aj+r,

as a consequence of linearized stability of constant states, where aj denote theeigenvalues of A := dF 1 and a⇤j those of A⇤; this is closely related to argumentsgiven by Yong (see, e.g., [Yo]) in the relaxation case. The subcharacteristic relationgoes the “wrong way” in the relaxation case, and so one cannot conclude a resultanalogous to Corollary 1.8; for further discussion, see [MaZ.1, MaZ.5].

4. In the special case r = 1, there holds d+

= d� = 1 whenever there is aconnection, and so profiles are always of Lax type n� i = r�d = �1. This recoversthe observation of Pego [P.2] in the case of one-dimensional isentropic gas dynamicsthat smooth (undercompressive type) phase-transitional shock profiles cannot occurunder the e↵ects of viscosity alone, even for a van der Waals-type equation of state.(They can occur, however, when dispersive, capillary pressure e↵ects are taken intoaccount; see, e.g., [Sl.1–5, B-G.1–2].)


1.3. Classical (inviscid) stability analysis.

The classical approach to the study of stability of shock waves, as found in themathematical physics literature, consists mainly in the study of the related problemsof structural stability, or existence and stability of profiles (1.2) as solutions of ODE(1.3), and dynamical stability, or hyperbolic stability of the corresponding idealshock

(1.21) U(x, t) = ¯U(x1

� st), ¯U(x1

) :=⇢

U� for x1

< 0,

U+

for x1

� 0,

as a solution of the inviscid equations (1.4); see, for example, the excellent surveyarticles [BE] and [MeP]. These may be derived formally by matched asymptoticexpansion, either in the vanishing-viscosity limit ⌫ ! 0, or, as discussed in Section1.3, [Z.3], in the low-frequency, or large-space–time limit

(1.22) x := "x, t := "t, "! 0,

with ⌫ held fixed and t varying on a bounded interval [0, T ]. We summarize therelevant results below.

1.3.1. Structural stability. The viscous profile problem, or “inner” problemfrom the matched asymptotic expansion point of view, yields first of all the Rankine–Hugoniot conditions

(RH) s[U ] = [F 1(U)],

where [H] := H(U+

) � H(U�) denotes jump across the shock in quantity H, asthe requirement that U± must both be rest points of the traveling-wave ODE (1.3).These arise more directly in the inviscid theory, as the conditions that mass beconserved across an ideal shock discontinuity (1.21); see, e.g., [La, Sm].

In matched asymptotic expansion, solvability of the inner problem serves as theboundary (or free boundary, in this case) condition for the “outer” problem, whichin this case (see Section 1.3, [Z.3]) is just the inviscid equations (1.4). By Remark1.9.1, the Rankine–Hugoniot conditions are su�cient for existence in the vicinityof a transverse Lax- or overcompressive-type shock profile, while in the vicinity ofa transverse undercompressive shock profile, existence is equivalent to (RH) plusan additional q boundary conditions, sometimes known as “kinetic conditions”,where q = n + 1 � i (in the notation of the previous section) is the degree ofundercompressivity. Thus, local to an existing transverse profile, the outer, inviscidproblem may in the Lax or overcompressive case be discussed without referenceto the inner problem, appealing only to conservation of mass, (RH). As discussedcogently in [vN] (roundtable discussion), this remarkable fact is in sharp contrast tothe situation in the case of a solid boundary layer, and is at the heart of hyperbolicshock theory.

16 KEVIN ZUMBRUN

Of course, there remains the original question of existence in the first place of anytransverse smooth shock profile, which is a priori far from clear given the incompleteparabolicity of equations (1.1). Here, the genuine coupling condition (A2) plays akey role. For example, in the “completely decoupled” case that dF j

12

⌘ 0, the firstequation of (1.1) is first-order hyperbolic, so supports only discontinuous traveling-wave solutions (1.21) [Sm]. On the other hand, (A2) precludes such discontinuoussolutions, at least for small shock amplitude [U ] := U

+

� U�. For, a distributionalsolution (1.21) of (1.1), ⌫ = 1, may be seen to satisfy not only the Rankine–Hugoniot conditions (RH), but also, using the block structure assumption (1.8) inthe symmetric, W coordinates, the condition [wII ] = 0, or [W ] 2 kerB11.9 For [U ]small, however, (RH) implies that [U ] is approximately parallel to an eigenvector ofdF 1, hence by (A2) cannot lie in kerB11(U�), and therefore (using smallness again)[W ] cannot lie in kerB11, a contradiction. Viewed another way, this is an exampleof the time-asymptotic smoothing principle introduced in [HoZi.1–2] in the specificcontext of compressible Navier–Stokes equations, which states that (A2) impliessmoothness of time-asymptotic states as in the strictly parabolic case, despite in-complete smoothing for finite times: in particular, of stationary or traveling-wavesolutions.

Indeed, Pego [P.1] has shown by a center-manifold argument generalizing thatof [MP] in the strictly parabolic case that the small-amplitude existence theory forhyperbolic–parabolic systems satisfying assumption (A2) is identical to that of thestrictly parabolic case, as we now describe.

First, recall the Lax structure theorem:

Proposition 1.10 [La, Sm]. Let F 1 2 C2 be hyperbolic (�(dF 1) real and semisim-ple) for U in a neighborhood of some base state U

0

, and let ap(U0

) be a simple eigen-value, where a

1

· · · ak denote the eigenvalues of dF 1(U), and rj = rJ(U) theassociated eigenvectors. Then, there exists a smooth function Hp : (U�, ✓) ! U

+

,✓ 2 R1, with Hp(U�, 0) ⌘ U� and (@/@✓)Hp(U�, 0) ⌘ rp(U�), such that, forU± lying su�ciently close to U

0

and [U ] := U+

� U� lying su�ciently close to|U

+

� U�|rp(U0

), the triple (U�, U+

, s) satisfies the Rankine–Hugoniot conditions(RH) if and only if U

+

= Hp(U�, ✓) for ✓ su�ciently small.

Proof. See [Sm], pp. 328–329, proofs of Theorem 17.11 and Corollary 17.12. ⇤Note that, for [U ] su�ciently small and U su�ciently near U

0

, (RH) impliesthat [U ] lies approximately parallel to some rj(U0

); thus, the assumption that[U ] lie nearly parallel to rp(U0

) is no real restriction. For fixed U�, the image ofHp is known as the (pth) Hugoniot curve through U�. For gas dynamics, undermild assumptions on the equation of state10 the Hugoniot curves extend globally,describing the full solution set of (RH).

9Without symmetric, block form, we cannot make conclusions regarding the product B11Ux1

of a discontinous function and a possibly point measure. However, (1.8)–(1.9) together allow usto eliminate the possibility of a jump in wII .

10For example, existence of a global convex entropy, plus the “weak condition” of [MeP],


Definition 1.11. For U lying on the pth Hugoniot curve through U�, denoteby �(U�, U) the associated shock speed defined by (RH). Then, the shock triple(U�, U

+

= Hp(U�, ✓+

), s = �(U�, U+

)) is said to satisfy the strict Liu-Oleinikadmissibility condition if and only if

(LO) �(U�, U(✓⇤)) > �(U�, U+

) = s for all ✓⇤(✓⇤ � ✓+) < 0,

i.e., on the segment of the Hugoniot curve between U� and U+

, �(U�, U) takes onits minimum value at U

+

.

A general small-amplitude existence theory may now be concisely stated as fol-lows, with the strictly parabolic theory subsumed as a special case.

Proposition 1.12 [P.1]. Let F 1 2 C2 hyperbolic and B11 2 C2 of form (1.5)–(1.6) satisfy the genuine coupling condition (1.10) for U in a neighborhood of somebase state U

0

, and let ap(U0

) be a simple eigenvalue, where a1

· · · ak denotethe eigenvalues of dF 1(U). Then, for a su�ciently small neighborhood U of U

0

,and for (U�, U

+

, s) lying su�ciently close to (U0

, U0

, ap(U0

)) and satisfying thenoncharacteristic condition a⇤p(U±) 6= s, there exists a traveling-wave connection(1.2) lying in U if and only if the triple (U�, U

+

, s) satisfies both the Rankine–Hugoniot condtions (RH) and the strict Liu–Oleinik admissibility condition (LO).Moreover, such a local connecting orbit, if it exists, is transverse, hence unique upto translation.

Proof. See [P.1], or the related [Fre.4]. ⇤In the case of “standard” gas dynamics, for which the equation of state admits a

global convex entropy, there is a correspondingly simple global existence theory. Inthe contrary case of “nonstandard”, or “real” (e.g., van der Waals) gas dynamics,the global existence problem so far as we know is open.

Proposition 1.13[Gi, MeP]. For gas dynamics with a global convex entropy, atriple (U�, U

+

, s) admits a traveling-wave connections (viscous profile) (1.2) if andonly if it satisfies both (RH) and (LO). Moreover, connections if they exist aretransverse.

Proof. See [Gi] in the “genuinely nonlinear” case raprp 6= 0, [MeP], AppendixC in the general case. Transversality is automatic, by Remark 1.9.1. ⇤

Remarks 1.14. 1. In the case that eigenvalues aj retain their order relativeto ap, e.g., ap remains simple, along the Hugoniot curve, the Liu–Oleinik condition(LO) may be seen to be a strengthened version of the classical Lax characteristicconditions [La]

(1.23) ap�1

(U�) < 0 < ap(U�), ap(U+

) < 0 < ap+1

(U+

),

satisfied for all known examples possessing a convex entropy (in particular, ideal gas dynamicsand the larger classes of equations considered by Bethe, Weyl, Gilbarg, Wendro↵, and Liu [Be,We, Gi, Wen.1–2, L.4]); see [MeP].

18 KEVIN ZUMBRUN

where 1 p n is the principal characteristic speed associated with the shock, or,equivalently (by preservation of order),

(1.24) ap(U�) > s > ap(U+

),

by ap(U�) = �(U�, U�) > s and symmetry with respect to ± of (LO). In particular,the profiles described in Propositions 1.12 and 1.13 are always of classical, Laxtype. Small-amplitude profiles bifurcating from a multiple eigenvalue ap, or large-amplitude profiles of general hyperbolic–parabolic systems may be of arbitrary type;see [FreS.1] and [CS], respectively.

In the genuinely nonlinear caseraprp 6= 0, (LO) is equivalent along the Hugoniotcurve U

+

= Hp(U�, ✓) through U� to condition (1.24), which in turn is equivalentto sgn ✓ = � sgn raprp(U�) 6= 0 [La, Sm]. Here, Hp denotes an arbitrarynondegenerate extension of the local parametrization defined in Proposition 1.10.

2. Condition (LO) yields existence and uniqueness of admissible scale-invariantsolutions of (1.4) for Riemann initial data

(1.25) U0

=⇢

U� for x < 0,U

+

for x � 0,

for small amplitudes |U+

� U�| and general equations [L.4], and, under mild ad-ditional assumptions (the “medium condition” of [MeP]), for large amplitudes inthe case of gas dynamics with a global convex entropy [MeP]. Such Riemann so-lutions represent possible time-asymptotic states (after renormalization (1.22)) forthe viscous equations (1.1) with “asymptotically planar” initial data in the sensethat limx1!±1 U

0

(x) = U±; see, e.g., [AMPZ.1] for further discussion.3. For large amplitudes, the genuine coupling condition (A2) does not in gen-

eral appear to preclude discontinuous traveling-wave solutions of (1.1) analogousto “subshocks” in the relaxation case [L.5, Wh], but rather must be replaced bythe nonlinear version [W ] 62 kerB11(U�) = kerB11(U

+

) for (U�, U+

, s) satisfying(RH). For compressible Navier–Stokes equations, this condition is satisfied glob-ally for classical equations of state. Without loss of generality taking shock speeds = 0, we find that failure corresponds to a jump in density with velocity, pres-sure, and temperature held fixed; see the discussion of gas-dynamical structurebelow Corollary 6.5, Appendix A.1. Thus, it can occur only for a nonmonotone,van der Waals-type pressure law, and corresponds to an undercompressive, phase-transitional type shock. Recall that such shocks do not possess smooth profilesunder the influence of viscosity alone, Remark 1.9.4.

1.3.2. Dynamical stability. Hyperbolic stability concerns the bounded-timestability of (1.21) as a solution of inviscid equations (1.4) augmented with the free-boundary condition of solvability of the viscous profile problem, i.e., well-posednessof the “outer problem” arising through formal matched asymptotic expansion. Thatis, it is Hadamard-type stability, or well-posedness, that is the relevant notion in this


case, in keeping with the homogeneous nature of (1.4). In view of Remark 1.14.1, wefocus our attention on the classical, Lax case, for which the free-boundary conditionsare simply the Rankine–Hugoniot conditions (RH) arising through conservation ofmass.

Postulating a perturbed front location X(x, t), x =: (x1

, x), following the stan-dard approach to stability of fluid interfaces (see, e.g., [Ri, Er.1, FD, M.1–4]), andcentering the discontinuity by the change of independent variables

(1.26) z := x1

�X(x, t),

we may convert the problem to a more conventional fixed-boundary problem atz ⌘ 0, with front location X carried as an additional dependent variable. Thisdevice eliminates the di�culty of measures arising in the linearized equations (andeventual nonlinear iteration) through di↵erentiation about a moving discontinuity.

Necessary inviscid stability condition. Linearizing the resulting equationsabout (1.21), assuming without loss of generality s = 0, we obtain constant-coe�cient problems

(1.27) Ut + A1

±Uz +dX

2

Aj±Ux

j

= 0,

Aj± := dF j(U±), on z ? 0, linked by n transmission conditions given by the lin-

earized Rankine–Hugoniot conditions, which involve also t- and x- derivatives ofthe (scalar) linearized front location.

Linearized inviscid stability analysis by Laplace–Fourier transform of (1.27) yields,in the case of a Lax p-shock (defined by (1.23), or p := i

+

in the notation of Section1.2), the weak Kreiss–Sakamoto–Lopatinski stability condition

(1.28) �(e⇠,�) 6= 0, e⇠ 2 Rd�1, Re � > 0,

where e⇠ := (⇠2

, · · · , ⇠d) is the Fourier wave number in transverse spatial directionsx := (x

2

, · · · , xd), � is the Laplace frequency in temporal direction t, and

(1.29) � := det (r�1

, · · · , r�p�1

, i[F e⇠] + �[U ], r+

p+1

, · · · , r+

n ),

with Fe⇠(U) :=

Pj 6=1

⇠jF j(U), [g] := g(U+

)� g(U�), and {r+

p+1

, . . . , r+

n }(e⇠,�) and{r�

1

, . . . , r�p�1

}(e⇠,�) denoting (analytically chosen) bases for the unstable/resp. sta-ble subspaces of matrices

(1.30) A±(e⇠,�) := (�I + dFe⇠(U±))(dF 1(U±))�1;

for details, see, e.g. [Se.2–4, ZS, Me.5]. Condition (1.28) is obtained from therequirement that the inhomogeneous equation

(1.31) (r�1

, · · · , r�n�i�, i[F e⇠] + �[U ], r+

i++1

, · · · , r+

n )↵ = f

20 KEVIN ZUMBRUN

arising through Laplace–Fourier transform of the linearized problem have a uniquesolution for every datum f , i.e. the columns of the matrix on the lefthand side forma basis of Rn.

This result was first obtained by Erpenbeck in the case of gas dynamics11 [Er.1–2], and in the general case by Majda [M.1–4]. Condition (1.28) is an example of ageneral type of stability condition introduced by Kreiss and Sakamoto in their pio-neering work on hyperbolic initial-boundary value problems [Kr,Sa.1-2]; as pointedout by Majda, it plays a role analogous to that of the Lopatinski condition in ellip-tic theory. Zeroes of � correspond to normal modes e�teie⇠·exw(x

1

) of the linearizedequations, hence (1.28) is necessary for linearized inviscid stability. Evidently, �is positive homogeneous (i.e., homogeneous with respect to positive reals), degreeone, as are the inviscid equations; thus, instabilities if they occur are of Hadamardtype, corresponding to ill-posedness of the linearized problem.

Su�cient inviscid stability condition. A fundamental contribution of Majda[M.1–4], building on the earlier work of Kreiss [Kr], was to point out the importanceof the uniform (or “strong”) Kreiss–Sakamoto–Lopatinski stability condition

(1.32) �(e⇠,�) 6= 0, e⇠ 2 Rd�1, Re � � 0, (e⇠,�) 6= (0, 0),

extending (1.28) to imaginary �, as a su�cient condition for nonlinear stability.Assuming (1.32), a certain “block structure” condition on matrix A(⇠, ⌧) defined in(1.30) (recall: implied by (H4)), and appropriate compatibility conditions on theperturbation at the shock (ensuring local structure of a single discontinuity, i.e.,precluding the formation of shocks in other characteristic fields), Majda establishedlocal existence of perturbed ideal shock solutions

(1.33) U(x, t) =: ¯U(x1

�X(x, t)) + U(x1

�X(x, t), x, t),

¯U as in (1.21), for initial perturbation su�ciently small in Hs, s su�ciently large,with optimal bounds on kUkHs

(x,t) and kXkHs

(x,t). Moreover, using pseudodif-ferential techniques, he established a corresponding result for curved shock frontsunder the local version of the planar uniform stability condition, establishing bothexistence and stability of curved shock solutions.

These result have since been significantly sharpened and extended by Metivierand coworkers; see [Me.5], and references therein. In particular, Metivier [Me.1]has obtained uniform local existence and stability for vanishing shock amplitudesunder assumptions (A1) and (H2) plus the additional nondegeneracy conditionsthat principal characteristic ap(U, ⇠) be: (i) simple and genuinely nonlinear in thenormal direction to the shock, without loss of generality ⇠ = (1, 0, . . . , 0), and (ii)strictly convex in ⇠ in a vicinity of ⇠ = (1, 0, . . . , 0). In the existence result of

11See also related, earlier work in [Be,D,Ko.1-2,Free.1-2,Ro,Ri].


Majda, the time of existence vanished with the shock strength, due to the singular(i.e., characteristic) nature of the small-amplitude limit.

Remark 1.15. Failure of (1.32) on the boundary � = i⌧ , ⌧ real, i.e., weakbut non-uniform stability, corresponds to existence of a one-parameter family of(nondecaying) transverse traveling-wave solutions ei(⌧+e⇠·ex)w(z) of the linearizedequations, 2 R, moving with speed � = ⌧/|e⇠| in direction ⇠/|⇠|, and is in generalassociated with loss of smoothness in the front X(·, ·) and loss of derivatives in theiteration used to prove nonlinear stability.12 When w has finite energy, w 2 L2,such solutions are known as surface waves; however, this is not always the case. Inparticular, w 2 L2 only if

(1.34) a±j±(⇠1

, ⇠) < ⌧ < a±k±(⇠1

, ⇠)

for some j±, k± (without loss of generality ak±(⇠) = �aj±(�⇠)): see Proposition 3.1of [BRSZ]. For ⇠

1

= 0, this implies that the traveling wave is “subsonic” in the sensethat � lies between the j±th and k±th hyperbolic characteristic speeds in direction⇠/|⇠| at endstates U±. It is easily seen that the set determined by (1.34) is boundedby the glancing set defined in Definition 1.3; indeed, the complement of the closureof this set is the union G of the components of (⇠, ⌧) = (0,±1) in the complementof the glancing set. A more fundamental division is between the larger hyperbolicdomain R, defined as the set of (⇠, ⌧) for which the stable and unstable subspacesof A(⇠, i⌧) in (1.30) admit real bases, and its complement. (Note: G ⇢ R.) On R,the eigenvalues of A(⇠, i⌧) in (1.30) determining exponential rates of spatial decayin z are pure imaginary, and so surface waves cannot occur. The set R like G isbounded by the glancing set, at which certain eigenvalues of A(⇠, i⌧) bifurcate awayfrom the imaginary axis.

As discussed in [BRSZ], these various cases are in fact quite di↵erent. In partic-ular, (i) in the surface wave case, the linearized equations are “weakly well-posed”in L2, permitting a degraded version of the energy estimates in the strongly stablecase, whereas in the complementary case they are not even weakly well-posed, and(ii) zeroes (⇠, ⌧) lying in the closure of Rc generically mark a boundary betweenregions of strong stability and strong instability in parameter space (U�, U

+

, s),whereas zeroes lying in the interior of R generically persist, marking an open set inparameter space. The explanation for the latter, at first surprising fact is that forreal (⇠, ⌧) 2 R, it may be arranged by choosing r±j real that �(⇠, ·) take the imag-inary axis to itself; thus, zeroes may leave the imaginary axis only by coalescing ata double (or higher multiplicity) root, either finite,

(1.35) �(e⇠, i⌧) = 0, (@/@�)�(e⇠, i⌧) = 0,

or at infinity (recall, by complex symmetry, that roots escape to infinity in complexconjugate pairs). The latter possibility corresponds to one-dimensional instability

12An exception is the scalar case; see [M.1–4].

22 KEVIN ZUMBRUN

�(0, 1) = 0, as discussed in [Se.2, BRSZ, Z.3].13 The important category of weaklystable shocks possessing a zero �(e⇠, i⌧) = 0 with (e⇠, ⌧) in R was denoted in [BRSZ]as “weakly stable of real type”, and forms a third open (i.e., generic) type besidesthe strongly stable and strongly unstable ones.

Evaluation in one dimension. In one-dimension, e⇠ = 0, the basis vectorsr±j may be fixed as the outgoing characteristic directions for A1

± = dF 1(U±), i.e.,eigenvectors with associated eigenvalues a±j ? 0, yielding �(0,�) = ��(0, 1). Thus,both weak and uniform stability conditions reduce to

(1.36) �(0, 1) = det (r�1

, · · · , r�p�1

, [U ], r+

p+1

· · · , r+

n ) 6= 0,

Condition (1.36) may be regarded as a sharpened form of the Lax characteristiccondition (1.23), which asserts only that the number of columns of the matrix onthe lefthand side of (1.36) is correct (i.e., equal to n). Condition (1.36) can berecognized also as the condition that the associated Riemann problem (U�, U

+

) belinearly well-posed (transverse), since otherwise there is a nearby family of Riemannsolutions possessing the same (final) end states U±, but for which the data at thep-shock is modified by infinitesimal perturbations dU± such that the linearizedRankine–Hugoniot relations remain satisfied:

(1.37) �ds[U ] = F 1(U+

)dU+

� dF 1(U�)dU�.

That is, failure of one-dimensional inviscid stability is generically associated withthe phenomenon of wave-splitting; see, e.g., [Er.1, ZH, Z.3], for further details.

As observed by Majda [M.1–4], in the case of a simple eigenvalue ap(U⇤), the Laxstructure theorem, Theorem 1.10, ensures that one-dimensional stability holds forsu�ciently small amplitude shocks of general systems. One-dimensional stabilityholds for arbitrary amplitude shocks of the equations of gas dynamics with a “stan-dard” equation of state (e.g., possessing a global convex entropy and satisfying the“medium condition” of [MeP]), but may fail at large amplitudes for more generalequations of state; see, e.g., [Er.1–2, M.1–4, MeP].

Remark 1.16. The set of parameters (U�, U+

, s) on which the one-dimensionaldynamical stability condition (1.36) fails is of codimension one, hence genericallyconsists of isolated points along a given Hugoniot curve Hp(U�, ·). In other words,the one-dimensional dynamical stability condition yields essentially no informationfurther than the Lax structure already imposed on the problem by the condition ofstructural stability.

13By positive homogeneity, 0 ⌘ �j

(e⇠j

, i⌧j

) = �j

(e⇠j

/|⌧j

|, i sgn ⌧j

) ! �(0,±i) as j ! 1,where �

j

are the Lopatinski conditions for a sequence of converging parameter values indexed by

j, with zeroes (e⇠j

, i⌧j

) such that �j

= ⌧j

/|⇠j

|!1 as j !1, and � = lim �j

is the Lopatinskicondition for the limiting parameter values as j !1. Hence, � vanishes at one of (0,±i) (in factboth, by complex symmetry of the spectrum), yielding one-dimensional instability.


Evaluation in multiple dimensions. Evaluation of the stability conditionsin multi-dimensions is significantly more di�cult than in the one-dimensional case,even the computation of� being in general nontrivial. Nonetheless, we obtain againa small-amplitude result for general systems and an arbitrary-amplitude result forgas dynamics, both with appealingly simple formulations.

Namely, under the assumptions of [Me.1]– (A1) and (H2) plus the conditions thatprincipal characteristic ap(U, ⇠) be (i) simple and genuinely nonlinear in the normaldirection to the shock, without loss of generality ⇠ = (1, 0, . . . , 0), and (ii) strictlyconvex in ⇠ in a vicinity of ⇠ = (1, 0, . . . , 0)– we have the simple small-amplituderesult, generalizing Majda’s observation in one dimension, that su�ciently small-amplitude shocks are strongly (“uniformly”) stable. This is an indirect consequenceof the detailed partial di↵erential equation (PDE) estimates carried out by Metivierin the proof of existence and stability of small-amplitude shocks [Me.1]. We give adirect proof in Appendix B, in the spirit of Metivier’s analysis but carried out inthe much simpler linear-algebraic setting.

A detailed analysis of the large-amplitude gas-dynamical case recovering thephysical stability critera of Erpenbeck–Majda [Er.1–2, M.1–4] is presented in Ap-pendix C.14 This includes in particular Majda’s theorem that for ideal gas dynamics,shocks of arbitrary amplitude are strongly (uniformly) stable. Discussion of moregeneral equations of state may be found, e.g., in [Er.1–2, BE, MeP]. In particular,uniform stability holds for equations of state possessing a global convex entropy andsatisfying the “medium condition” of [MeP], but in general may fail, depending onthe geometry of the associated Hugoniot curve.

Remark 1.17. For gas dynamical shocks, direct calculation shows that weakmulti-dimensional inviscid stability when it occurs is always of real type, withR = G. Moreover, transition from weak multi-dimensional inviscid stability ofreal type to strong multi-dimensional inviscid instability (failure of (1.28)) canoccur only through the infinite speed limit � = ⌧/|e⇠| ! 1 corresponding to one-dimensional inviscid instability, �(0, 1) = 0; see for example discussions in [Er.1](p. 1185), [Fo].15 Likewise (see Remark 1.15), the transition from weak multi-dimensional inviscid stability of real type to strong multidimensional inviscid sta-bility can occur only through collision of (⇠, ⌧) with the glancing set {(⇠, ⌘(⇠))}defined in Definition 1.3: easily calculated for gas dynamics; see, e.g., [Z.3], exam-ple near (3.12), or Appendix C of this article. By rotational invariance and homo-geneity, this transition may be detected by the single condition �(⇠

0

, ⌘(⇠0

)) 6= 0,⇠0

= (1, 0, . . . , 0), similarly as in the previous case. These two observations greatlysimplify the determination of the various inviscid shock stability regions once �has been determined; see, e.g., [BRSZ] or Section 6.5 of [Z.3].16 From our point

14Contributed by K. Jenssen and G. Lyng.15Equivalently, (@/@�)�(e⇠, i⌧) and �(e⇠, i⌧) do not simultaneously vanish for e⇠, ⌧ real (indeed,

(@/@�)�(e⇠, i⌧) does not vanish for (e⇠, ⌧) in R); see Appendix C and Remark 1.15.16Precisely, together with Metivier’s theorem guaranteeing uniform stability of small-amplitude

24 KEVIN ZUMBRUN

of view, however, the main point is simply that, for gas dynamics, transition fromstrong inviscid stability to instability, if it occurs, occurs via passage through theopen region of weak inviscid stability of real type, on which (inviscid) stability isindeterminate.

Remark 1.18. (nonclassical case) Freistuhler [Fre.1–2] has extended the Majdaanalysis to the nonclassical, undercompressive case, obtaining in place of (1.28)–(1.29) and (1.32) the extended, (n+q)⇥(n+q) dimensional determinant conditions

(1.38) �(⇠,�) 6= 0, for all ⇠ 2 Rd�1, Re � > 0 (resp. Re � � 0 and � 6= 0),

�(⇠,�) := det

0

B@

@g

@U�A1

��1

r�1

, . . . ,@g

@U�A1

��1

r�n�i�,

r�1

, . . . , r�n�i�,

@g

@U+

A1

+

�1

r+

i++1

, . . .@g

@U+

A1

+

�1

r+

n , i@g

@!⇠ +

@g

@s�

ri++1

, . . . , r+

n , i[F ˜⇠(U)] + �[U ]

1

CA ,

whereg(U�, U

+

,!, s) = 0, g 2 Rq

encodes the q kinetic conditions needed along with (RH) to determine existence ofan undercompressive connection in direction ! 2 Sd�1 (i.e., a traveling-wave solu-tion U = U(x ·!� st)), and all partial derivatives are taken at the base parameters(U�, U

+

, e1

, 0) corresponding to the one-dimensional stationary profile U = U(x1

).This condition has been evaluated in [B-G.1–2] for the physically interesting case ofphase boundaries, with the result of strong (i.e., uniform) stability for su�cientlysmall but positive viscosities.17 Applied to the overcompressive case, on the otherhand, the analysis of Majda yields automatic instability. For, (1.31) is in this caseill-posed for all e⇠, �, since the matrix on the lefthand side has fewer than n columns.

1.4. Viscous stability analysis.

The formal, matched asymptotic expansion arguments underlying the classical,inviscid stability analysis leave unclear its precise relation to the original, physicalstability problem (1.1). In particular, rigorous validation of matched asymptoticexpansion generally proceeds through stability estimates on the inner solution (vis-cous shock profile), which in the planar case is somewhat circular (recall: the inner

shocks, we recover by these observations the results of the full Nyquist diagram analysis carriedout in Appendix C. (In fact, these two approaches are closely related; see discussion of [Z.3],Section 6.5.)

17Recall that, in the undercompressive case, higher-order derivative terms influence the inviscidproblem through their e↵ect on the connection conditions g.


problem is the planar evolution problem, with additional terms comprising theformal truncation error); hence, we cannot rigorously conclude even necessity ofthese conditions for viscous stability. This is more than a question of mathematicalnicety: in certain physically interesting cases arising in connection with nonclas-sical under- and overcompressive shocks, the classical stability analysis is knownto give wrong results both positive and negative; see Remarks 1.18, 1.20.1, and1.22.5, or Section 1.3, [Z.3]. Moreover, even in the classical Lax case, it is notsu�ciently sharp to determine the transition from stability to instability, either inone or multiple dimensions; see Remarks 1.16 and 1.17.

These and related issues may be resolved by a direct stability analysis at the levelof the viscous equations (1.1), as we now informally describe. Rigorous statementsand proofs will be established throughout the rest of the article. In particular,the viscous theory not only refines (and in some cases corrects) the inviscid theorybut also completes it by the addition of a third, “strong spectral stability” con-dition augmenting the classical conditions of structural and dynamical stability.Strong spectral stability is somewhat weaker than linearized stability– in the one-dimensional case, for example, it is more or less equivalent to stability with respectto zero-mass perturbations [ZH]– and may be recognized as the missing conditionneeded to validate the formal asymptotic expansion; see, for example, the relatedinvestigations [GX, GR, R, MeZ.1, GMWZ.2–4].

1.4.1. Connection with classical theory. The starting point for the viscousanalysis is a linearized, Laplace–Fourier transform analysis very similar in spirit tothat of the inviscid case. This yields the Evans function condition

(1.39) D(e⇠,�) 6= 0, e⇠ 2 Rd�1, Re � > 0,

as a necessary condition for linearized viscous stability, where the Evans functionD (defined in Section 2.2, below) is a determinant analogous to the Lopatinskideterminant �(·, ·) of the inviscid theory. Similarly as in the inviscid case, zeroesof D(·, ·) correspond to growing modes v = e�teie⇠·exw(x

1

) of the linearized equationsabout the background profile U(·).

Definition 1.19. We define weak spectral stability (necessary but not su�cientfor linearized viscous stability) as (1.39). We define strong spectral stability (byitself, neither necessary nor su�cient for linearized stability) as

(1.40) D(e⇠,�) 6= 0 for (e⇠,�) 2 Rd�1 ⇥ {Re � � 0} \ {0, 0}.

The fundamental relation between viscous and inviscid theories, quantifying theformal rescaling argument alluded to in (1.22), Section 1.3, is given by

Result 1 [ZS, Z.3]. (Thm 3.5) D(e⇠,�) ⇠ ��(e⇠,�) as (e⇠,�) ! (0, 0), where � is aconstant measuring transversality of the connection U(·) in (1.1), i.e., D is tangent

26 KEVIN ZUMBRUN

to � in the low frequency limit. Here, � in the Lax or undercompressive case isgiven by (1.29) or (1.38), respectively, and in the overcompressive case by a modifieddeterminant correcting the inviscid theory, as described in equations (3.9)–(3.10),Theorem 3.5.

That is, in the low-frequency limit (⇠ large-space–time, by the standard dualitybetween spatial and frequency variables), higher-derivative terms become neglible in(1.1), and behavior is governed by its homogeneous first-order (inviscid) part (1.4).The departure from inviscid theory in the overcompressive case reflects a subtlebut important distinction between low-frequency/large-space–time and vanishing-viscosity limits, as discussed in Section 1.3, [Z.3].18

1.4.2. One-dimensional stability. The most dramatic improvement in theviscous theory over the inviscid theory comes in the one-dimensional case, for whichthe inviscid dynamical stability condition yields essentially no information, Remark1.16. From Result 1, we find that vanishing of the one-dimensional Lopatinski de-terminant�(0, 1) generically corresponds to crossing from the stable to the unstablecomplex half-plane Re � > 0 of a root of the Evans function D. Indeed, we have

Result 2 [Z.3–4]. (Props. 4.2, 4.4, and 4.5) With appropriate normalization, thecondition ��(0, 1) > 0 is necessary for one-dimensional linearized viscous stability.

More precisely, we have the following sharp one-dimensional stability criterion.

Result 3 [MaZ.3–4]. (Thms. 4.8 and 4.9) Necessary and su�cient conditionsfor linearized and nonlinear one-dimensional viscous stability are one-dimensionalstructural and dynamical stability, ��(0, 1) 6= 0, plus strong spectral stability,D(0,�) 6= 0 on {Re � � 0} \ {0}.

Remarks 1.20. 1. Result 2 replaces the codimension one inviscid instabilitycondition ��(0, 1) = 0 with an open condition ��(0, 1) > 0 of which it is theboundary. That is, one-dimensional inviscid instability theory marks the bound-ary between one-dimensional viscous stability and instability. The strengthened,viscous condition ��(0, 1) > 0 in particular resolves the problem of physical se-lection discussed in [MeP], p. 110, for which multiple Riemann solutions exist,each satisfying the classical conditions of dynamical and structural stability, butonly one– the one satisfying the one-dimensional viscous stability condition– is se-lected by numerical experiment; see related discussion, Section 6.3, [Z.3]. Likewise,it yields instability of undercompressive, pulse-type (i.e., homoclinic) profiles aris-ing in phase-transitional flow [GZ, Z.5], which, again, satisfy the classical inviscidstability criteria but numerically appear to be unstable [AMPZ.1]. Result 3 as

18Briefly, initial data scales as " in the low-frequency limit, but in the vanishing-viscosity limitis fixed, order one. For "-order data, underdetermination of the inner problem associated withnonuniqueness of overcompressive orbits can balance with overdetermination of the outer problem;for general data, the e↵ects occur on di↵erent scales. For related discussion outside the matchedasymptotic setting, see [Fre.3], [L.2], or [FreL].


claimed completes the inviscid stability conditions with the addition of the thirdcondition of strong spectral stability. Spectral stability has been verified for small-amplitude, Lax-type profiles in [HuZ, PZ], and for special large-amplitude profilesin [MN.1] and [Z.5]. E�cient algorithms for numerical testing of spectral stabilityare described, e.g., in [Br.1–2, BrZ, KL, BDG].

2. It is an interesting and physically important open question which of the threecriteria of structural, dynamical, or spectral stability is the more restrictive in dif-ferent regimes; in particular, it would be very interesting to know whether, or underwhat circumstances, spectral stability can be the determining factor in transition toinstability as |✓| is gradually increased along the Hugoniot curve U

+

= Hp(U�, ✓)defined in Section 1.3.1. The latter question is closely related to the following gener-alized Sturm–Liouville question: Do the global symmetry assumptions (1.11)–(1.9),or perhaps the stronger condition that the system (1.1) possess a global convex en-tropy, as discussed in Section 1.1, preclude the existence of nonzero pure imaginaryeigenvalues? For, as pointed out in [ZH,Z.3], the same calculations used to es-tablish the signed stability condition show that eigenvalues can cross through theorigin from stable to unstable complex half-plane only if �� = 0, i.e., � vanishesor transversality is lost. Thus, beginning with small-amplitude profiles (stable, by[HuZ,PZ]) and continuously varying the endstate U

+

, we find that loss of stability,if it occurs before structural or dynamical stability fail, must correspond to a pairof complex-conjugate eigenvalues crossing the imaginary axis, presumably signalinga Poincare–Hopf-type bifurcation19 to a family of time-periodic solutions.

Recall that such “galloping” instabilities do in fact occur in the closely relatedsituation of detonation (see, e.g., [Ly.1,LyZ.1–2]); thus, the question of whether ornot or under what circumstances they can occur for shock waves seems physicallyquite interesting. If they cannot occur for the class of systems we consider, then therelatively simple conditions of structural and dynamical stability– i.e., the classicalconditions of inviscid stability– completely determine the transition from viscousstability to instability, a very satisfactory conclusion. If, on the other hand, they canoccur, then this implies that heretofor neglected viscous e↵ects play a far strongerrole in stability than we have imagined, and this result would be philosophicallystill more significant.

1.4.3. Multi-dimensional stability. A deficiency of the inviscid multi-dimensional stability theory is that transition from strong inviscid stability to stronginviscid instability typically occurs through the open region of weak inviscid stabil-ity of real type, on which stability is indeterminate; see Remarks 1.15 and 1.17, orSection 1.4, [Z.3]. This indeterminacy may be remedied by a next-order correctionin the low-frequency limit including second-order, viscous e↵ects. There is alsosome question [BE] whether the physical transition to instability might sometimesoccur before the passage to weak inviscid instability, i.e., for parameters within the

19Degenerate, because of the well-known absence [Sat] of a spectral gap between � = 0 andthe essential spectrum of the linearized operator about the wave.

28 KEVIN ZUMBRUN

region of strong inviscid stability. This likewise can be determined, at least in prin-ciple, by the inclusion of neglected viscous e↵ects, this time in the (global) form ofthe strong spectral stability condition.

Definition 1.21. We define weak refined dynamical stability as inviscid weakstability (1.28) augmented with the second-order condition that Re �(e⇠, i⌧) � 0for any real e⇠, ⌧ such that � (hence also D

e⇠,i⌧ (⇢) := D(⇢e⇠, ⇢i⌧), by Result 1) isanalytic at (e⇠, i⌧), �(e⇠, i⌧) = 0, and (@/@�)�(e⇠, i⌧) 6= 0, where

(1.41) �(e⇠, i⌧) :=(@/@⇢)`+1D(⇢e⇠, ⇢i⌧)|⇢=0

(@/@�)�(e⇠, i⌧)=

(@/@⇢)`+1D(⇢e⇠, ⇢�)(@/@�)(@/@⇢)`D(⇢e⇠, ⇢�) |

⇢=0,�=i⌧

.

We define strong refined dynamical stability as inviscid weak stability (1.28) aug-mented with the conditions that � be analytic at (e⇠, i⌧), (@/@�)�(e⇠, i⌧) 6= 0,

(1.42) {r�1

, . . . , r�p�1

, r+

p+1

, . . . , r+

n }(⇠, i⌧)

be independent for r±j defined as in (1.29) (automatic for “extreme” shocks p = 1or n),20 and Re �(e⇠, i⌧) > 0 for any real e⇠, ⌧ such that �(e⇠, i⌧) = 0. Condition��(e⇠, i⌧) 6= 0 implies that such imaginary zeroes are confined to a finite union ofsmooth curves

(1.43) ⌧ = ⌧j(e⇠).

Result 4 [Z.3]. (Cor. 5.2) Given one-dimensional inviscid and stuctural stability,��(0, 1) 6= 0, weak refined dynamical stability is a necessary condition for multi-dimensional linearized viscous stability (indeed, for weak spectral stability as well).

Result 5 (New). (Thms. 5.4–5.5) Structural and strong refined dynamical stabil-ity together with strong spectral stability are su�cient for linearized viscous stabilityin dimensions d � 2 and for nonlinear viscous stability in dimensions d � 3 (Laxor overcompressive case) or d � 4 (undercompressive case). In the strongly invis-cid stable Lax or overcompressive case, we obtain nonlinear viscous stability in alldimensions d � 2.

Remarks 1.22. 1. By the parenthetical comment in Result 4, we may substitutein Result 5 for strong refined dynamical stability the condition of weak inviscidstability together with the nondegeneracy conditions that for any real e⇠, ⌧ forwhich �(e⇠, i⌧) = 0, e⇠, ⌧ lie o↵ of the glancing set (⌧ 6= ⌘q(e⇠) in the notation ofDefinition 1.3), (@/@�)�(e⇠, i⌧) 6= 0, and �(e⇠, i⌧) 6= 0. This gives a formulationmore analogous to that of Result 3 in the one-dimensional case.

20This condition is for Lax or overcompressive shocks, and must be replaced by a modifiedversion in the undercompressive case [Z.3].


2. Results 4 and 5 together resolve the indeterminacy associated with transitionto instability through the open region of weak inviscid stability of real type, atleast in dimensions d � 3.21 For, � is analytic at (e⇠, i⌧) for any real (e⇠, ⌧) not lyingin the glancing set described in Definition 1.3; see discussion below Remark 1.4.Thus, referring to Remark 1.15, we find that the transition from refined dynamicalstability to instability must occur at one of three codimension-one sets: the bound-ary between strong inviscid stability and weak inviscid stability, marked either byone-dimensional inviscid instability �(0, 1) = 0 or coalescence of two roots (e⇠, i⌧)at a double zero of �; the boundary between weak and strong inviscid stability,at which (e⇠, ⌧) associated with a root (e⇠, i⌧) of � strikes the glancing set; or elsethe (newly defined) boundary between refined dynamical stability and instability,which (e⇠, ⌧) associated with a root (e⇠, i⌧) of � strikes the set �(e⇠, i⌧) = 0 within theinterior of R. In the scalar case, �(e⇠, i⌧) is uniformly positive and, moreover, � islinear, hence globally analytic with simple roots; indeed, viscous stability holds forprofiles of arbitrary strength [HoZ.3–4, Z.3]. More generally, � can be expressed interms of a generalized Melnikov integral, as described in [BSZ.2, B-G.4]; evaluation(either numerical or analytical) of this integral in physically interesting cases is animportant area of current investigation.

3. The condition in the definition of strong refined stability that � be analyticat roots on the imaginary boundary is generically satisfied in dimension d = 2,or for rotationally invariant systems such as gas dynamics. However, it typicallyfails for general systems in dimensions d � 3. A weaker condition appropriateto this general case is to require that the curve of imaginary zeroes intersect theglancing set transversely. At the expense of further complications, all of our analysisgoes through with this relaxed assumption [Z.4]. Moreover, it fails again on acodimension-one boundary.

4. Multi-dimensional spectral stability has been verified for small-amplitudeLax-type profiles in [FreS.2, PZ], completing the verification of viscous stability byResult 5 and the small-amplitude structural and dynamical stability theorems ofPego and Metivier. Classification (either numerical or analytical) of large-amplitudestability, and the associated question of whether refined dynamical stability ormulti-dimensional spectral stability in practice is the determining factor, remainoutstanding open problems. In particular, large-amplitude multidimensional spec-tral stability has so far not been verified for a single physical example. (It isautomatic in the scalar case, by the maximum principle [HoZ.3–4]. It has beenverified in [PZ] for a class of artificial systems constructed for this purpose.)

5. Freistuhler and Zumbrun [FreZ.2] have recently established for the over-compressive version of � defined in (3.9)–(3.10), Theorem 3.5, a weak version ofMetivier’s theorem analogous to that presented for the Lax case in Appendix B, as-

21Stability is more delicate in the critical dimension d = 2 or the undercompressive case, andrequires a refined analysis as discussed respectively in Remark 2, end of Section 5.2.3, and Section5.6, [Z.3].

30 KEVIN ZUMBRUN

serting uniform low-frequency stability under the assumptions of Metivier plus theadditional nondegeneracy condition of invertibility of an associated one-dimensional“mass map” introduced in [FreZ.1] taking potential time-asymptotic states to cor-responding initial perturbation mass. This agrees with numerical and analyticalevidence given in [Fre.3, FreL, L.2, FreP.1–2, DSL], in contrast to the result ofautomatic inviscid instability described in Remark 1.18. The corresponding inves-tigation of small-amplitude spectral stability remains an important open problemin both the over- and undercompressive case.

Plan of the paper. In Section 2, we recall the general Evans function andinverse Laplace transform (C0-semigroup) machinery needed for our analysis. InSection 3, we construct the Evans function and establish the key low-frequencylimit described in Result 1. In Section 4, we carry out the one-dimensional viscoustheory described in Results 2 and 3, and in Section 5 the multi-dimensional viscoustheory described in Results 4 and 5.

2. PRELIMINARIES.

Consider a profile U of a hyperbolic–parabolic system (1.1), satisfying assump-tions (A1)–(A3), (H0)–(H3). By the change of coordinate frame x

1

! x1

� st ifnecessary, we may arrange that U ⌘ U(x

1

) be a standing-wave solution, i.e., anequilibrium of (1.1). We make this normalization throughout the rest of the paper,fixing s = 0 once and for all.

Linearizing (1.1) about the stationary solution U(·) gives

(2.1) Ut = LU :=X

j,k

(BjkUxk

)xj

�X

j

(AjU)xj

,

where

(2.2) Bjk := Bjk(U(x1

))

and

(2.3) AjU := dF j(U(x1

))U � dBj1(u(x1

))(U, Ux1)

are Cs�1 functions of x1

alone. In particular,

(2.4) (Aj11

, Aj12

) = (dF j11

, dF j12

).

Taking now the Fourier transform in transverse coordinates ex = (x2

, · · · , xd), weobtain

(2.5)bUt = Le⇠

bU :=

L0 bUz }| {(B11 bU 0)0 � (A1 bU)0�i

X

j 6=1

Aj⇠j bU

+ iX

j 6=1

Bj1⇠j bU 0 + iX

k 6=1

(B1k⇠k bU)0 �X

j,k 6=1

Bjk⇠j⇠k bU,


where “ 0 ” denotes @/@x1

, and bU = bU(x1

, e⇠, t) denotes the Fourier transform ofU = U(x

1

, x, t). Hereafter, where there is no danger of confusion we shall drop thehats and write U for bU . Operator Le⇠ reduces at e⇠ = 0 to the linearized operatorL

0

associated with the one-dimensional stability problem.A necessary condition for stability is that the family of linear operators Le⇠ have

no unstable point spectrum, i.e. the eigenvalue equations

(2.6) (Le⇠ � �)U = 0

have no solution U 2 L2(x1

) for e⇠ 2 Rd�1, Re � > 0. For, unstable (Lp) pointspectrum of Le⇠ corresponds to unstable (Lp) essential spectrum of the operatorL for p < 1, by a standard limiting argument (see e.g. [He, Z.1]), and unstablepoint spectrum for p = 1. This precludes Lp ! Lp stability, by the Hille–Yosidatheorem (see, e.g. [He, Fr, Pa, Z.1]). Moreover, standard spectral continuity results[Kat, He, Z.1] yield that instability, if it occurs, occurs for a band of e⇠ values, fromwhich we may deduce by inverse Fourier transform the exponential instability of(2.1) for test function initial data U

0

2 C10

, with respect to any Lp, 1 p 1.We shall see later that linearized stability of constant solutions U ⌘ U± (embodiedin (A1)–(A3)) together with convergence at x

1

! ±1 of the coe�cients of L,implies that all spectrum of Le⇠ is point spectrum on the unstable (open) complexhalf-plane Re � > 0;22 thus, we are not losing any information by restricting topoint spectrum in (2.6). Of course, condition (2.6) is not su�cient for even neutral,or bounded, linearized stability of (2.1).

2.1. Spectral resolution formulae.

We begin by deriving spectral resolution, or inverse Laplace Transform formulae

(2.7) eLe⇠

t� =1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�t(Le⇠ � �)�1� d�,

and

(2.8) Ge⇠(x1

, t; y1

) =1

2⇡iP.V.

Z ⌘+i1

⌘�i1eiktGe⇠,�(x1

, y1

) d�,

for the solution operator eLe⇠

t and Green distribution Ge⇠(x1

, t; y1

) := eLe⇠

t�y1(x1

)associated with the Fourier-transformed linearized evolution equations (2.5), whereGe⇠,�(x1

, y1

) denotes the resolvent kernel (Le⇠ � �)�1�y1(x1

), and ⌘ is any real con-stant greater than or equal to some ⌘

0

(e⇠).

22In the strictly parabolic case Re �P

⇠j

⇠k

Bjk > 0, or for any asymptotically constant op-erator L of nondegerate type, this follows by a standard argument of Henry; see [He, Z.1]. Moregenerally, it holds whenever the eigenvalue equation may be expressed as a nondegenerate first-order ODE [MaZ.3].

32 KEVIN ZUMBRUN

Lemma 2.1. Operator Le⇠ satisfies

(2.9)|U |H1 + |B11U |H2 C(e⇠)(|Le⇠U |L2 + |U |L2),

|SU |L2 |�� ⇤|�1|S(Le⇠ � �)U |L2 ,

for all U 2 H := {f : f 2 H1, B11f 2 H2}, for some well-conditioned coordinatetransformation U ! S(x

1

)U , supx1|S||S�1| C(e⇠), and all real � greater than

some value �⇤ = �⇤(e⇠).

Proof. Consider the resolvent equation (Le⇠ ��)U = f . By the lower triangularcoordinate transformation U ! S

1

U ,

(2.10) S1

:=✓

I 0�b11

1

(b11

2

)�1 I

◆,

we may convert to the case that b11

1

⌘ 0, with conjugation error EU = E1

Ux+E0

U ,

E1

=✓

0 0e1

e2

◆,

where b11

2

and A1

⇤ := A1

11

� A1

12

b11

1

(b11

2

)�1 = A1

11

retain their former values, andthus their former properties of strict parabolicity and hyperbolicity with constantmultiplicity, respectively. By a further, block-diagonal transformation, we mayarrange also that b11

2

be strictly positive definite and A1

11

symmetric; moreover, thismay be chosen so that the combined transformation is uniformly well-conditioned.

Setting � = 0 and taking the L2 inner product of (A1

⇤uIx, b11

2

uIIxx) against the

resulting equation (Le⇠ + E)U = f , we obtain after some rearrangement the bound

C�1(|uIx|L2 + |uII

xx|L2) |A1

⇤uIx|L2 + |b11

2

uIIxx|L2 C(|U |L2 + |uII |H1),

which in the original coordinates implies (2.9)(i). Note that we have so far onlyused nondegeneracy of the respective leading order coe�cients A1

⇤ and b11

2

in theuI and uII equations.

Taking the real part of the L2 inner product of U against the full resolventequation (Le⇠ + E � �)U = f , integrating by parts, using the assumptions on A1

11

,b11

2

, and bounding terms of order |U |L2 |uIIx |L2 by Youngs inequality as O(|U |2L2) +

✏|uIIx |2L2 with ✏ as small as needed, we obtain after some rearrangement the estimate

(2.11) Re �|U |2L2 + O(|U |2L2) + ✓|uIIx |2L2 |f |L2 |U |L2 ,

whence we obtain the second claimed bound

(2.12) |U |L2 |�� ⇤|�1|f |L2 = |�� ⇤|�1|(Le⇠ � �)U |L2

for all � > �⇤ on the real axis, and �⇤ su�ciently large. ⇤


Corollary 2.2. Operator Le⇠ is closed and densely defined on L2 with domain H,generating a C0 semigroup eLe

⇠

t satisfying |eLe⇠

t|L2 Ce!t for some real ! = !(e⇠).

Proof. The first assertion follows in standard fashion from bound (2.9)(i);see, e.g., [Pa] or [Z.1]. The bound (2.9)(ii) applies also to the limiting, constant-coe�cient operators Le⇠± as x ! ±1, whence the spectra of these operators isconfined to Re � �⇤. Because the eigenvalue equation (Le⇠ � �)U = 0, after thecoordinate transformation described in the proof of Lemma 2.1 above, may evi-dently be written as a nondegenerate ODE in (U, uII0), we may conclude by thegeneral theory described in Section 2.4 below that the essential spectrum of Le⇠ isalso confined to the set Re � �⇤; see also Proposition 4.4. of [MaZ.3]. (Thisobservation generalizes a standard result of Henry ([He], Lemma 2, pp. 138–139)in the case of a nondegenerate operator L.) Since (2.9) precludes point spectrumfor real � > �⇤, we thus find that all such � belong to the resolvent set ⇢(Le⇠), withthe resolvent bound (after the coordinate transformation described above)

(2.13) |(Le⇠ � �)�1|L2 |�� ⇤|�1.

But, this is a standard su�cient condition that a closed, densely defined operatorLe⇠ generate a C0 semigroup, with |eLe

⇠

t|L2 Ce!t for all real ! > �⇤ (see e.g. [Pa],Theorem 5.3, or [Fr,Y]). ⇤

Corollary 2.3. For Le⇠, eLe⇠

t as in Corollary 2.2, the inverse Laplace Transform(spectral resolution) formula (2.7) holds for any real ⌘ greater than some ⌘

0

, forall � 2 D(Le⇠ · Le⇠), where domain D(Le⇠ · Le⇠) is defined as in [Pa], p. 1. Likewise,(2.8) holds in the distributional sense.23

Proof. The first claim is a general property of C0 semigroups (see, e.g. [Pa],Corollary 7.5). The second follows from the first upon pairing with test functions� (see, e.g., [MaZ.3], Section 2). ⇤

Remarks 2.4. 1. By standard semigroup properties, the Green distributionGe⇠(x, t; y) satisfies

(i) (@t � Le⇠)Ge⇠(·, ·; y1

) = 0 in the distributional sense, for all t > 0, and(ii) Ge⇠(x1

, t; y1

) * �(x� y) as t ! 0.Here, Ge⇠ is to be interpreted in (i) as a distribution in the joint variables (x

1

, y1

, t),and in (ii) as a distribution in (x

1

, y1

), continuously parametrized by t. We shallsee by explicit calculation that Ge⇠ so defined (uniquely, by uniqueness of weaksolutions of (2.5) within the class of test function initial data), is a measure but not

23Note: From the definition in [Pa], we find that the domain D(Le⇠

) of Le⇠

satisfies H2 ⇢ H ⇢D(Le

⇠

). Likewise, the domain D(Le⇠

·Le⇠

), consisting of those functions V such that Le⇠

V 2 D(Le⇠

),

satisfies H4 ⇢ D(Le⇠

· Le⇠

).

34 KEVIN ZUMBRUN

a function. Note that, on the resolvent set ⇢(L), the resolvent kernel Ge⇠,� likewisehas an alternative, intrinsic characterization as the unique distribution satisfying

(2.14) (Le⇠ � �)Ge⇠,�(x, y) = �(x, y)

and taking f 2 L2 to hGe⇠,�(x, y), f(y)iy 2 L2. This is the characterization that weshall use in constructing Ge⇠,� in Section 2.4.

2. A semigroup on even a still more restricted function class than L2 would havesu�ced in this construction, since we are constructing objects in the very weakclass of distributions. Later, by explicit computation, we will verify that Le⇠ in factgenerates a C0 semigroup in any Lp. Note also that (2.8) implies the expected,standard solution formula eLe

⇠

tf = hGe⇠(x, t; y), f(y)iy, or, formally:

(2.15) eLe⇠

tf =Z

Ge⇠(x, t; y)f(y)dy,

for f in any underlying Banach space on which eLe⇠

t is defined, in this case on Lp,1 p 1.

3. As noted in [MaZ.1, MaZ.3], there is another, more concrete route to (2.7)–(2.8) and (2.15), generalizing the approach followed in [ZH] for the parabolic case.Namely, we may observe that, in each finite integral in the approximating sequencedefined by the principal value integral (2.8), we may exchange orders of integrationand distributional di↵erentiation, using Fubini’s Theorem together with the fact,established in the course of our analysis, that Ge⇠,� is uniformly bounded on thecontours under consideration (in fact, we establish the much stronger result thatGe⇠,� decays exponentially in |x� y|, with uniform rate), to obtain

(2.16)(@t � Le⇠)

Z ⌘+iK

⌘�iK

e�tGe⇠,�(x, y) d� = �(x� y)Z ⌘+iK

⌘�iK

e�t d�

* �(x� y)⌦ �(t)

as K ! 1, for all t � 0. So, all that we must verify is: (i’) the Principal Valueintegral (2.8) in fact converges to some distribution Ge⇠(x, t; y) as K !1; and, (ii’)Ge⇠(x, t; y) has a limit as t ! 0. For, distributional limits and derivatives commute(essentially by definition), so that (i’)–(ii’) together with (2.16) imply (i)–(ii) above.Facts (i’)–(ii’) will be established by direct calculation in the course of our analysis,thus verifying formula (2.8) at the same time that we use it to obtain estimateson Ge⇠. Note that we have made no reference in this argument to the semigroupmachinery cited above.


2.2. Evans function framework.

We next recall the basic Evans function/asymptotic ODE tools that we will need.

2.2.1. The gap/conjugation lemma. Consider a family of first-order systems

(2.17) W 0 = A(x,�)W,

where � varies within some domain ⌦ ⇢ Ck and x varies within R1. Equations(2.17) are to be thought of as generalized eigenvalue equations, with parameter �representing a suite of frequencies. We make the basic assumption:

(h0) Coe�cient A(·,�), considered as a function from ⌦ into L1(x) is analyticin �. Moreover, A(·,�) approaches exponentially to limits A±(�) as x ! ±1, withuniform exponential decay estimates

(2.18) |(@/@x)k(A� A±)| C1

e�✓|x|/C2 , for x ? 0, 0 k K,

Cj , ✓ > 0, on compact subsets of ⌦.

Then, there holds the following conjugation lemma of [MeZ.1], a refinement ofthe “gap lemma” of [GZ,KS], relating solutions of the variable-coe�cient ODE(2.17) to solutions of its constant-coe�cient limiting equations as

(2.19) Z 0 = A±(�)Z

as x ! ±1.

Lemma 2.5[MeZ.1]. Under assumption (h0) alone, there exists locally to any given�

0

2 ⌦ a pair of linear transformations P+

(x,�) = I + ⇥+

(x,�) and P�(x,�) =I +⇥�(x,�) on x � 0 and x 0, respectively, ⇥± analytic in � as functions from⌦ to L1[0,±1), such that:

(i) |P±| and their inverses are uniformly bounded, with

(2.20) |(@/@�)j(@/@x)k⇥±| C(j)C1

C2

e�✓|x|/C2 for x ? 0, 0 k K + 1,

j � 0, where 0 < ✓ < 1 is an arbitrary fixed parameter, and C > 0 and the size ofthe neighborhood of definition depend only on ✓, j, the modulus of the entries of Aat �

0

, and the modulus of continuity of A on some neighborhood of �0

2 ⌦.(ii) The change of coordinates W := P±Z reduces (2.17) on x � 0 and x 0,

respectively, to the asymptotic constant-coe�cient equations (2.19).

Equivalently, solutions of (2.17) may be conveniently factorized as

(2.21) W = (I +⇥±)Z±,

where Z± are solutions of the constant-coe�cient equations (2.19), and ⇥± satisfybounds (2.20).

36 KEVIN ZUMBRUN

Proof. As described in [MeZ.1], for j = k = 0 this is a straighforward corollaryof the gap lemma as stated in [Z.3], applied to the “lifted” matrix-valued ODEfor the conjugating matrices P±; see also Appendix A.3, below. The x-derivativebounds 0 < k K + 1 then follow from the ODE and its first K derivatives.Finally, the �-derivative bounds follow from standard interior estimates for analyticfunctions. ⇤

Definition 2.6. Following [AGJ], we define the domain of consistent splittingfor problem (2.17) as the (open) set of � such that the limiting matrices A

+

and A�are hyperbolic (have no center subspace), and the dimensions of their stable (resp.unstable) subspaces S

+

and S� (resp. unstable subspaces U+

and U�) agree.

Lemma 2.7. On any simply connected subset of the domain of consistent splitting,there exist analytic bases {V

1

, . . . , Vk}± and {Vk+1

, . . . , VN}± for the subspaces S±and U± defined in Definition 2.6.

Proof. By spectral separation of U±, S±, the associated (group) eigenprojec-tions are analytic. The existence of analytic bases then follows by a standard resultof Kato; see [Kat], pp. 99–102. ⇤

By Lemma 2.5, on the domain of consistent splitting, the subspaces

(2.22) S+ = Span{W+

1

, . . . ,W+

k } := Span{P+

V +

1

, . . . , P+

V +

k }

and

(2.23) U� := Span{W�k+1

, . . . ,W�N } := Span{P�V �k+1

, . . . , P�V �N }

uniquely determine the stable manifold as x ! +1 and the unstable manifold asx ! �1 of (2.17), defined as the manifolds of solutions decaying as x ! ±1,respectively, independent of the choice of P±.

Definition 2.8. On any simply connected subset of the domain of consistentsplitting, let V +

1

, . . . , V +

k and V �k+1

, . . . , V �N be analytic bases for S+

and U�, asdescribed in Lemma 2.7. Then, the Evans function for (2.17) associated with thischoice of limiting bases is defined as

(2.24)D(�) := det

⇣W+

1

, . . . ,W+

k ,W�k+1

, . . . ,W�N

⌘

|x=0,�

= det⇣P

+

V +

1

, . . . , P+

V +

k , P�V �k+1

, . . . , P�V �N⌘

|x=0,�,

where P± are the transformations described in Lemma 2.5.

Remark 2.9. Note that D is independent of the choice of P±; for, by uniquenessof stable/unstable manifolds, the exterior products (minors) P

+

V +

1

^ · · · ^ P+

V +

k

and P�V �k+1

^ · · · ^ P�V �N are uniquely determined by their behavior as x ! +1,�1, respectively.


Proposition 2.10. Both the Evans function and the stable/unstable subspaces S+

and U� are analytic on the entire simply connected subset of the domain of consis-tent splitting on which they are defined. Moreover, for � within this region, equation(2.17) admits a nontrivial solution W 2 L2(x) if and only if D(�) = 0.

Proof. Analyticity follows by uniqueness, and local analyticity of P±, V ±k .

Noting that the first k columns of the matrix on the righthand side of (2.24) are abasis for the stable manifold of (2.17) at x ! +1, while the final N � k columnsare a basis for the unstable manifold at x ! �1, we find that its determinantvanishes if and only if these manifolds have nontrivial intersection, and the secondassertion follows. ⇤

Remarks 2.11. 1. In the case that (2.17) describes an eigenvalue equationassociated with an ordinary di↵erential operator L, � 2 C1, Proposition 2.10 impliesthat eigenvalues of L agree in location with zeroes of D. In [GJ.1–2], Gardner andJones [GJ.1–2] have shown that they agree also in multiplicity; see also Lemma 6.1,[ZH], or Proposition 6.15 of [MaZ.3].24

2. If, further, L is a real-valued operator (i.e., has real coe�cients), or, moregenerally, A has complex symmetry A(x, �) = A(x,�), where bar denotes complexconjugate, then D may be chosen with the same symmetry

(2.25) D(�) = D(�).

For, all steps in the construction of the Evans function respect complex symmetry:projection onto stable/unstable subspaces by the spectral resolution formula

P±(�) =1

2⇡i

I

�±

(A±(�)� µ)�1 dµ

(�± denoting fixed contours enclosing stable/unstable eigenvalues of A±), choice ofasymptotic bases V±(�) by solution of the analytic ODE prescribed by Kato [Kat],and finally conjugation to variable-coe�cient solutions by a contraction mappingthat likewise respects complex symmetry.

2.2.2. The tracking/reduction lemma. Next, consider the complementarysituation of a family of equations of form

(2.26) W 0 = A✏(x,�)W,

on an (✏,�)-neighborhood for which the coe�cient A✏ does not exhibit uniformexponential decay to its asymptotic limits, but instead is slowly varying. Thisoccurs quite generally for rescaled eigenvalue equations arising in the study of thelarge-frequency regime; see, e.g., [GZ,ZH,MaZ.1,Z.3].

24The latter result applies specifically to ordinary di↵erential operators of degenerate type.

38 KEVIN ZUMBRUN

In this situation, it frequently occurs that not only A✏ but also certain of itsinvariant (group) eigenspaces are slowly varying with x, i.e., there exist matrices

(2.27) L✏ =✓

L✏1

L✏2

◆(x), R✏ = (R✏

1

R✏2

) (x)

for which L✏R✏(x) ⌘ I and |LR0| = |L0R| is small relative to

(2.28) M✏ := L✏A✏R✏(x) =✓

M ✏1

00 M ✏

2

◆(x),

where “0” as usual denotes @/@x. In this case, making the change of coordinatesW ✏ = R✏Z, we may reduce (2.26) to the approximately block-diagonal equation

(2.29) Z✏0 = M✏Z✏ + �✏⇥✏Z✏,

where M✏ is as in (2.28), ⇥✏(x) are uniformly bounded matrices, and �✏(x) �(✏)is a (relatively) small scalar.

Let us assume that such a procedure has been successfully carried out, and,moreover, that the approximate flows (i.e., solution operators) Fy!x

j generated bythe decoupled equations

(2.30) Z 0 = M ✏j Z

are uniformly exponentially separated to order 2⌘(✏), in the sense that

(2.31) |Fy!x1

||(Fy!x2

)�1| Ce�2⌘(✏)|x�y| for x � y.

Remark 2.12. A su�cient condition for and the usual means of verification of(2.31) is existence of an approximate uniform spectral gap:

(2.32) max�(Re (M ✏1

))�min�(Re (M ✏2

)) �2⌘(✏) + ↵✏(x)

for all x, ⌘(✏) > 0, where ↵✏ is uniformly integrable,R|↵✏(x)| dx C, or, alter-

natively, existence of flows conjugate to Fy!xj and satisfying this condition. Here,

Re (M) := (1/2)(M⇤ + M) denotes the symmetric part of of a matrix or linearoperator M . Note that an exact uniform spectral gap (2.32)) with ↵✏ ⌘ 0 maybe achieved from (2.32) by the coordinate change Z✏

1

! !Z✏1

, where ! is a well-conditioned (scalar) exponential weight defined by !0 = �↵✏!, !(0) = 1. From theexact version, we may obtain (2.31) by standard energy estimates, with C = 1.

Then, there holds the following reduction lemma, a refinement established in[MaZ.1] of the “tracking lemma” given in varying degrees of generality in [GZ,ZH,Z.3].


Proposition 2.13[MaZ.1]. Consider a system (2.29) satisfying the exponentialseparation assumption (2.31), with ⇥✏ uniformly bounded for ✏ su�ciently small.If, for 0 < ✏ < ✏

0

, the ratio �(✏)/⌘(✏) of formal error vs. spectral gap is su�cientlysmall relative to the bounds on ⇥, in particular if �(✏)/⌘(✏) ! 0, then, for all0 < ✏ ✏

0

, there exist (unique) linear transformations �✏1

(x,�) and �✏2

(x,�),possessing the same regularity with respect to the parameters ✏, � as do coe�cientsM✏ and �(✏)⇥✏, for which the graphs {(Z

1

,�✏2

Z1

)} and {(�✏1

Z2

, Z2

)} are invariantunder the flow of (2.29), and satisfying

(2.33) |�✏j |, |@x�✏j | C�(✏)/⌘(✏) for all x.

Proof. As described in Appendix C, [MaZ.1], this may be established by acontraction mapping argument carried out for the projectivized “lifted” equationsgoverning the flow of exterior forms; see, e.g., [Sat] for a corresponding argumentin the case that M

1

, M2

are scalar. We give a more direct, “matrix-valued” versionof this argument in Appendix A.3, below, from which one may obtain further anexplicit (but rather complicated) Neumann series for �✏j in powers of �/⌘. ⇤

From Proposition 2.13, we obtain reduced flows

(2.34) Z✏1

0 = M ✏1

Z✏1

+ �✏(⇥11

+⇥✏12

�✏2

)Z✏1

= M ✏1

Z✏1

+O(�✏(x))Z✏1

and

(2.35) Z✏2

0 = M ✏2

Z✏2

+ �✏(⇥22

+⇥✏21

�✏1

)Z✏2

= M ✏2

Z✏2

+O(�✏(x))Z✏2

on the two invariant manifolds described. Let us focus on the flow (2.34), assumingwithout loss of generality (by a centering exponential weighting if necessary) that

(2.36) |Fy!x1

| Ce�⌘(✏)|x�y| for x y.

Corollary 2.14 [MaZ.1]. Assuming (2.36), the flow Fy!x1

of the reduced equation(2.34) satisfies

(2.37) Fy!x1

= Fy!x1

+JX

j=1

(�/⌘)jEj(x, y, ✏) +O(�/⌘)J+1e�˜✓⌘|x�y|,

for x y, for any 0 < ✓ < 1, where Fy!x1

is the flow of the associated decoupledsystem (2.30); the iterated integrals(2.38)

Ej+1

:= ⌘

Z x

y

Fz!x1

�⇥

11

+⇥12

�2

�(z, ✏)Ej(z, y, ✏) dz, E

0

(x, y, ✏) := Fy!x1

,

40 KEVIN ZUMBRUN

satisfy the uniform exponential decay estimates

(2.39) |Ej(x, y, ✏)| Cje�˜✓⌘|x�y|,

and are as smooth in x and y and as smooth in ✏ as is�⇥

11

+⇥12

�2

�; and Cj > 0

and O(·) are uniform in ✏, depending only on ✓, Mj, and the constant C in (2.33).Symmetric bounds hold for the flow Fy!x

2

of (2.35).

Proof. This follows by a contraction mapping argument similar to (but simplerthan) that used in the proof of Proposition 2.13, where expansion (2.37) is justthe Jth-order Neumann expansion with remainder associated with the contractionmapping; see Appendix C of [MaZ.1], or Appendix A.3 of this paper. ⇤

2.2.3. Formal block-diagonalization. To complete our discussion of ODEwith slowly varying coe�cients, we now supply a formal block-diagonalization pro-cedure to be used in tandem with the rigorous error bounds of Corollary 2.14.Consider (2.17) with coe�cients of the form

(2.40) A✏(x,�) = A("x, "�/|�|),

✏ small, �/|�| (and possibly other indexing parameters) restricted to a compact setdepending on ✏, where A has formal Taylor expansion

(2.41) A(y, ") =pX

k=0

"kAk(y) + O("p+1).

Assume:(h0) A decays uniformly exponentially as y ! ±1 to limits A±.(h1) Aj 2 Cp+1�j(y) for 0 j p, with derivatives uniformly bounded for

all y 2 R, and O(·) 2 C0 and uniformly bounded for y 2 R, " "0

, some "0

> 0.(h2) A

0

(y) is block-diagonalizable to form

(2.42) D0

(y) = T0

A0

T�1

0

(y) = diag {d0,1, . . . , d0,s}, dj 2 Cn

j

⇥nj ,

with spectral separation between blocks (i.e., complex distance between their eigen-values) uniformly bounded below by some � > 0, for " "

0

, y 2 R.Then, we have:

Proposition 2.15[MaZ.1]. Given (h1)–(h2), there exists a uniformly well-conditionedchange of coordinates W = TW such that

(2.43) W 0 = D("x, ")W + O("p+1)W

uniformly for x 2 R, " "0

, with

(2.44) D(y, ") =pX

k=0

"kDk(y) + O("p+1)


and

(2.45) T (y, ") =pX

k=0

"kTk(y)

where Dj, Tj 2 Cp+1�j(y) with uniformly bounded derivatives for all 0 j p,O(✏p+1) 2 C0(y) uniformly bounded, and each Dj has the same block-diagonal form

(2.46) Dj(y) = diag {dj,1, . . . , dj,s},

dj,k 2 Cnk

⇥nk , as does D

0

.If there holds (h0) as well, then it is possible to choose T

0

(·) in such a way that,also, D

0

+"D1

is determined simply by the block-diagonal part of T�1

0

(A0

+"A1

)T0

,or, equivalently,

(2.47) T�1

0

(@/@y)T0

⌘ 0.

Proof. See Propositions 4.4 and 4.8 of [MaZ.1], or Appendix A of this paper.Note that our slightly weakened version of (h1) is what is actually used in the proof,and not the stronger version given in [MaZ.1]. This distinction was unimportant inthe relaxation case, but will be important here. ⇤

Remarks 2.16. 1. An intuitive way to see Proposition 2.15 is to express T asa product T

0

T1

· · ·Tp and solve the resulting succession of nearby diagonalizationproblems; a more e�cient procedure is given in [MaZ.1]. From this point of view,it is clear that one derivative in y must be given up for each power of ✏, since eachtransformation Tj introduces a diagonalization error of form Tj(@/@y)T�1

j ; thisexplains also the origin of condition (2.47). The class of perturbation series (h1) ispreserved under slowly varying changes of coordinates, hence quite natural for theproblem at hand. In particular, the series D obtained by diagonalization is in thesame class, and so the procedure is convenient for iteration. If desired, W j,1(y)may be substituted for Cj(y) everywhere.

2. We call attention to the important statement in Proposition 2.15 that any ad-ditional regularity with respect to " in the error term O("p+1) in (2.41) translates tothe same regularity Cq("! C0(y)) for the error terms O("p+1) in (2.44)-(2.45). Inparticular, if the remainder term is C!("! C0(y)), as often happens (for example,for both relaxation and real viscous profiles), then the error term has regularityC!(" ! C0(y)). This observation is used in conjunction with Proposition 2.13and Corollary 2.14 above; specifically, it gives regularity with respect to ✏ in er-ror terms ⇥jk, and the resulting graphs �j , and thus in terms Fy!x, Ej , andO(�/⌘)J+1e�˜✓|x�y| of expansion (2.37). Likewise, we point out, in the typical casethat (2.40) is obtained by rescaling, that analyticity in � in the original (unrescaled)coordinates, if it holds, may also preserved by appropriate choice of T

0

, even though

42 KEVIN ZUMBRUN

we have expressed A in terms of the nonanalytic parameter �/|�|, and this propertyalso is inherited in the fixed-point constructions of Proposition 2.13 and Corollary2.14; for details, see Remarks 4.11–4.12, [MaZ.1].

2.2.4. Splitting of a block-Jordan block. Augmenting the above discussion,we briefly consider the complementary case of an expansion (2.41) for which (h0)–(h2) hold, but (h3) is replaced by

(h3’)(i) A0

is a standard block-Jordan block of order s,

(2.48) A0

(y) = Js,q + µ(y)Isq, Js,q :=

0

BBBB@

0 Iq · · · 00 0 Iq · · ·...

... · · ·...

0 · · · 0 Iq

0 · · · · · · 0

1

CCCCA;

and (ii) for each y, the spectrum of the lower lefthand block M := (A1

)s1 of A1

liesin a compact set on which z ! z1/s is analytic; in particular, det (M) 6= 0.

Such an expansion typically arises as a single block dj(y, ✏) :=Pp

k=0

✏kdk,j(y) ofthe matrix D(y, ✏) obtained by the block-diagonalization procedure just described.Indeed, the situation may be expected to occur in general for operators of degenerate(mixed) type, indicating the simultaneous presence of multiple scales. (Clearly, stillmore degenerate situations may occur, and could be treated on a case-by-case basis.)

Remark 2.17. We have assumed that A0

has already been put in form (2.48) be-cause (h3’)(ii) is not preserved under general coordinate changes, due to dynamicale↵ects T�1

0

(@/@y)T0

. On the other hand a calculation similar to that of Proposition4.8, [MaZ.1] shows that (T�1

0

(@/@y)T0

)s1 = 0 for transformations preserving theblock form (2.48) (namely, block upper-triangular matrices with diagonal blocksequal to some common invertible ↵), and so (h3’)(ii) is independendent of themanner in which (2.48) is achieved from (2.41).

Performing the “balancing” transformation A ! B�1AB,

(2.49) B := diag {1, ✏1/s, . . . , ✏(s�1)/s},

we may convert this to an expansion

(2.50) A(y, ✏) = µI +pX

k=1

✏k/sAk(y) +O(✏(p+1)/s)

in powers of ✏1/s, Aj 2 Cp+1�j , O(✏(p+1)/s) 2 C0, where

(2.51) A1

=

0

BBBB@

0 Iq · · · 00 0 Iq · · ·...

... · · ·...

0 · · · 0 Iq

M · · · · · · 0

1

CCCCA,


and M satisfies the spectral criterion (h3’)(ii).Then, the spectrum (�M)1/s of A

1

separates into s groups of q, correspond-ing to the s di↵erent analytic representatives of z1/s, and so A

1

is (q ⇥ q) block-diagonalizable, reducing to the previous case with ✏ replaced by ✏1/s and A

0

shiftedto A

1

. This corresponds to the identification of a new, slow scale ✏1/s in the prob-lem. Note that di↵erentiation still brings down a full factor of ✏ << ✏1/s, so thatthe first step in the diagonalization process can still be carried out, despite theshift in the series. Alternatively, we could remove the µI term by an exponentialweighting, and rescale y to remove the initial factor ✏1/s, thus converting to thestandard situation described in Section 2.2.3.

2.2.5. Approximation of stable/unstable manifolds. Proposition 2.15and Corollary 2.14 together give a recipe for the estimation of the stable/unstablemanifolds of ODE with slowly varying coe�cients, given (h0)–(h2): namely, expandformally to order � = ✏p such that �/⌘ ! 0, then apply Corollary 2.14 to obtaina rigorous approximation in terms of the resulting block-decoupled system. In thecase relevant to traveling waves that all Aj (not just A

0

) converge exponentially iny to constants A±

j as y ! ±1, we have, provided that (i) d0,j are scalar multiples

of the identity; and (ii) there holds the necessary condition of neutral separation tozeroth order, Re �(d

0,j) 0 Re �(d0,k), j S < k, for all y; that this procedure

is possible if and only if the corresponding eigenspaces of the original matrix A✏, orequivalently of ✏pDp, have a uniform spectral gap of order ✏p at the limits x = ±1,i.e., there holds the easily checkable condition

(2.52) Re �(dp,j±) �✓ < 0 < ✓ Re �(dp,k)±, j S < k,

for some uniform ✓ > 0. For, this condition is clearly necessary by Lemma 2.5.On the other hand, suppose that the condition holds. The diagonalizing construc-tion described in Proposition 2.15 preserves the property of exponential decay toconstant states. Thus, making a change of coordinates such that

Re d±p,j �✓/2 < 0 < ✓/2 Re d±p,k, j S < k,

we obtain by zeroth order neutrality (preserved, by assumption (i)) plus exponentialdecay, that (2.32) holds with integrable error ↵(x) = O(✏e�✓✏|x|), verifying (2.31).

We remark that d0,j scalar is equivalent to constant multiplicity of the associated

eigenvalue of A0

. For neutrally zeroth-order stable (resp. unstable) blocks, this isa standard structural assumption, corresponding in the critical regime �/|�| pureimaginary to local well-posedness of the underlying evolution equations. For strictlyzeroth-order stable (resp. unstable) blocks, diagonalizability is not necessary eitherfor our construction or for local well-posedness.

Likewise, uniform spectral gap at ±1 for a set of problems ✏, (�/|�|)(✏) is roughlyequivalent to asymptotic direction lim(�/|�|) pointing into the domain of consistentsplitting as |�| ! 1, or equivalently ✏ ! 0. Thus, the requirement of exponential

44 KEVIN ZUMBRUN

separation is in practice no restriction for applications to stability of traveling waves,the domain of feasibility lying in general asymptotic to the domain of consistentsplitting. The only real requirement is su�cient regularity in the coe�cients of thelinearized equations to carry out the formal diagonalization procedure to the nec-essary order p. Whether or not our regularity requirement is sharp is an interestingtechnical question, to which we do not know the answer. We point out only thatthe regularity required in the sectorial case is just C0+↵, as is correct; see [ZH], orRemark 5.4, [MaZ.3].

2.3. Hyperbolic–parabolic smoothing.

We now recall some important ideas of Kawashima et al concerning the smooth-ing e↵ects of hyperbolic–parabolic coupling. The following results assert that hyper-bolic e↵ects can compensate for degenerate viscosity B, as revealed by the existenceof a compensating matrix K.

Lemma 2.18 ([KSh]). Assuming A0, A, B symmetric, A0 > 0, and B � 0, thegenuine coupling condition

(GC) No eigenvector of A lies in kerB

is equivalent to either of:

(K1) There exists a smooth skew-symmetric matrix function K(A0, A, B) suchthat

(2.53) Re�K(A0)�1A + B

�(U) > 0.

(K2) For some ✓ > 0, there holds

(2.54) Re �(�i⇠(A0)�1A� |⇠|2(A0)�1B) �✓|⇠|2/(1 + |⇠|2),

for all ⇠ 2 R.

Proof. These and other useful equivalent formulations are established in [KSh];see also [MaZ.4, Z.4]. ⇤Corollary 2.19. Under (A1)–(A3), there holds the uniform dissipativity condition

(2.55) Re �(X

j

i⇠jAj �

X

j,k

⇠j⇠kBjk)± �✓|⇠|2/(1 + |⇠|2), ✓ > 0.

Moreover, there exist smooth skew-symmetric “compensating matrices” K±(⇠), ho-mogeneous degree one in ⇠, such that

(2.56) Re⇣X

j,k

⇠j⇠kBjk �K(⇠)(A0)�1

X

k

⇠kAk⌘

±� ✓ > 0


for all ⇠ 2 Rd \ {0}.

Proof. By the block–diagonal structure of Bjk (GC) holds also for Aj± and

Bjk := (A0)�1Re Bjk, since

kerX

j,k

⇠j⇠kBjk = kerX

j,k

⇠j⇠kRe Bjk = kerX

⇠j⇠kBjk = kerX

⇠j⇠kBjk.

Applying Lemma 2.18 to

A0 := A0

±, A :=⇣(A0)�1

X

k

⇠kAk⌘

±, B :=

⇣(A0)�1

X

j,k

⇠j⇠kRe Bjk⌘

±,

we thus obtain (2.56) and

(2.57) Re �h(A0)�1

��X

j

i⇠jAj �

X

j,k

⇠j⇠kRe Bjk�i

± �✓

1

|⇠|2/(1 + |⇠|2),

✓1

> 0, from which we readily obtain

(2.58)��X

j

i⇠jAj �

X

j,k

⇠j⇠kBjk�± �✓2|⇠|

2/(1 + |⇠|2).

Observing that M > ✓1

, (A0)�1/2

± M(A0)�1/2

± > ✓ and �(A0)�1/2

± M(A0)�1/2

± >

✓ , �(A0)�1

± M > ✓, together with S > ✓ , �S > ✓ for S symmetric, we obtain(2.55) from (2.58). Because all terms other than K in the lefthand side of (2.56) arehomogeneous, it is evident that we may choose K(·) homogeneous as well (restrictto the unit sphere, then take homogeneous extension). ⇤

Remark 2.20 [GMWZ.4]. In the special case A0 = I, B = block-diag {0, b},and Re b > 0, (GC) is equivalent to the condition that no eigenvector of A

11

lie in the kernel of A21

. If, also, A11

is a scalar multiple of the identity, therefore,Re A

12

A21

> 0. Thus, on any compact set of such A, B satisfying (GC), K(I,A, B)may be taken as a linear function

(2.59) K(A) = ✓

✓0 A

12

�A21

0

◆

of A alone, for ✓ > 0 su�ciently small. This implies, in particular, for systems (1.1)that can be simultaneously symmetrized, and for which each Aj

⇤ is a scalar multipleof the identity, there is a linear, or “di↵erential”, choice K(

Pj ⇠jA

j) =P

j ⇠jKj

such thatP

jk(KjAk + Bjk)⇠j⇠k � 0 on any compact set of U . In particular,this holds for the standard Navier–Stokes equations of gas dynamics and MHD, aspointed out in [Kaw]; more generally, it holds for all of the variants described inAppendix A.1: that is, for all examples considered in this article.

46 KEVIN ZUMBRUN

2.4. Construction of the resolvent kernel.

We conclude these preliminaries by deriving explicit representation formulae forthe resolvent kernel Ge⇠,�(x, y) using a variant of the classical construction (see e.g.[CH], pp. 371–376) of the Green distribution of an ordinary di↵erential operator interms of decaying solutions of the homogeneous eigenvalue equation (Le⇠��)U = 0,or

(2.60)

L0Uz }| {(B11U 0)0 � (A1U)0 � i

X

j 6=1

Aj⇠jU + iX

j 6=1

Bj1⇠jU0

+ iX

k 6=1

(B1k⇠kU)0 �X

j,k 6=1

Bjk⇠j⇠kU � �U = 0,

matched across the singularity x = y by appropriate jump conditions, in the processobtaining standard decay estimates on the resolvent kernel (see (2.85), Proposition2.23, below) suitable for analysis of intermediate frequencies �. Here, Aj andBjk are as defined as in (2.1)–(2.4), U 2 Rn, and 0 as usual denotes d/dx

1

. Ourtreatment follows the general approach introduced in [MaZ.3] to treat the case of anordinary di↵erential operator, such as L above, that is of degenerate type, i.e., thecoe�cient of the highest-order derivative is singular, but for which the eigenvalueequation can nonetheless be written as a nondegenerate first-order ODE (2.17) inan appropriate reduced phase space: in this case,

(2.61) W 0 = A(x1

, e⇠,�)W

with W = (U, b11

1

uI0 + b11

2

uII0) 2 Cn+r. For related analyses in the nondegeneratecase, see, e.g., [AGJ, K.1–2, ZH, MaZ.1].

2.4.1. Domain of consistent splitting. Define

(2.62) ⇤e⇠ := \nj=1

⇤±j (e⇠)

where ⇤±j (e⇠) denote the open sets bounded on the left by the algebraic curves�±j (⇠

1

, e⇠) determined by the eigenvalues of the symbols�⇠2B±�i⇠A± of the limitingconstant-coe�cient operators

(2.63) L±w := B±w00 �A±w0

as ⇠1

is varied along the real axis, with e⇠ held fixed. The curves �±j (·, e⇠) comprisethe essential spectrum of operators Le⇠±.


Lemma 2.21. Assuming (A1)–(A3), (H0)–(H1), the eigenvalue equation may bewritten as first-order equation (2.61), for which the set ⇤e⇠ is equal to the componentcontaining real +1 of the domain of consistent splitting. Moreover,

(2.64) ⇤e⇠ ⇢ {� : Re � > �⌘(|Im �|2 + |e⇠|2)/(1 + |Im �|2 + |e⇠|2)}, ⌘ > 0.

Proof. We do not need to explicitly calculate the matrix A 2 C(n+r)⇥(n+r) thatresults, but only to point out that its existence follows by invertibility of A1

⇤ andb11

2

, (H1), once we rewrite (2.60) as the lower triangular system of equations(2.65)

u0 = (A1

⇤)�1(�A1

12

(b11

2

)�1z � (A1

0

11

+ �)u�A1

0

12

v)� iX

j 6=1

(Aj11

u + Aj12

v),

v0 = (b11

2

)�1z � (b11

2

)�1b11

1

u0,

z0 = (A1

21

�A1

22

(b11

2

)�1b11

1

)u0 + A1

22

(b11

2

)�1z + A1

0

21

u + (A1

0

22

+ �)v +e⇠ termsz}|{· · · ,

with (u, v, z) := (uI , uII , b11

1

u0 + b11

2

v0). (We will need to carry out explicit compu-tations only for our treatment in Section 4.3.2 of one-dimensional high-frequencybounds.) Note, as follows from the second equation, the important fact that ev-ery solution of (2.65) indeed corresponds to a solution U = (u, v) of (2.60), withz = b11

1

u0 + b11

2

v0.The final assertion, bound (2.64), follows easily from the bound (2.55) on the

dispersion curves �±j (e⇠). To establish the second assertion, we must show that, onthe set ⇤, the limiting eigenvalue equations

(2.66) (Le⇠± � �)w = 0

have no center manifold and the stable manifold at +1 and the unstable manifoldat �1 have dimensions summing to the full dimension n; moreover, that theseproperties do not hold together on the boundary of ⇤e⇠.

The fundamental modes of (2.66) are of form eµxV , where µ, V satisfy thecharacteristic equation

(2.67)

2

4µ2B11

± + µ(�A1

± + iX

j 6=1

Bj1± ⇠j + i

X

k 6=1

B1k⇠k)

�(iX

j 6=1

Aj⇠j +X

jk 6=1

Bjk⇠j⇠k + �I)

3

5V = 0.

The existence of a center manifold thus corresponds with existence of solutionsµ = i⇠

1

, V of (2.67), ⇠1

real, i.e., solutions of the dispersion relation

(2.68) (�X

j,k

Bjk± ⇠j⇠k � i

X

j

Aj±⇠j � �I)V = 0.

48 KEVIN ZUMBRUN

But, � 2 �(�B⇠⇠ � iA⇠±) implies, by definition (2.62), that � lies outside of ⇤e⇠,

establishing nonexistence of a center manifold. Moreover, it is clear from the sameargument that a center manifold does exist on the boundary of ⇤, since this corre-sponds to existence of pure imaginary eigenmodes.

Finally, nonexistence of a center manifold, together with connectivity of ⇤, im-plies that the dimensions of stable/unstable manifolds at +1/�1 are constant on⇤. Taking �! +1 along the real axis, with e⇠ ⌘ 0, we find that these dimensionssum to the full dimension n + r as claimed. For, Fourier expansion about ⇠

1

= 1of the one-dimensional (e⇠ = 0) dispersion relation (see Appendix A.4, below) yieldsn� r “hyperbolic” modes

�j = �i⇠1

a⇤j + . . . , j = 1, . . . , n� r,

where a⇤j denote the eigenvalues of A1

⇤ and r “parabolic” modes

�n�r+j = �bj⇠2

1

+ . . . , j = 1, . . . r,

where bj denote the eigenvalues of b11

2

; here, we have suppressed the ± indicesfor readability. Inverting these relationships to solve for µ := i⇠

1

, we find, for�!1, that there are n� r hyperbolic roots µj ⇠ ��/a⇤j , and 2r parabolic rootsµ±n�r+j ⇠ ±

p�/bj . By assumption (H1)(iii), the former yield a fixed number

k/(n� r� k) of stable/unstable roots, independent of x1

, and thus of ±. Likewise,(H1)(i) implies that the latter yields r stable, r unstable roots. Combining, we findthe desired consistent splitting, with (k + r)/ (n� k) stable/unstable roots at both±1. ⇤

A standard fact, for asymptotically constant-coe�cient ordinary di↵erential op-erators (see, e.g. [He], Lemma 2, pp. 138–139) is that components of the domainof consistent splitting agree with components of the complement of the essentialspectrum of the variable-coe�cient operator L. Thus, ⇤ is a maximal domain inthe essential spectrum complement, consisting of the component containing real,plus infinity. Moreover, provided that the coe�ents of L approach their limits atintegrable rate, it can be shown (see, e.g. [AGJ,GZ,ZH,MaZ.1]) that each connectedcomponent consists either entirely of eigenvalues, or else entirely of normal points,defined as resolvent points or isolated eigenvalues of finite multiplicity. The latterfact will be seen directly in the course of our construction.

2.4.2. Basic construction. We now carry out the resolvent construction atpoints � in ⇤, more generally, at any point in the domain of consistent splitting, bythe method of [MaZ.3]. Our starting point is a duality relation between solutionsU of the eigenvalue equation and solutions U of the adjoint eigenvalue equation,which may be obtained by the observation that E defined by

(d/dx1

)E := hU , (Le⇠ � �)Ui � h(L⇤e⇠ � �)U , Ui = hU , Le⇠Ui � hL⇤e⇠U , Ui


is a conserved quantity whenever U and U are solutions of the eigenvalue andadjoint eigenvalue equations, respectively.

Since hU , Le⇠Ui � hL⇤e⇠U , Ui is a perfect derivative involving derivatives of U , U

of order strictly less than the maximum order of Le⇠, the quantity E may always beexpressed as a quadratic form in phase space, in this case

(2.69) E(x1

) = hU , B11U 0i+ hU , (�A1 + iB1

e⇠ + iBe⇠1)Ui+ hU 0,�B11Ui,

or

(2.70) E(x1

) =D✓

UU 0

◆,S e⇠

✓UU 0

◆E(x

1

),

where

(2.71) S e⇠ :=✓�A1 + iB1

e⇠ + iBe⇠1 B11

�B11 0

◆.

In general, we obtain a cross-banded matrix S, with elements of the jth bandconsisting of alternating ± multiples of the jth coe�cient of the operator Le⇠, theanti-diagonal being filled with multiples of the principal coe�cient, and the bandsbelow the anti-diagonal being filled with zero blocks. Thus, for nondegenerateoperators, it is immediate that S is invertible, and we obtain the characterization[ZH,MaZ.1] of solutions of the eigenvalue equation as those functions U with theproperty that E(x

1

) is preserved for any solutions U of the adjoint equation.For degenerate operators, S is of course not invertible, and we must work instead

in a reduced phase space. In the present case, we may rewrite (2.69) as

(2.72) E(x1

) =D✓

UZ

◆, S

✓UZ

◆E,

where

(2.73) S e⇠ :=

0

@ �A1 + iB1

e⇠ + iBe⇠1

✓0Ir

◆

(�(b11

2

)�1b11

I �Ir ) 0

1

A

and

(2.74) Z := (b11

1

, b11

2

)U 0, Z := (0, b11

2

⇤)U 0.

Applying an elementary column operation (subtracting from the first column thesecond column times (b11

2

)�1b11

1

), we find that

(2.75) det S e⇠ ⌘ det A1

⇤ 6= 0,

by (H1)(i). Thus, we may conclude, similarly as in the nondegenerate case:

50 KEVIN ZUMBRUN

Lemma 2.22. W = (U,Z) satisfies eigenvalue equation (2.60) if and only if

(2.76) W ⇤SW ⌘ constant

for all W = (U , Z) satisfying the adjoint eigenvalue equation, and vice versa.

For future reference, we note the representation

(2.77) (S e⇠)�1 =

0

@�(A1

⇤)�1 0 (A1

⇤)�1A(e⇠)12

(b11

2

)�1b11

1

(A1

⇤)�1 0 �(b11

2

)�1b11

1

(A1

⇤)�1A(e⇠)12

� I

�A(e⇠)(A1

⇤)�1 I �A(e⇠)22

+ A(e⇠)(A1

⇤)�1A12

1

A ,

A1

⇤ := A1

11

�A1

12

(b11

2

)�1b11

1

,

A(e⇠) := ↵21

(e⇠)� ↵22

(e⇠)(b11

2

)�1b11

1

,

↵(e⇠) := A1 � iBe⇠1 � iB1

e⇠,

which may be obtained by the above-mentioned column operation, followed bystraightforward row reduction.

Let

(2.78) �+

j := P+

v+

j , j = 1 . . . , k,

and

(2.79) ��j , j = k + 1, . . . , n + r

denote the locally analytic bases of the stable manifold at +1 and the unstablemanifold at �1 of solutions of the variable-coe�cient equation (2.60) that arefound in Section 3.1, and set

(2.80) �+ := (�+

1

, . . . ,�+

k ), �� := (��k+1

, . . . ,��n+r),

and

(2.81) � := (�+,��).

Define the solution operator from y1

to x1

of (2.60), denoted by Fy1!x1 , as

(2.82) Fy1!x1 = �(x1

;�)��1(y1

;�)

and the projections ⇧±y1on the stable manifolds at ±1 as

(2.83)⇧+

y1= (�+(y

1

;�) 0 )��1(y1

;�) and ⇧�y1= ( 0 ��(y

1

;�) )��1(y1

;�).

Then, we have the following universal result (see, e.g. [ZH,MaZ.1] in the nonde-generate case, or [MaZ.3] for the one-dimensional hyperbolic–parabolic case).


Proposition 2.23. With respect to any Lp, 1 p 1, the domain of consistentsplitting consists entirely of normal points of Le⇠, i.e., resolvent points, or isolatedeigenvalues of constant multiplicity. On this domain, the resolvent kernel Ge⇠,� ismeromorphic, with representation

(2.84) Ge⇠,�(x1

, y1

) =

((In, 0)Fy1!x1 ⇧+

y1(S e⇠)�1(y

1

)(In, 0)tr x1

> y1

,

�(In, 0)Fy1!x1 ⇧�y1(S e⇠)�1(y

1

)(In, 0)tr x1

< y1

,

(S e⇠)�1 as described in (2.77). Moreover, on any compact subset K of ⇢(Le⇠) \ ⇤(⇢(Le⇠) denoting resolvent set), there holds the uniform decay estimate

(2.85) |Ge⇠,�(x1

, y1

)| Ce�⌘|x1�y1|,

where C > 0 and ⌘ > 0 depend only on K, Le⇠.

Proof. As discussed further in [Z.1,MaZ.1], to show that � is in the resolvent setof Le⇠, with respect to any Lp, 1 p 1, it is enough to: (i) construct a resolventkernel Ge⇠,�(x1

, y1

) satisfying

(2.86) (Le⇠ � �)Ge⇠,� = �y1(x1

)

and obeying a uniform decay estimate

(2.87) |Ge⇠,�(x1

, y1

)| Ce�⌘|x1�y1|;

and, (ii) show that there are no Lp solutions of (Le⇠��)W = 0, i.e., � is not an eigen-value of Le⇠, or, equivalently, a zero of the Evans function D(·) (For p < 1, theseare necessary as well as su�cient, [Z.1]). For, then, the Hausdor↵–Young inequalityyields that the distributional solution formula (Le⇠ � �)�1f :=

RG�(x1

, y1

)f(y1

)dy

yields a bounded right inverse of (Le⇠ � �) taking Lp to Lp, while nonexistence ofeigenvalues implies that this is also a left inverse.

On the domain of consistent splitting, we shall show that (ii) implies (i) (theyare in fact equivalent [Z.1,MaZ.1]), which immediately implies the first assertionby the properties of analytic functions and the correlation between eigenvalues andzeroes of D, Remark 2.11. It remains, then, to establish existence of a solution of(2.86) satisfying the bound (2.87), under the assumption that D(�) 6= 0.

Note, by Lemma 2.5, that the stable manifold �+ decays x1

� 0 with uniformrate Ce⌘|x1|, ⌘ > 0. Moreover, if D 6= 0, then �+ does not decay at �1, whereuponwe may conclude from Lemma 2.5 that it grows exponentially as x

1

! �1, so thatit in fact decays uniformly exponentially in x

1

on the whole line. Likewise, ��decays exponentially as x

1

decreases, uniformly on the whole line. Thus, we findthat (2.84) satisfies (2.85) as claimed. Clearly, it also satisfies (2.86) away from the

52 KEVIN ZUMBRUN

singular point x1

= y1

. Finally, we note that, by the construction of S e⇠, we havethe general fact that Z y1

+

y1�Le⇠Ge⇠,� = (In, 0)S e⇠.

Substituting from (2.84), we obtain

[Ge⇠,�][(b11

1

, b11

2

)G0e⇠,�]

!= (In, 0)S e⇠Fy1!y1 (S e⇠)�1(In, 0)tr = In,

validating the jump condition across x1

= y1

, and completing the proof. ⇤Proposition 2.23 includes a satisfactory intermediate-frequency bound (2.85) on

the resolvent kernel. More careful analyses will be required in the large- and small-freqency limits.

Remark 2.24. Similarly as in the strictly parabolic case [ZH], formula (2.84)extends to the full phase-variable representation(2.88)✓

Ge⇠,�(x1

, y1

) (@/@y1

)Ge⇠,�(x1

, y1

)(0, (b11

2

)t)t

(b11

1

, b11

2

)(@/@x1

)Ge⇠,�(x1

, y1

) (b11

1

, b11

2

)(@/@x1

)(@/@y1

)Ge⇠,�(x1

, y1

)(0, (b11

2

)t)t

◆=

(Fy1!x1 ⇧+

y1(S e⇠)�1(y

1

) x1

> y1

,

�Fy1!x1 ⇧�y1(S e⇠)�1(y

1

) x1

< y1

.

This formula (though not the specific phase variable) is also universal, and followsfrom (2.84), which states that the upper lefthand corner is correct, together with thefact that the columns of the righthand side satisfy the forward eigenvalue equation(written as a first-order system) with respect to x

1

, by inspection, while the rowssatisfy the adjoint eigenvalue equation (again, written as a first-order system) withrespect to y

1

, by duality relation (2.76), a simple consequence of which is that thecolumns and rows do respect the phase-variable formulation and so we may deducefrom correctness of the upper lefthand corner that all entries are correct. Formula(2.88) is important in the development of the e↵ective spectral theory given inSection 4.3.3, below; see discussion, proof of Proposition 4.38.

2.4.3. Generalized spectral decomposition. For the treatment of low-frequency behavior, we develop a modified representation of the resolvent kernelconsisting of a scattering decomposition in solutions of the forward and adjointeigenvalue equations.

From (2.76), it follows that if there are k independent solutions �+

1

, . . . ,�+

k of(Le⇠ ��I)W = 0 decaying at +1, and n� k independent solutions ��k+1

, . . . ,��n ofthe same equations decaying at �1, then there exist n� k independent solutions +

k+1

, . . . , +

n of (L⇤e⇠ � �I)W = 0 decaying at +1, and k independent solutions

�1

, . . . , �k decaying at �1. More precisely, setting

(2.89) +(x1

;�) = ( +

k+1

(x1

;�) · · · +

n (x1

;�) ) 2 Rn⇥(n�k),


(2.90) �(x1

;�) = ( �1

(x1

;�) · · · �k (x1

;�) ) 2 Rn⇥k,

and

(2.91) (x1

;�) = ( �(x1

;�) +(x1

;�) ) 2 Rn⇥n,

where ±j are exponentially growing solutions obtained through Lemma 2.5, wemay define dual exponentially decaying and growing solutions ±j and �±j via

(2.92) ( � )⇤± Se⇠ ( � )± ⌘ I.

Then, we have:

Proposition 2.25. The resolvent kernel may alternatively be expressed as(2.93)

Ge⇠,�(x1

, y1

) =⇢

(In, 0)�+(x1

;�)M+(�) �⇤(y1

;�)(In, 0)tr x1

> y1

,

�(In, 0)��(x1

;�)M�(�) +⇤(y1

;�)(In, 0)tr x1

< y1

,

where

(2.94) M(�) := diag(M+(�),M�(�)) = ��1(z;�)(S e⇠)�1(z) �1⇤(z;�),

:= ( � + ). (Note: the righthand side of (2.94) is independent with respectto z, as a consequence of Lemma 2.22.)

Proof. Immediate, by rearrangement of (2.86). ⇤Remarks. 1. Representation (2.84) reflects the classical duality principle (see,

e.g. [ZH], Lemma 4.2) that the transposition G⇤e⇠,�(y1

, x1

) of the Green distributionGe⇠,�(x1

, y1

) associated with operator (Le⇠ ��) should be the Green distribution forthe adjoint operator (L⇤e⇠ � �).

2. As before, it is possible to represent the matrix Ge⇠,� by means of intrinsicobjects such as solution operators and projections on stable manifolds(2.95)

Ge⇠,�(x1

, y1

) =

((In, 0)Fz!x1 ⇧+

z (S e⇠)�1(z) ⇧�z Fz!y1 (In, 0)tr x1

> y1

,

�(In, 0)Fz!x1 ⇧�z (S e⇠)�1(z) ⇧+

z Fz!y1 (In, 0)tr x1

< y1

,

Fz!y1 := �1(z;�) (y1

;�), ⇧+

z := �1

✓0 +

◆(z;�) ⇧�z := �1

✓ �0

◆(z;�).

The exponential decay asserted in Proposition 2.23 is somewhat more straightfor-ward to see in this dual formulation, by judiciously choosing z in (2.95) and usingthe exponential decay of forward and adjoint flows on x

1

? 0 alone.

From Proposition 2.25, we obtain the following scattering decomposition, gener-alizing the Fourier transform representation in the constant–coe�cient case.

54 KEVIN ZUMBRUN

Corollary 2.26 [Z.3]. On ⇤ \ ⇢(Le⇠), there hold

(2.96) Ge⇠,�(x1

, y1

) =X

j,k

M+

jk(�)�+

j (x1

;�) e �k (y1

;�)⇤

for y1

0 x1

,

(2.97) Ge⇠,�(x1

, y1

) =X

j,k

d+

jk(�)��j (x1

;�) e �k (y1

;�)⇤ �X

k

�k (x1

;�) e �k (y1

;�)⇤

for y1

x1

0, and

(2.98) Ge⇠,�(x1

, y1

) =X

j,k

d�jk(�)��j (x1

;�) e �k (y1

;�)⇤ +X

k

��k (x1

;�)��k (y1

;�)⇤

for x1

y1

0, with

(2.99) M+ = (�I, 0) (�+ �� )�1 �

and

(2.100) d± =�0, I

�(�+ �� )�1 �.

Symmetric representations hold for y1

� 0.

Proof. The matrix M+ in (2.94) may be expanded using duality relation (2.76)as

(2.101)

M+ = (�I, 0) (�+ �� )�1A�1 ( � �� )⇤�1

✓I0

◆

|z

= (�I, 0) (�+ �� )�1 ( � �� )✓

I0

◆

|z

= (�I, 0) (�+ �� )�1 �|z

,

yielding (2.99) for x1

� y1

, in particular for y1

0 x1

.Next, expressing �±j (x

1

;�) as a linear combination of basis elements at �1, weobtain the preliminary representation

(2.102) G�(x1

, y1

) =X

j,k

d+

jk(�)��j (x1

;�) e �k (y1

;�)⇤+X

j,k

e+

jk �j (x

1

;�) e �k (y1

;�)⇤,

valid for y1

x1

0. Duality, (2.76), with (2.84), and the fact that ⇧+

= I �⇧�,gives

(2.103)

�✓

d+

e+

◆= ( �� )⇤A⇧

+

�|x1

= � (�� )�1 [I � ( 0 �� ) (�+ �� )�1] �

=✓

0�Ik

◆�✓

0 In+r�k

0 0

◆(�+ �� )�1 �,


yielding (2.97) and (2.100) for y1

x1

0. Relations (2.98) and (2.100) follow forx

1

y1

0 in similar, but more straightforward fashion from (2.76) and (2.84). ⇤Remark 2.27. In the constant-coe�cient case, with a choice of common bases

± = �⌥ at ±1, (2.96)–(2.100) reduce to the simple formula

(2.104) Ge⇠,�(x1

, y1

) =

(�Pn+r

j=k+1

�+

j (x1

;�)�+⇤j (y

1

;�) x1

> y1

,Pk

j=1

��j (x1

;�)��⇤j (y1

;�) x1

< y1

,

where, generically, �±j , �±j may be taken as pure exponentials

(2.105) �±j (x1

)�±⇤j (y1

) = eµ±j

(�)(x1�y1)V ±j (�)V ±⇤

j (�).

Note that, moving individual modes �±j �±j in the spectral resolution formula (2.8),

using Cauchy’s Theorem, to contours µj(�) ⌘ i⇠1

lying along corresponding dis-persion curves � = �j(⇠), we obtain the standard decomposition of eLe

⇠

t into eigen-modes of continuous spectrum: in this (constant-coe�cient) case, just the usualrepresentation obtained by Fourier transform solution.

3. THE EVANS FUNCTION, AND ITS LOW-FREQUENCY LIMIT.

We next explore the key link between the Evans function D and the Lopatinskideterminant � in the limit as frequency goes to zero, establishing Result 1 of theintroduction.

3.1. The Evans function.

We first recall the following important result, established by Kreiss in the strictlyhyperbolic case (see [Kr], Lemma 3.2, and discussion just above; see also [CP],Theorem 3.5, p. 431, and [ZS], Lemma 3.5) and by Metivier [Me.4] in the constantmultiplicity case. The result holds, more generally, for systems satisfying the blockstructure condition of Kreiss–Majda [Kr,M.1–4].

Lemma 3.1. Let there hold (A1) and (H1)–(H2), and (H4), or, more generally,�(dF ⇠(U±)) real, semisimple, and of constant multiplicity for ⇠ 2 Rd \ {0} anddet dF 1 6= 0. Then, the vectors {r�

1

, · · · , r�n�i�}, {r+

i++1

, · · · , r+

n } (defined as in(1.30), Section 1.3), and thus �(e⇠,�) may be chosen to be homogeneous (degreezero for r±j , degree one for �), analytic on e⇠ 2 Rd�1, Re � > 0 and continuous atthe boundary e⇠ 2 Rd�1 \ {0 , Re � = 0.

Proof. See Exercises 4.23–4.24 and Remark 4.25, Section 4.5.2 of [Z.3] for aproof in the general case (three alternative proofs, based respectively on [Kr], [CP],and [ZS]). In the case of main interest, when (H5) holds as well, we will see this

56 KEVIN ZUMBRUN

later in the course of the explicit computations of Section 5.3; see Remark 5.20.⇤

Remark 3.2. Typically, � has a conical singularity at (e⇠,�) = (0, 0), in thesense that the level set through the origin is a cone and not a plane. That is, it isdegree one homogeneous but not linear, with a gradient discontinuity at the origin.This reflects the fact that r±j are degree zero homogeneous but are not constantunless all dF j commute.

Following the construction of Section 2.2, we may define an Evans function

(3.1)D(e⇠,�) := det

⇣W+

1

, . . . ,W+

k ,W�k+1

, . . . ,W�n+r

⌘

|x=0,�

= det⇣P

+

V +

1

, . . . , P+

V +

k , P�V �k+1

, . . . , P�V �n+r

⌘

|x=0,�,

associated with the linearized operator Le⇠ about the viscous shock profile, on itsdomain of consistent splitting, in particular on the set

⇤ := {(e⇠,�) : � 2 ⇤e⇠}= {(e⇠,�) : Re � > �⌘(|Im �|2 + |e⇠|2)/(1 + |Im �|2 + |e⇠|2)},

⇤e⇠ as defined as in (2.62), where P± are the transformations described in Lemma2.5, W±

j denote solutions of variable-coe�cient eigenvalue problem (Le⇠ � �)U = 0written as a first-order system in phase-variables W = (U, z0

2

), z2

= b11

1

uI + b11

2

uII ,and V ±

j solutions of the associated limiting, constant-coe�cient systems as x1

!±1. By construction, D is analytic on ⇤.

For comparison with the inviscid case, it is convenient to introduce polar coor-dinates

(3.2) (e⇠,�) =: (⇢e⇠0

, ⇢�0

),

⇢ 2 R1, (e⇠0

,�0

) 2 Rd�1 ⇥ {Re � � 0} \ {(0, 0)}, and consider W±j , V ±

j as functionsof (⇢, e⇠

0

,�0

.

Lemma 3.3 [MeZ.2]. Under assumptions (A1)–(A3) and (H0)–(H3), the functionsV ±

j may be chosen within groups of r “fast modes” bounded away from the centersubspace of coe�cient A±, analytic in (⇢, e⇠

0

,�0

) for ⇢ � 0, e⇠0

2 Rd�1, Re �0

� 0,and n “slow modes” approaching the center subspace as ⇢! 0, analytic in (⇢, e⇠

0

,�0

)for ⇢ > 0, e⇠

0

2 Rd�1, Re �0

� 0 and continuous at the boundary ⇢ = 0, with limits

(3.3) V ±j (0, e⇠

0

,�0

) =✓

(A1

±)�1r±j (e⇠0

,�0

)0

◆,

r±j defined as in Lemma 3.1.


Proof. We here carry out the case Re �0

> 0. In the case of interest, that (H5)also holds, we shall later obtain the case Re � = 0 as a straightforward consequenceof the more detailed computations in Section 5.3; see Remark 5.20. The generalcase (without (H5)) follows by the analysis of [MeZ.2].

Substituting v = eµx1v into the limiting eigenvalue equations (Le⇠ � �)v = 0written in polar coordinates, we obtain the polar characteristic equation,

(3.4)

2

4µ2B11

± + µ(�A1

± + i⇢X

j 6=1

Bj1± ⇠j + i⇢

X

k 6=1

B1k⇠k)

�(i⇢X

j 6=1

Aj⇠j + ⇢2

X

jk 6=1

Bjk⇠j⇠k + ⇢�I)

3

5v = 0,

where for notational convenience we have dropped subscripts from the fixed param-eters e⇠

0

, �0

. At ⇢ = 0, this simplifies to

�µ2B11

± � µA1

±�v = 0,

which, by the analysis in Appendix A.2 of the linearized traveling-wave ordinarydi↵erential equation

�µB11

± �A1

±�v = 0, has n roots µ = 0, and r roots Re µ 6= 0.

The latter, “fast” roots correspond to stable and unstable subspaces, which extendanalytically as claimed by their spectral separation from other modes; thus, weneed only focus on the bifurcation as ⇢ varies near zero of the n-dimensional centermanifold associated with “slow” roots µ = 0.

Positing a first-order Taylor expansion

(3.5)⇢

µ = 0 + µ1⇢+ o(⇢),v = v0 + v1⇢+ o(⇢),

and matching terms of order ⇢ in (3.4), we obtain:

(3.6) (�µ1A1

± � iX

j 6=1

Aj⇠j � �I)v0 = 0,

or �µ1 is an eigenvalue of (A1)�1(�+ iAe⇠) with associated eigenvector v0.

For Re � > 0, (A1)�1(� + iAe⇠) has no center subspace. For, substituting µ1 =

i⇠1

in (3.6), we obtain � 2 �(iA⇠), pure imaginary, a contradiction. Thus, thestable/unstable spectrum splits to first order, and we obtain the desired analyticextension by standard matrix perturbation theory, though not in fact the analyticityof individual eigenvalues µ. ⇤

58 KEVIN ZUMBRUN

Corollary 3.4. Denoting W = (U, b11

1

uI + b11

2

uII) as above, we may arrange at⇢ = 0 that all W±

j satisfy the linearized traveling-wave ODE (B11U 0)0�(A1U)0 = 0,with constant of integration

(3.7) B11U±0j �A1U±

j ⌘⇢ 0, for fast modes,

r±j , for slow modes,

r±j as above, with fast modes analytic at the ⇢ = 0 boundary and independent of(e⇠

0

,�0

) for ⇢ = 0, and slow modes continuous at ⇢ = 0.

Proof. Immediate ⇤

3.2. The low-frequency limit.

We are now ready to establish the main result of this section, validating Result 1of the introduction. This may be recognized as a generalization of the basic Evansfunction calculation pioneered by Evans [E.4], relating behavior near the origin togeometry of the phase space of the traveling wave ODE and thus giving an explicitlink between PDE and ODE dynamics. The corresponding one-dimensional resultwas established in [GZ]; for related calculations, see, e.g., [J, AGJ, PW].

Theorem 3.5 [ZS, Z.3]. Under assumptions (A1)–(A3) and (H0)–(H4), thereholds

(3.8) D(e⇠,�) = ��(e⇠,�) +O(|e⇠|+ |�|)`+1,

where � is given in the Lax case by the inviscid stability function described in(1.29), in the undercompressive case by the analogous undercompressive inviscidstability function (1.38) with g appropriately chosen,25 and in the overcompressivecase by the special, low-frequency stability function

(3.9) �(e⇠,�) := det (r�1

, · · · , r�n�i�+1

,m�1 , · · · ,m�`

, r+

i++1

, · · · , r+

n ),

where

(3.10)m(e⇠,�, �) := �

Z 1

�1(U�(x)� U(x))dx

+ i

Z 1

�1(F e⇠(U�(x))� F

e⇠(U(x)))dx.

In each case the factor � is a constant measuring transversality of the intersection ofunstable/resp. stable manifolds of U�/U

+

, in the phase space of the traveling-waveODE (� 6= 0 , transversality), while the constant ` as usual denotes the dimension

25Namely, as a Melnikov separation function associated with the undercompressive connection;see [ZS] for details.


of the manifold {U�}, � 2 U ⇢ R` of connections between U± (see (H3), Section1.1, and discussion, Section 1.2). In the Lax or undercompressive case, ` = 1 and{U�} = {U(·� �)} is simply the manifold of translates of U .

That is, D(·, ·) is tangent to �(·, ·) at (0, 0); equivalently, �(·, ·) describes thelow-frequency behavior of D(·, ·). Note that �(·, ·) is evidently homogeneous degree` in each case (recall, ` = 1 for Lax, undercompressive cases), hence O(|e⇠|+ |�|)`+1

is indeed a higher order term.

Remark 3.6. In the one-dimensional setting, m(0, 1, �) has an interpretationas “mass-map”, see [FreZ.1]; likewise, �(0, 1) (� as in (3.10)) arises naturally indetermining shock shift/distribution of mass resulting from a given perturbationmass.

Proof. We will carry out the proof in the Lax case only. The proofs for theunder- and overcompressive cases are quite similar, and may be found in [ZS].We are free to make any analytic choice of bases, and any nonsingular choice ofcoordinates, since these a↵ect the Evans function only up to a nonvanishing analyticmultiplier which does not a↵ect the result. Choose bases W±

j as in Lemma 3.4,W = (U, z0

2

), z2

= b11

1

uI + b11

2

uII . Noting that L0

U 0 = 0, by translation invariance,we have that U 0 lies in both Span{U+

1

, · · · , U+

K} and Span(U�K+1

, · · · , U�n+r) for⇢ = 0, hence without loss of generality

(3.11) U+

1

= U�n+r = U 0,

independent of e⇠, �. (Here, as usual, “ 0 ” denotes @/@x1

).More generally, we order the bases so that W+

1

, . . . ,W+

k and W�n+r�k�1

, . . . ,W�n+r

are fast modes (decaying for ⇢ = 0) and W+

k+1

, . . . ,W+

K and W�K+1

, . . . ,W�n+r�k�2

are slow modes (asymptotically constant for ⇢ = 0), fast modes analytic and slowmodes continuous at ⇢ = 0 (Corollary 3.4).

Using the fact that U+

1

and U�n+r are analytic, we may express

U+

1

(⇢) = U+

1

(0) + U+

1,⇢⇢+ o(⇢),

U�n+r(⇢) = U�n+r(0) + U�n+r,⇢⇢+ o(⇢).

Writing out the eigenvalue equation in polar coordinates,

(3.12)

(B11U 0)0 = (A1U)0 � i⇢X

j 6=1

Bj1⇠jU0

� i⇢(X

k 6=1

B1k⇠kU)0 + i⇢X

j 6=1

Aj⇠jU + ⇢�U

� ⇢2

X

j,k 6=1

Bjk⇠j⇠kU,

60 KEVIN ZUMBRUN

we find that Y + := U+

1,⇢(0) and Y � := U�n+r,⇢(0) satisfy the variational equations

(3.13)

(B11Y 0)0 = (A1Y )0 � iX

j 6=1

Bj1⇠jU00

� i⇣X

k 6=1

B1k⇠kU 0⌘0

+ iX

j 6=1

Aj⇠jU0 + �U 0,

with boundary conditions Y +(+1) = Y �(�1) = 0. Integrating from +1, �1respectively, we obtain therefore

(3.14)B11Y ±0 �A1Y ± = iF

e⇠(U)� iB1

e⇠(U)U 0

� iBe⇠1(U)U 0 + �U �

hiF

e⇠(U±) + �U±i,

hence Y := (Y � � Y +) satisfies

(3.15) B11Y 0 �A1Y = i[F e⇠] + �[u].

By hypothesis (H1),✓

A1

11

A1

12

b11

1

b11

2

◆is invertible, hence

(3.16)(U, z0

2

) ! (z2

,�z1

, z02

� (A1

21

+ b11

0

1

, A1

22

+ b11

0

2

)U)

= (z2

, B11U 0 �A1U)

is a nonsingular coordinate change, where✓

z1

z2

◆:=

✓A1

11

A1

12

b11

1

b11

2

◆U.

Fixing e⇠0

, �0

, and using W+

1

(0) = W�n+r(0), we have

(3.17)

D(⇢) = det⇣W+

1

(0) + ⇢W+

1

⇢

(0) + o(⇢), · · · ,W+

K (0) + o(1),

W�K+1

(0) + o(1), · · · , W�n+r(0) + ⇢W�

n+r⇢

(0) + o(⇢)⌘

= det⇣W+

1

(0) + ⇢W+

1

⇢

(0) + o(⇢), · · · ,W+

K (0) + o(1),

W�K+1

(0) + o(1), · · · , ⇢Y (0) + o(⇢)⌘

= det⇣W+

1

(0) · · · ,W+

K (0) ,W�K+1

(0) , · · · , ⇢Y (0)⌘

+ o(⇢)

Applying now coordinate change (3.16) and using (3.7) and (3.15), we obtain

(3.18)


D(⇢) = Cdet

0

BBBBBB@

fastz }| {z+

2,1 · · · , z+

2,k,

slowz }| {⇤, · · · , ⇤, ⇤, · · · , ⇤,

0, · · · , 0, r+

i++1

, · · · , r+

n , r�1

, · · · , r�n�i�

fastz }| {z�2,n+r�k�1

, · · · , z�2,n+r�1

, ⇤

0, · · · , 0, i[F e⇠(U)]+�[U ]

1

CCCA

|x1=0

+ o(⇢)

= ��(e⇠,�) + o(⇢)

as claimed, where

� := Cdet⇣z+

2,1, · · · , z+

2,k, z�2,n+r�k�1

, · · · , z�2,n+r�1

⌘

x1=0

.

Noting that {z+

2,1, · · · , z+

2,k} and {z�2,n+r�k�1

, · · · , z�2,n+r} span the tangent man-

ifolds at U(·) of the stable/unstable manifolds of traveling wave ODE (1.17) atU

+

/U�, respectively, with z+

2,1 = z�2,n+r = (b11

1

, b11

2

)U 0 in common, we see that �indeed measures transversality of their intersection; moreover, � is constant, byCorollary 3.4. ⇤

4. ONE-DIMENSIONAL STABILITY.

We now focus attention on the one-dimensional case, establishing results 2 and 3of the introduction. For further discussion/applications, see, e.g., [GS, BSZ.1, ZH,MaZ.4] and [Z.3], Section 6.

4.1. Necessary conditions: the stability index.

In the one-dimensional case e⇠ ⌘ 0, we obtain by the construction of Section 3 anEvans function D(�) depending only on the temporal frequency, associated withthe one-dimensional linearized operator L := L

0

. Since L is real-valued, we obtainby Remark 2.11.2 that D(·) may be chosen with complex symmetry

D(�) = D(�),

where bar denotes complex conjugate, so that, in particular, D(�) is real-valuedfor � real. Further, the energy estimates of Lemma 2.1 show that (since there is nospectrum there) D is nonzero for � real and su�ciently large, so that

(4.1) sgn D(+1) := lim�!real +1

sgn D(�)

62 KEVIN ZUMBRUN

is well-defined. Finally, Theorem 3.5 gives (d/d�)`D(0) = `!��(0, 1).

Definition 4.1. Combining the above observations, we define the stability index

(4.2) � := sgn (d/d�)`D(0)D(+1) = sgn ��(0, 1)D(+1),

where �, � as defined in Section 1.4 denote transversality and inviscid stabilitycoe�cients, and ` as defined in Section 1.2 the dimension of the manifold {U�} ofconnections between U±. In the Lax or undercompressive case, ` = 1.

Proposition 4.2. The number of unstable eigenvalues � 2 {� : Re � > 0} haseven parity if � > 0 and odd parity if � < 0. In particular, � > 0 is necessaryfor one-dimensional linearized viscous stability with respect to L1\L1 or even testfunction (C1

0

) initial data.

Proof. By complex symmetry (2.25), nonreal eigenvalues occur in conjugatepairs, hence do not a↵ect parity. On the other hand, the number of real rootsclearly has the parity claimed. This establishes the first assertion, from which itfollows that � � 0 is necessary for stability with respect to L1 \ L1 data (recallthat zeroes of D on {Re � > 0} correspond to exponentially decaying eigenmodes).The assertion that ��(0, 1) = 0 implies linearized instability, and instability withrespect to test function initial data follow from the more detailed calculations ofSection 4.3.4. ⇤

Proposition 4.2 gives a weak version of Result 2 of the introduction, the defi-ciency being that the normalizing factor sgn D(+1) is a priori unknown. Asdescribed in Section 6, [Z.3], the basic definition (4.2) may nonetheless yield usefulconclusions as a measure of spectral flow/change in stability under homotopy ofmodel or shock parameters. In particular, in the Lax case for which stability ofsmall-amplitude shocks is known, one may conclude that, moving along the Hugo-niot curve Hp(U�, ✓) in direction of increasing |✓|, instability occurs when ��(0, 1)first changes sign.

On the other hand, formula (4.2) is of little use as a measure of absolute stability,e.g., for nonclassical shocks, or Lax-type shocks occurring on disconnected compo-nents of the Hugoniot curve. It is useful therefore, as described with increasinglevels of generality in [GS, BSZ.1, Z.3], to give an alternative, “absolute” versionof the stability index in which sgn D(+1) is explicitly evaluated.

Lemma 4.3 [Z.3]. Let matrix A be symmetric, invertible, and matrix B positivesemidefinite, Re (B) � 0. Then, the cones S(A�1B)�(N(A)\kerB) and U(A) aretransverse, where S(M), U(M) refer to stable/unstable subspaces of M and N(M)to the cone {v : Re hv,Mvi 0}.

Proof. Suppose to the contrary that x0

6= 0 lies both in the cone S(A�1B) �(N(A) \ kerB) and in U(A), i.e.,

x0

= x1

+ x2


where x1

2 S(A�1B), x2

2 (N(A) \ kerB), and x0

2 U(A). Define x(t) bythe ordinary di↵erential equation x0 = A�1Bx, x(0) = x

0

. Then x(t) ! x2

ast ! +1 and thus limt!+1hx(t), Ax(t)i 0. On the other hand,

hx,Axi0 = 2hA�1Bx, Axi = 2hBx, xi � 0

by assumption, hence hx0

, Ax0

i 0, contradicting the assumption that x0

belongsto U(A). ⇤Proposition 4.4 [Z.3]. Given (A1)–(A3) and (H0)–(H3), we have

(4.3) sgn D(�) = sgn det (S+, U+)det (⇡Z+, "S+)det ("U�,⇡Z�)|�=0 6= 0,

for su�ciently large, real �, where ⇡ denotes projection of Z = (z1

, z2

, z02

) onto(z

1

, z2

) with

(4.4)✓

z1

z2

◆:=

✓A

11

A22

b1

b2

◆U,

" denotes extension of (z1

) to (z1

, 0), and S(x1

), U(x1

) are bases of the sta-ble/unstable subspaces of A1

⇤ = (A1

11

�A12

(b11

2

)�1b11

1

) (note: (n� r) dimensional).

Proof. Working in standard coordinates W = (uI , uII , z02

), we show equivalentlythat

(4.5) sgn D(�) = sgn det (S+, U+)det (⇡W+, "S+)det ("U�,⇡W�)|�=0 6= 0,

where ⇡ denotes projection of W = (uI , uII , z02

) onto (uI , uII) components, z2

:=b11

1

uI + b11

2

uII , "uI := (uI ,�(b11

2

)�1b11

1

uII) denotes extension, and S(x1

), U(x1

)are bases of the stable/unstable subspaces of A1

⇤Though it is not immediately obvious, we may after a suitable coordinate change

arrange that A1

± be symmetric, B11

± =✓

0 00 b11

2

◆

±, and Re (b11

2

)± > 0. For, as

pointed out in [KSh], we may without loss of generality take A0

± block-diagonal in(1.11), by multiplying coe�cients on the left by T t and on the right by T with T asuitably chosen upper block-triangular matrix; for further discussion, see the proofof Proposition 6.3, Appendix A.1. Multiplying on the left and right by (A0

±)�1/2

then reduces the equation to the desired form, in coordinate V = (A0

±)1/2W .It is su�cient to show that quantity (4.5) does not vanish in the class (A1), (A3).

For, since D(�) does not vanish either, for real � su�ciently large, we can thenestablish the result by homotopy of the symmetric matrix A1

± to an invertible realdiagonal matrix (straightforward, using the unitary decomposition A1 = UDU⇤,U⇤U = I, and the fact that the unitary group is arcwise connected26) and of B11

±

26See, e.g., [Se.5], Chapter 7.

64 KEVIN ZUMBRUN

to✓

0 00 Ir

◆(e.g., by linear interpolation of the positive definite b11

II± to Ir), in

which case it can be seen by explicit computation. (Note that the endpoint ofthis homotopy is on the boundary of but not in the Kawashima class (A2), sinceeigenvectors of A1 are in the kernel of B.)

As to behavior at � = 0, a bifurcation analysis as in Section 3 of the limitingconstant-coe�cient equations at ±1 shows that the projections ⇧ of slow modesof W

+

may be chosen as the unstable eigenvectors r+

j of A1

+

, corresponding tooutgoing characteristic modes, and the projections of fast modes as the stable (i.e.Re µ < 0) solutions of

(4.6) (A1 � µB11)+

✓uI

uII

◆=✓

00

◆,

or without loss of generality

(4.7)✓

A11

A12

A21

A22

� µb2

◆

+

✓uI

uII

◆=✓

00

◆,

and thus of form

(4.8)✓�(A

11

)�1A12

uII

uII

◆

+

,

where

(4.9)�b�1

2

(A22

�A21

(A11

)�1A12

)� µI�+

uII = 0.

Likewise, using b11+

1

= 0, we find from the definitions of S, " that stable solutionsS+ are in the stable subspace of A1

11+

, with " S+ =�S+

0

�, hence vectors "S+ lie in the

intersection of the stable subspace of A1

+

and the kernel of B11

+

. Our claim is thatthese three subspaces are independent, spanning Cn. Rewording this assumption,we are claiming that the stable subspace of (A1)�1B11

+

, the center subspace kerB11

+

intersected with the stable subspace of A1

+

, and the unstable subspace of A1

+

aremutually independent. (Note: that dimensions are correct follows by consistentsplitting, a consequence of (H1)(i)). But, this follows by Lemma 4.3, with A := A1

+

and B := B11

+

, hence det (⇡Z+, "S+) 6= 0. A symmetric argument shows thatdet ("U�,⇡Z�)|

�=0 6= 0, completing the result. ⇤Formula (4.5) gives in principle an explicit evaluation of �, involving only in-

formation that is linear-algebraic or concerning the dynamics of the traveling-waveODE; however, it is still quite complicated. In the extreme Lax shock case, theformula simplifies considerably.


Proposition 4.5. In the case of an extreme right (n-shock) Lax profile,

(4.10) � = sgn �2det (r�1

, . . . , r�n�1

, [U ])det (r�1

, . . . , r�n�1

, U 0/|U 0|(�1)) > 0

is necessary for one-dimensional stability. A symmetric formula holds for extremeleft (1-shock) Lax profiles.

Proof. From Corollary 1.8, we may deduce that � for an extreme right (i.e.,n-shock) Lax profile consists of a Wronskian involving only modes from the +1side, and is therefore explicitly evaluable; in particular, working in (z

1

, z2

, z02

) co-ordinates, we obtain � as a determinant of z

2

components only. Moreover, theexpression (4.5) simplifies greatly. For, in (z

1

, z2

) coordinates, we have that U = ;,S is full dimension n� r, and ✏S consists of vectors of the simple form (z, 0). Thismeans that det (S, U) simplifies to just det S, while det (Z�, ✏S+) simplifies to theproduct of det S+ and �. Therefore, this term, similarly as in the strictly paraboliccase, combines with like term � in the computation of the stability index. Finally,det (✏U0,⇧Z�) simplifies to det (r�

1

, . . . , r�n�1

, U 0), completing the result. ⇤Remark 4.6. Condition (4.10), our strongest version of Result 2, is identical

with that of the strictly parabolic case. The only very weak information requiredfrom the connection problem is the orientation of u0 as x ! �1, i.e., the directionin which the profile leaves along the one-dimensional unstable manifold. In the caser = 1, for example for isentropic gas dynamics, the traveling-wave ODE is scalar,and so the orientation of U 0 is determined by the direction of the connection. Inthis case, (4.10) gives a full evaluation.

4.2 Su�cient conditions.

By (H3), there exists an `-parameter family of stationary solutions {U�} of(1.1) near U , ` � 1. Indeed, by Lemma 1.6, these lie arbitrarily close to U inL1\C4(x

1

), precluding one-dimensional asymptotic stability of U for any reasonableclass of initial perturbation (say, test function initial data). The appropriate notionof stability of shock profiles in one dimension is, rather, “orbital” stability, orconvergence to the manifold of stationary solutions {U�}.

Definition 4.7. We define nonlinear orbital stability as convergence of U(·, t)as t !1 to U�(t), where �(·) is an appropriately chosen function, for any solutionU of (1.1) with initial data su�ciently close in some norm to the original profile U .Likewise, we define linearized orbital stability as convergence of U(·, t) to (@U�/@�) ·�(t) for any solution U of the linearized equations (2.1) with initial data boundedin some chosen norm.

Note that di↵erentiation with respect to �j of the standing-wave ODE F 1(U�)x1 =(B11(U�)U�

x1)x1 yields L

0

@U�/@�j = 0, so that (@U�/@�j) represent stationary so-lutions of (2.1), with Span{@U�/@�j)} lying tangent to the stationary manifold{U�}. In the Lax or undercompressive case, ` = 1, and the stationary manifold

66 KEVIN ZUMBRUN

is just the set of translates U�(x1

) := U(x1

� �) of U , with the tangent station-ary manifold Span{Ux1} corresponding to the usual translational zero eigenvaluearising in the study of stability of traveling waves.

Result 3 of the introduction is subsumed in the following two results, to beestablished throughout the rest of the section.

Theorem 4.8 [MaZ.4]. (Linearized stability) Let U be a shock profile (1.2) of(1.1), under assumptions (A1)–(A3) and (H0)–(H3). Then, U is L1 \ Lp ! Lp

linearly orbitally stable in dimension d = 1, for all p > 1, if and only if it satisfiesthe conditions of structural stability (transversality), dynamical stability (inviscidstability), and strong spectral stability. More precisely, for initial data U

0

2 L1\Lp,p � 1, the solution U(x, t) of equations (1.1) linearized about U satisfies

(4.11) |U(·, t) + �(t)U 0(·)|Lp C(1 + t)�12 (1�1/p)|U

0

|L1\Lp .

Theorem 4.9. (Nonlinear stability) Let U be a Lax-type profile (1.2) of a gen-eral real viscosity model (1.1), satisfying (A1)–(A3), (H0)–(H3), and the neces-sary conditions of structural, dynamical, and strong spectral stability. Then, U isL1 \H3 ! Lp \H3 nonlinearly orbitally stable in dimension d = 1, for all p � 2.More precisely, the solution U(x, t) of (1.1) with initial data U

0

satisfies

(4.12)|U(x, t)� U(x� �(t))|Lp C(1 + t)�

12 (1�1/p)|U

0

|L1\H3 ,

|U(x, t)� U(x� �(t))|H3 C(1 + t)�14 |U

0

|L1\H3

for initial perturbations U0

:= U0

� U that are su�ciently small in L1 \H3, for allp � 2, for some �(t) satisfying �(0) = 0,

(4.13) |�(t)| C(1 + t)�12 |U

0

|L1\H3 ,

and

(4.14) |�(t)| C|U0

|L1\H3 .

Remarks. 1. For slightly more regular initial perturbation U0

2 L1 \ H4,we may obtain by the same argument, using higher-derivative Green distributionbounds (described, e.g., in [MaZ.2, MaZ.4]), the derivative estimate

|U(x, t)� U(x� �(t))|W 1,p C(1 + t)�12 (1�1/p)|U

0

|L1\H4

generalizing (4.12), which yields in turn the extension of (4.12) to low norms Lp,1 p 2, by a simple bootstrap argument (described in the Remark just belowthe proof of Theorem 4.9).

2. Theorem 4.8 and the extension of 4.9 just described contain in particular theresults of L1 ! L1 linear and L1\H4 ! L1\H4 nonlinear global bounded stability,indicating that L1 and L1 \H4, respectively, are natural norms for the linearizedand nonlinear viscous shock perturbation problem in one dimension.


4.2.1. Linearized estimates.

Theorem 4.8 (and to some extent Theorem 4.9) is obtained as a consequenceof detailed, pointwise bounds on the Green distribution G(x, t; y) of the linearizedevolution equations (2.1), which we now describe. For readability, we defer therather technical proof to Section 4.3. In the remainder of this section, for easeof notation, we drop superscripts on A1, F 1 and B11, and subscripts on x

1

andy1

, denoting them simply as A, F , B, x, and y. We also make the simplifyingassumption

(P0) Either A± are strictly hyperbolic, or else A± and B± are simultaneouslysymmetrizable.As described in Remark 4.13 below, the bounds needed for our stability analysispersist in the general case, but the full description of the Green’s function does not.

Let a±j , j = 1, . . . (n) denote the eigenvalues of A(±1), and l±j and r±j associatedleft and right eigenvectors, respectively, normalized so that l±t

j r±k = �jk. In case

A± is strictly hyperbolic, these are uniquely defined up to a scalar multiplier.Otherwise, we require further that l±j , r±j be left and right eigenvectors also ofP±

j B±P±j , P±

j := R±j L±t

j , where L±j and R±

j denote m±j ⇥ m±

j left and righteigenblocks associated with the m±

j -fold eigenvalue a±j , normalized so that L±j R±

j =Im±

j

. (Note: The matrix P±j B±P±

j ⇠ L±tj B±R±

j is necessarily diagonalizable, bysimultaneous symmetrizability of A±, B±).

Eigenvalues a±j , and eigenvectors l±j , r±j correspond to large-time convectionrates and modes of propagation of the degenerate model (1.1). Likewise, let a⇤j (x),j = 1, . . . , (n� r) denote the eigenvalues of

(4.15) A⇤ := A11

�A12

B�1

22

B21

,

A := dF (U(x)), and l⇤j (x), r⇤j (x) 2 Rn�r associated left and right eigenvectors,normalized so that l⇤tj rj ⌘ �j

k. More generally, for an m⇤j -fold eigenvalue, we choose

(n�r)⇥m⇤j blocks L⇤j and R⇤j of eigenvectors satisfying the dynamical normalization

L⇤tj @xR⇤j ⌘ 0,

along with the usual static normalization L⇤tj Rj ⌘ �jkIm⇤

j

; as shown in Lemma 4.9,[MaZ.1], this may always be achieved with bounded L⇤j , R⇤j . Associated with L⇤j ,R⇤j , define extended, n⇥m⇤

j blocks

(4.16) L⇤j :=✓

L⇤j0

◆, R⇤j :=

✓R⇤j

�B�1

22

B21

R⇤j

◆.

Eigenvalues a⇤j and eigenmodes L⇤j , R⇤j correspond, respectively, to short-time hy-perbolic characteristic speeds and modes of propagation for the reduced, hyperbolicpart of degenerate system (1.1).

68 KEVIN ZUMBRUN

Define time-asymptotic, e↵ective di↵usion coe�cients

(4.17) �±j :=�ltjBrj

�± , j = 1, . . . , n,

and local, mj ⇥mj dissipation coe�cients

(4.18) ⌘⇤j (x) := �L⇤tj D⇤R⇤j (x), j = 1, . . . , J n� r,

where

(4.19)D⇤(x) :=

A12

B�1

22

hA

21

�A22

B�1

22

B21

+ A⇤B�1

22

B21

+ B22

@x(B�1

22

B21

)i

is an e↵ective dissipation analogous to the e↵ective di↵usion predicted by formal,Chapman–Enskog expansion in the (dual) relaxation case.

At x = ±1, these reduce to the corresponding quantities identified by Zeng[Ze.1,LZe] in her study by Fourier transform techniques of decay to constant solu-tions (u, v) ⌘ (u±, v±) of hyperbolic–parabolic systems, i.e., of limiting equations

(4.20) Ut = L±U := �A±Ux + B±Uxx.

These arise naturally through Taylor expansion of the one-dimensional (frozen-coe�cient) Fourier symbol (�i⇠A� ⇠2B)±, as described in Appendix A.4; in par-ticular, as a consequence of dissipativity, (A2), we obtain (see, e.g., [Kaw, LZe,MaZ.3], or Lemma 2.18)

(4.21) �±j > 0, Re �(⌘⇤±j ) > 0 for all j.

However, note that the dynamical dissipation coe�cient D⇤(x) does not agree withits static counterpart, possessing an additional term B

22

@x(B�1

22

B21

), and so wecannot conclude that (4.21) holds everywhere along the profile, but only at theendpoints. This is an important di↵erence in the variable-coe�cient case; see Re-marks 1.11-1.12 of [MaZ.3] for further discussion.

Proposition 4.10[MaZ.3]. In spatial dimension d = 1, under assumptions (A1)–(A3), (H0)–(H3), and (P0), for Lax-type shock profiles satisfying the conditionsof transversality, hyperbolic stability, and strong spectral stability, the Green dis-tribution G(x, t; y) associated with the linearized evolution equations (2.1) may bedecomposed as

(4.22) G(x, t; y) = H + E + S + R,

where, for y 0:

(4.23)

H(x, t; y) :=JX

j=1

a⇤�1

j (x)a⇤j (y)R⇤j (x)⇣⇤j (y, t)�x�a⇤j

t(�y)L⇤tj (y)

=JX

j=1

R⇤j (x)O(e�⌘0t)�x�a⇤j

t(�y)L⇤tj (y),


(4.24)

E(x, t; y) :=X

a�k

>0

[c0

k,�]U 0(x)l�tk

0

@errfn

0

@y + a�k tq4��k t

1

A� errfn

0

@y � a�k tq4��k t

1

A

1

A ,

and(4.25)S(x, t; y) := �{t�1}

X

a�k

<0

r�k l�kt(4⇡��k t)�1/2e�(x�y�a�

k

t)2/4��k

t

+ �{t�1}X

a�k

>0

r�k l�kt(4⇡��k t)�1/2e�(x�y�a�

k

t)2/4��k

t

✓e�x

ex + e�x

◆

+ �{t�1}X

a�k

>0, a�j

<0

[cj,�k,�]r�j l�k

t(4⇡��jkt)�1/2e�(x�z�jk

)

2/4

¯��jk

t

✓e�x

ex + e�x

◆,

+ �{t�1}X

a�k

>0, a+j

>0

[cj,+k,�]r+

j l�kt(4⇡�+

jkt)�1/2e�(x�z+jk

)

2/4

¯�+jk

t

✓ex

ex + e�x

◆

denote hyperbolic, excited, and scattering terms, respectively, ⌘0

> 0, and R denotesa faster decaying residual (described in Proposition 4.39 below). Symmetric boundshold for y � 0.

Here, the averaged convection rates a⇤j = a⇤j (x, t) in (4.23) denote the time-averages over [0, t] of a⇤j (x) along backward characteristic paths z⇤j = z⇤j (x, t) definedby

(4.26) dz⇤j /dt = a⇤j (z⇤j ), z⇤j (t) = x,

and the dissipation matrix ⇣⇤j = ⇣⇤j (x, t) 2 Rm⇤j

⇥m⇤j is defined by the dissipative flow

(4.27) d⇣⇤j /dt = �⌘⇤j (z⇤j )⇣⇤j , ⇣⇤j (0) = Imj

.

Similarly, in (4.25),

(4.28) z±jk(y, t) := a±j

✓t� |y|

|a�k |

◆

and

(4.29) �±jk(x, t; y) :=|x±||a±j t|

�±j +|y||a�k t|

a±ja�k

!2

��k ,

represent, respectively, approximate scattered characteristic paths and the time-averaged di↵usion rates along those paths. In all equations, aj, a⇤±j , lj, L⇤±j ,

70 KEVIN ZUMBRUN

rj, R⇤±j , �±j and ⌘⇤j are as defined just above, and scattering coe�cients [cj,ik,�],

i = �, 0,+, are uniquely determined by

(4.30)X

a�j

<0

[cj,�k,�]r�j +

X

a+j

>0

[cj, +k,�]r+

j + [c0

k,�]�U(+1)� U(�1)

�= r�k

for each k = 1, . . . n, and satisfying

(4.31)X

a�k

>0

[c0

k,�]l�k =X

a+k

<0

[c0

k,+]l+k = ⇡,

where the constant vector ⇡ is the left zero e↵ective eigenfunction of L associatedwith the right eigenfunction U 0. Similar decompositions hold in the over- and un-dercompressive case.

Proposition 4.10, the variable-coe�cient generalization of the constant-coe�cientresults of [Ze.1, LZe], was established in [MaZ.3] by Laplace transform (i.e., semi-group) techniques generalizing the Fourier transform approach of [Ze.1–2, LZe]; fordiscussion/geometric interpretation, see [Z.6, MaZ.1–2]. In our stability analysis,we will use only a small part of the detailed information given in the proposition,namely Lp ! Lq estimates on the time-decaying portion H + S + R of the Greendistribution G (see Lemma 4.1, below). However, the stationary portion E of theGreen distribution must be estimated accurately for an e�cient stability analysis.

Remark 4.11[MaZ.2]. The function H described in Proposition 4.10 satisfies

(4.32) H(x, t; y)⇧2

⌘ 0

and

(4.33) ⇧2

(A0(x))�1H(x, t; y) ⌘ 0,

where ⇧2

:=✓

0 00 Ir

◆denotes projection onto final r coordinates, or equivalently

L⇤trj ⇧2

⌘ 0 and ⇧2

(A0)�1R⇤j ⌘ 0, 1 j J .These may be seen by the intrinsic property of L⇤j and R⇤j (readily obtained from

our formulae) that they lie, respectively, in the left and right kernel of B. This givesthe first relation immediately, by the structure of B given in (A3). Likewise, wefind that

(4.34)0 ⌘ BR⇤j :=

✓0 00 b

◆(A0)�1R⇤j

=✓

0b⇧

2

(A0)�1R⇤j

◆,


yielding the second result by invertibility of b (see condition (A3)). This quantifiesthe observation that, in W = (wI , wII)t coordinates, data in the wII coordinate issmoothed under the evolution of (2.1), whereas data in the wI coordinate is not;likewise, the “parabolic” variable wII experiences smoothing while the “hyperbolic”coordinate wI does not. (Recall that W = (A0)�1U).

Remark 4.12. Following [Ho], we may recover the hyperbolic evolution equa-tions (4.23), (4.26)–(4.27) by direct calculation, considering the evolution of a jumpdiscontinuity along curve x = z(t) in the original equation Ut + (AU)x = (BUx)x;see also the related calculation of Section 1.3, [MaZ.1] in the relaxation case. Forsimplicity, write U = (u, v). Introducing the parabolic variable v := b

1

u + b2

v, anddenoting jump across z of a variable f by [f ], we may write (2.1) as the pair ofequations

(4.35) ut + (A⇤u)x + (A12

b�1

2

v)x = 0

and(4.36)vt +

⇣(A

21

�A22

b�1

2

b1

+ b1,x� b

2,xb�1

2

b1

)u⌘

x+⇣(A

22

b�1

2

+ b2,xb�1

2

)v⌘

x� (vx)x = 0.

Applying the Rankine–Hugoniot conditions, we obtain relations

(A⇤ � z)[u] = 0, [v] = 0, and

(4.37)[vx] =

⇣A

21

�A22

b�1

2

b1

+ zb�1

2

b1

+ b2

@x(b�1

2

b1

)⌘[u] =

⇣A

21

�A22

b�1

2

b1

+ b�1

2

b1

A⇤ + b2

@x(b�1

2

b1

)⌘[u] =: d⇤[u],

yielding zj = aj(zj(t)), [u] = R⇤j (zj(t))⇣j(t), [v] ⌘ 0, and [vx] = d⇤R⇤j (zj(t))⇣j(t),⇣j 2 R. Substituting these Ansatze into (4.35) yields

[ut] + a⇤j [ux] + a⇤j,x[u] + A12

b�1

2

[vx] + (A12

b�1

2

)x[v] = 0,

and thus˙[u] + a⇤j,x[u] + D⇤[u] = 0,

where all coe�cients are evaluated at zj(t) and D⇤ = A12

b�1

2

d⇤ as defined in (4.19).Taking inner product with L⇤j (zj(t)) and rearranging using

L⇤tj R⇤j = � ˙L⇤tj R⇤j = �a⇤j (@xL⇤tj )R⇤j ,

we therefore obtain the characteristic evolution equations

(4.38) ⇣j =�a⇤j (@xL⇤tj )R⇤j � @xaj � ⌘j

�⇣j ,

72 KEVIN ZUMBRUN

⌘j = L⇤tj D⇤R⇤j as defined in (4.18). Under the normalization (@xL⇤tj )R⇤j ⌘ 0 (note:automatic in the gas-dynamical case considered in [Ho], for which L⇤j and R⇤j arescalar), this yields in the original coordinate U precisely the solution operatora⇤�1

j (zj)a⇤j (y)R⇤j (zj)⇣j(y, t)L⇤j (y) described in (4.23).

Remark 4.13. In the case that A± is nonstrictly hyperbolic and B± are notsimultaneously symmetrizable, the simple description of scattering terms breaksdown, and much more complicated behavior appears to occur. However, the excitedterm E remains much the same, with scalar di↵usion replaced by a multi-mode heatkernel and associated multi-mode error function. Likewise, a review of the point-wise Green’s function arguments of Section 4.3.3 shows that they still yield sharpmodulus bounds on the scattering term even though we do not know the leading-order behavior, and so all bounds relevant to the ultimate stability argument gothrough.

4.2.2 Linearized stability.

As described in [MaZ.2–3], linearized orbital stability follows immediately fromthe pointwise bounds of Proposition 4.10. We reproduce the argument here, bothfor completeness and to motivate the nonlinear argument to follow in Section 4.2.4.Similarly as in [Z.6, MaZ.1–3], define the linear instantaneous projection:

(4.39)'(x, t) :=

Z+1

�1E(x, t; y)U

0

(y) dy

=: ��(t)U 0(x),

where U0

denotes the initial data for (2.1), and U = U(x) as usual. The amplitude� may be expressed, alternatively, as

�(t) = �Z

+1

�1e(y, t)U

0

(y) dy,

where

(4.40) E(x, t; y) =: U 0(x)e(y, t),

i.e.,

(4.41) e(y, t) :=X

a�k

>0

[C0

k,�]l�tk

0

@errfn

0


1

A� errfn

0


1

A

1

A

for y 0, and symmetrically for y � 0.


Then, the solution U of (2.1) satisfies

(4.42) U(x, t)� '(x, t) =Z

+1

�1(H + G)(x, t; 0)U

0

(y) dy,

where

(4.43) G := S + R

is the regular part and H the singular part of the time-decaying portion of theGreen distribution G.

Lemma 4.14 [MaZ.2]. In spatial dimension d = 1, under assumptions (A1)–(A3),(H0)–(H3), for Lax-type shock profiles satisfying the conditions of transversality,hyperbolic stability, and strong spectral stability, G and H satisfy

(4.44) |Z

+1

�1G(·, t; y)f(y)dy|Lp C(1 + t)�

12 (1/q�1/p)|f |Lq ,

(4.45) |Z

+1

�1Gy(·, t; y)f(y)dy|Lp C(1 + t)�

12 (1/q�1/p)�1/2|f |Lq + Ce�⌘t|f |Lp ,

and

(4.46) |Z

+1

�1H(·, t; y)f(y)dy|Lp Ce�⌘t|f |Lp ,

(4.47) |Z

+1

�1Hx(·, t; y)f(y)dy|Lp Ce�⌘t|f |W 1,p ,

for all t � 0, some C, ⌘ > 0, for any 1 q p (equivalently, 1 r p) andf 2 Lq \W 1,p, where 1/r + 1/q = 1 + 1/p.

Proof. Assuming (P0), bounds (4.44)–(4.45) follow by the Hausdor↵-Younginequality together with bounds (4.25) and the more stringent bounds on R andits derivatives given in Proposition 4.39 below; see [MaZ.2–3] for further details.Bound (4.46) follows by direct computation and the fact that particle paths z⇤j (x, t)satisfy uniform bounds

1/C |(@/@x)zj | < C,

for all x, t, by the fact that characteristic speeds aj(z) converge exponentially asz ! ±1 to constant states. Finally, (4.47) follows from the relations

�x�a⇤j

t(�y) = a⇤j (y)�1�

t�

Z x

y

dz

a⇤j (z)

!

74 KEVIN ZUMBRUN

and✓@

@x

◆�

t�

Z x

y

dz

a⇤j (z)dz

!= a⇤j (y)a⇤j (x)�1

✓@

@y

◆�

t�

Z x

y

dz

a⇤j (z)

!,

where a⇤j = a⇤j (x, t) is defined as the average of a⇤j (z) along the backward char-acteristic z⇤j (x, t) defined by (4.26), which are special cases of the more generalrelations

�h(x,t)(y) = fy(x, y, t)��f(x, y, t)

�

and(@/@x)�(f(x, y, t)) = (fx/fy)(@/@y)�(f(x, y, t)),

where h(x, t) is defined by f(x, h(x, t), t) ⌘ 0. Noting that the same modulusbounds, though not the pointwise description, persist also when (P0) fails, Remark4.13, we obtain the general case as well. ⇤

Proof of Theorem 4.8. It is equivalent to show that, for initial data U0

2L1 \ Lp, the solution U(x, t) of (2.1) satisfies

(4.48) |U(·, t)� '(·, t)|Lp C(1 + t)�12 (1�1/p)(|U

0

|L1 + |U0

|Lp).

But, this follows immediately from (4.42) and bounds (4.44) and (4.46), with q = p.The estimate (4.11) gives su�ciency of structural, dynamical, and strong spectralstability for viscous linearized orbital stability. Necessity will be established by aseparate, and somewhat simpler computation in Section 4.3.5; see Proposition 4.43and Corollary 4.44. ⇤

4.2.3 Auxiliary energy estimate.

Along with Proposition 4.10, the proof of Theorem 4.9 relies on the follow-ing auxiliary energy estimate, generalizing estimates of Kawashima [Kaw] in theconstant–coe�cient case. Following [MaZ.2], define nonlinear perturbation

(4.49) U(x, t) := U(x + �(t), t)� U(x),

where the “shock location” � is to be determined later; for definiteness, fix �(0) = 0.Evidently, decay of U is equivalent to nonlinear orbital stability as described in(4.12).

Proposition 4.15[MaZ.4]. Under the hypotheses of Theorem 4.9 let U0

2 H3,and suppose that, for 0 t T , both the supremum of |�| and the W 2,1 norm ofthe solution U = (uI , uII)t of (1.1), (4.49) remain bounded by a su�ciently smallconstant ⇣ > 0. Then, for all 0 t T ,

(4.50) |U(t)|2H3 C|U(0)|2H3e�✓t + C

Z t

0

e�✓2(t�⌧)(|U |2L2 + |�|2)(⌧) d⌧.


Remarks 4.16. 1. Estimate (4.50) asserts that the H3 norm is controlledessentially by the L2 norm, that is, it reveals strong hyperbolic-parabolic smoothingof the initial data. Weaker versions of this estimate may be found in [Kaw, MaZ.1–2,MaZ.4] (used there, as here, to control higher derivatives in the nonlinear iterationyielding time-asymptotic decay).

2. As described in [MaZ.2], Hs energy estimates may for small-amplitude profilesbe obtained essentially as a perturbation of the constant–coe�cient case, usingthe fact that coe�cients are in this case slowly varying; in particular, hypothesis(H1) is not needed in this case. In the large-amplitude case, there is a key newingredient in the analysis beyond that of the constant-coe�cient case, connectedwith the “upwind” hypothesis (H1)(ii) requiring that hyperbolic convection ratesbe of uniform sign relative to the shock speed s. This allows us to control terms ofform

RO(|Ux|)|v|2 that were formerly controlled by the small-amplitude assumption

|Ux| = O("), " := |U+

� U�| << 1, instead by a “Goodman-type” weighted normestimate in the spirit of [Go.1–2], thus closing the argument.

Proof of Proposition 4.15. A straightforward calculation shows that |U |Hr ⇠|W |Hr ,

(4.51) W = W � W := W (U)�W (U),

for 0 r 3 provided |U |W 2,1 remains bounded, hence it is su�cient to prove acorresponding bound in the special variable W . We first carry out a complete proofin the more straightforward case that the equations may be globally symmetrizedto exact form (1.7), i.e., with conditions (A1)–(A3) replaced by the following globalversions, indicating afterward by a few brief remarks the changes needed to carryout the proof in the general case.

(A1’) Aj , Ajk⇤ := Ajk

11

, A0 are symmetric, A0 > 0.

(A2’) No eigenvector ofP⇠jdF j(U) lies in the kernel of

P⇠j⇠kBjk(U), for

all nonzero ⇠ 2 Rd. (Equivalently, no eigenvector ofP⇠jAj(A0)�1(W ) lies in the

kernel ofP⇠j⇠kBjk(W ).)

(A3’) Bjk =✓

0 00 bjk

◆, with Re

P⇠j⇠k bjk(W ) � ✓|⇠|2 for some ✓ > 0, for

all W and all ⇠ 2 Rd, and G ⌘ 0.

Substituting (4.49) into (1.11), we obtain the quasilinear perturbation equation

(4.52)⇣A0Wt � A0Wt

⌘+⇣AWx � AWx

⌘�⇣BWx � BWx

⌘

x= �(t)A0Wx,

where(4.53)A0 := A0(W ), A0 := A0(W ); A := A(W ), A := A(W ); B := B(W ), B := B(W ),

76 KEVIN ZUMBRUN

using the quadratic Leibnitz relation

(4.54) A2

W2

�A1

W1

= A2

(W2

�W1

) + (A2

�A1

)W1

,

and recalling the block structure assumption (A3), we obtain the alternative per-turbation equation:

(4.55) A0Wt + AWx � (BWx)x = M1

Wx + (M2

Wx)x + �(t)A0Wx + �(t)A0Wx,

where

(4.56) M1

= M1

(W, W ) := A� A =✓Z

1

0

dA(W + ✓W ) d✓

◆W,

and

(4.57) M2

= M2

(W, W ) := B � B =✓

0 00 (

R1

0

db(W + ✓W ) d✓)W

◆.

We now carry out a series of successively higher order energy estimates of thethe type formalized by Kawashima [Kaw]. The origin of this approach goes backto [Kan,MNi] in the context of gas dynamics; see, e.g., [HoZ.1] for further discus-sion/references. The novelty here is that the stationary background state W is notconstant in x, nor even slowly varying as in [MaZ.2]. We overcome this di�culty byintroducing the additional ingredient of a Goodman-type weighted norm in orderto control hyperbolic modes, assumed for this purpose (see hypothesis (H1)(ii)) tobe uniformly transverse to the background shock profile.

Let K denote the skew-symmetric matrix described in Lemma 2.18 associatedwith A0, A, B, satisfying

K(A0)�1A + B > 0.

Then, regarding A0, K, we have the bounds(4.58)

A0

x = dA0(W )Wx, Kx = dK(W )Wx, Ax = dA(W )Wx, Bx = dB(W )Wx,

A0

t = dA0(W )Wt, Kt = dK(W )Wt, At = dA(W )Wt, Bx = dB(W )Wt,

and (from the defining equations):

(4.59) |Wx| = |Wx + Wx| |Wx|+ |Wx|

and

(4.60)

|Wt| C(|Wx|+ |wIIxx|+ |�||Wx|)

C(|Wx|+ |Wx|+ |wIIxx|+ |wII

xx|+ |�||Wx|+ |�||Wx|) C(|Wx|+ |Wx|+ |wII

xx|).


Thus, in particular

(4.61)|�|, |A0

x|, |A0

xx|, |Kx|, |Kxx|, |Ax|, |Axx|, |Bx|, |Bxx|, |A0

t |, |Kt|, |At|, |Bt| C(⇣ + |Ux|).

Finally, we introduce the weighted norms and inner product

(4.62) |f |↵ := |↵1/2f |L2 , |f |Hs

↵

:=sX

r=0

|@rxf |↵, hf, gi↵ := h↵f, giL2 ,

↵(x) scalar, uniformly positive, and uniformly bounded. For the remainder of thissection, we shall for notational convenience omit the subscript ↵, referring alwaysto ↵-norms or -inner products unless otherwise specified. For later reference, wenote the commutator relation

(4.63) hf, gxi = �hfx + (↵x/↵)f, gi,

and the related identities

(4.64) hf, Sfxi = �(1/2)hf,�Sx + (↵x/↵)S

�fi,

(4.65) hf, (Sf)xi = (1/2)hf,�Sx � (↵x/↵)S

�fi,

valid for symmetric operators S.By (H1)(ii), we have that A

11

(A0

11

)�1 has real spectrum of uniform sign, withoutloss of generality negative, so that the similar matrix

(A0

11

)�1/2A11

(A0

11

)�1/2 = (A0

11

)�1/2A11

(A0

11

)�1(A0

11

)1/2

has real, negative spectrum as well. (Recall, A0

11

is symmetric negative definite asa principal minor of the symmetric negative definite matrix A0.) It follows thatA

11

itself is uniformly symmetric negative definite, i.e.,

(4.66) A11

�✓ < 0.

Defining ↵, following Goodman [Go.2], by the ODE

(4.67) ↵x = C⇤|Ux|↵, ↵(0) = 1,

where C⇤ > 0 is a large constant to be chosen later, we have by (4.66)

(4.68) (↵x/↵)A11

�C⇤✓|Ux|.

78 KEVIN ZUMBRUN

Note, because |Ux| Ce�✓|x| (see Lemma 1.6), that ↵ is indeed positive andbounded from both zero and infinity, as the solution of the simple scalar exponentialgrowth equation (4.67). In what follows, we shall need to keep careful track of thedistinguished constant C⇤.

Computing

(4.69) �hW, AWxi = (1/2)hW, (Ax + (↵x/↵)A)W iand expanding A = A + O(⇣), Ax = O(|Ux| + ⇣), we obtain by (4.68) the keyproperty

(4.70)

�hW, AWxi = (1/2)hu, (↵x/↵)A11

ui

+O⇣h|Ux||W |, |W |i+ h(↵x/↵)|W |, ⇣|W |+ |wII |i

⌘

�(C⇤✓/3)h|wI |, |Wx||wI |i+ C⇣|wI |2 + C(C⇤)|wII |2,by which we shall control transverse modes, provided C⇤ is chosen su�ciently large,or, more generally,

(4.71) �h@kxW, A@k

xWxi �(C⇤✓/3)h|@kxu|, |Ux||@k

xu|i+ C⇣|@kxu|2 + C(C⇤)|@k

xv|2.Here and below, C(C⇤) denotes a suitably large constant depending on C⇤, whileC denotes a fixed constant independent of C⇤: likewise, O(·) indicates a boundindependent of C⇤.

Zeroth order “Friedrichs-type” estimate. We first perform a standard,zeroth- and first-order “Friedrichs-type” estimate for symmetrizable hyperbolicsystems [Fri]. Taking the ↵-inner product of W against (4.55), we obtain afterrearrangement, integration by parts using (4.63)–(4.64), and several applications ofYoung’s inequality, the energy estimate(4.72)12hW, A0Wit = hW, A0Wti+

12hW, A0

t Wi

= �hW, AWxi+ hW, (BWx)xi+ hW,M1

Uxi+ hW, (M2

Wx)xi

+ �(t)hW, A0Wxi+ �(t)hW, A0Uxi+12hW, A0

t Wi

=12hW, (Ax + (↵x/↵)A)W i � hWx � (↵x/↵)W, BWxi+ hW,M

1

Uxi

� hWx + (↵x/↵)W,M2

Uxi �12�(t)hW, (A0

x + (↵x/↵)A0)W i

+ �(t)hW, A0Uxi+12hW, A0

t Wi

�hWx, BWxi+ C(C⇤)Z↵⇣(|Wx|+ |Ux|)|W |2

+ |wIIx ||W |(|Wx|+ |Ux|) + |�||W ||Ux|

⌘

�✓|vx|2 + C(C⇤)⇣|W |2L2 + |�|2

⌘.


Here, we have freely used the bounds (2.18), as well as (4.61). We have alsoused in a crucial way the block-diagonal form of M

2

in estimating |hWx,M2

Uxi| CR|vx||W ||Ux| in the first inequality.

First order “Friedrichs-type” estimate. For first and higher derivativeestimates, it is crucial to make use of the favorable terms (4.71) a↵orded by theintroduction of ↵-weighted norms. Di↵erentiating (4.55) with respect to x, takingthe ↵-inner product of Wx against the resulting equation, and substituting theresult into the first term on the righthand side of

(4.73)12hWx, A0Wxit = hWx, (A0Wt)xi � hWx, A0

xWti+12hWx, A0

t Wxi,

we obtain after various simplifications and integrations by parts:(4.74)12hWx, A0Wxit = �hWx, (AWx)xi+ hWx, (BWx)xxi+ hWx, (M

1

Wx)xi

+ hWx, (M2

Wx)xxi+ �(t)hWx, (A0Wx)xi

+ �(t)hWx, (A0Wx)xi � hWx, A0

xWti+12hWx, A0

t Wxi

= �hWx, AWxxi � hWx, AxWxi � hWxx + (↵x/↵)Wx, BWxx + BxWxi+ hWx, (M

1

Ux)xi � hWxx + (↵x/↵)Wx, (M2

Ux)xi

+12�(t)hWx,

�A0

x � (↵x/↵)A0

�Wxi+ �(t)hWx, A0

xWxi

+ �(t)hWx, A0Wxxi � hWx, A0

xWti+12hWx, A0

t Wxi.

Estimating the first term on the righthand side of (4.74) using (4.71), k = 1, andsubstituting (A0)�1 times (4.55) into the second to last term on the righthand sideof (4.74), we obtain by (4.61) plus various applications of Young’s inequality the

80 KEVIN ZUMBRUN

next-order energy estimate:(4.75)12hWx, A0Wxit �hWx, AWxxi � hWxx, BWxxi

+ C(C⇤)Z

(|W |2 + |�|2)(|Uxx|+ |Ux|)

+ C(C⇤)h|W IIx |+ ⇣|Wx|, (|W |+ |Wx|)|Ux|+ |wII

xx|i+ Ch(|Wx|+ |wII

xx|), |Ux|(|W |+ |Wx|)i

�✓|wIIxx|2 � (C⇤✓/3)h|Ux||wI

x|, |wIx|i+ C⇣|wI

x|2 + C(C⇤)|wIIx |2

+ C(C⇤)⇣(|W |2L2 + |�|2

⌘

+ C(C⇤)h|vx|+ ⇣|Wx|, (|W |+ |Wx|)|Ux|+ |wIIxx|i

+ Ch(|Wx|+ |wIIxx|), |Ux|(|W |+ |Wx|)i

�(✓/2)|wIIxx|2 � (C⇤✓/4)h|Wx||wI

x|, |wIx|i+ C(C⇤)⇣|wI

x|2 + C(C⇤)|wIIx |2

+ C(C⇤)⇣(|W |2L2 + |�|2

⌘,

provided C⇤ is su�ciently large and ⇣ su�ciently small.First order “Kawashima-type” estimate. Next, we perform a “Kawashima-

type” derivative estimate. Taking the ↵-inner product of Wx against K(A0)�1 times(4.55), and noting that (integrating by parts, and using skew-symmetry of K)

(4.76)

12hWx, KWit =

12hWx, KWti+

12hWxt, KWi+

12hWx, KtWi

=12hWx, KWti �

12hWt, KWxi

� 12hWt,

�Kx + (↵x/↵)

�W i+

12hWx, KtWi

= hWx, KWti+12hW,

�Kx + (↵x/↵)

�Wti+

12hWx, KtWi,

we obtain by calculations similar to the above the auxiliary energy estimate:

(4.77)

12hWx, KWit �hWx, K(A0)�1AWxi

+ C(C⇤)|wIIx |2 + Ch(|Ux|+ ⇣ + ⇣)|wI

x|, |wIx|i

+ C ⇣�1|wIIxx|2 + C(C⇤)(|W |2L2 + |�(t)|2),

where ⇣ > 0 is an arbitrary constant arising through Young’s inequality. (Here, wehave estimated term hAUx, (↵x/↵)Ui arising in the middle term of the righthandside of (4.76) using (4.64) by C(C⇤)

R|Ux||U |2 C(C⇤)|U |2L1 .)


Combined, weighted H1 estimate. Choosing ⇣ << ⇣ << 1, adding (4.77) tothe sum of (4.72) and (4.75) times a suitably large positive constant M(C⇤, ⇣) >>⇣�1, and recalling (2.53), we obtain, finally, the combined first-order estimate

(4.78)

12

⇣M(C⇤, ⇣)hW, A0Wi+ hWx, KWi+ M(C⇤, ⇣)hWx, A0Wxi

⌘

t

�✓(|Wx|2 + |wIIxx|2) + C(C⇤)

⇣|W |2L2 + |�|2

⌘,

✓ > 0, for any ⇣, ⇣(⇣, C⇤) su�ciently small, and C⇤, C(C⇤) su�ciently large.

Higher order estimates. Performing the same procedure on the twice- andthrice-di↵erentiated versions of equation (4.55), we obtain, likewise, Friedrichs es-timates(4.79)

12h@q

xW,A0@q

xW it �(✓/2)|@q+1

x wII |2 � (C⇤✓/4)h|Wx||@qxwI |, |@q

xwI |i

+ C(C⇤)�⇣|@q

xwI |2 + |@qxwII |2 + |Wx|Hq�2

↵

+ |W |2L2 + |�|2�,

and Kawashima estimates(4.80)

12h@q

xW, K@q�1

x W it �h@qxW, K(A0)�1A@q

xW i

+ C(C⇤)|@qxwII |2 + Ch(|Wx|+ ⇣ + ⇣)|@q

xu|, |@qxu|i

+ C ⇣�1|@q+1

x wII |2 + C(C⇤)(|Wx|Hq�2↵

+ |W |2L2 + |�(t)|2,

for q = 2, 3, provided ⇣, ⇣(⇣, C⇤) are su�ciently small, and C⇤, C(C⇤) are su�-ciently large. The calculations are similar to those carried out already; see also theclosely related calculations of Appendix A, [MaZ.2].

Final estimate. Adding M(C⇤, ⇣)2 times (4.78), M(C⇤, ⇣) times (4.79), and(4.80), with q = 2, where M is chosen still larger if necessary, we obtain

(4.81)

12

⇣M(C⇤, ⇣)3hW, A0Wi+ M(C⇤, ⇣)2hWx, KWi+ M(C⇤, ⇣)3hWx, A0Wxi

+ h@2

xW, K@xW i+ M(C⇤, ⇣)h@2

xW, A0@2

xi⌘

t

�✓(|Wx|2H1↵

+ |wIIx |2H2

↵

) + C(C⇤)⇣|W |2L2) + |�|2

⌘.

Adding now M(C⇤, ⇣)2 times (4.81), M(C⇤, ⇣) times (4.79), and (4.80), with

82 KEVIN ZUMBRUN

q = 3, we obtain the final higher-order estimate

(4.82)

12


+ M(C⇤, ⇣)2h@2

xW, K@xW i+ M(C⇤, ⇣)3h@2

xW, A0@2

xi

+ h@3

xW, K@2

xW i+ M(C⇤, ⇣)h@3

xW, A0@3

xi⌘

t

�✓(|Wx|2H2↵

+ |wIIx |3H2

↵

) + C(C⇤)⇣|W |2L2) + |�|2

⌘.

�✓|W |2H3↵

+ C(C⇤)⇣|W |2L2) + |�|2

⌘.

Denoting

E(W ) :=12


+ M(C⇤, ⇣)2h@2

xW, K@xW i+ M(C⇤, ⇣)3h@2

xW, A0@2

xi

+ h@3

xW, K@2

xW i+ M(C⇤, ⇣)h@3

xW, A0@3

xi⌘,

we have by Young’s inequality that E1/2 is equivalent to norms H3 and H3

↵, hence(4.78) yields

Et �✓2E + C(C⇤)⇣|W |2L2) + |�|2

⌘.

Multiplying by integrating factor e�✓2t, and integrating from 0 to t, we thus obtain

E(t) e�✓2tE(0) + C(C⇤)Z t

0

e�✓2s⇣|W |2L2) + |�|2

⌘(s) ds.

Multiplying by e�✓2t, and using again that E1/2 is equivalent to H3, we obtain theresult.

The general case. It remains only to discuss the general case that hypotheses(A1)–(A3) hold as stated and not everywhere along the profile, with G possiblynonzero. These generalizations requires only a few simple observations. The firstis that we may express matrix A in (4.55) as

(4.83) A = A + (|Ux|+ ⇣)✓

0 O(1)O(1) O(1)

◆,

where A is a symmetric matrix obeying the same derivative bounds as described forA, identical to A in the 11 block and obtained in other blocks jk by smoothly in-terpolating over a bounded interval [�R,+R] between A(W�1)jk and A(W

+1)jk.Replacing A by A in the qth order Friedrichs-type bounds above, we find that theresulting error terms may be expressed as (integrating by parts if necessary)

h@qxO(|Ux|+ ⇣)|W |, |@q+1

x wII |i


plus lower-order terms, hence absorbed using Young’s inequality to recover the sameFriedrichs-type estimates obtained in the previous case. Thus, we may relax (A1’)to (A1).

The second observation is that, because of the favorable terms

�(C⇤✓/4)h|Ux||@qxwI |, |@q

xwI |i

occurring in the righthand sides of the Friedrichs-type estimates, we need theKawashima-type bound only to control the contribution to |@q

xwI |2 coming fromx near ±1; more precisely, we require from this estimate only a favorable term

(4.84) �✓h�1�O(|Ux|+ ⇣ + ⇣)

�|@q

xwI |, |@qxwI |i

rather than �✓|@qxwI |2 as in (4.77) and (4.80). But, this may easily be obtained

by substituting for K a skew-symmetric matrix-valued function K defined to beidentically equal to K(+1) and K(�1) for |x| > R, and smoothly interpolatingbetween K(±1) on [�R,+R], and using the fact that

�K(A0)�1A + B

�± � ✓ > 0,

hence �K(A0)�1A + B

�� ✓(1�O(|Ux|+ ⇣)).

Thus, we may relax (A2’) to (A2).Finally, notice that the term G� G in the perturbation equation may be Taylor

expanded as

(4.85)✓

0g(Wx, Ux) + g(Ux, Wx)

◆+✓

0O(|Wx|2)

◆

The first, linear term on the righthand side may be grouped with term A0Wx

and treated in the same way, since it decays at plus and minus spatial infinityand vanishes in the 1-1 block. The (0,O(|Wx|2) nonlinear term may be treatedas other source terms in the energy estimates Specifically, the worst-case termsh@3

xW,K@2

xO(|Wx|2)i and h@3

xW, @3

x(0,O(|Wx|2))i = h@4

xwII , @2

xO(|Wx|2)i may bebounded, respectively, by |W |W 2,1 |W |2H3 and |W |W 2,1 |wII |H4 |W |H3 . Thus, wemay relax (A3’) to (A3), completing the proof of the general case (A1)–(A3) andthe theorem. ⇤

Remark 4.17. Given Aj , Bjk, G 2 C2r, we may obtain by the same argumenta corresponding result in H2r�1 or H2r, r � 1, assuming that W r,1 remainssu�ciently small.

84 KEVIN ZUMBRUN

4.2.4 Nonlinear stability.

We are now ready to carry out our proof of nonlinear stability. We give a sim-plified version of the basic iteration scheme of [MaZ.1-2, Z.6, Z.4]; for precursors ofthis scheme, see [Go.2, K.1–2, LZ.1–2, ZH, HZ.1–2]. For this stage of the argument,it will be convenient to work again with the conservative variable

(4.86) U := U � U ,

writing (4.52) in the more standard form:

(4.87) Ut � LU = Q(U,Ux)x + �(t)(Ux + Ux),

where

(4.88)Q(U,Ux) = O(|U |2 + |U ||Ux|),

Q(U,Ux)x = O(|U |2 + |Ux|2 + |U ||Uxx|),

so long as |U |, |Ux| remain bounded.By Duhamel’s principle, and the fact that

(4.89)Z 1

�1G(x, t; y)Ux(y)dy = eLtUx(x) = Ux(x),

we have, formally,

(4.90)

U(x, t) =Z 1

�1G(x, t; y)U

0

(y)dy

+Z t

0

Z 1

�1Gy(x, t� s; y)(Q(U,Ux) + �U)(y, s)dyds

+ �(t)Ux.

Defining, by analogy with the linear case, the nonlinear instantaneous projection:

(4.91)

'(x, t) := ��(t)Ux

:=Z 1

�1E(x, t; y)U

0

(y)dy

�Z t

0

Z 1

�1Ey(x, t� s; y)(Q(U,Ux) + �U)(y, s)dy,

or equivalently, the instantaneous shock location:

(4.92)�(t) = �

Z 1

�1e(y, t)U

0

(y)dy

+Z t

0

Z+1

�1ey(y, t� s)(Q(U,Ux) + �U)(y, s)dyds,


where E, e are defined as in (4.24), (4.41), and recalling (4.43), we thus obtain thereduced equations:

(4.93)

U(x, t) =Z 1

�1(H + G)(x, t; y)U

0

(y)dy

+Z t

0

Z 1

�1H(x, t� s; y)(Q(U,Ux)x + �Ux)(y, s)dy ds

�Z t

0

Z 1

�1Gy(x, t� s; y)(Q(U,Ux) + �U)(y, s)dy ds,

and, di↵erentiating (4.92) with respect to t,

(4.94)�(t) = �

Z 1

�1et(y, t)U

0

(y)dy

+Z t

0

Z+1

�1eyt(y, t� s)(Q(U,Ux) + �U)(y, s)dyds.

Note: In deriving (4.94), we have used the fact that ey(y, s) * 0 as s ! 0, asthe di↵erence of approaching heat kernels, in evaluating the boundary term

(4.95)Z

+1

�1ey(y, 0)(Q(U,Ux) + �U)(y, t)dy = 0.

(Indeed, |ey(·, s)|L1 ! 0; see Remark 2.6, below).

The defining relation �(t)ux := �' in (4.91) can be motivated heuristically by

U(x, t)� '(x, t) ⇠ U = U(x + �(t), t)� U(x)

⇠ U(x, t) + �(t)Ux(x),

where U denotes the solution of the linearized perturbation equations, and U thebackground profile. Alternatively, it can be thought of as the requirement that theinstantaneous projection of the shifted (nonlinear) perturbation variable U be zero;see [HZ.1–2].

Lemma 4.18 [Z.6]. The kernel e satisfies

(4.96) |ey(·, t)|Lp , |et(·, t)|Lp Ct�12 (1�1/p),

(4.97) |ety(·, t)|Lp Ct�12 (1�1/p)�1/2,

for all t > 0. Moreover, for y 0 we have the pointwise bounds

(4.98) |ey(y, t)|, |et(y, t)| Ct�12X

a�k

>0

⇣e�

(y+a

�k

t)2

Mt + e�(y�a

�k

t)2

Mt

⌘,

86 KEVIN ZUMBRUN

(4.99) |ety(y, t)| Ct�1

X

a�k

>0

⇣e�

(y+a

�k

t)2

Mt + e�(y�a

�k

t)2

Mt

⌘,

for M > 0 su�ciently large, and symmetrically for y � 0.

Proof. For definiteness, take y 0. Then, (4.41) gives

(4.100) ey(y, t) :=X

a�k

>0

[c0

k,�]l�tk

�K(y + a�k t, t,��k )�K(y � a�k t, t,��k )

�

(4.101)et(y, t) :=

X

a�k

>0

[c0

k,�]l�tk

�(K + Ky)(y + a�k t, t,��k )� (K + Ky)(y � a�kt, t,��k )

�,

(4.102)ety(y, t) :=

X

a�k

>0

[c0

k,�]l�tk

�(Ky + Kyy)(y + a�k t, t)� (Ky + Kyy)(y � a�k t, t,��k )

�,

where

(4.103) K(y, t,�) :=e�y2/4�t

p4⇡�t

denotes a heat kernel with di↵usion coe�cient �. The pointwise bounds (4.98)–(4.99) follow immediately for t � 1 by properties of the heat kernel, in turn yielding(4.96)–(4.97) in this case. The bounds for small time t 1 follow from estimates(4.104)

|Ky(y + at, t,�)�Ky(y � at, t,�)| =��Z y�at

y+at

Kyy(z, t,�) dz

��

Ct�3/2

Z y�at

y+at

e�z

2Mt dz Ct�1/2e�

y

2Mt ,

and, similarly,

(4.105)|Kyy(y + at, t,�)�Kyy(y � a, t,�)| =

��Z at

�at

Kyyy(z, t,�) dz

��

Ct�2

Z y�at

y+at

e�z

2Mt dz Ct�1e�

y

2Mt .

The bounds for |ey| are again immediate. Note that we have taken crucial accountof cancellation in the small time estimates of et, ety. ⇤

Remark 4.19: For t 1, a calculation analogous to that of (4.104) yields|ey(y, t)| Ce�

y

2Mt , and thus |ey(·, s)|

L

1 ! 0 as s ! 0.


With these preparations, we are ready to carry out our analysis:

Proof of Theorem 4.9. Define(4.106)

⇣(t) := sup0st, 2p1

h|U(·, s)|Lp(1 + s)

12 (1�1/p) + |�(s)|(1 + s)1/2 + |�(s)|

i.

We shall establish:Claim. For all t � 0 for which a solution exists with ⇣ uniformly bounded by

some fixed, su�ciently small constant, there holds

(4.107) ⇣(t) C2

(|U0

|L1\H3 + ⇣(t)2).

From this result, it follows by continuous induction that, provided

|U0

|L1\H3 < 1/4C2

2

,

there holds

(4.108) ⇣(t) 2C2

|U0

|L1\H3

for all t � 0 such that ⇣ remains small. For, by standard short-time theory/localwell-posedness in H3, and the standard principle of continuation, there exists asolution U 2 H3(x) on the open time-interval for which |U |H3 remains bounded,and on this interval ⇣ is well-defined and continuous. Now, let [0, T ) be the maximalinterval on which |U |H3

(x)

remains strictly bounded by some fixed, su�ciently smallconstant � > 0. By Proposition 4.15, and the one-dimensional Sobolev bound|U |W 2,1 C|U |H3 , we have

(4.109)|U(t)|2H3 C|U(0)|2H3e�✓t + C

Z t

0

e�✓2(t�⌧)(|U |2L2 + |�|2)(⌧) d⌧

C2

�|U(0)|2H3 + ⇣(t)2

�(1 + t)�1/2,

and so the solution continues so long as ⇣ remains small, with bound (4.108), atonce yielding existence and the claimed sharp Lp \H3 bounds, 2 p 1.

Thus, it remains only to establish the claim above.

Proof of Claim. We must show that each of the quantities |U |Lp(1 + s) 12 (1�1/p),

|�|(1 + s)1/2, and |�| is separately bounded by

(4.110) C(|U0

|L1\H3 + ⇣(t)2),

88 KEVIN ZUMBRUN

for some C > 0, all 0 s t, so long as ⇣ remains su�ciently small. By (4.93)–(4.94), we have

(4.111)

|U |Lp(t) |Z 1

�1(H + G)(x, t; y)U

0

(y)dy|Lp

+ |Z t

0

Z 1

�1H(x, t� s; y)(Q(U,Ux)x + �Ux)(y, s)dy ds|Lp

+ |Z t

0

Z 1

�1Gy(x, t� s; y)(Q(U,Ux) + �U)(y, s)dy ds|Lp

=: Ia + Ib + Ic,

(4.112)

|�|(t) |Z 1

�1et(y, t)U

0

(y)dy|

+ |Z t

0

Z+1

�1eyt(y, t� s)(Q(U,Ux) + �U)(y, s)dyds|

=: IIa + IIb,

and

(4.113)

|�|(t) |Z 1

�1e(y, t)U

0

(y)dy|

+ |Z t

0

Z+1

�1ey(y, t� s)(Q(U,Ux) + �U)(y, s)dyds|

=: IIIa + IIIb.

We estimate each term in turn, following the approach of [Z.6, MaZ.1]. Thelinear term Ia satisfies bound

(4.114) Ia C|U0

|L1\Lp(1 + t)�12 (1�1/p),

as already shown in the proof of Theorem 4.8. Likewise, applying the bounds ofLemma 4.14 together with (4.88), (4.109), and definition (4.106), we have

(4.115)

Ib = |Z t

0

Z 1

�1H(x, t� s; y)(Q(U,Ux)x + �Ux)(y, s)dy ds|Lp

C

Z t

0

e�⌘(t�s)(|U |L1 + |Ux|L1 + |�|)|U |W 2,p(s)ds

C⇣(t)2Z t

0

e�⌘(t�s)(1 + s)�1/2ds

C⇣(t)2(1 + t)�1/2,


and (taking q = 2 in (4.45))

(4.116)

Ic = |Z t

0

Z 1

�1Gy(x, t� s; y)(Q(U,Ux) + �U)(y, s)dy ds|Lp

C

Z t

0

e�⌘(t�s)(|U |L1 + |Ux|L1 + |�|)|U |Lp(s)ds

+ C

Z t

0

(t� s)�3/4+1/2p(|U |L1 + |�|)|U |H1(s)ds

C⇣(t)2Z t

0

e�⌘(t�s)(1 + s)�12 (1�1/p)�1/2ds

+ C⇣(t)2Z t

0

(t� s)�3/4+1/2p(1 + s)�3/4ds

C⇣(t)2(1 + t)�12 (1�1/p).

Summing bounds (4.114)–(4.116), we obtain (4.110), as claimed, for 2 p 1.Similarly, applying the bounds of Lemma 4.18 together with definition (4.106),

we find that

(4.117)

IIa =��Z 1

�1et(y, t)U

0

(y)dy��

|et(y, t)|L1(t)|U0

|L1

C|U0

|L1(1 + t)�1/2

and

(4.118)

IIb = |Z t

0

Z+1

�1eyt(y, t� s)(Q(U,Ux) + �U)(y, s)dyds|

Z t

0

|eyt|L2(t� s)(|U |L1 + |�|)|U |H1(s)ds

C⇣(t)2Z t

0

(t� s)�3/4(1 + s)�3/4ds

C⇣(t)2(1 + t)�1/2,

while

(4.119)IIIa =

��Z 1

�1e(y, t)U

0

(y)dy��

|e(y, t)|L1(t)|U0

|L1

C|U0

|L1

90 KEVIN ZUMBRUN

and

(4.120)

IIIb =��Z t

0

Z+1

�1ey(y, t� s)�U(y, s)dyds

��

Z t

0

|ey|L2(t� s)(|U |L1 + |�|)|U |H1(s)ds

C⇣(t)2Z t

0

(t� s)�1/4(1 + s)�3/4ds

C⇣(t)2.

Summing (4.117)–(4.118) and (4.119)–(4.120), we obtain (4.110) as claimed.This completes the proof of the claim, and the result. ⇤Remark. (Low norm stability analysis, Lp, 1 p 2) The source term �Ux

appearing in the reduced equations is convenient for high norm estimates Lp, p � 2,but not for low norm estimates Lp, 1 p 2. To treat low norms, we may redefineU := U(x, t) � U(x � �(t)), which has the e↵ect of replacing �Ux in the reducedequations with “centering errors”

S(U,Ux, �)x := �((A(U(x� �)�A(U(x))U)x + ((B(U(x� �)�B(U(x))Ux)x,

satisfying|S|L1 C|U |W 1,1 , |S|W 2,1 C|U |W 3,1 C|U |H4 .

This does not a↵ect the previously-obtained Lp and Hs estimates, since these normsare invariant under spatial shift.

Repeating our high-norm analysis for more regular initial data U0

2 L1\H4, weobtain (using higher-derivative analogs of Green distribution bounds (4.44)–(4.45))

|U |W 1,1 C(1 + t)�1/2|U0

|L1\H4 , |U |H4 C(1 + t)�1/4|U0

|L1\H4 .

Estimating Ia–Ic using this new information, taking q = 1 now in (4.45) and esti-mating |Q|W 1,p C|U |2H3 C

2

(1 + t)�1/2|U0

|L1\H4 , we obtain the sharp rate ofdecay

|U(x, t)� U(x)|Lp C(1 + t)�(1/2)(1�1/p)|U0

|L1\H4 ,

for all 1 p < 1. (Indeed, we could have carried out the original nonlineariteration in these “uncentered” coordinates, for all 2 p p < 1, using q = 1 in(4.45).) We omit the details, as lying outside the scope of our main development.

4.3 Proof of the linearized bounds.

Finally, we carry out in this subsection the proof of Proposition 4.10, completingthe analysis of the one-dimensional case. For simplicity, we restrict to the case thatA± have distinct eigenvalues; as discussed in [MaZ.3–4], the general case can betreated in the same way, at the expense only of some further bookkeeping. Followingthe notation of [MaZ.3], we abbreviate by (D) the triple conditions of structural,dynamical, and strong spectral stability.27

27This refers to an alternate characterization in terms of the Evans function D.


4.3.1 Low-frequency bounds on the resolvent kernel.

We begin by estimating the resolvent kernel in the critical regime |�| ! 0, cor-responding to large time behavior of the Green distribution G, or global behaviorin space and time. From a global perspective, the structure of the linearized equa-tions is that of two nearly constant-coe�cient regions separated by a thin shocklayer near x = 0; accordingly, behavior is essentially governed by the two, limitingfar-field equations:

(4.121) Ut = L±U,

coupled by an appropriate transmission relation at x = 0. We begin by examiningthese limiting systems.

Lemma 4.20[MaZ.3]. Let dimension d = 1 and assume (A1)–(A3), (H0)–(H3),and (P0). Then, for |�| su�ciently small, the eigenvalue equation (L± � �)U = 0associated with the limiting, constant-coe�cient operator L± has an analytic basisof n + r solutions U±

j consisting of r “fast” modes

(4.122)|(@/@x)↵U±

j (x)| Ce�✓|x|, x � 0, j = 1, . . . , k±,

|(@/@x)↵U±j (x)| Ce�✓|x|, x 0, j = k± + 1, . . . , r,

✓ > 0, 0 |↵| 1, and n “slow” modes U±r+j = eµxV ±

j , j = 1, . . . n, where

(4.123)µ±r+j(�) := ��/a±j + �2�±j /a±j

3 +O(�3),

V ±j (�) := r±j +O(�),

with a±j , r±j , and �±j as in Proposition 4.10. Likewise, the adjoint eigenvalueequation (L± � �)⇤U = 0 has an analytic basis consisting of r fast solutions

(4.124)|(@/@x)↵ ¯Uj

±(x)| Ce�✓|x|, x 0, j = 1, . . . , k±,

|(@/@x)↵ ¯Uj

±(x)| Ce�✓|x|, x � 0, j = k± + 1, . . . , r,

✓ > 0, 0 |↵| 1, and n slow solutions ¯U±r+j = e�µ±

j

(�)xVj(�),

(4.125) V ±j (�) = l±j +O(�),

j = 1, . . . , n, where l±j are as in Proposition 4.10. Moreover, for ✓ > 0, 0 |↵| 1,

(4.126)

��(@/@y)↵kX

j=1

Uj(x) ¯U j(y)��± Ce�✓|x�y|, x � y,

��(@/@y)↵rX

j=k+1

U+

j (x) ¯U+

j (y)��± Ce�✓|x�y|, x y.

92 KEVIN ZUMBRUN

Proof. By Lemma 3.3 and Corollary 3.4, there exist analytic bases of claimeddimension for the “strong”, or “fast” unstable and stable subspaces of the coe�cientmatrix of the eigenvalue ODE written as a first-order system, defined as the totaleigenspaces of spectra with real part uniformly bounded away from zero as |�|! 0,and these satisfy the standard constant-coe�cient bounds (4.122) and (4.124). Thebounds (4.126) may likewise be seen from standard constant-coe�cient bounds,together with the fact (see Section 2.4.3, and especially relation (2.92)) that

⇣ kX

j=1

Wj(x) ¯W j(y)⌘

±=⇣Fy!x ⇧

fast

(S0)�1

⌘

±

W := (U, b11U 0), W := (U , (0, b11Tr

2

)U 0), where Fy!x is the solution operator ofthe limiting eigenvalue ODE written as a first-order system W 0 = A±W , ⇧

fast± isthe projection onto the strongly stable subspace of A±, well-conditioned thanks tospectral separation, and S0

± is as defined in (4.25), also well-conditioned by (2.75).There exists also an n-dimensional manifold of slow solutions, of which the stable

and unstable manifolds in the general, multidimensional case, are only continuousas |�| ! 0. In the one-dimensional case, however, they may be seen to varyanalytically, with expansions (4.123) and (4.125), by inversion of the expansions

�j(i⇠) = �ia⇤±j ⇠ � �⇤±j ⇠2 +O(⇠3)

carried out in Appendix A.4 for the the dispersion curves near ⇠ = 0, togetherwith the fundamental relation µ = i⇠ between roots of characteristic and dispersionequations (recall (2.67)–(2.68) of Section 3.1). Alternatively, one can carry outdirectly the associated matrix bifurcation problem as in the proof of Lemma 3.4.A symmetric argument applies to the adjoint problem. ⇤

Remark 4.21 [MaZ.3]. Under the simplifying assumption that matrices

N± := ( df21

df22

)✓

df11

df12

b1

b2

◆�1

✓0Ir

◆

|(U±)

governing fast modes (see discussion of traveling-wave ODE, Appendix A.2) havedistinct eigenvalues, then fast modes may be taken as pure exponential solutionsU = eµxV , U = e�µxV , and (4.126) follows immediately from (4.122) and (4.124).

Our main result of this subsection is then:

Proposition 4.22[MaZ.3]. Let d = 1 and assume (A1)–(A3), (H0)–(H3), (P0),and (D). Then, for r > 0 su�ciently small, the resolvent kernel G� has a mero-morphic extension onto B(0, r) ⇢ C, which may in the Lax or overcompressive casebe decomposed as

(4.127) G� = E� + S� + R�


where, for y 0:

(4.128) E�(x, t; y) := ��1

X

a�k

>0

[cj,0k,�](@U�/@�j)l�t

k e(�/a±k

��2�±k

/a±k

3)y,

(4.129) S�(x, t; y) :=X

a�k

>0, a+j

>0

[cj,+k,�]r+

j l�kte(��/a+

j

+�2�+k

/a+k

3)x+(�/a�

k

��2��k

/a�k

3)y

for y 0 x,

(4.130)

S�(x, t; y) :=X

a�k

>0

r�k l�kte(��/a�

k

+�2��k

/a�k

3)(x�y)

+X

a�k

>0, a�j

<0


te(��/a�

j

+�2��j

/a�j

3)x+(�/a�

k

��2��k

/a�k

3)y

for y x 0, and

(4.131)

S�(x, t; y) :=X

a�k

<0

r�k l�kte(��/a�

k

+�2��k

/a�k

3)(x�y)

+X

a�k

>0, a�j

<0


te(��/a�

j

+�2��j

/a�j

3)x+(�/a�

k

��2��k

/a�k

3)y

for x y 0, and R� = RE� + RS

� is a faster-decaying residual satisfying(4.132)

(@/@y)↵RE� = O(e�✓|x�y|) +

X

a�k

>0

e�✓|x|e(�/a±k

��2�±k

/a±k

3)y

⇥⇣�↵�1O(eO(�3

)y � 1) + �↵�1O(eO(�3)x � 1) +O(�↵)

⌘,

(4.133)

(@/@y)↵RS�(x, t; y) =

X

a�k

>0, a+j

>0

e(��/a+j

+�2�+k

/a+k

3)x+(�/a�

k

��2��k

/a�k

3)y

⇥⇣�↵O(eO(�3

)y � 1) + �↵O(eO(�3)x � 1) + �↵O(e�✓|x|) +O(�)

⌘

94 KEVIN ZUMBRUN

for y 0 x,

(4.134)

(@/@y)↵RS�(x, t; y) =

X

a�k

>0

e(��/a�k

+�2��k

/a�k

3)(x�y)

⇥⇣�↵O(eO(�3

)y � 1) + �↵O(eO(�3)x � 1) + �↵O(e�✓|x|) +O(�)

⌘

+X

a�k

>0, a�j

<0

e(��/a�j

+�2��j

/a�j

3)x+(�/a�

k

��2��k

/a�k

3)y

⇥⇣�↵O(eO(�3

)y � 1) + �↵O(eO(�3)x � 1) + �↵O(e�✓|x|) +O(�)

⌘

for y x 0, and

(4.135)

(@/@y)↵RS�(x, t; y) =

X

a�k

<0

e(��/a�k

+�2��k

/a�k

3)(x�y)

⇥⇣�↵O(eO(�3

)y � 1) + �↵O(eO(�3)x � 1) + �↵O(e�✓|x|) +O(�)

⌘

+X

a�k

>0, a�j

<0

e(��/a�j

+�2��j

/a�j

3)x+(�/a�

k

��2��k

/a�k

3)y

⇥⇣�↵O(eO(�3

)y � 1) + �↵O(eO(�3)x � 1) + �↵O(e�✓|x|) +O(�)

⌘

for x y 0, 0 ↵ 1, with each O(·) term separately analytic in �, and where[cj,0

k,i, i = �, 0,+, are constants to be determined later. Symmetric bounds hold fory � 0. Similar, but more complicated formulae hold in the undercompressive case(see Remark 6.9, [MaZ.1] or Section 3, [Z.6] for further discussion).

Normal modes (behavior in x). We begin by relating the normal modes ofthe variable coe�cient eigenvalue equation (2.60) to those of the limiting, constant-coe�cient equations (4.121).

Lemma 4.23[MaZ.3]. Let d = 1 and assume (A1)–(A3), (H0)–(H3), and (P0).Then, for � 2 B(0, r), r su�ciently small, there exist solutions U±

j (x;�) of (2.60),C1 in x and analytic in �, satisfying, for slow modes, the bounds

(4.136)U±

r+j(x;�) = V ±j (x;�)eµ±

j

x

(@/@�)kV ±j (x;�) = (@/@�)kV ±

j (�) +O(e�˜✓|x||V ±j (�)|), x ? 0,

for any k � 0 and 0 < ✓ < ✓, where ✓ is the rate of decay given in (2.18), µ±j (�),V ±

j (�) are as in Lemma 4.20 above, and O(·) depends only on k, ✓, and, for fastmodes, the bounds

(4.137)|U+

j (x)| Ce�✓|x|, x � 0, j = 1, . . . , k+

,

|U+

j (x)| Ce�✓|x|, x � 0, j = k+

+ 1, . . . , r,


(4.138)|U�j (x)| Ce�✓|x|, x 0, j = k� + 1, . . . , r,

|U�j (x)| Ce�✓|x|, x 0, j = 1, . . . , k�,

(4.139)

��k+X

j=1

U+

j (x)U+

j (y)�� Ce�✓|x�y|, 0 y x,

��rX

j=k++1

U+

j (x)U+

j (y)�� Ce�✓|x�y|, 0 x y.

and

(4.140)

��rX

j=k�+1

U�j (x)U�j (y)�� Ce�✓|x�y|, x y 0.

��k�X

j=1

U�j (x)U�j (y)�� Ce�✓|x�y|, y x 0.

Proof. An immediate consequence of Lemma 4.20 together with Lemma 2.5.(Note the helpful factorization

XWj(x)Wj(y) = P (x)

⇣XWj(x) ¯W j(y)

⌘(S0)�1P�1S0(y)

linking solutions W and W of the associated first-order systems, where P is theconjugating matrix of Lemma 2.5 and S0, S0 are as defined in (4.25), both uniformlywell-conditioned.) ⇤

The bases �±j , ±j defined in Section 2.4 may evidently be chosen from amongU±

j , yielding an analytic choice of bases in �, with the detailed description (4.136).It follows that the dual bases �±j , ±j defined in Section 2.4 are also analytic in �and satisfy corresponding bounds with respect to the dual solutions (4.124)–(4.125).With this observation, we have immediately, from Definition (3.1) and Corollary2.26:

Corollary 4.24[MaZ.3]. Let d = 1 and assume (A1)–(A3), (H0)–(H3). Then, theEvans function D(�) admits an analytic extension onto B(0, r), for r su�cientlysmall, and the resolvent kernel G�(x, y) admits a meromorphic extension, with anisolated pole of finite multiplicity at � = 0.

Remark 4.25. Analytic extendability of the Evans function past the origin(more generally, into the essential spectrum of L) is a special feature of the one-dimensional, or scalar multidimensional case, and reflects the simple geometry of

96 KEVIN ZUMBRUN

hyperbolic propagation in these settings; see [HoZ.3–4, Z.3] for discussion of thescalar multidimensional case. The simple hyperbolic propagation makes possible amuch more detailed description of the Green’s function in these cases, for whichanalytic extension is a key tool

For general systems in multidimensions, D(e⇠,�) has a conical singularity at theorigin; see Proposition 3.5.

Refined derivative bounds. We have close control on the (x, y) behaviorof G� through the spectral decomposition formulae of Corollary 2.26. For slow,dual modes, the bounds (4.136) (in particular, the consequent bounds on first spa-tial derivatives) can be considerably sharpened, provided that we appropriatelyinitialize our bases at � = 0. This observation will be quite significant in theLax or overcompressive case. Likewise, fast-decaying forward modes can be well-approximated near � = 0 by their representatives at � = 0, using only the basicbounds (4.136). These two categories comprise the modes determining behavior ofthe Green distribution to lowest order.

Specifically, due to the special, conservative structure of the underlying evolu-tion equations, the adjoint eigenvalue equation L ⇤ U = 0 at � = 0 admits ann-dimensional subspace of constant solutions

(4.141) U ⌘ constant;

this is equivalent to the observation that integral quantities in variable U are con-served under time evolution for general conservation laws. Thus, at � = 0, we maychoose, by appropriate change of coordinates if necessary, that slow-decaying dualmodes �±j and slow-growing dual modes ±j be identically constant, Note that thisdoes not interfere with our previous choice in Lemma 4.23, since that concernedonly the choice of limiting solutions U

±j of the asymptotic, constant coe�cient

equations at x ! ±1, and not the particular representatives U±j that approach

them (which, in the case of slow modes, are specified only up to the addition of anarbitrary fast-decaying mode).

Remark 4.26. The prescription of constant dual bases just described requiresthat we reverse our previous approach, choosing dual bases first using the conju-gation Lemma, then defining forward bases using duality. Alternatively, we mayrewrite the forward eigenvalue equation at � = 0 as

(4.142)⇣✓A

11

A12

b1

b2

◆U 0⌘0

=⇣✓ 0 0

A12

A22

◆U⌘0

to see, after integration from x = ±1, that fast-decaying modes satisfy the lin-earized traveling-wave ODE (cf. Appendix A.2)

(4.143) U 0 =✓

A11

A12

b1

b2

◆�1

✓0 0

A12

A22

◆U.


By duality relation (2.76), requiring slow dual modes to be constant is equivalent tochoosing fast-growing (forward) modes as well as fast-decaying modes from amongthe solutions of (4.143). (Recall: though fast-decaying modes are uniquely deter-mined as a subspace, fast-growing modes are only determined up to the additionof faster decaying modes.)

Lemma 4.27 [MaZ.1]. In dimension d = 1, assuming (A1)–(A3), (H0)–(H3), and(P0), with the above choice of bases at � = 0, and for � 2 B(0, r) and r su�cientlysmall, slow modes U±

j satisfy

(4.144) U±j (y;�) = e�µ(�)y ¯V

±j (0) + �⇥±j (y;�),

where

(4.145)|⇥±j | C|e�µ(�)y|

|(@/@y)⇥±j | C|e�µ(�)y|(|�|+ e�✓|y|),

✓ > 0, as y ! ±1, and ¯V±j ⌘ constant.28 Similarly, fast-decaying (forward) modes

U±j satisfy

(4.146) U±j (x;�) = U±

j (x; 0) + �⇥±j (x;�),

where

(4.147) |⇥±j |, |(@/@x)⇥±j | Ce�✓|x|

as x ! ±1, for some ✓ > 0.

Proof. Applying Lemma 2.5 to the augmented variables

U±j (y;�) :=

✓U±

j

U±0j

◆(y;�) =:

✓U±

j

U±0j

◆(y;�)

= e�µ(�)y

✓V ±

j

�µ±j V ±j + V ±0

j

◆(y;�)

and

U±j (x;�) :=

✓U±

j

U±j

0

◆(x;�) =: eµ(�)xV±

j (x;�)

= e�µ(�)y

✓V ±

j

µ±j V ±j + V ±0

j

◆(y;�)

28In particular, this includes |(@/@y)U±j

(y; �)| C|�eµ

±j

(�)y | in place of the general spatial-

derivative bound |(@/@y)U±j

(y; �)| C(|µV ±j

|+ |(@/@y)V ±j

|)eµ

±j

(�)y ⇠ C|eµ

±j

(�)y |.

98 KEVIN ZUMBRUN

we obtain bounds

(4.148)U±

j (x;�) = V±j (x;�)e�µ±

j

(�)x

(@/@�)kV±j (x;�) = (@/@�)kV±

j (�) +O(e�˜✓|x||V±j (�)|), x ? 0,

and

(4.149)U±

j (x;�) = V±j (x;�)eµ±

j

(�)x

(@/@�)kV±j (x;�) = (@/@�)kV±

j (�) +O(e�˜✓|x||V±j (�)|), x ? 0,

✓ > 0, analogous to (4.136), valid for � 2 B(0, r), where

V±j (�) =

✓V (�)

�µ±j V (�)

◆

andV±

j (�) =✓

V (�)µ±j V (�).

◆

(Note: an appropriate phase space in which to apply the conjugation lemma is,e.g., (U,U 0, (bU 0)0) for the forward equations, and similarly for the dual equations.)

By Taylor’s Theorem with di↵erential remainder, applied to V, we have:(4.150)

U±j (y,�) = e�µ±

j

(�)y

✓V±

j (y; 0) + �(@/@�)V±j (y; 0) +

12�2(@/@�)2V±

j (y;�⇤)◆

,

for some �⇤ on the ray from 0 to �, where, recall, (@/@�)V±j (y; ·) and (@/@�)2V±

j (y; ·)are uniformly bounded in L1[0,±1] for � 2 B(0, r). Together with the choiceV ±

j (y, 0) ⌘ constant, this immediately gives the first (undi↵erentiated) bound in(4.145).

Applying now the bound (4.149) with k = 1, we may expand the second coordi-nate of (4.150) as(4.151)

(@/@y)U±j (y,�) = e�µ±

j

(�)y⇣� µ±j V ±

j (y; 0) + V ±0j (y; 0)

� ��(@/@�)(µ±j V ±

j )(0) +O(e�✓|y|)�

+O(�2)⌘

= e�µ±j

(�)y⇣��

�(@/@�)µ±j (0)V ±

j (0) +O(e�✓|y|)�

+O(�2)⌘

,

and subtracting o↵ the corresponding Taylor expansion

(4.152)

(@/@y)�e�µ±

j

(�)yV ±j (y, 0)

�

= µ±j (�)e�µ±j

(�)yV ±j (y, 0)

= e�µ±j

(�)y⇣�µ±j (0)V ±

j (y; 0)� �(@/@�)µ±j (0)V ±j (y; 0) +O(�2)

⌘

= e�µ±j

(�)y⇣��(@/@�)µ±j (0)V ±

j (y; 0) +O(�2)⌘

,


we obtain

(4.153) (@/@y)⇥±j (y,�) = e�µ±j

(�)y⇣�O(e�✓|y|) +O(�2)

⌘,

as claimed.Similarly, we may obtain (4.146)–(4.147) by Taylor’s Theorem with di↵erential

remainder applied to U = eµ(�)x, and the Leibnitz calculation

(@/@�)U = (dµ/d�)xeµxV + eµx(@/@�)V,

together with the observation that |xe�✓|x| Ce�✓|x|/2. ⇤This leaves only the problem of determining behavior in � through the study of

coe�cients M , d±. To this end, we make the following further observations in theLax or overcompressive case, generalizing the corresponding observation of Lemma4.30, [Z.3], in the strictly parabolic case:

Lemma 4.28[MaZ.3]. For transverse (� 6= 0) Lax and overcompressive shocks,with the above-specified choice of basis at � = 0, fast-growing modes +

j , �j are fast-decaying at �1, +1, respectively. Equivalently, fast-decaying modes +

j , �j arefast-growing at �1, +1: i.e., the only bounded solutions of the adjoint eigenvalueequation are constant solutions.

Proof. Noting that the manifold of solutions of the r-dimensional ODE (4.143)that decay at either x ! +1 or x ! �1 is by transversality, together withLemma 1.7, exactly d

+

+ d� � ` = r, we see that all solutions of this ODE in factdecay at least at one infinity. This implies the first assertion, by the alternativecharacterization of our bases described in Remark 4.26 above; the second followsby duality, (2.76). Since the manifold of fast-decaying dual modes is uniquelydetermined, independent of the choice of basis, this implies that nonconstant dualmodes that are bounded at one infinity must blow up at the other, hence boundedsolutions must be constant as claimed. ⇤

Remark 4.29. As noted in [LZ.2, ZH, Z.3, Z.6], the property of Lax and over-compressive shocks that the adjoint eigenvalue equation has only constant solutionshas the interpretation that the only L1 time-invariants of the evolution of the one-dimensional linearized equations about (U , v)(·) are given by conservation of mass,see discussion [LZ.2]. This distinguishes then from undercompressive shocks, whichdo have additional L1 time-invariants. The presence of additional time-invariantsfor undercompressive shocks has significant implications for their behavior, (see dis-cussions [LZ.2] Section 3 and [ZH], Section 10): in particular, the time-asymptoticlocation of a perturbed undercompressive shock (generically) evolves nonlinearly,and is not determinable by any linear functional of the initial perturbation. By con-trast, the time-asymptotic location of a perturbed (stable) Lax or overcompressiveshock may be determined by the mass of the initial perturbation alone.

100 KEVIN ZUMBRUN

Scattering coe�cients (behavior in �). We next turn to the estimation ofscattering coe�cients Mjk, d±jk defined in Corollary 2.26. Consider coe�cient Mjk.Expanding (2.101) using Cramer’s rule, and setting z = 0, we obtain

(4.154) M±jk = D�1C±

jk,

where

(4.155) C+ := (I, 0)✓�+ ��+0 ��0

◆adj

✓ � �0

◆

|z=0

and a symmetric formula holds for C�. Here, P adj denotes the adjugate matrix ofa matrix P , i.e. the transposed matrix of minors. As the adjugate is polynomial inthe entries of the original matrix, it is evident that |C±| is uniformly bounded andtherefore

(4.156) |M±jk| C

1

|D�1| C2

��`

by (D), where C1

, C2

> 0 are uniform constants.However, the crude bound (4.156) hides considerable cancellation, a fact that

will be crucial in our analysis. Relabel the {'±} so that, at � = 0,

(4.157) '+

n+r�j+1

⌘ '�j = (@/@�j)✓

u�

v�

◆, j = 1, · · · , `.

With convention (4.157), we have the sharpened bounds:

Lemma 4.30[MaZ.3]. In dimension d = 1, let there hold (A1)–(A3), (H0)–(H3),and (D), and let �±j be labeled as in (4.157). Then, for |�| su�ciently small, therehold

(4.158) |M±jk|, |d

±jk| C

⇢��1 for j = 1, · · · , `,

1 otherwise,

where M±, d± are as defined in Corollary 2.26. Moreover,

(4.159) Residue �=0

M+

n+r�j+1,k = Residue �=0

d±j,k,

for 1 j `, all k.

That is, blowup in Mjk occurs to order �`�1|D�1| rather than |D�1|, and, moreimportantly, only in (fast-decaying) stationary modes (@/@�j)(u�, v�). Moreover,each stationary mode (@/@�j)(u�, v�) has consistent scattering coe�cients with dualmode �k , to lowest order, across all of the various representations of G� given inCorollary 2.26.


Proof. Formula (4.155) may be rewritten as

(4.160) C+

jk = det

0

@'+

1

, · · · ,'+

j�1

, �k ,'+

j , · · · ,'+

n , ��

'+

1

0, · · · ,'+

j�1

0, �k0,'+

j0, · · · ,'+

n , ��0

1

A

|z=0

,

from which we easily obtain the desired cancellation in M+ = C+D�1. For exam-ple, for j > `, we have

(4.161)C+

jk = det

0

@'+

1

+ �'+

1

�

+ · · · , · · · , '+

n + �'+

n�

+ · · ·

· · · , , · · · , '+

n0 + �'+

n�

0 + · · ·

1

A

= O(�`) C|D|,

yielding |Mjk| = |Cjk||D|�1 C as claimed, by elimination of ` zero-order terms,using linear dependency among fast modes at � = 0. For 1 j `, there is only an`�1 order dependency, and we obtain instead the bound |Cjk| C|�|`�1, or |Mjk| =|Cjk||D|�1 C|�|�1. The bounds on |djk| follow similarly; likewise, (4.159) followseasily from the observation that, since columns +

j and �n+r�j+1

agree, and �jis held fixed, the expansions of the various representations by Cramer’s rule yielddeterminants which at ��1 order di↵er only by a transposition of columns. ⇤

In the Lax or overcompressive case, we can say a bit more:

Lemma 4.31[MaZ.3]. In dimension d = 1, let there hold (A1)–(A3), (H0)–(H3),and (D), with |�| su�ciently small. Then, for Lax and overcompressive shocks,with appropriate basis at � = 0 (i.e. slow dual modes taken identically constant),there hold

(4.162) |Mjk|, |djk| C

if e k is a fast mode, and

(4.163) |Mjk|, |djk| C|�|

if, additionally, 'j is a slow mode.

As we shall see in the following section, this result has the consequence that onlyslow, constant dual modes play a role in long-time behavior of G.

Proof. Transversality, � 6= 0, follows from (D2), so that the results of Lemma4.28 hold. We first establish the bound (4.162). By Lemma 4.30, we need onlyconsider j = 1, . . . , `, for which ��j = �+

j . By Lemma 4.28, all fast-growing modes +

k , �k lie in the fast-decaying manifolds Span('�1

, · · · ,'�i�), Span('+

1

, · · · ,'+

i+)

102 KEVIN ZUMBRUN

at �1, +1 respectively. It follows that in the righthand side of (4.160), there isat � = 0 a linear dependency between columns

(4.164) '+

1

, · · · ,'+

j�1

, �k ,'+

j+1

,'+

i+and '�j = '+

j

i.e. an `-fold dependency among columns

(4.165) '+

1

, · · · , �k , · · · ,'+

i+ and '�1

, · · · ,'�` .

It follows as in the proof of Lemma 4.30 that |Cjk| C�` for � near zero, givingbound (4.162) for Mjk. If '+

j is a slow mode, on the other hand, then the sameargument shows that there is a linear dependency in columns

(4.166) '+

1

, · · · ,'+

j�1

, �k ,'+

j+1

,'+

i+

and an (` + 1)-fold dependency in (4.165), since the omitted slow mode '+

j playsno role in either linear dependence; thus, we obtain the bound (4.163), instead.Analogous calculations yield the result for djk, as well. ⇤

Proof of Proposition 4.22. The proof of Proposition 4.22 is now just a matterof collecting the bounds of Lemmas 4.23–4.31, and substituting in the representa-tions of Corollary 2.26. More precisely, approximating fast-decaying dual modes�+

1

, . . . ,�+

` and ��n+r�`+1

, . . . ,��n+r by stationary modes (@/@�j)(u�, v�), and slowdual modes by e�µ(�)y ¯V (�) as described in Lemma 4.27, truncating µ(�) at secondorder, and keeping only the lowest order terms in the Laurent expansions of scatter-ing coe�cients M±, d± we obtain E� and S�, respectively, as order ��1 and orderone terms, except for negligible O(e�✓|x�y|) terms arising through (4.137)–(4.138)and (4.139)–(4.140), which we have accounted for in error term RE

� .Besides the latter, we have accounted in term RE

� for: (i) the truncation er-rors involved in the aforementioned approximations (di↵erence between stationarymodes and actual fast modes, and between constant-coe�cient approximant andactual slow dual modes, plus truncation errors in exponential rates); and, (ii) ne-glected fast-decaying forward/slow-decaying dual mode combinations (of order �,by Lemma 4.30). A delicate point is the fact that term O(e�✓|x�y|)O(e�✓|y|) arisingin the dual truncation of E through the �O(e�✓|y|) portion of estimate (4.145) for(@/@y)⇥±j , absorbs in the (leading) O(e�✓|x�y|) term of (4.132). It was to obtainthis key reduction that we carried out expansion (4.145) to such high order.

In RS� , we have accounted for truncation errors in the slow/slow pairings approxi-

mated by term S� (di↵erence between constant-coe�cient approximants and actualslow forward modes, corresponding to an additional e�✓|x| term in the truncationerror factor, and between constant-coe�cient approximants and actual slow dualmodes, plus truncation errors in exponential rates). We have also accounted forthe remaining, slow-decaying forward/fast-decaying dual modes, which accordingto Lemma 4.31 have scattering coe�cient of order � and therefore may be groupedwith the di↵erence between dual modes and their constant-coe�cient approximants(of smaller order by a factor of e�✓|y|) in the term O(�). ⇤


4.3.2 High-frequency bounds on the resolvent kernel.

We now turn to the estimation of the resolvent kernel in the high frequencyregime |�| ! +1. According to the usual duality between spatial and frequencyvariables, large frequency � corresponds to small time t, or local behavior in spaceand time. Therefore, we begin by examining the frozen-coe�cient equations

(4.167) Ut = Lx0U := B(x0

)Uxx �A(x0

)Ux

at each value x0

of x, for simplicity taking b(x0

) diagonalizable (for motivationonly: we make no such assumption in the variable-coe�cient case).

Lemma 4.32[MaZ.3]. In dimension d = 1, let b2

(x0

) be diagonalizable, and letA⇤(U(x

0

)) have distinct, real eigenvalues. Then, for |�| su�ciently large, the eigen-value equation (Lx0 � �)U = 0 associated with the frozen-coe�cient operator Lx0

at x0

has a basis of n + r solutions

{�+

1

, . . . , �+

k , ��k+1

, . . . , ��n+r}(x;�, x0

), �±j = eµj

(�,x0)xVj(�, x0

),

of which n� r are “hyperbolic modes,” analytic in 1/� and satisfying

(4.168)µ±j (�, x

0

) = ��/a⇤j (x0

)� ⌘⇤j (x0

)/a⇤j (x0

) +O(1/�),Vj(�, x

0

) = R⇤j (x0

)/a⇤j (x0

) +O(1/�),

a⇤j (·), ⌘⇤j (·), and R⇤j (·) as defined in Section 4.2.1 (now with @x(b�1

2

b1

) ⌘ 0), and2r are “parabolic modes,” analytic in 1/

p� and satisfying

(4.169)

µ±j (�, x0

) = �q�/�j(x0

) +O(1/�),

Vj(�, x0

) =

0

@0n�r

sj

µjsj

1

A (x0

) +O(1/�),

�j(·), sj(·), the eigenvalues and right eigenvectors of b2

(U(x0

)). Likewise, the ad-joint eigenvalue equation (Lx0 � �)⇤U = 0 has a basis of solutions

{¯��1

, . . . , ¯��k , ¯�

+

k+1

, . . . , ¯�+

n+r}(x;�, x0

), ¯�±j = e�µ

j

(�,x0)xVj(�, x0

),

with

(4.170) Vj(�, x0

) = L⇤j (x0

) +O(1/�), Vj(�, x0

) =

0

@��1

j btr1

tjtj

�µjtj

1

A (x0

) +O(1/�),

respectively, for hyperbolic and parabolic modes, L⇤j (·) as defined in Section 4.2.1and tj(·) the left eigenvectors of b

2

(U(x0

)).

104 KEVIN ZUMBRUN

The expansions (4.168)–(4.170) hold also in the nonstrictly hyperbolic case, with�j,

¯�j and R⇤j , L⇤j now denoting n ⇥ m⇤j blocks, and µj, ⌘j denoting m⇤

j ⇥ m⇤j

matrices, j = 1, . . . , J , where m⇤j is the multiplicity of eigenvalue a⇤j of A⇤(x0

).

Proof. This follows by inversion of the expansions about ⇠ = 1 of the eigenval-ues �j(⇠) and eigenvectors Vj(⇠) of the one-dimensional frozen-coe�cient Fouriersymbol �i⇠A(x

0

)� ⇠2B(x0

), using the basic relation µj = i⇠ relating characteristicand dispersion equations (2.67) and (2.68). The expansions of the one-dimensionalFourier symbol are carried out in Appendix A.4. ⇤

By (2.104), Lemma 4.32 gives an expression for the constant-coe�cient resolventkernel G�;x0 at x

0

of, to lowest order, a “hyperbolic” part,(4.171)

H�;x0 ⇠( Pk

j=1

a⇤j (x0

)�1e(��/a⇤j

�⌘⇤j

/a⇤j

)(x0)(x�y)R⇤j (x0

)L⇤tj (x0

) x < y,

�PJ

j=k+1

a⇤j (x0

)�1e(��/a⇤j

�⌘⇤j

/a⇤j

)(x0)(x�y)R⇤j (x0

)L⇤tj (x0

) x > y,

with next-order contribution a “parabolic” part P�;x0 , analytic in ��1/2 (i.e., for �away from the negative real axis) and satisfying

(4.172) P�;x0 = O(|�|�1/2e�✓|�|1/2|x�y|), ✓ > 0,

on some sector

(4.173) ⌦P := {� : Re � � �✓1

| Im �|+ ✓2

}, ✓j > 0.

Our main result of this section will be to establish on an appropriate unboundedsubset of the resolvent set ⇢(L) that the variable coe�cient solutions �±j , �±j , andthus the resolvent kernel, satisfy analogous formulae, with static quantities replacedby the corresponding dynamical ones defined in Section 4.2.1. More precisely, define

(4.174) ⌦ := {� : �⌘1

Re �},

with ⌘1

> 0 su�ciently small that ⌦ \ B(0, r) is compactly contained in the set ofconsistent splitting ⇤ (defined in (2.62)), for some small r > 0 to be chosen later;this is possible, by Lemma 2.21(i).

Proposition 4.33[MaZ.3]. In dimension d = 1, assume (A1)–(A3) and (H0)–(H3), plus strict hyperbolicity of A⇤(·). Then, for R > 0 su�ciently large, thereholds ⌦\B(0, R) ⇢ ⇤\⇢(L); moreover, there holds on ⌦\B(0, R) the decomposition

(4.175) G�(x, y) = H�(x, y) + P�(x, y) +⇥�(x, y),

(4.176)

H�(x, y) :=

(�PJ

j=k+1

a⇤j (x)�1eR

x

y

(��/a⇤j

�⌘⇤j

/a⇤j

)(z) dzR⇤j (x)L⇤tj (y) x > y,Pk

j=1

a⇤j (x)�1eR

x

y

(��/a⇤j

�⌘⇤j

/a⇤j

)(z) dzR⇤j (x)L⇤tj (y) x < y,


(4.177) ⇥�(x, y) = ��1B(x, y;�) + ��1(x� y)C(x, y;�) + ��2D(x, y;�),

where

(4.178) B(x, y;�) =

( PJj=k+1

e�R

x

y

�/a⇤j

(z) dzb+

j (x, y) x > y,Pk

j=1

e�R

x

y

�/a⇤j

(z) dzb�j (x, y) x < y,

(4.179)C(x, y;�) =

(�PJ

i,j=k+1

meanz2[x,y]

e�R

z

y

�/a⇤i

(s)ds�Rx

z

�/a⇤j

(s)dsc+

i,j(x, y; z), x > y,Pk

i,j=1

meanz2[x,y]

e�R

z

y

�/a⇤i

(s)ds�Rx

z

�/a⇤j

(s)dsc�i,j(x, y; z), x < y,

with

(4.180) |b±j |, |c±i,j | Ce�✓|x�y|

and

(4.181) D(x, y;�) = O(e�✓(1+Re �)|x�y| + e�✓p|�||x�y|),

for some uniform ✓ > 0 independent of x, y, z, each described term separatelyanalytic in �, and P� is analytic in � on a (larger) sector ⌦P as in (4.173), with✓1

su�ciently small, and ✓2

su�ciently large, satisfying uniform bounds

(4.182) (@/@x)↵(@/@y)�P�(x, y) = O(|�|(|↵|+|�|�1)/2e�✓|�|1/2|x�y|), ✓ > 0,

for |↵|+ |� 2 and 0 |↵|, |�| 1.Likewise, there hold derivative bounds

(4.183)(@/@x)⇥�(x, y) =

�B0

x(x, y;�) + (x� y)C0

x(x, y;�)�

+ ��1

�B1

x(x, y;�)

+ (x� y)C1

x(x, y;�) + (x� y)2D1

x(x, y;�)�

+ ��3/2Ex(x, y;�)

and(4.184)(@/@y)⇥�(x, y) =

�B0

y(x, y;�) + (x� y)C0

y(x, y;�)�

+ ��1

�B1

y(x, y;�)

+ (x� y)C1

y(x, y;�) + (x� y)2D1

y(x, y;�)�

+ ��3/2Ey(x, y;�),

where B↵� and C↵

� satisfy bounds of form (4.178) and (4.179), D1

� now denotes theiterated integral(4.185)D1

�(x, y;�) =8>>>>><

>>>>>:

�PJ

h,i,j=k+1

meanywzxe�R

w

y

�/a⇤h

(s)ds�Rz

w

�/a⇤i

(s)ds�Rx

z

�/a⇤j

(s)ds

⇥d�,+h,i,j(x, y; z), x > y,

Pkh,i,j=1

meanxzwye�R

w

y

�/a⇤h

(s)ds�Rz

w

�/a⇤i

(s)ds�Rx

z

�/a⇤j

(s)ds

⇥d�,�h,i,j(x, y; z), x < y,

106 KEVIN ZUMBRUN

with |d�,�h,i,j | Ce�✓|x�y|, and E� satisfies a bound of form (4.181).

The bounds (4.177)–(4.184) hold also in the nonstrictly hyperbolic case, with(4.176) replaced by(4.186)

H�(x, y) =

(�PJ

j=K+1

a⇤j (x)�1e�R

x

y

�/a⇤j

(z) dzR⇤j (x)⇣j(x, y)L⇤tj (y) x > y,PK

j=1

a⇤j (x)�1e�R

x

y

�/a⇤j

(z) dzR⇤j (x)⇣j(x, y)L⇤tj (y) x < y,

where ⇣j(x, y) 2 Rm⇤j

⇥m⇤j denotes a dissipative flow similar to ⇣j in (4.27), but with

respect to variable x: i.e.,

(4.187) d⇣j/dx = ⌘⇤j (x)⇣j(x, y)/a⇤j (x), ⇣j(y, y) = I,

or, equivalently, ⇣j(x, y) = ⇣j(x, ⌧) for ⌧ such that zj(x, 0) = y, where zj(x, s) asin (4.26) denotes the backward characteristic path associated with a⇤j , zj(x, ⌧) := x;R⇤j , L⇤j now denote n⇥m⇤

j blocks; and µj, ⌘⇤j denote m⇤j⇥m⇤

j matrices, j = 1, . . . ,K,j = K + 1, . . . , J , where m⇤

j is the multiplicity of eigenvalue a⇤j of A⇤(x0

), with

a⇤1

· · · a⇤K < 0 < a⇤K+1

· · · a⇤J .

(Note that we have not here assumed (D)).

Proof. Our argument is similar in spirit to the proof of Proposition 4.4 [MaZ.1]in the hyperbolic relaxation case. However, the mixed hyperbolic–parabolic natureof the underlying equations, and the associated presence of multiple scales, makesthe analysis more delicate. In accordance with the behavior we seek to identify,it is convenient to express the eigenvalue equation in “local” variables u := A⇤u,v := b

1

u + b2

v similar to those used in proving Lemma 2.1, yielding

(4.188)

ux = ��A�1

⇤ u� (A12

b�1

2

v)x,

(vx)x =h((A

21

�A22

b�1

2

b1

+ b2

@x(b�1

2

b1

))A�1

⇤ u)x

+ (A22

+ @x(b2

))b�1

2

v)x + �b�1

2

b1

A�1

⇤ u + �b�1

2

vi.

Similarly as in the relaxation case, the substitution of A⇤u for u converts the hy-perbolic part of the equation from divergence to nondivergence form, simplifyingslightly the flow along characteristics in the hyperbolic term H�; for related discus-sion, see Remark 4.12.

Following standard procedure (e.g., [AGJ, GZ, ZH, Z.3]), we perform the rescal-ing

(4.189) x := |�|x, � := �/|�|.


suggested by the leading order hyperbolic behavior of µ(x0

), (4.168), to obtain aftersome rearrangement the perturbation equation

(4.190) Y 0 = A(x, |�|�1)Y, Y := (u, v, v0)tr,

where

(4.191) A(x, |�|�1) = A0

(x) + |�|�1A1

(x) +O(|�|�2),

(4.192)

A0

(x) :=

0

@��A�1

⇤ 0 �A12

b�1

2

0 0 Ir

0 0 0

1

A , A1

(x) :=

0

@0 @x(A

12

b�1

2

) 00 0 0

��d⇤A�2

⇤ ��b�1

2

e⇤b�1

2

1

A ,

(4.193)d⇤ := A

21

�A22

b�1

2

b1

+ b�1

2

b1

A⇤ + b2

@x(b�1

2

b1

),

e⇤ := A22

+ d⇤A�1

⇤ A12

+ @x(b2

),

and “0” denotes @/@x. Here, expansion (4.191) is to be considered as a continuousfamily of one-parameter perturbation equations, indexed by � 2 S1. The form ofe⇤ is not important, and is included only for completeness; d⇤, on the other hand,may be recognized as the important quantity defined in (4.37).

We seek to develop corresponding expansions of the resolvent kernel in ordersof |�|�1, or, equivalently, by representation (2.84), expansions of stable/unstablemanifolds of (4.190), and the reduced flows therein. This we will accomplish by themethods described in Sections 2.2.2–2.2.5, first reducing to an approximately block-diagonal system segregating spectrally separated (in particular, stable/unstable)modes, with formal error of a suitably small order, then converting the formal errorinto rigorous error bounds using Proposition 2.13 together with Corollary 2.14.An interesting aspect of this calculation is the block-diagonalization of parabolicmodes, which are not uniformly spectrally separated in this (hyperbolic) scaling,but rather comprise a block-Jordan block of order s = 2 associated with eigenvalueµ = 0. Accordingly, we treat them separately, by the method sketched in Section2.2.4, after the initial block-diagonalization has been carried out. This results ina slower decaying error series for these blocks, converging in powers of |�|�1/2;parabolic modes are well-behaved, however, and so the loss of accuracy is harmless.

We will carry out in detail the basic estimate (4.177), indicating the extensionto derivative bounds (4.183)–(4.184) by a few brief remarks. First, observe that inthe modified coordinates Y = QW , (2.84) just becomes

(4.194) G�(x, y) =

8><

>:

(In, 0)Q�1(x)Fy!xY ⇧�YQS

1(y)(In, 0)tr x < y,

�(In, 0)Q�1(x)Fy!xY ⇧+

YQS�1(y)(In, 0)tr x > y,

108 KEVIN ZUMBRUN

where ⇧±Y and Fy!xY denote projections and flows in Y -coordinates, and the lower

triangular matrices

(4.195) Q =

0

@In�r 0 0

0 Ir 00 0 |�|�1Ir

1

A

0

@A⇤ 0 0b1

b2

0@x(b

1

) @x(b2

) Ir

1

A ,

(4.196) Q�1 =

0

@A�1

⇤ 0 0�b�1

2

b1

A�1

⇤ b�1

2

0�b

2

@x(b�1

2

b1

)A�1

⇤ �@x(b2

)b�1

2

Ir

1

A

0

@In�r 0 0

0 Ir 00 0 |�|Ir

1

A

relate Y coordinates to the original coordinates W = (u, v, b1

ux + b2

vx)tr of (2.84):

(4.197) Y = QW, W = Q�1Y.

It is readily checked that Q 2 Cp+1 for f , B 2 Cp+2, whence Q 2 C2 under theregularity hypothesis (H0) (p = 3).

Initial diagonalization. For f, B 2 Cp+2, we may check further that Aj 2Cp�j+1 as required in the theory of Section 2.2.3, suitable for approximate block-diagonalization to formal error O(|�|�p�1): formal error O(|�|�4) in the presentcase p = 3. Applying the formal diagonalization procedure of Proposition 2.15to (4.190), and choosing the special initialization (2.47), at the same time takingcare as described in Remark 2.16(3) to preserve analyticity with respect to ��1 inoriginal coordinates, we obtain the approximately block-diagonalized system

(4.198) Z 0 = D(x, |�|�1)Z, TZ := Y, D := TAT�1,

(4.199) T (x, |�|�1) = T0

(x) + |�|�1T1

(x) + · · ·+ |�|�3T3

(x),

(4.200) D(x, |�|�1) = D0

(x) + |�|�1D1

(x) + · · ·+ D3

(x)|�|�3 +O(|�|�4),

where without loss of generality (since T0

is uniquely determined up to a constantlinear coordinate change)(4.201)

T�1

0

=

0

@Ltr⇤ 0 ��1Ltr

⇤ A⇤A12

b�1

2

0 Ir 00 0 Ir

1

A , T0

=

0

@R⇤ 0 ��1A⇤A12

b�1

2

0 Ir 00 0 Ir

1

A ,

L⇤ := (L⇤1

, . . . , L⇤J), R⇤ := (R⇤1

, . . . , R⇤J), L⇤j , R⇤j as defined in Section 4.2.1, and so(4.202)

D0

=

0

@�diag {�Im⇤

j

/a⇤j} 0 00 0 Ir

0 0 0

1

A , D1

=

0

@�diag {⌘⇤j Im⇤

j

/a⇤j} 0 00 0 00 ��b�1

2

⇤

1

A ,


a⇤j and m⇤j as defined in (4.15), and ⌘⇤j as defined in (4.18)–(4.19). Here, we have

used the simple block upper-triangular form of A0

to deduce the form of (4.201),then calculated (4.202) from the relations D

0

= T�1

0

A0

T0

, D1

= T�1

0

A1

T0

, thesecond a consequence of the special normalization (2.47).

The parabolic block. At this point, we have diagonalized into J m⇤j ⇥ m⇤

j

hyperbolic blocks corresponding to eigenvalues µ = ��/a⇤j of A0

, and a 2r ⇥ 2rparabolic block-Jordan block (the lower righthand corner)

(4.203) N :=✓

0 Ir

0 0

◆+ |�|�1

✓0 0

��b�1

2

⇤

◆+O(|�|�2),

corresponding to eigenvalue µ = 0 of A0

. We now focus on the latter block, applyingthe procedure of Section 2.2.4. “Balancing” by transformation B := {1, |�|1/2}yields

(4.204) M := B�1NB = |�|�1/2M1

+O(|�|�1), M1

:=✓

0 Ir

��b�1

2

0

◆,

or |�|�1/2 times the standard rescaled expansion of the strictly parabolic case [ZH].

Proceeding as in that case, we observe that �(M1

) = ±q�(�b�1

2

) has a uniformspectral gap of order one, by (H1)(i), on the sector ⌦P defined in (4.173), for ✓

1

> 0su�ciently small. Thus (see, e.g. Proposition A.9, p. 361 of [St]), there exists awell-conditioned coordinate change S = S(M

1

(y)) depending continuously on M1

,such that

M1

= diag {M�1

, M+

1

}(x) := S�1MS(x)

with M�1

uniformly negative definite and M+

1

uniformly positive definite. Applyingthis coordinate change, and noting that the “dynamic error” S�1(@/@x)S is of order(@/@x)M

1

= O(|�|�1), we obtain, finally, the formal expansion

(4.205) M(x, |�|�1) = |�|�1/2 diag {M�1

, M+

1

}+O(|�|�1).

On the sector ⌦P , blocks ± are exponentially separated to order |�|�1/2, by Remark2.12. Thus, we may apply Proposition 2.13 to find that, on ⌦P , there is a furthertransformation S = I

2r +O(|�|�1/2) converting (4.205) to fully diagonalized form

(4.206) M(x, |�|�1) = |�|�1/2 diag {M�1

,M+

1

},

where M±1

= M±1

+ O(|�|�1/2) are still uniformly positive/negative definite. Wewill define P� to be the part of the Green distribution associated to the flow of thisdiagonal block. Note that it has been obtained independent of any separation (orlack thereof) from hyperbolic modes, so is analytic on all of ⌦p

110 KEVIN ZUMBRUN

More precisely, denote by

(4.207) Z := T Y :=✓

In�r 00 SSB

◆(T

0

+ |�|�1T1

)Y

the concatenation of coordinate changes made in the course of the above reductions,Z = (⇣

1

, . . . , ⇣J , ⇢�, ⇢+

)tr. Then, we have, similarly as in (4.194), the representation(4.208)

G�(x, y) =

8><

>:

(In, 0)Q�1T �1(x)Fy!xZ ⇧�ZT QS

1(y)(In, 0)tr x < y,

�(In, 0)Q�1T �1(x)Fy!xZ ⇧+

ZT QS�1(y)(In, 0)tr x > y,

where ⇧±Z and Fy!xZ denote projections and flows in Z-coordinates, and Z satisfies

the approximately decoupled system of equations

(4.209)⇣ 0j = �(�/a⇤j � |�|�1⌘⇤j /a⇤j + cj |�|�2 + dj |�|�3)⇣j +O(|�|�4)(⇢, ⇣),

⇢0± = |�|�1/2M±1

⇢± +O(|�|�7/2)⇣.

(Note: the degraded error estimate in the ⇢-equations is due to the ill-conditioningof the balancing transformation B.) Approximating ⇧±Z and Fy!x

Z by the corre-sponding projections ⇧±Z = ⇧H,±

Z + ⇧P,±Z and flows Fy!x

Z for the associated decou-pled equations, where ⇧H,±

Z and ⇧P,±Z denote the subprojections on hyperbolic and

parabolic blocks, we obtain for x < y the approximation

(4.210) G�(x, y) := (In, 0)trQ�1T �1(x)Fy!xZ (⇧H,�

Z + ⇧H,�Z )T QS1

(y)(In, 0),

and similarly for x > y. Defining

(4.211)P�(x, y) := (In, 0)Q�1T �1(x)Fy!x

Z ⇧P,�Z T QS1

(y)(In, 0)tr

= (In, 0)Q�1T �1(x)Fy!xZ diag {0n�r, Ir, 0r}T QS

1(y)(In, 0)tr,

and untangling the various coordinate changes, we find that P� satisfies bounds(4.182), similarly as in the strictly parabolic case [ZH]. Specifically, we have bydirect calculation that

(4.212) (In, 0)Q�1 =✓

A�1

⇤ 0 0�b�1

2

b1

A�1

⇤ b�1

2

0

◆= O(1)

and

(4.213) QS�1(In, 0)tr =

0

@�In�r 0

0 0O(|�|�1) |�|�1Ir

1

A ,


whence (by the special form of T0

)

(4.214) (T0

+ |�|�1T1

)QS�1(In, 0)tr =

0

@O(1) O(1)

0 0O(|�|�1) |�|�1Ir

1

A

and therefore

(4.215) ⇧P,�Z T QS�1(In, 0)tr = O(|�|�1/2).

Combining with the evident bound

(4.216) |Fy!xZ ⇧P,�

Z | Ce�✓|�|�1/2|x�y| = Ce�✓|�|

1/2|x�y|,

for some ✓ > 0, coming from M�1

�✓|�|�1/2, we obtain (4.182) for |↵| = 0.Derivative bounds then follow from the fact that x and y derivatives of all trans-formations are at worst order one, while

|(@/@(x, y))↵Fy!xZ ⇧P,�

Z | = |�||↵||(@/@(x, y))↵Fy!xZ ⇧P,�

Z | = O(|�|�|↵/2|)

by inspection of the decoupled equation ⇢0� = |�|�1/2M�1

⇢�. For closely related butmuch simpler calculations in the strictly parabolic case, see the proof of Proposition7.3, [ZH] (discussion just below (7.24)).

Approximation error. Let us now return to the approximately decoupledequations (4.209), or, converting back to the original spatial coordinates:

(4.217)⇣ 0j = �(�/a⇤j � ⌘⇤j /a⇤j + cj�

�1 + dj��2)⇣j +O(|�|�3)(⇢, ⇣),

⇢0± = �1/2M±1

⇢± +O(|�|�5/2)⇣.

By condition (2.52) together with (4.21) to the ⇣j equations and condition (2.32) tothe ⇢± equations, we find that the stable and unstable manifolds of the decoupledflow, consisting respectively of ⇢� together with ⇣j , a⇤j > 0 and ⇢

+

together with ⇣j ,a⇤j > 0, are exponentially separated to order ✓min{|�|1/2, 1+Re �}, for some ✓ > 0.Applying Proposition 2.13 together with Corollary 2.14, therefore, yields just thedecoupled flows with error terms of order |�|�5/2(e�✓|�|

1/2|x�y|e�✓(1+Re �)|x�y|),and diagonalization error O(|�|�5/2).

Recalling representation (4.208), we find after some rearrangement that G� isgiven by the decoupled approximant G� of (4.210) plus error terms (coming fromflow plus diagonalization error) of form |�|�5/2(e�✓|�|

1/2|x�y|e�✓(1+Re �)|x�y|), ab-sorbable in term D. The contribution to G� from decoupled parabolic modes, asalready discussed, is exactly P�. Having separated o↵ the parabolic modes, we havenow essentially reduced to the purely hyperbolic case treated in [MaZ.1], and weproceed accordingly. Focusing on the decoupled hyperbolic equations

(4.218) ⇣ 0j = �(�/a⇤j � ⌘⇤j /a⇤j )⇣j + (cj��1 + dj�

�2)⇣j ,

112 KEVIN ZUMBRUN

and applying Corollary 2.14 once again, considering the second term of the right-hand side as a perturbation, we find, substituting into (4.210), that the contributionto G� from decoupled hyperbolic modes is H� plus terms of form ⇥� as claimed.We omit the details of this latter calculation, which are entirely similar to thoseof the relaxation case. We note in passing that exponential separation impliesthat the stable/unstable manifolds of (4.217) are uniformly transverse for � 2 ⌦,|�| su�ciently large, yielding ⌦ \ B(0, R) ⇢ ⇢(L) \ ⇤, for R su�ciently large, asclaimed.

Derivative estimates. Derivative estimates (4.183)–(4.184) now follow instraighforward fashion, by di↵erentiation of (4.208), noting from the approximatelydecoupled equations that di↵erentiation of the flow brings down a factor (to ab-sorbable error) of � in hyperbolic modes, �1/2 in parabolic modes. ⇤

4.3.3 Extended spectral theory.

Finally, we cite without proof the extended spectral theory of [ZH, MaZ.3] thatwe shall need in order to establish necessity of condition (D). In the case that (D)holds, this reduces to the simple relations (4.159). Let Cj

⌘ denote the space of Cj

functions f(x) satisfying

(4.219) |(d/dx)kf(x)e�⌘|x|| C, 0 k j.

Definition 4.34. Let L be a (possibly degenerate type) rth order linear ordinarydi↵erential operator for which the associated eigenvalue equation may be writtenas a first-order system with bounded Cq coe�cients (so that L : Cq+r

⌘ ! Cq⌘), and

let G� denote the Green distribution of L��I, G� 2 Cq+1

�⌘/2

(x, y) away from y = x.Further, let ⌦ be an open, simply connected domain intersecting the resolvent setof L, on which G� has a (necessarily unique) meromorphic extension considered asa function from ⌦ to L1

loc

. Then, for �0

2 ⌦, we define the e↵ective eigenprojectionP�0 : Cq

⌘ ! Cq�⌘/2

by

P�0f(x) =Z

+1

�1P�0(x, y)f(y) dy,

where

(4.220) P�0(x, y) = Res�0G�(x, y)

and Res�0 denotes residue at �0

. (See [ZH], p. 44, for a proof that P�0 : Cq⌘ !

Cq�⌘/2

.) We will refer to P�0(x, y) as the projection kernel. Likewise, we define thee↵ective eigenspace ⌃0�0

(L) ⇢ Cq�⌘ by

⌃0�0(L) = Range (P�0),


and the e↵ective point spectrum �0p(L) of L in ⌦ to be the set of � 2 ⌦ such thatdim ⌃0�0

(L) 6= 0.

Definition 4.35. Let L, ⌦, �0

be as above, and K be the order of the pole of(L � �I)�1 at �

0

. For �0

2 ⌦, and k any integer, we define Q�0,k : Cq⌘ ! Cq

�⌘/2

by

Q�0,kf(x) =Z

+1

�1Q�0,k(x, y)f(y) dy,

whereQ�0,k(x, y) = Res�0(�� 0

)kG�(x, y).

For 0 k K, we define the e↵ective eigenspace of ascent k by

⌃0�0,k(L) = Range (Q�0,K�k)

With the above definitions, we obtain the following, modified Fredholm Theory.

Proposition 4.36 [ZH, MaZ.3]. Let L, �0

, ⌦ be as in Definition 4.34, and K bethe order of the pole of G� at �

0

. Then,

(i) The operators P�0 , Q�0,k : Cq⌘ ! Cq

�⌘ are L-invariant, with

(4.221) Q�0,k+1

= (L� �0

I)Q�0,k = Q�0,k(L� �0

I)

for all k 6= �1, and

(4.222) Q�0,k = (L� �0

I)kP�0

for k � 0.

(ii) The e↵ective eigenspace of ascent k satisfies

(4.223) ⌃0�0,k(L) = (L� �0

I)⌃0�0,k+1

(L).

for all 0 k K, with

(4.224) {0} = ⌃0�0,0(L) ⇢ ⌃0�0,1(L) ⇢ · · · ⇢ ⌃0�0,K(L) = ⌃0�0(L).

Moreover, each containment in (4.224) is strict.

(iii) On P�1

�0(Cq

⌘), P�0 , Q�0,k all commute (k � 0), and P�0 is a projection. Moregenerally, P�0f = f for any f 2 CKr

⌘ ) such that (L � �0

)Kf = 0, hence each⌃0�0,k(L) contains all f 2 Ckr

⌘ such that (L� �0

)kf = 0.

(iv) The multiplicity of the eigenvalue �0

, defined as dim⌃0�0(L), is finite and bounded

by Kn, where n is the dimension of the phase space for the first-order eigenvalueODE. Moreover, for all 0 k K,

(4.225) dim⌃0�0,k(L) = dim⌃0�0,k(L⇤).

114 KEVIN ZUMBRUN

Further, the projection kernel can be expanded as

(4.226) P�0 =X

j

'j(x)⇡j(y),

where {'j}, {⇡j} are bases for ⌃0�0(L), ⌃0�⇤0 (L

⇤), respectively.

Proof. Residue calculations similar in spirit to those of [Kat] in the classicalcase of an isolated eigenvalue of finite multiplicity, but substituting everywhere theresolvent kernel for the resolvent. See [ZH] for details. ⇤

Remark 4.37. For �0

in the resolvent set, P�0 agrees with the standard defi-nition, hence ⌃0�0,k(L) agrees with the usual Lp eigenspace ⌃�0,k(L) of generalizedeigenfunctions of ascent k, for all p < 1, since C1⌘ is dense in Lp, p > 1,and ⌃0�0,k(L) is closed. In the context of stability of traveling waves (more gen-erally, whenever the coe�cients of L exponentially approach constant values at±1), ⌃0�0,k(L) lies between the Lp subspace ⌃�0,k(L) and the corresponding Lp

Loc

subspace.

Now, suppose further (as holds in the case under consideration) that, on ⌦: (i) Lis a di↵erential operator for which both the forward and adjoint eigenvalue equationsmay be expressed in appropriate phase spaces as nondegenerate first-order ODEof the same order, and for which the reduced quadratic form S of Lemma 2.22is invertible; and, (ii) there exists an analytic choice of bases for stable/unstablemanifolds at +1/�1 of the associated eigenvalue equation. In this case, we candefine an analytic Evans function D(�) as in (3.1), which we assume does notvanish identically. It is easily verified that the order to which D vanishes at any �

0

is independent of the choice of analytic bases �±. The following result establishedin [ZH, MaZ.3] generalizes to the extended spectral framework the standard resultof Gardner and Jones [GJ.1–2] in the classical setting.

Proposition 4.38 [ZH, MaZ.3]. Let L, �0

be as above. Then,

(i) dim⌃0�0(L) is equal to the order d to which the Evans function DL vanishes

at �0

.(ii) P�0(L) =

Pj 'j(x)⇡j(y), where 'j and ⇡j are in ⌃0�0

(L) and ⌃0�0(L⇤),

respectively, with ascents summing to K + 1, where K is the order of the pole ofG� at �

0

.

Proof. Direct calculation using the augmented resolvent kernel formula (2.88)of Remark 2.24, Section 2.4.2, and working in a Jordan basis similarly as in [GJ.1].See [ZH] for details. ⇤

4.3.4 Bounds on the Green distribution.

With these preparations, we may now establish pointwise bounds on the Greendistribution G and consequently su�ciency and necessity of condition (D) for lin-earized orbital stability.


Proposition 4.39. In dimension d = 1, assuming (A1)–(A3), (H0)–(H3), and(D), the Green distribution G(x, t; y) associated with (2.1) may be decomposed asin Theorem 4.10, where, for y 0:(4.227)R(x, t; y) = O(e�⌘te�|x�y|2/Mt))

+nX

k=1

O⇣(t + 1)�1/2e�⌘x+

+ e�⌘|x|⌘

t�1/2e�(x�y�a�k

t)2/Mt

+X

a�k

>0, a�j

<0

�{|a�k

t|�|y|}O((t + 1)�1/2t�1/2)e�(x�a�j

(t�|y/a�k

|))2/Mte�⌘x+,

+X

a�k

>0, a+j

>0

�{|a�k

t|�|y|}O((t + 1)�1/2t�1/2)e�(x�a+j

(t�|y/a�k

|))2/Mte�⌘x� ,

(4.228)

Ry(x, t; y) =JX

j=1

O(e�⌘t)�x�a⇤j

t(�y) +O(e�⌘te�|x�y|2/Mt))

+nX

k=1

O⇣(t + 1)�1/2e�⌘x+

+ e�⌘|x|⌘

t�1e�(x�y�a�k

t)2/Mt

+X

a�k

>0, a�j

<0

�{|a�k

t|�|y|}O((t + 1)�1/2t�1)e�(x�a�j

(t�|y/a�k

|))2/Mte�⌘x+

+X

a�k

>0, a+j

>0

�{|a�k

t|�|y|}O((t + 1)�1/2t�1)e�(x�a+j

(t�|y/a�k

|))2/Mte�⌘x� ,

for some ⌘, M > 0, where a⇤j , a⇤j are as in Theorem 4.10, x± denotes the posi-tive/negative part of x, and indicator function �{|a�

k

t|�|y|} is one for |a�k t| � |y|and zero otherwise. Symmetric bounds hold for y � 0.

Proof. Our starting point is the representation

(4.229) G(x, t; y) =1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�tG�(x, y)d�,

valid by Proposition 2.3 for any ⌘ su�ciently large.

Case I. |x � y|/t large. We first treat the simpler case that |x � y|/t � S, Ssu�ciently large. Fixing x, y, t, set � = ⌘ + i⇠. Applying Proposition 4.33, we

116 KEVIN ZUMBRUN

obtain, for ⌘ > 0 su�ciently large, the decomposition(4.230)

G(x, t; y) =1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�tH�(x, y)d�+

12⇡i

P.V.Z ⌘+i1

⌘�i1e�t⇥H

� (x, y)d�

+1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�tP�(x, y)d�+

12⇡i

P.V.Z ⌘+i1

⌘�i1e�t⇥P

� (x, y)d�

=: I + II + III + IV,

where(4.231)

H�(x, y) =

(�PJ

j=K+1

a⇤j (y)�1e�R

x

y

�/a⇤j

(z) dzR⇤j (x)⇣⇤j (x, y)L⇤tj (y) x > y,PK

j=1

a⇤j (y)�1e�R

x

y

�/a⇤j

(z) dzR⇤j (x)⇣⇤j (x, y)L⇤tj (y) x < y,

(4.232) ⇥H� (x, y) := ��1B(x, y;�) + ��1(x� y)C(x, y;�),

and

(4.233) ⇥P� (x, y) := ��2D(x, y;�),

with ⇣j , B, C, D as defined in Proposition 4.33. For definiteness taking x > y, weestimate each term in turn.

Term I. Term I of (4.230) contributes to integral (4.229) the explicitly evaluablequantity(4.234)

12⇡i

P.V.Z ⌘+i1

⌘�i1e�tH�(x, y)d� =

JX

j=K+1

✓a⇤j (x)�1P.V.

Z+1

�1ei⇠�t�R

x

y

(1/a⇤j

(z)) dz�

d⇠

◆

⇥ e⌘�t�R

x

y

(1/a⇤j

(z)) dz�rj(x)⇣j(x, y)ltj(y)

=JX

j=K+1

✓a⇤j (x)�1�

�t�

Z x

y

(1/a⇤j (z)) dz�◆

rj(x)⇣j(x, y)ltj(y)

=JX

j=K+1

a⇤j (x)�1a⇤j (y)�x�a⇤j

t(�y)rj(x)⇣⇤j (y, t)ltj(y)

= H(x, t; y),

where a⇤j , ⇣⇤j , and H are as defined in Theorems 4.10 and 4.39. Note: in the secondto last equality of (4.234) we have used the general fact that

(4.235) fy(x, y, t)��f(x, y, t)

�= �h(x,t)(y),


provided fy 6= 0, where h(x, t) is defined by f(x, h(x, t), t) ⌘ 0. This term clearlyvanishes for x outside [z

1

(y, t), zJ(y, t)], hence makes no contribution for S su�-ciently large.

Term II. Similar calculations show that the “hyperbolic error term” II in (4.230)also vanishes. For example, the term e�t��1B(x, y;�) contributes(4.236)1

2⇡i

KX

j=K+1

✓P.V.

Z+1

�1(⌘ + i⇠)�1ei⇠

�t�R

x

y

(1/a⇤j

(z)) dz�

d⇠

◆e⌘�t�R

x

y

(1/a⇤j

(z)) dz�b+

j (x, y).

The factor

(4.237) P.V.Z

+1

�1(⌘ + i⇠)�1ei⇠

�t�R

x

y

(1/a⇤j

(z)) dz�

d⇠,

though not absolutely convergent, is integrable and uniformly bounded as a prin-cipal value integral, for all real ⌘ bounded from zero, by explicit computation. Onthe other hand,

(4.238) e⌘�t�R

x

y

(1/a⇤j

(z)) dz� Ce�⌘ d(x,[z1(y,t),z

J

(y,t)])/ min

j,x

|a⇤j

(x)| ! 0

as ⌘ ! +1, for each K + 1 j J , since a⇤j > 0 on this range. Thus, taking⌘ ! +1, we find that the product (4.236) goes to zero, giving the result. Likewise,the contributions of terms e�tC(x, y;�) and e�tD(x, y;�) split into the product ofa convergent, uniformly bounded integral in ⇠, a (constant) factor depending onlyon (x, y), and a factor ↵(x, y, t, ⌘) going to zero as ⌘ ! 0 at rate (4.238). Thus,each of these terms vanishes also as ⌘ ! +1, as claimed.

Term III. The parabolic term III may be treated exactly as in the strictly par-abolic case [ZH]. Namely, using analyticity of P� within the sector ⌦P defined in(4.173), we may for any fixed ⌘ deform the contour in the principal value integralto

(4.239)Z

�1[�2

e�tP�(x, y) d�,

where �1

:= @B(0, R) \ ⌦P and �2

:= @⌦P \ B(0, R) plus an error term of ordernot greater than

|⇠|�1/2

Z �1

⌘

ewtdw ! 0

as |⇠|!1, where

(4.240) R := ↵2, ↵ :=✓|x� y|

2t,

118 KEVIN ZUMBRUN

✓ is as in (4.182).By estimate (4.182) we have for all � 2 �

1

[ �2

that

(4.241) |P�(x, y)| C|��1/2|e�✓|�1/2||x�y|

Further, we have

(4.242)Re� R(1� ⌘!2), � 2 �

1

,

Re� Re�0

� ⌘(| Im �|� | Im �0

|), � 2 �2

for R su�ciently large, where ! is the argument of � and �0

and �⇤0

are the twopoints of intersection of �

1

and �2

, for some ⌘ > 0 independent of ↵.Combining (4.241), (4.242), and (4.240), we obtain

|Z

�1

e�tP�d�| Z

�1

C|��1/2| eRe�t�✓|�1/2||x�y| d�

Ce�↵2t

Z+L

�L

R�1/2e�R⌘!2t Rd!

Ct�1/2e�↵2t.

Likewise,

(4.243)

|Z

�2

e�tP�d�| Z

�2

C|��1/2|CeRe�t�✓|�1/2||x�y| d�

CeRe(�0)t�✓|�1/20 ||x�y|

Z

�2

|��1/2|e(Re�)�Re�0)t |d�|

Ce�↵2t

Z

�2

|Im�|�1/2e�⌘|Im��Im�0|t |d Im�|

Ct�1/2e�↵2t.

Combining these last two estimates, and recalling (4.240), we have

(4.244) III Ct�1/2e�↵2t/2e�✓

2(x�y)

2/8t Ct�1/2e�⌘te�(x�y)

2/Mt,

for ⌘ > 0, M > 0 independent of ↵. Observing that

|x� at|2t

|x� y|t

2|x� at|t

for any bounded a, for |x� y|/t su�ciently large, we find that III can be absorbedin the residual term O(e�⌘te�|x�y|2/Mt) for t � ✏, any ✏ > 0, and by any summandO(t�1/2(t + 1)�1/2e�(x�y�a±

k

t)2/Mt)e�⌘x±e�⌘y± for t small.


Likewise, di↵erentiating within the absolutely convergent integral (4.239), wefind that |IIIy| can be absorbed in the corresponding summand of (4.228).

Note. A delicate point about the derivative bounds to first move the contourto (4.239) and then di↵erentiate. Di↵erentiating within the original principal valueintegral yields an integral that is not obviously convergent, and for which it is notclear that the contour may be so deformed.

Term IV. Similarly as in the treatment of term III, the principal value integralfor the “parabolic error term” IV may be shifted to ⌘ = R = ↵2, ↵ as in (4.240),at the cost of an error term that vanishes as |⇠|!1. But, this yields an estimate

|IV | Ce�↵2t

Z+1

�1|⌘

0

+ i⇠|�2d⇠ e�↵2t

that by (4.244) may be absorbed in O(e�⌘te�|x�y|2/Mt) for all t. Derivatives IVx

and IVy may be similarly absorbed, while IVxy may be absorbed in

(@/@y)O(e�⌘te�|x�y|2/Mt).

Case II. |x�y|/t bounded. We now turn to the critical case that |x�y|/t Sfor some fixed S. In this regime, note that any contribution of order e✓t, ✓ > 0,may be absorbed in the residual (error) term R (resp. Rx, Ry); we shall use thisobservation repeatedly.

Decomposition of the contour. We begin by converting contour integral(4.229) into a more convenient form decomposing high, intermediate, and low fre-quency contributions.

Observation 4.40. In dimension d = 1, assuming (A1)–(A3), (H0)–(H3), thereholds the representation

(4.245)

G(x, t; y) = Ia + Ib + Ic + IIa + IIb

:=1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�tH�(x, y)d�

+1

2⇡iP.V.

Z �⌘1�iR

�⌘1�i1+Z �⌘1+i1

�⌘1+iR

!e�t(G� �H� � P�)(x, y)d�

+1

2⇡i

I

�2

e�tP�(x, y)d�

+1

2⇡i

I

�

e�tG�(x, y)d�� 12⇡i

Z �⌘1+iR

�⌘1�iR

e�tH�(x, y)d�,

(4.246) � := [�⌘1

� iR, ⌘ � iR] [ [⌘ � iR, ⌘ + iR] [ [⌘ + iR,�⌘1

+ iR],

120 KEVIN ZUMBRUN

(4.247) �2

:= @⌦P \ ⌦

⌦P as defined in (4.173), for any ⌘ > 0 such that (4.229) holds, R su�ciently large,and �⌘

1

< 0 as in (4.174). (Note that we have not here assumed (D)).

Proof. We first observe that, by Proposition 4.33, L has no spectrum on theportion of ⌦ lying outside of the rectangle

(4.248) R := {� : �⌘1

Re � ⌘, �R Im � R}

for ⌘ > 0, R > 0 su�ciently large, hence G� is analytic on this region. Since,also, H� is analytic on the whole complex plane, contours involving either one ofthese contributions may be arbitrarily deformed within ⌦ \R without a↵ecting theresult, by Cauchy’s Theorem. Likewise, P� is analytic on ⌦P \ R, and so contoursinvolving this contribution may be arbitrarily deformed within this region.

Recalling, further, that

|G� �H� � P�| = O(��1)

by Proposition 4.33, we find that

12⇡i

P.V.Z ⌘+i1

⌘�i1e�t(G� �H� � P�)(x, y)d�

may be deformed to

12⇡i

P.V.I

@(⌦\R)

e�t(G� �H� � P�)(x, y)d�.

And, similarly as in the treatment of term III of the large |x � y|/t case, we findusing bounds (4.182) that

12⇡i

P.V.Z ⌘+i1

⌘�i1e�tP�(x, y)d�

may be deformed to the absolutely convergent integral

12⇡i

I

@(⌦

P

\R)

e�tP�(x, y)d�.

Noting, finally, that

+1

2⇡i

I

@Re�tH�(x, y)d� = 0,


by Cauchy’s Theorem, we obtain, finally,(4.249)

12⇡i

P.V.Z ⌘+i1

⌘�i1e�tG�(x, y)d� =

12⇡i

Z ⌘+i1

⌘�i1e�tH�(x, y)d�+

12⇡i

P.V.Z ⌘+i1

⌘�i1e�t(G� �H� � P�)(x, y)d�

+1

2⇡i

I

@(⌦

P

\R)

e�tP�(x, y)d�

=1

2⇡iP.V.

Z ⌘+i1

⌘�i1e�tH�(x, y)d�+

12⇡i

P.V.I

@(⇤\R)

e�t(G� �H� � P�)(x, y)d�

+1

2⇡i

I

@Re�tH�(x, y)d�+

12⇡i

I

@(⌦

P

\R)

e�tP�(x, y)d�,

from which the result follows by combining the second third and fourth contourintegrals along their common edges �. ⇤Observation 4.41. In dimension d = 1, assuming (A1)–(A3), (H0)–(H3), and(D), we may replace (4.245) by

G(x, t; y) = Ia + Ib + Ic + IIa + IIb + III,

where Ia, Ib, Ic, and IIb are as in (4.245), and

IIa :=12⇡

Z �⌘1�ir/2

�⌘1�iR

+Z �⌘1+iR

�⌘1+ir/2

!e�tG�(x, y)dxi,

(4.250) III :=1

2⇡i

I

˜

�

e�tG�(x, y)d�,

and

(4.251) � := [�⌘1

� ir/2, ⌘ � ir/2] [ [⌘ � ir/2, ⌘ + ir/2] [ [⌘ + ir/2,�⌘1

+ ir/2],

for any ⌘, r > 0, and ⌘1

su�ciently small with respect to r.

Proof. By assumption (D), L has no spectrum on the region between contour� and the union of contour � and the contour of term IIa, hence G� is analytic onthat region, and

IIa = IIa + III

by Cauchy’s Theorem, giving the result. ⇤

122 KEVIN ZUMBRUN

Using the final decomposition (4.250), we shall estimate in turn the high fre-quency contributions Ia, Ib, and Ic, the intermediate frequency contributions IIa

and IIb, and the low frequency contributions III.

High frequency contribution. We first carry out the straightforward esti-matation of the high-frequency terms Ia, Ib, and Ic. The principal term Ia hasalready been computed in (4.234) to be H(x, t; y). Likewise, calculations similar tothose of (4.236)–(4.238) show that the error term

Ib =1

2⇡iP.V.

Z �⌘1�R

�⌘1�i1

Z �⌘1+i1

�⌘1+iR

!e�t⇥�(x, y)d�

=1

2⇡iP.V.

Z �⌘1�R

�⌘1�i1

Z �⌘1+i1

�⌘1+iR

!e�t

��1B(x, y;�)

+ ��1(x� y)C(x, y;�) + ��2D(x, y;�)�d�

is time-exponentially small.For example, for x > y the term e�t��1B(x, y;�) contributes

(4.252)

KX

j=K+1

12⇡i

P.V.

Z �R

�1

Z+1

R

!(�⌘

1

+ i⇠)�1ei⇠�t�R

x

y

(1/a⇤j

(z)) dz�

d⇠

⇥ e�⌘1�t�R

x

y

(1/a⇤j

(z)) dz�b+

j (x, y),

where

(4.253)1

2⇡iP.V.

Z �R

�1

Z+1

R

!(⌘ + i⇠)�1ei⇠

�t�R

x

y

(1/a⇤j

(z)) dz�

d⇠ < 1,

and

(4.254) e⌘1R

x

y

(1/a⇤j

(z)) dzb(x, y) C1

e⌘1R

x

y

(1/a⇤j

(z)) dz�✓|x�y| C2

,

for ⌘1

su�ciently small. This may be absorbed in the first term of R, (4.227).Likewise, the contributions of terms e�tC(x, y;�) and e�tD(x, y;�) split into theproduct of a convergent, uniformly bounded integral in ⇠, a bounded factor analo-gous to (4.254), and the factor e�⌘1t, giving the result.

The term Ic may be estimated exactly as was term III in the large |x � y|/tcase, to obtain contribution O(t�1/2e�⌘1t) absorbable again in the residual termO(e�⌘te�|x�y|2/Mt) for t � ✏, any ✏ > 0, and by any summand O(t�1/2(t +1)�1/2e�(x�y�a±

k

t)2/Mt)e�⌘x±e�⌘y± for t small.Derivative bounds. Derivatives (@/@y)Ib may be treated in identical fashion using

(4.184) to show that they are absorbable in the estimates given for Ry. We point


out that the integral arising from term B0

y of (4.184) corresponds to the first, delta-function term in (4.228), while the integral arising from term (x � y)C0

y vanishes,as the product of (x�y) and a delta function �z(y) with z 6= x. Derivatives of termIc may be treated exactly as were derivatives of term III in the large |x� y|/t case.

Intermediate frequency contribution. Error term IIb is time-exponentiallysmall for ⌘

1

su�ciently small, by the same calculation as in (4.252)–(4.254), hencenegligible. Likewise, term IIa by the basic estimate (2.85) is seen to be time-exponentially small of order e�⌘1t for any ⌘

1

> 0 su�ciently small that the associ-ated contour lies in the resolvent set of L.

Low frequency contribution. It remains to estimate the low-frequency termIII, which is of essentially the same form as the low-frequency contribution ana-lyzed in [ZH, Z.3, Z.6] in the strictly parabolic case, in that the contour is the sameand the resolvent kernel G� satisfies identical bounds in this regime. Thus, we mayconclude from these previous analyses that III gives contribution E + S + R, asclaimed, exactly as in the strictly parabolic case. For completeness, we indicate themain features of the argument here.

Case t 1. First observe that estimates in the short-time regime t 1 aretrivial, since then |e�tG�(x, y)| is uniformly bounded on the compact set �, andwe have |G(x, t; y)| C e�✓t for ✓ > 0 su�ciently small. But, likewise, E andS are uniformly bounded in this regime, hence time-exponentially decaying. Asobserved previously, all such terms are negligible, being absorbable in the errorterm R. Thus, we may add E + S and subtract G to obtain the result

Case t � 1. Next, consider the critical (long-time) regime t � 1. For definiteness,take y x 0; the other two cases are similar. Decomposing(4.255)

G(x, t; y) =1

2⇡i

I

˜

�

e�tE�(x, y)d�+1

2⇡i

I

˜

�

e�tS�(x, y)d�+1

2⇡i

I

˜

�

e�tR�(x, y)d�,

with E�, S�, and R� as defined in Proposition 4.22, we consider in turn each of thethree terms on the righthand side.

E� term. Let us first consider the dominant term

(4.256)1

2⇡i

I

˜

�

e�tE�(x, y)d�,

which, by (4.128), is given by

(4.257)X

a�k

>0

[cj,0k,�](@U�/@�)(x) l�t

k ↵k(x, t; y),

where

(4.258) ↵k(x, t; y) :=1

2⇡i

I

˜

�

��1e�te(�/a�k

��2��k

/a�k

3)y d�.

124 KEVIN ZUMBRUN

Using Cauchy’s Theorem, we may move the contour � to obtain(4.259)

↵k(x, t; y) =12⇡

P.V.Z

+r/2

�r/2

(i⇠)�1ei⇠te(i⇠/a�k

+⇠2��k

/a�k

3)y d⇠

+1

2⇡i

Z �ir/2

�⌘1�ir/2

+Z �⌘1+ir/2

ir/2

!��1e�te(�/a�

k

��2��k

/a�k

3)y d�

+12Residue �=0

e�te(�/a�k

��2��k

/a�k

3)y,

or, rearranging and evaluating the final, residue term:(4.260)

↵k(x, t; y) =✓

12⇡

P.V.Z

+1

�1(i⇠)�1ei⇠(t+y/a�

k

)e⇠2(��

k

/a�k

3)y d⇠ +

12

◆

� 12⇡

Z �r/2

�1+Z

+1

r/2

!(i⇠)�1ei⇠(t+y/a�

k

)e⇠2(��

k

/a�k

3)y d⇠

+1

2⇡i

Z �ir/2

�⌘1�ir/2

+Z �⌘1+ir/2

ir/2

!��1e�te(�/a�

k

��2��k

/a�k

3)y d�.

The first term in (4.260) may be explicitly evaluated to give

(4.261) errfn

0

@ y + a�k tq4��k |y/a�k |

1

A ,

where

(4.262) errfn (z) :=12⇡

Z z

�1e�y2

dy,

whereas the second and third terms are clearly time-exponentially small for t C|y|and ⌘

1

su�ciently small relative to r (see detailed discussion of similar calculationsbelow, under R� term). In the trivial case t � C|y|, C > 0 su�ciently large, we cansimply move the contour to [�⌘

1

� ir/2,�⌘1

+ ir/2] to obtain (complete) residue 1plus a time-exponentially small error corresponding to the shifted contour integral,which result again agrees with (4.261) up to a time-exponentially small error.

Expression (4.261) may be rewritten as

(4.263) errfn

0

@ y + a�k tq4��k |y/a�k |

1

A


plus error

(4.264)

errfn

0

@ y + a�k tq4��k |y/a�k |

1

A� errfn

0


1

A

⇠ errfn 0

0


1

A✓�4��k

2(y + a�k t)2(4��k t)�3/2

◆

= O(t�1e(y+a�k

t)2/Mt),

for M > 0 su�ciently large, and similarly for x- and y-derivatives. Multiplying by

[cj,0k,�](@U�/@�)(x) l�t

k = O(e�✓|x|),

we find that term (4.263) gives contribution

(4.265) [cj,0k,�](@U�/@�)(x) l�t

k errfn

0

@ y + a�k tq4��k |y/a�k |

1

A ,

whereas term (4.264) gives a contribution absorbable in R (resp. Rx, Ry); seeRemark 7.5 below for detailed discussion of a similar calculation.

Finally, observing that

(4.266) [cj,0k,�]l�t

k errfn

0


1

A

is time-exponentially small for t � 1, since a�k > 0, y < 0, and (@/@�j)(u�, v�) Ce�✓|x|, ✓ > 0, we may subtract and add this term to (4.265) to obtain a total ofE(x, t; y) plus terms absorbable in R (resp. Rx, Ry).

S� term. Next, consider the second-order term

(4.267)1

2⇡i

I

˜

�

e�tS�(x, y)d�,

which, by (4.130), is given by

(4.268)X

a�k

>0

r�k l�kt↵k(x, t; y) +

X

a�k

>0, a�j

<0


t↵jk(x, t; y)

where

(4.269) ↵k(x, t; y) :=1

2⇡i

I

˜

�

e�te(��/a�k

+�2��k

/a�k

3)(x�y) d�.

126 KEVIN ZUMBRUN

and

(4.270) ↵jk(x, t; y) :=1

2⇡i

I

˜

�

e�te(��/a�j

+�2��j

/a�j

3)x+(�/a�

k

��2��k

/a�k

3)y d�.

Similarly as in the treatment of the E� term, just above, by deforming the contour� to

(4.271) �0 := [�⌘1

� ir/2,�ir/2] [ [�ir/2,+ir/2] [ [+ir/2,�⌘1

+ ir/2],

these may be transformed modulo time-exponentially decaying terms to the ele-mentary Fourier integrals

(4.272)12⇡

P.V.Z

+1

�1ei⇠(t�(x�y)/a�

k

)e⇠2(��

k

/a�k

3)(x�y) d⇠

= (4⇡��k t)�1/2e�(x�y�a�k

t)2/4��k

t

and

(4.273)

12⇡

P.V.Z

+1

�1ei⇠((t�|y/a�

k

|)�x/a�j

)e�⇠2⇣(��

k

/a�k

3)|y|+(��

j

/a�j

3)|x|

⌘

d⇠

= (4⇡��jkt)�1/2e�(x�z�jk

)

2/4

¯��jk

t,

respectively, where �jk�+� and z±jk are as defined in (4.29) and (4.28). Thesecorrespond to the first and third terms in expansion (4.25), the latter of which hasan additional factor e�x/(ex + e�x). Noting that the second and fourth terms of(4.25) are time-exponentially small for t � 1, y x 0, and that

|(4⇡��jkt)�1/2e�(x�z�jk

)

2/4

¯��jk

t(1� e�x/(ex + e�x))|

|(4⇡��jkt)�1/2e�(x�z�jk

)

2/4

¯��jk

te�✓|x|

for some ✓ > 0, so is absorbable in error term R, we find that the total contributionof this term, modulo terms absorbable in R, is S.

R� term. Finally, we briefly discuss the estimation of error term

(4.274)1

2⇡i

I

�

e�tR�(x, y)d�,

which decomposes into the sum of integrals involving the various terms of RE�

and RS� given in (4.132) and (4.134). Since each of these are separately analytic,

they may be split up and estimated sharply via the Riemann saddlepoint method(method of steepest descent), as described at great length in [ZH, HoZ.4]. That is,


for each summand ↵�(x, y) ⇠ e�(�)x+�(�)y in R� we deform the contour � to a new,contour in ⇤ that is a mini-max contour for the modulus

m↵(x, y,�) := |e�t↵�(x, y)| = eRe �t+Re �x+Re �y,

passing through an appropriate saddlepoint/critical point of m↵(x, y, ·): necessarilylying on the real axis, by the underlying complex symmetry resulting from realityof operator L.

Since terms of each type appearing in R have been sharply estimated in [ZH],we shall omit the details, only describing two sample calculations to illustrate themethod:

Example 1: e�✓|x�y|. Contour integrals of form

(4.275)1

2⇡i

I

˜

�

e�te�✓|x�y|d�,

arising through the pairing of fast forward and fast dual modes, may be deformedto

(4.276)1

2⇡i

Z �⌘1+ir/2

�⌘1�ir/2

e�te�✓|x�y|d�

and estimated as O(e�⌘1t�✓|x�y|), a negligible, time-exponentially decaying contri-bution. ⇤

Example 2: �re(��/a�k

+�2��k

/a�k

3)(x�y). Contour integrals of form

(4.277)1

2⇡i

I

˜

�

e�t�re(��/a�k

+�2��k

/a�k

3)(x�y)d�,

a�k > 0, arising through the pairing of slow forward and slow dual modes, may bedeformed to contour

(4.278) �0 := [�⌘1

� ir/2, ⌘⇤� ir/2][ [⌘⇤� ir/2, ⌘⇤+ ir/2][ [⌘⇤+ ir/2,�⌘1

+ ir/2],

where saddlepoint ⌘⇤ is defined as

(4.279) ⌘⇤(x, y, t) :=

(↵p if | ↵p | "

±" if ↵p ? ",

with

(4.280) ↵ :=x� y � a�k t

2t, p :=

��k (x� y)(a�k )2t

> 0,

128 KEVIN ZUMBRUN

and ⌘1

, " > 0 are chosen su�ciently small with respect to r, to yield, modulotime-exponentially decaying terms, the estimate(4.281)

e�(x�y�a�k

t)2/4��t

Z+1

�1O(|⌘⇤|r + |⇠|r)e�✓⇠2t d⇠ = O(t�(r+1)/2e�(x�y�a�

k

t)2/Mt)

if | ↵p | ", and

(4.282) e�"t/M

Z+1

�1O(|⌘⇤|r + |⇠|r)e�✓⇠2t d⇠ = O(t�(r+1)/2e�⌘t).

if | ↵p | � ", ✓ > 0. In either case, the main contribution lies along the central portion[⌘⇤ � ir/2, ⌘⇤ + ir/2] of contour �0.

To see this, note that |x�y| and t are comparable when |↵/p| is bounded, whencethe evident spatial decay e�✓⇠

2|x�y| of the integrand along contour � = ⌘⇤+ i⇠ maybe converted to the e�⇠

2t decay displayed in (4.281)–(4.282); likewise, temporalgrowth in e�t on the horizontal portions of the contour, of order e(|⌘1|+|"|)t, isdominated by the factor e�✓⇠

2= e�✓r

2/4, provided |⌘1

| + |"| is su�ciently smallwith respect to r2. For |↵/p| large, on the other hand, ⌘⇤ is uniformly negative,and also |x � y| is negligible with respect to t, whence the estimate (4.282) holdstrivially.

We point out that ⌘⇤ is easily determined, as the minimal point on the real axisof the quadratic function

(4.283) fx,y,t(�) := �t(��/a�k + �2��k /a�k3)(x� y),

the argument of the integrand of (4.277). ⇤Other terms may be treated similarly: All “constant-coe�cient” terms �k�⇤k or

k ⇤k are of either the form treated in Example 1 (fast modes) or in example 2(slow modes). Scattering pairs involving slow forward and slow dual modes fromdi↵erent families (i.e., terms with coe�cients Mjk, d±jk) may be treated similarlyas in Example 2. Scattering pairs involving two fast modes are of the form alreadytreated in Example 1, since both modes of a scattering pair are decaying. Scatteringpairs involving one fast and one slow mode may be factored as the product of aterm of the form treated in example 2 and a term that is uniformly exponentiallydecaying in either x or y; factoring out the exponentially decay, we may treat suchterms as in Example 2.

Scattering coe�cients. The relation (4.30) may now be deduced, a posteriori,from conservation of mass in the linearized flow (2.1). For, by inspection, all termssave E and S in the decomposition (4.22) of G decay in L1, whence these termsmust, time-asymptotically, carry exactly the mass in U of the initial perturbation�y(x)In, i.e.,

(4.284) limt!1

Z+1

�1

�E(x, t; y) + S(x, t; y)

�dx = In.


Taking y 0 for definiteness, and right-multiplying both sides of (4.284) by r�k , wethus obtain the result from definitions (4.24)–(4.25).

Moreover, rewriting (4.30) as

(4.285) (r�1

, . . . , r�n�i�, r+

i+, . . . , r+

n ,m1

, . . . ,m`)

0

BBBBBBBBBBBBBBBBB@

[c1,�k,�]...

[cn�i�,�k,� ][ci+,+

k,� ]...

[cn,+k,� ]

[c1,0k,�]...

[c`,0k,�]

1

CCCCCCCCCCCCCCCCCA

= r�k ,

where

(4.286) mj :=Z

+1

�1(@/@�j)U�(x) dx,

we find that, under assumption (D), it uniquely determines the scattering coe�-cients [cj,±

k,�]. For, the determinant

(4.287) det (r�1

, . . . , r�n�i�, r+

i+, . . . , r+

n ,m1

, . . . ,m`)

of the matrix on the righthand side of (4.285) is exactly the inviscid stability coef-ficient � defined in [ZS, Z.3], hence is nonvanishing by the equivalence of (D2) and(D2) (recall discussion of Section 1.2, just below Theorem 4.8). Note that (4.287)reduces to the inviscid stability determinant (1.36) in the Lax case, ` = 1, withU�(x) parametrized as U�(x) := U(x� �

1

x), for which m1

reduces to [U ].Relation (4.31) follows from the observation that

P�(x, y) := Residue �=0

G�(x, y) =X

j=1

(@/@�j)U�(x)⇡j ,

with ⇡j defined by the first, or second expressions appearing in (4.31), according asy is less than or equal to/greater than or equal to zero. For, the extended spectraltheory of Section 4.3.3 then implies that ⇡j are the left e↵ective eigenfunctions as-sociated with right eigenfunctions (@/@�j)U�. Alternatively, (4.31) may be deducedby linear-algebraic manipulation directly from (4.285) and its counterpart for y � 0(the same identity with k,� everywhere replaced by k,+). ⇤

130 KEVIN ZUMBRUN

Remark 4.42 (The undercompressive case). In the undercompressive case,the result of Lemma 4.28 is false, and consequently the estimates of Lemma 4.31do not hold. This fact has the implication that shock dynamics are not governedsolely by conservation of mass, as in the Lax or overcompressive case, but by morecomplicated dynamics of front interaction; for related discussion, see [LZ.2, Z.6].At the level of Proposition 4.22, it means that the simple representations of E�and S� in terms of slow dual modes alone (corresponding to characteristics that areincoming to the shock) are no longer valid in the undercompressive case, and thereappear new terms involving rapidly decaying dual modes ⇠ e�✓|y| related to innerlayer dynamics. Though precise estimates can nonetheless be carried out, we havenot found a similarly compact representation of the resulting bounds as that of theLax/overcompressive case, and so we shall not state them here. We mention onlythat this rapid variation in the y-coordinate precludes the Lp stability argumentsused here and in [Z.6], requiring instead detailed pointwise bounds as in [HZ.2].See [LZ.1–2, ZH, Z.6] for further discussion of this interesting case.

4.3.5. Inner layer dynamics.

Similarly as in [ZH], we now investigate dynamics of the inner shock layer, in thecase that (D) does not necessarily hold, in the process establishing the necessity of(D) for linearized orbital stability.

Proposition 4.43. In dimension d = 1, given (A1)–(A3) and (H0)–(H3), thereexists ⌘ > 0 such that, for x, y restricted to any bounded set, and t su�cientlylarge,

(4.288) G(x, y; t) =X

�2�0p

(L)\{Re(�)�0}e�t

X

k�0

tk(L� �I)kP�(x, y) +O(e�⌘t),

where P�(x, y) is the e↵ective projection kernel described in Definition 4.34, and�0�(L) the e↵ective point spectrum.

Proof. Similarly as in the proof of Proposition 4.39, decompose G into termsIa, Ib, Ic, IIa, and IIb of (4.245). Then, the same argument (Case II. |x � y|/tbounded) yields that Ia = H(x, t; y), while Ib and IIb are time-exponentially small.Observing that H = 0 for x, y bounded, and t su�ciently large, we have reducedthe problem to the study of

(4.289) IIa :=1

2⇡i

I

�

e�tG�(x, y)d�,

where, recall,

(4.290) � := [�⌘1

� iR, ⌘ � iR] [ [⌘ � iR, ⌘ + iR] [ [⌘ + iR,�⌘1

+ iR].


Note that the high-frequency bounds of Proposition 4.33 imply that L has no pointspectrum in ⌦ outside of the rectangle R enclosed by � [ [�⌘

1

� iR,�⌘ + iR].Choosing ⌘

1

su�ciently small, therefore, we may ensure that no e↵ective pointspectrum lies within the strip �⌘

1

Re � < 0, by compactness of R together withthe fact that e↵ective eigenvalues are isolated from one another, as zeroes of theanalytic Evans function.

Recall that G� is meromorphic on ⌦. Thus, we may express (4.289), usingCauchy’s Theorem, as

(4.291)1

2⇡i

Z �⌘1+iR

�⌘1�iR

e�tG�(x, y)d�+ Residue �2R e�tG�(x, t; y).

By Definition 4.34 and Proposition 4.36,(4.292)

Res�2{Re ��0}e�tG�(x, y)

=X

�02�0p

(L)\{Re ��0}e�0tRes�0e

(��0)tG�(x, y)

=X

�02�0p

(L)\{Re ��0}e�0t

X

k�0

(tk/k!)Res�0(�� 0

)kG�(x, y)

=X

�02�0p

(L)\{Re ��0}e�0t

X

k�0

(tk/k!)Q�0,k(x, y)

=X

�02�0p

(L)\{Re ��0}e�0t

X

k�0

(tk/k!)(L� �0

I)kP�0(x, y)).

On the other hand, for x, y bounded and t su�ciently large, t dominates |x| and|y| and we obtain from Propositions 2.23 and 4.22 that

|G�(x, t; y)| C

for all � 2 ⌦, for ⌘1

su�ciently small, hence

(4.293)1

2⇡i

Z �⌘1+iR

�⌘1�iR

e�tG�(x, y)d� 2CRe�⌘1t.

Combining (4.291), (4.292), and (4.293), we obtain the result. ⇤Corollary 4.44. Let dimension d = 1, and assume (A1)–(A3) and (H0)–(H3).Then, (D) is necessary for linearized orbital stability with respect to compactly sup-ported initial data, as measured in any Lp norm, 1 p 1.

Proof: From (4.288), we find that U(·) is linearly orbitally stable only if P� = 0for all � 2 {Re � � 0} \ {0} and Range P

0

= Span {(@/@�j)U�}. By Propositions4.36 and 4.38, this is equivalent to (D). ⇤

This completes our treatment of the one-dimensional case

132 KEVIN ZUMBRUN

5. MULTI-DIMENSIONAL STABILITY.

We now focus on the multi-dimensional case, establishing results 4 and 5 of theintroduction.

5.1. Necessary conditions.

Define the reduced Evans function as

(5.1) �(e⇠,�) := lim⇢!0

⇢�`D(⇢e⇠, ⇢�).

By the results of the previous section, the limit � exists and is analytic, with

(5.2) � = ��,

for shocks of pure type (indeed, such a limit exists for all types). Evidently, �(·, ·)is homogeneous, degree `.29

Recall that the restriction

(5.3) D(�) := D(0,�)

is the one-dimensional Evans function considered in [GZ, ZH].

Lemma 5.1 [ZS]. Let �(0, 1) 6= 0. Then, near any root (e⇠0

,�0

) of �(·, ·), thereexists a continuous branch �(e⇠), homogeneous degree one, of solutions of

(5.4) �(e⇠,�(e⇠)) ⌘ 0

defined in a neighborhood V of e⇠0

, with �(e⇠0

) = �0

. Likewise, there exists a contin-uous branch �⇤(e⇠) of roots of

(5.5) D(e⇠,�⇤(e⇠)) ⌘ 0,

defined on a conical neighborhood V⇢0 := {e⇠ = ⇢⌘ : ⌘ 2 V, 0 < ⇢ < ⇢0

}, ⇢0

> 0su�ciently small, “tangent” to �(·) in the sense that

(5.6) |�⇤(e⇠)� �(e⇠)| = o(|e⇠|)

as |e⇠|! 0, for e⇠ 2 V⇢0 .

Proof. The statement (5.4) follows by Rouche’s Theorem, provided �(e⇠0

, ·) 6⌘0, since �(e⇠

0

, ·) are a continuous family of analytic functions. For, otherwise,restricting � to the positive real axis, we have by homogeneity that

0 = lim�!+1

�(e⇠0

,�) = lim�!+1

�(e⇠0

/�, 1) = �(0, 1),

29Here, and elsewhere, homogeneity is with respect to the positive reals, as in most cases shouldbe clear from the context. Recall that � (and thus �) is only defined for real e⇠, and Re � � 0.


in contradiction with the hypothesis. Clearly we can further choose �(·) homoge-neous degree one, by homogeneity of �. Similar considerations yield existence of abranch of roots �(e⇠, ⇢) of the family of analytic functions

(5.7) ge⇠,⇢(�) := ⇢�`D(⇢e⇠, ⇢�),

for ⇢ su�ciently small, since ge⇠,0 = �(e⇠, ·). Setting �⇤(e⇠) := |e⇠|�(e⇠/|e⇠|, |e⇠|), we have

D(e⇠,�⇤(e⇠)) = |e⇠|`ge⇠/|e⇠|,|e⇠|(�⇤) ⌘ 0,

as claimed. “Tangency,” in the sense of (5.6), follows by continuity of � at ⇢ = 0,the definition of �⇤, and the fact that �(e⇠) = |e⇠|�(e⇠/|e⇠|), by homogeneity of �. ⇤Corollary 5.2 [ZS]. Given one-dimensional inviscid stability, �(0, 1) 6= 0, weakrefined dynamical stability (Definition 1.21) is necessary for viscous multi-dimensionalweak spectral stability, and thus for multi-dimensional linearized viscous stabilitywith respect to test function (C1

0

) initial data.

Proof. By the discussion just preceding Section 2.1, if is su�cient to prove thefirst assertion, i.e., that failure of weak refined dynamical stability implies existenceof a zero D(⇠,�) = 0 for ⇠ 2 Rd�1, Re � > 0. Failure of weak inviscid stability, or�(e⇠,�) = 0 for e⇠ 2 Rd�1, Re � > 0, implies immediately the existence of such aroot, by tangency of the zero-sets of D and � at the origin, Lemma 5.1. Thus, itremains to consider the case that weak inviscid stability holds, but there exists aroot D(⇠, i⌧) for ⇠, ⌧ real, at which � is analytic, �� 6= 0, and �(⇠, i⌧) < 0, where� is defined as in (4.17).

Recalling that D(⇢e⇠, ⇢�) vanishes to order ` in ⇢ at ⇢ = 0, we find by L’Hopital’srule that

(@/@⇢)`+1D(⇢e⇠, ⇢�)|⇢=0,�=i⌧ = (1/`!)(@/@⇢)ge⇠,i⌧ (0)

and(@/@� )D(⇢e⇠, ⇢�)|⇢=0,�=i⌧ = (1/`!)(@/@�)ge⇠,i⌧ (0),

where ge⇠,⇢(�) := ⇢�`D(⇢e⇠, ⇢�) as in (5.7), whence

(5.8) � =(@/@⇢)ge⇠,�(0)(@/@� )ge⇠,�(0)

,

with (@/@� )ge⇠,�(0) 6= 0.By the (analytic) Implicit Function Theorem, therefore, �(e⇠, ⇢) is analytic in e⇠, ⇢

at ⇢ = 0, with

(5.9) (@/@⇢) �(e⇠, 0) = ��,

134 KEVIN ZUMBRUN

where �(e⇠, ⇢) as in the proof of Lemma 5.1 is defined implicitly by ge⇠,�(⇢) = 0,

�(e⇠, 0) := i⌧ . We thus have, to first order,

(5.10) �(e⇠, ⇢) = i⌧ � �⇢+O(⇢2).

Recalling the definition �⇤(e⇠) := |e⇠|�(e⇠/|e⇠|, |e⇠|), we have then, to second order,the series expansion

(5.11) �⇤(⇢e⇠) = i⇢⌧ � �⇢2 +O(⇢3),

where �⇤(e⇠) is the root of D(e⇠,�) = 0 defined in Lemma 5.1. It follows that thereexist unstable roots of D for small ⇢ > 0 unless Re � � 0. ⇤

Remarks 5.3. When �� 6= 0, � = i⌧ is a simple root of �(e⇠, ·) and, likewise,�⇤(e⇠) defined in (5.11) represents an isolated branch of the zeroes of D(e⇠, ·). Thequantity �� gives the curvature of the zero level set of D tangent to the level set{⇢e⇠, ⇢i⌧} for �. The value � also represents the e↵ective di↵usion coe�cient forthe transverse traveling waves associated with this frequency; see [Z.3], Section 3.3.

5.2. Su�cient Conditions for Stability.

Result 5 of the introduction is subsumed in the following two theorems, to beestablished throughout the remainder of the section.

Theorem 5.4. (Linearized stability) Under assumptions (A1)–(A3), (H0)–(H5),structural and strong refined dynamical stability together with strong spectral sta-bility are su�cient for linearized viscous stability in L2 with respect to initial per-turbations U

0

2 L1 \ L2, or in Lp, p � 2, with respect to initial perturbationsU

0

2 L1 \H [

d�12 ]+2, for all dimensions d � 2, with rate of decay

(5.12)kU(t)kL2 C(1 + t)�

(d�1)4 +�✏|U

0

|L1\L2 ,

kU(t)kLp C(1 + t)�(d�1)

2 (1�1/p)+�✏|U0

|L1\H[ d�1

2 ]+2

for all t � 0, where

(5.13) � =⇢

0 for uniformly inviscid stable (U.I.S.) shocks,1 for weakly inviscid stable (W.I.S.) shocks,

✏ > 0 is arbitrary, and C = C(✏) is independent of p.

Theorem 5.5. (Nonlinear stability) Under assumptions (A1)–(A3), (H0)–(H5),structural and strong refined dynamical stability together with strong spectral stabil-ity are su�cient for nonlinear viscous stability in Lp \Hs�1, p � 2 with respect toinitial perturbations U

0

:= U0

� U that are su�ciently small in L1 \Hs�1, where


s � s(d) is as defined in (H0), Section 1.1, for dimensions d � 2 in the case ofa uniformly inviscid stable Lax or overcompressive shock, d � 3 in the case of a aweakly inviscid stable Lax or overcompressive shock, and d � 4 in the case of anundercompressive shock, with rate of decay

(5.14)kU(·, t)� UkLp C(1 + t)�

d�12 (1�1/p)+�✏kU

0

kL1\Hs�1 ,

kU(·, t)� UkHs�1 C(1 + t)�d�14 +�✏kU

0

kL1\Hs�1

for all t � 0, where ✏ > 0 is arbitrary, C = C(✏), and � = 1 if dimension d = 2and p > 2, or in the weakly inviscid stable (W.I.S.) case, and otherwise � = 0. (Inparticular, � = 0 for uniformly inviscid stable Lax or overcompressive shocks, indimension d � 3 or in dimension d = 2 with p = 2).

Remarks 5.6. 1. Stability here is in the usual sense of asymptotic stability,and not only bounded or orbital stability as in the one-dimensional case, withrate of decay in dimension d equal to that of a (d � 1)-dimensional heat kernel,corresponding to transverse di↵usion along the front. This reflects the fact thatthe class of L1 perturbations is much more restrictive in multiple than in singledimensions; in particular, conservation of mass precludes convergence of an L1

perturbation of a planar shock front to any translate of the shock front other thanthe initial front itself [Go.3]. Recall that our necessary stability results concernedthe still more restrictive class of test function initial data, so that the problems areconsistent.

2. The di↵usive decay rate given here is sharp, neglecting error term �, for theweakly inviscid stable case; indeed, the basic iteration scheme [Z.3] was motivatedby the scalar analysis of [Go.3] (recall: scalar shocks are always weakly inviscidstable). However, uniformly inviscid stable shocks, for which transverse front dis-turbances are strongly damped, are expected to decay at the faster rate governingfar-field behavior, of a d-dimensional heat kernel. That is, our analysis here as in[Z.3] is focused on the W.I.S. regime, and the transition from viscous stability toinstability; see again Remark 1.22.2.

5.2.1. Linearized estimates.

Theorems 5.4 and 5.5 are obtained using the following Lq ! Lp bounds on thelinearized solution operator, analogous to the one-dimensional bounds describedin Lemma 4.14 (proof deferred to Section 5.3). Note that we no longer requiredetailed Green distribution bounds as described in Proposition 4.10, nor, becauseof the more complicated geometry of characteristic surfaces in multi-dimensions,does there exist such a simple description of the propagation of solutions; see, e.g.,[HoZ.1–2] for further discussion in the constant–coe�cient case.

Proposition 5.7. Let (A1)–(A3), (H0)–(H5) hold, together with structural, strongrefined dynamical, and strong spectral stability. Then, solution operator S(t) := eLt

136 KEVIN ZUMBRUN

of the linearized equations (2.1) may be decomposed into S = S1

+ S2

satisfying

(5.15) kS1

(t)@�xFkLp

(x)

C(1 + t)�(d�1)

2 (1�1/p)�(1�↵)

|�|2 +�✏kFkL1

(x)

for 0 |�| 1, 2 p 1, and t � 0, where ↵ = 0 for Lax and overcompressiveshocks and ↵ = 1 for undercompressive shocks, � = 0 for uniformly inviscid stable(U.I.S.) shocks and � = 1 for weakly inviscid stable (W.I.S.) shocks, ✏ > 0 isarbitrary, and C = C(d, ✏) is independent of p, and, for 0 |�

1

| 1, 0 |�|,

(5.16) k@�1x1@�xS2

(t)FkL2 Ce�⌘tkFkH|�1|+|�| .

Remarks 5.8. 1. Operators S1

and S2

represent low- and high-frequency partsof S. Though we do not state them, S

2

satisfies refined bounds analogous to (4.32)–(4.33) in the one-dimensional case.

2. The time-asymptotic decay rate of solutions is determined by the bounds onS

1

, which may be recognized as those for a (d � 1)-dimensional heat equation. Inconstrast to the situation for the heat equation, x-di↵erentiation of solution U = SFdoes not improve the rate of decay for the variable-coe�cient shock problem, due tocommutator terms arising in the di↵erentiated equations (that is, terms for whichderivatives fall on coe�cients), nor in general does di↵erentiation with respect tosource F . A single derivative of the source does improve the rate of decay in theLax or overcompressive case, but further derivatives would not (and are not neededin our later argument). This is a slight di↵erence from the constant-coe�cientcase considered in, e.g., [Kaw, HoZ.1–2], and requires some modification in thearguments. However, in spirit our nonlinear stability analysis in dimension d followsthat of [Kaw, HoZ.1–2] in one lower dimension, d� 1.

Proof of Theorem 5.4. Applying (5.15)–(5.16), we have

|U(t)|L2 |S1

(t)U0

|L2 + |S2

(t)U0

|L2

C(1 + t)�(d�1)

4 +�✏|U0

|1

+ Ce�⌘t|U0

|L2

C(1 + t)�(d�1)

4 +�✏|U0

|L1\L2 .

Likewise, we have, using the Sobolev embedding

|f |Lp |f |L2\L1 |f |H[(d�1)/2]+1(x;H1

(x1))

for 2 p 1, that

|U(t)|Lp |S1

(t)U0

|Lp + |S2

(t)U0

|H[(d�1)/2]+1(x;H1

(x1))

C(1 + t)�(d�1)

2 (1�1/p)+�✏|U0

|1

+ Ce�⌘t|U0

|H[d/2]+2

C(1 + t)�(d�1)

2 (1�1/p)+�✏|U0

|L1\H[d/2]+2 .

⇤


5.2.2. Auxiliary energy estimate.

In our analysis of nonlinear stability, we shall make use also of the followingauxiliary energy estimate, analogous to that of the one-dimensional case.

Proposition 5.9. Under the hypotheses of Theorem 5.5, suppose that, for 0 t T , the Hs�1 norm of the perturbation U := U � U = (uI , uII)t remains bounded bya su�ciently small constant, where U = U + U denotes a solution of (1.1). Then,for all 0 t T ,

(5.17) |U(t)|2Hs�1 C|U(0)|2Hs�1e�✓t + C

Z t

0

e�✓2(t�⌧)|U |2L2(⌧) d⌧.

Proof of Proposition 5.9. The proof follows almost exactly the argument ofthe one-dimensional case; see Remark 4.17. Indeed, it is somewhat simpler, sincethere are no front location terms �Ux, We therefore omit most details, pointingout only two small points in which the argument must be modified for the multi-dimensional case.

The first is the standard issue that “good” terms

�X

jk

h@`xWxj

, Bjk@`xWxk

i↵ = �X

jk

h@`xwIIx

j

, bjk@`xwIIx

k

i↵

arising in the `th order Friedrichs estimate must now be estimated indirectly, usingGarding inequality

(5.18)X

jk

h@`xwIIx

j

, bjk@`xwIIx

k

i↵ � ✓|@`+1

x wII |2↵ � C|@`xwII |L2

(a consequence of uniform ellipticity, (1.6)). Recall that hf, gi↵ := hf,↵gi, where ↵is a scalar weight function. Recall that O(|@`x|2L2) terms already appear in the `thorder estimate with arbitrarily large constant, and so the second, lower-order errorterm in (5.18) is harmless.

A second, related issue, is the treatment of Kawashima estimates. In the case (asin gas dynamics and MHD) that there exists a linear (i.e., di↵erential) compensatingmatrix K(⇠) :=

Pj Kj⇠j for A(⇠) :=

Pj Aj⇠j and B(⇠) :=

Pjk Bjk⇠j⇠k in the

sense that

(5.19) (K(⇠)(A0)�1A(⇠) + B(⇠)) � ✓ > 0,

the `th order Kawashima estimate takes the form

(5.20) (d/dt)12h@`�1

x W,�X

j

K@xj

@`�1

x W i↵,

138 KEVIN ZUMBRUN

and yields a “good” term

h@`�1

x W,X

jk

K(A0)�1Ak@xj

@xk

@`�1

x W i↵,

which may be again estimated by an appropriate Garding inequality as

�✓|@`xwI |2↵ + C|@`xwII |2↵ + C(C⇤)|@`�1

x W |2L2

for some uniform ✓, C > 0, and C(C⇤) depending on the details of weight ↵, andused as before to compensate the Friedrichs estimate near plus and minus spatialinfinity. Recall that lower-order derivative terms are harmless for the Kawashimaestimate, whether in |wI |H`�1 or |wII |H`�1 .

Of course, the compensating matrix K(⇠) in general depends on ⇠ in pseudodif-ferential rather than di↵erential fashion (that is, it is homogeneous degree one andnot linear in ⇠), and in this case the argument must be modified. A convenientmethod is to consider the variables W±

↵ := �±↵1/2W , where �± = �±(x1

) aresmooth, scalar cuto↵ functions supported on x

1

� L and x1

�L, respectively.For L su�ciently large, the equations for W±

↵ have coe�cients that are constantup to an arbitrarily small error. Taking the Fourier transform now in x, we mayperform an energy estimate in the frequency domain, treating coe�cient errors andnonlinear terms alike as a small source, to obtain a similar ↵-weighted Kawashimaestimate, this time explicitly localized near plus and minus spatial infinity (essen-tially, the pseudodi↵erential Garding inequality in a simple case). We omit thedetails, which may be found in [Z.4].

Finally, we estimate nonlinear source terms more systematically using the factthat Hr is an algebra, |fg|Hr |f |Hr |g|Hr for r > d/2, in particular for r = s� 1(Moser’s inequality and Sobolev embedding; see, e.g. [M.1–4, Kaw, GMWZ.2, Z.4]).With these changes, the argument goes through as before, yielding the result. See[Z.4] for further details. ⇤

5.2.3. Nonlinear stability.

Using the linearized bounds of Section 5.2.1 together with the energy estimateof Section 5.2.2, it is now straightforward to establish L1 \ Hs�1 ! Lp \ Hs�1

asymptotic nonlinear stability, p � 2.

Proof of Theorem 5.5. We present in detail the case of a uniformly invis-cid stable Lax or overcompressive shock in dimension d � 3. Other cases followsimilarly; see, e.g., [Z.3–4].

Hs�1 stability. We first carry out a self-contained stability analysis in Hs�1.Afterwards, we shall establish sharp L1 decay rates (and thus, by interpolation,Lp rates for 2 p 1) by a bootstrap argument. Defining

(5.21) U := U � U ,


and Taylor expanding as usual, we obtain the nonlinear perturbation equation

(5.22) Ut � LU =X

j

Qj(U, @xU)xj

,

where

(5.23)Qj(U, @xU) = O(|U |2 + |U ||@xU |)

@xj

Qj(U, @xU) = O(|U ||@xU |+ |U ||@2

xU |+ |@xU |2)

so long as |U | remains bounded by some fixed constant. Applying Duhamel’s prin-ciple, and integrating by parts, we can thus express

(5.24) U(x, t) = S(t)U(0) +Z t

0

S(t� s)X

j

@xj

Qj(s)ds.

Define now

(5.25) ⇣(t) := sup0st

kU(s)kL2(1 + s)(d�14 ).

By standard, short-time existence theory (see, e.g., [Kaw], or [Z.4]) and the principleof continuation, there exists a solution U 2 Hs�1(x), Ut 2 Hs�3(x) ⇢ L2(x) onthe open time-interval for which |U |Hs�1 remains bounded. On this interval, ⇣ iswell-defined and continuous.

Now, let [0, T ) be the maximal interval on which |U |Hs�1(x)

remains strictlybounded by some fixed, su�ciently small constant � > 0. By Proposition 5.9,

(5.26)|U(t)|2Hs�1 C|U(0)|2Hs�1e�✓t + C

Z t

0

e�✓2(t�⌧)|U(⌧)|2L2 d⌧

C2

�|U(0)|2Hs�1 + ⇣(t)2

�(1 + ⌧)�(d�1)/2.

Combining this with the bounds of Proposition 5.7 and |Q|L1\H1 |U |2Hs�1 , and

140 KEVIN ZUMBRUN

recalling that d � 3 and |U(0)|Hs�1 is small, we thus obtain(5.27)

|U(t)|L2 |S(t)U(0)|L2 +Z t

0

X

j

(S1

(t� s)@xj

)Qj(s)ds

+Z t

0

S2

(t� s)(X

j

@xj

Qj(s))ds

C(1 + t)�d�14 |U(0)|L1\L2 +

Z t

0

(1 + t� s)�d�14 � 1

2 |Qj(s)|L1ds

+Z t

0

e�✓(t�s)|X

j

@xj

Qj(s)|L2ds.

C(1 + t)�d�14 |U(0)|L1\L2 +

Z t

0

(1 + t� s)�d�14 � 1

2 |Qj(s)|L1\H1ds

C(1 + t)�d�14 |U(0)|L1\L2

+ C2

�|U(0)|2Hs�1 + ⇣(t)2

� Z t

0

(1 + t� s)�d�14 � 1

2 (1 + s)�d�12 ds

C3

(1 + t)�d�14

⇣|U(0)|L1\Hs�1 + ⇣(t)2

�,

or, dividing by (1 + t)�d�14 ,

(5.28) ⇣(t) C2

(|U(0)|L1\Hs�1 + ⇣(t)2).

Bound (5.28) together with continuity of ⇣ implies that

(5.29) ⇣(t) 2C2

|U(0)|L1\Hs�1

for t � 0, provided |U(0)|L1\Hs�1 < 1/4C2

2

. With (5.26), this gives

|U(t)|Hs�1 4C2

|U(0)|L1\Hs�1

and so we find that the maximum time of existence T is in fact +1, and (5.29)holds globally in time. Definition (5.25) then yields

(5.30)kU(t)kL2 2C

2

⇣(1 + t)�( d�14 ) C

3

⇣(t)

2C2

|U(0)|L1\Hs�1(1 + t)�( d�14 )

as claimed.

L1 stability. We now carry out the L1 estimate. Together with the L2 resultalready obtained, this yields rates (5.14) by interpolation, completing the proof.


Denoting by U1

and U2

the low- and high-frequency parts

(5.31) Uj(t) := S1

(t)U0

+Z t

0

Sj(t� s)(X

j

@xj

Qj(s))ds,

j = 1, 2 of U , we have, using the bounds of Proposition 5.7 together with thepreviously obtained Hs�1 bounds, Sobolev bound |f |L1 C|@�1x1

@�xf |H� for �1

= 1,� = [(d�1)/2)]+1 s�4, and the fact that Hr is an algebra, |fg|Hr |f |Hr |g|Hr

for r > d/2 (Moser’s inequality and Sobolev embedding; see, e.g. [M.1–4, Kaw,GMWZ.2, Z.4]), that(5.32)

|U1

(t)|L1 |S1

(t)U0

|L1 + |Z t

0

(S1

(t)@x)Q(s) ds|L1

C(1 + t)d�12 |U

0

|L1 + C

Z t

0

(1 + t� s)�d�12 � 1

2 |U(s)|2H1 ds

C⇣(1 + t)

d�12 +

Z t

0

(1 + t� s)�d�12 � 1

2 (1 + s)�d�12 ds

⌘|U

0

|L1\Hs�1

C(1 + t)d�12 |U

0

|L1\Hs�1

and

(5.33)

|U2

(t)|L1 C|@�1x1@�2x U

2

(t)|L2

|@�xS2

(t)U0

|L2 + |Z t

0

@�xS2

(t)(@xQ(s)) ds|L2

Ce�✓t|U0

|H� + C

Z t

0

e�✓(t�s)|@xQ(s)|H|�| ds

C⇣e�✓t + |

Z t

0

e�✓(t�s)(1 + s)�d�12 ds

⌘|U(0)|H|�|+2

C(1 + t)�d�12 |U(0)|Hs�1 ,

giving the claimed bound. ⇤Remarks. 1. Results of [HoZ.1] show that kGkLp for p 2 degrade for systems

in multi-dimensions, with blowup at p = 1. So, p � 2 is the optimal result.2. The restriction to d � 3 in the case of weakly inviscid stable Lax or overcom-

pressive shocks results from the degraded Green distribution bounds (5.15) obtainedin dimension d = 2, as indicated by the presence of error terms �. These, in turn,result from our nonsharp estimation of “pole” terms of order

R�

e⇠

(�� ⇤(e⇠))�1 oc-curring in the inverse Laplace transform of the resolvent via modulus estimatesR�

e⇠

|� � �⇤(e⇠)|�1; see (5.154)–(5.158), Section 5.3.3. To obtain optimal estimates,one should rather treat this as a residue term similarly as in the one-dimensionalcase, identifying cancellation in the nonintegrable 1/� singularity; see, e.g., thetreatment of the (always weakly inviscid stable) scalar case in [HoZ.3–4, Z.3].

142 KEVIN ZUMBRUN

5.3 Proof of the linearized estimates.

Finally, we carry out in this subsection the proof of Proposition 5.7, completingthe analysis of the multi-dimensional case.

5.3.1. Low-frequency bounds on the resolvent kernel.

Consider the family of elliptic Green distributions Ge⇠,�(x1

, y1

),

(5.34) Ge⇠,�(·, y1

) := (Le⇠ � �I)�1�y1(·),

associated with the ordinary di↵erential operators (Le⇠ � �I), i.e. the resolventkernel of the Fourier transformed operator Le⇠. The function Ge⇠,�(x1

, y1

) is theLaplace–Fourier transform in variables ex = (x

2

, . . . , xd) and t, respectively, of thetime-evolutionary Green distribution

(5.35) G(x, t; y) = G(x1

, ex, t; y1

, ey) := eLt�y(x)

associated with the linearized evolution operator (@/@t� L). Restrict attention tothe surface

(5.36) �e⇠ := @⇤e⇠ = {� : Re � = �✓1

(| Im �|2 + |e⇠|2)}

✓1

> 0 su�ciently small.Then, our main result, to be proved in the remainder of the subsection, is

Proposition 5.10. Under the hypotheses of Theorem 5.5, for � 2 �e⇠ (defined in(5.36)) and ⇢ := |(e⇠,�)|, ✓

1

> 0, and ✓ > 0 su�ciently small, there hold:

(5.37) |Ge⇠,�(x1

, y1

)| C�2

�1

(⇢�1e�✓|x1|e�✓⇢2|y1| + e�✓⇢

2|x1�y1|),

and

(5.38)|(@/@y

1

)Ge⇠,�(x1

, y1

)| C�2

�1

[⇢�1e�✓|x1|(⇢e�✓⇢2|y1| + ↵e�✓|y1|)

+ e�✓⇢2|x1�y1|(⇢+ ↵e�✓|y1|)],

where

(5.39) �1

(e⇠,�) :=

(1 in U.I.S. case,1 +

Pj [⇢

�1| Im �� i⌧j(e⇠)|+ ⇢]�1 in W.I.S. case,

(5.40) �2

(e⇠,�) := 1 +X

j,±[⇢�1| Im �� ⌘±j (e⇠)|+ ⇢]1/s

j

�1,


⌘j(·), sj(·) � 1 as in (H5) and ⌧j(·) as in (1.43), and

(5.41) ↵ :=⇢

0 for Lax or overcompressive case,1 for undercompressive case.

(Here, as above, U.I.S. and W.I.S. denote “uniform inviscid stable” and “weakinviscid stable,” as defined, repectively, in (1.32), Section 1.4, and (1.28), Section1.3; the classification of Lax, overcompressive, and undercompressive types is givenin Section 1.2). More precisely, t := 1�1/K

max

, where Kmax

:= maxK±j = max s±j

is the maximum among the orders of all branch singularities ⌘±j (·), s±j and ⌘±jdefined as in (H5); in particular, t = 1/2 in the (generic) case that only square-rootsingularities occur.30

Corollary 5.11. Under the hypotheses of Theorem 5.5, for � 2 �e⇠ (defined in(5.36)) and ⇢ := |(e⇠,�)|, ✓

1

> 0, and ✓ > 0 su�ciently small, there holds theresolvent bound

(5.42) |(Le⇠ � �)�1@�x1f |Lp

(x1) C�1

�2

⇢(1�↵)|�|�1|f |L1(x1)

for all 2 p 1, 0 |�| 1, where �j, ↵ are as defined in (5.39)–(5.41).

Proof. Direct integration, using the triangle inequality bound

|(Le⇠ � �)�1@↵x1f |Lp

(x1) supy1

|@↵y1Ge⇠,�(·, y1

)|Lp

(x1)|f |L1(x1).

⇤Remark 5.12. By standard considerations (see, e.g. [He, ZH, Z.1]), condition

(H3) implies that surface �e⇠ strictly bounds the essential spectrum of operatorLe⇠ to the right. That is, in this analysis, we shall work entirely in the resolventset of Le⇠. This is in marked contrast to our approach in the one-dimensionalcase, wherein bounds on the resolvent kernel G� within the essential spectrumwere used to obtain sharp pointwise estimates on G in the one-dimensional case.The nonoptimal bounds we obtain here could perhaps be sharpened by a similaranalysis; for further discussion, see [Z.3, HoZ.3–4].

Remark. Terms ⇢�1e�✓|x1|e�✓⇢2|y1| and e�✓⇢

2|x1�y1| in (5.37), respectively en-coding near and far-field behavior with respect to the front location x

1

= 0, are,

30Bound (5.37) corrects a minor error in [Z.3], where it was mistakenly stated as

|Ge⇠,�

(x1, y1)| C�2(�1⇢�1e�✓|x1|e�✓⇢

2|y1| + e�✓⇢

2|x1�y1|).

The source of this error was the normalization (4.137) of [Z.3] that '±1 agree to first order in⇢ at ⇢ = 0, which is incompatible with the assumption (used in the proof of the low-frequencyexpansion) that fast modes decay at x1 = ±1.

144 KEVIN ZUMBRUN

roughly speaking, associated respectively with point and essential spectrum of theoperator L. In particular, the simple pole ⇢�1 in the first term corresponds to asemisimple eigenvalue at ⇢ = 0, and the factors e�✓|x1| and e�✓⇢

2|y1| to associatedright and left eigenfunctions. However, this is clearly not the usual point spectrumencountered in the case of an operator with positive spectral gap. For example,note the lack of spatial decay/localization in the “left eigenfunction” term e�✓⇢

2|y1|

at ⇢ = 0. Also, though it is not apparent from the simple bound (5.37), it is a factthat the “left eigenspace” at ⇢ = 0 (defined by continuation along rays from ⇢ > 0following the approach of Section 4.3.3) depends on direction (e⇠

0

,�0

); that is, thereis a conical singularity at the origin in the spectral projection, reflecting that of theEvans function.

Remark 5.13. The ↵-terms appearing in (5.37)-(5.38) are not technical arti-facts, but in fact reflect genuinely new e↵ects arising in the undercompressive case,related to L1 time-invariants and conservation of mass. See [Z.6] for a detaileddiscussion in the one-dimensional case.

Normal modes. Paralleling our one-dimensional analysis, we begin by identi-fying normal modes of the limiting, constant-coe�cient equations (Le⇠±� �)U = 0.The key estimates of slow and super-slow mode decay will be seen to reduce to amatrix perturbation problem closely related to the one arising in the correspondinginviscid theory; the crucial issue here, as there, is the careful treatment of branchsingularities, corresponding to inviscid glancing modes.

Introduce the curves

(5.43) (e⇠,�)(⇢, e⇠0

, ⌧0

) :=�⇢e⇠

0

, ⇢i⌧0

� ✓1

⇢2

�,

where e⇠0

2 Rd�1 and ⌧0

2 R are restricted to the unit sphere Sd : |e⇠0

|2 + |⌧0

|2 = 1.Evidently, as (e⇠

0

, ⌧0

, ⇢) range in the compact set Sd ⇥ [0, �], (e⇠,�) traces out theportion of the surface �e⇠ contained in the set |e⇠|2 + |�|2 � of interest. As before,fixing e⇠

0

, ⌧0

, we denote by v±j (⇢), the solutions of the limiting, constant coe�cientequations at (e⇠,�)(⇢) from which '±j (⇢), ±j (⇢), etc. are constructed; it is thesesolutions that we wish to estimate.

Making as usual the Ansatz

(5.44) v± =: eµx1v,

and substituting � = i⇢⌧0

� ✓1

⇢2 into (3.4), we obtain the characteristic equation

(5.45)

2

4µ2B11

± + µ(�A1

± + i⇢X

j 6=1

Bj1± ⇠0j

+ i⇢X

k 6=1

B1k⇠0

k

)

�(i⇢X

j 6=1

Aj⇠0

j

+ ⇢2

X

jk 6=1

Bjk⇠0

j

⇠0

k

+ (⇢i⌧0

� ✓1

⇢2)I)

3

5v = 0.


Note that this agrees with (3.4) up to first order in ⇢, hence the (first-order) matrixbifurcation analyses of Lemma 3.3 applies for any ✓.

As in past sections, we first separate normal modes (5.44) into fast and slowmodes, the former having exponential growth/decay rate with real part boundedaway from zero, and the latter having growth/decay rate close to zero (indeed, van-ishing for ⇢ = 0). The fast modes haveO(1) decay rate, and are spectrally separatedby assumption (A1), hence admit a straightforward treatment. We now focus onthe slow modes, which we will further subdivide into intermediate- and super-slowtypes, corresponding respectively to elliptic and hyperbolic/glancing modes in theinviscid terminology.

Positing the Taylor expansion

(5.46)⇢

µ = 0 + µ1⇢+ · · · ,

v = v0 + · · ·

(or Puiseux expansion, in the case of a branch singularity), and matching terms oforder ⇢ in (5.45), we obtain

(�µ1A1

± � iX

j 6=1

Aj⇠0

j

� i⌧0

I)v0 = 0,

just as in (3.6), or equivalently

(5.47) [(A1)�1(i⌧0

+ iAe⇠0)� ↵

0

I]v0 = 0,

with µ1 =: �↵0

, which can be recognized as the equation occurring in the inviscidtheory on the imaginary boundary � = i⌧

0

. For eigenvalues ↵0

of nonzero real part,denoted as intermediate-slow modes, we have growth or decay at rate O(⇢), andspectral separation by assumption (A2); thus, they again admit straightforwardtreatment.

It remains to study super-slow modes, corresponding to pure imaginary eigenval-ues ↵

0

=: i⇠01 ; here, we must consider quadratic order terms in ⇢, and the viscous

and inviscid theory part ways. Using µ = �i⇢⇠01 + o(⇢), and dividing (5.45) by ⇢,

we obtain the modified equation

(5.48) [(A1)�1

�i⌧

0

+ iAe⇠0 + ⇢(B⇠0⇠0 � ✓

1

) + o(⇢)�� ↵I]v = 0,

where ↵ := �µ/⇢ and v denote exact solutions; that is, we expand the equationsrather than the solutions to second order. (Note that this derivation remains validnear branch singularities, since we have only assumed continuity of µ/⇢ and notanalyticity at ⇢ = 0.) Here, B⇠0⇠0 as usual denotes

PBjk⇠

0

j

⇠0

k

, where ⇠0

:=(⇠

01 , e⇠0). Equation (5.48) generalizes the (all-orders) perturbation equation

(5.49) [(A1)�1(i⌧0

+ iAe⇠0 + ⇢I)� ↵I]v = 0,

146 KEVIN ZUMBRUN

⇢ ! 0+, arising in the inviscid theory near the imaginary boundary � = i⌧0

[Kr,Me.5].Note that ⌧

0

is an eigenvalue of A⇠0 , as can be seen by substituting ↵0

= i⇠01 in

(5.47), hence |⌧0

| C|⇠0

| and therefore (since clearly also |⇠0

| � |e⇠0

|)

|⇠0

| � (1/C)(|(e⇠0

, ⌧0

)| = 1/C.

Thus, for B positive definite, and ✓1

su�ciently small, perturbation ⇢(B⇠0⇠0 � ✓1

),roughly speaking, enters (5.48) with the same sign as does ⇢I in (5.49). Indeed, foridentity viscosity Bjk := �j

kI, (5.48) reduces for fixed (e⇠0

, ⌧0

) exactly to (5.49), bythe rescaling ⇢! ⇢/(|⇠

0

|2 � ✓1

).For (e⇠

0

, ⌧0

) bounded away from the set of branch singularities [(e⇠, ⌘j(e⇠)), we maytreat (5.48) as a continuous family of single-variable matrix perturbation problemin ⇢, indexed by (e⇠

0

, ⌧0

); the resulting continuous family of analytic perturbationseries will then yield uniform bounds by compactness. For (e⇠

0

, ⌧0

) near a branchsingularity, on the other hand, we must vary both ⇢ and (e⇠

0

, ⌧0

), in general acomplicated multi-variable perturbation problem. Using homogeneity, however,and the uniform structure assumed in (H6), this can be reduced to a two-variableperturbation problem that again yields uniform bounds. For, noting that ⌘j(e⇠) ⌘ 0,we find that e⇠

0

must be bounded away from the origin at branch singularities; thus,we may treat the direction e⇠

0

/|e⇠0

| as a fixed parameter and vary only ⇢ and theratio |⌧

0

|/|e⇠0

|. Alternatively, relaxing the restriction of (e⇠0

, ⌧0

) to the unit sphere,we may fix e⇠

0

and vary ⇢ and ⌧0

, obtaining after some rearrangement the rescaledequation

(5.50) [(A1)�1

�i⌧

0

+ iAe⇠0 +[i�+⇢(B⇠0⇠0�✓

1

(|e⇠0

|2 + |⌧0

|2)+o(|⇢,�|)]��↵I]v = 0,

where � denotes variation in ⌧0

.We shall find it useful to introduce at this point the wedge, or exterior algebraic

product of vectors, determined by the properties of multilinearity and alternation.We fix once and for all a basis, thus determining a norm. (This allows us toconveniently factorize into minors, below.)

We make also the provisional assumption (P0) of Section 4, that either the invis-cid system is strictly hyperbolic, or else Aj and Bjk are simultaneously symmetriz-able. This is easily removed by working with vector blocks in place of individualmodes.

Lemma 5.14 [Z.3]. Under the hypotheses of Theorem 5.5, let ↵0

= i⇠01 be a pure

imaginary root of the inviscid equation (5.47) for some given e⇠0

, ⌧0

, i.e. det (A⇠0 +⌧0

) = 0. Then, associated with the corresponding root ↵ in (5.48), we have thefollowing behavior, for some fixed ✏, ✓ > 0 independent of (e⇠

0

, ⌧0

):(i) For (e⇠

0

, ⌧0

) bounded distance ✏ away from any branch singularity (e⇠, ⌘j(e⇠))involving ↵, the root ↵(⇢) in (5.48) such that ↵(0) := ↵

0

bifurcates smoothly into


m roots ↵1

, . . . ,↵m with associated vectors v1

, . . . ,vm, where m is the dimensionof ker(A⇠0 + ⌧

0

), satisfying

(5.51) |Re ↵j | � ✓⇢

and

(5.52) |v1

^ · · · ^ vm| � ✓ > 0

for 0 < ⇢ ✏, where the modulus of the wedge product is evaluated with respect toits coordinatization in some fixed reference basis.

(ii) For (e⇠0

, ⌧0

) lying at a branch singularity (e⇠, ⌘j(e⇠)) involving ↵, the root↵(⇢,�) in (5.50) such that ↵(0, 0) = ↵

0

bifurcates (nonsmoothly) into m groups ofs roots each:

(5.53) {↵1

1

, . . . ,↵1

s}, . . . , {↵m1

, . . . ,↵ms },

with associated vectors vjk, where m is the dimension of ker(A⇠0 +⌧

0

) and s is somepositive integer, such that, for 0 ⇢ ✏ and |�| ✏,

(5.54) ↵jk = ↵+ ⇡j

k + o(|�|+ |⇢|)1/s,

and, in appropriately chosen coordinate system,

(5.55) vjk =

0

BBBBBBBBBBBBBBBBBB@

0...01⇡j

k

(⇡jk)2...

(⇡jk)m�1

0...0

1

CCCCCCCCCCCCCCCCCCA

+ o(|�|+ |⇢|)1/s,

where

(5.56) ⇡jk := "ki(p� � iqj⇢)1/s,

" := 11/s, and the functions p(e⇠0

) and qj(e⇠0) are real-valued and uniformly boundedboth above and away from zero, with sgn p = sgn q. Moreover,

(5.57) |Re ↵jk| � ✓⇢

148 KEVIN ZUMBRUN

and the “group vectors” vj satisfy

(5.58) |v1 ^ · · · ^ vm| � ✓ > 0

with respect to some fixed reference basis.

Proof. Case (H2)(i): We first treat the considerably simpler strictly hyperboliccase, which permits a direct and relatively straightforward treatment. In this case,the dimension of ker(A⇠0+⌧

0

) is one, hence m is simply one and (5.52) and (5.58) areirrelevant. Let l(e⇠

0

, ⌧0

) and r(e⇠0

, ⌧0

) denote left and right zero eigenvectors of (A⇠0 +⌧0

) = 1, spanning co-kernel and kernel, respectively; these are necessarily real, since(A⇠0 + ⌧

0

) is real. Clearly r is also a right (null) eigenvector of (A1)�1(i⌧0

+ iA⇠0),and lA1 a left eigenvector.

Branch singularities are signalled by the relation

(5.59) lA1r = 0,

which indicates the presence of a single Jordan chain of generalized eigenvectorsof (A1)�1(i⌧

0

+ iA⇠0) extending up from the genuine eigenvector r; we denote thelength of this chain by s.

Observation 5.15 [Z.3]. Bound (2.55) implies that

(5.60) lB⇠0⇠0r � ✓ > 0,

uniformly in e⇠.Proof of Observation. In our present notation, (2.55) can be written as

Re �(�iA⇠0 � ⇢B⇠0⇠0) �✓1

⇢,

for all ⇢ > 0, some ✓1

> 0. (Recall: |⇠0

| � ✓2

> 0, by previous discussion). Bystandard matrix perturbation theory [Kat], the simple eigenvalue � = i⌧

0

of �iA⇠0

perturbs analytically as ⇢ is varied around ⇢ = 0, with perturbation series

�(⇢) = i⌧0

� ⇢lB⇠0⇠0r + o(⇢).

Thus,Re �(⇢) = �⇢lB⇠0⇠0r + o(⇢) �✓

1

⇢,

yielding the result. ⇤In case (i), ↵(0) = ↵

0

is a simple eigenvalue of (A1)�1(i⌧0

+iAe⇠0

), and so perturbsanalytically in (5.48) as ⇢ is varied around zero, with perturbation series

(5.61) ↵(⇢) = ↵0

+ ⇢↵1 + o(⇢),


where ↵1 = l(A1)�1r, l, r denoting left and right eigenvectors of (A1)�1(i⌧0

+Ae⇠0

).Observing by direct calculation that r = r, l = lA1/lA1r, we find that

↵1 = lB⇠0⇠0r/lA1r

is real and bounded uniformly away from zero, by Observation 5.15, yielding theresult (5.51) for any fixed (e⇠

0

, ⌧0

), on some interval 0 ⇢ ✏, where ✏ depends onlyon a lower bound for ↵1 and the maximum of �00(⇢) on the interval 0 ⇢ ✏. Bycompactness, we can therefore make a uniform choice of ✏ for which (5.51) is validon the entire set of (e⇠

0

, ⌧0

) under consideration.

In case (ii), ↵(0, 0) = ↵0

is an s-fold eigenvalue of (A1)�1(i⌧0

+ iAe⇠0

), corre-sponding to a single s⇥ s Jordan block. By standard matrix perturbation theory,the corresponding s-dimensional invariant subspace (or “total eigenspace”) variesanalytically with ⇢ and �, and admits an analytic choice of basis with arbitraryinitialization at ⇢, � = 0 [Kat]. Thus, by restricting attention to this subspacewe can reduce to an s-dimensional perturbation problem; moreover, up to linearorder in ⇢, �, the perturbation may be calculated with respect to the fixed, initialcoordinization at ⇢, � = 0.

Choosing the initial basis as a real, Jordan chain reducing the restriction (to thesubspace of interest) of (A1)�1(i⌧

0

+ iAe⇠0

) to i times a standard Jordan block, wethus reduce (5.50) to the canonical problem

(5.62),�iJ + i�M + ⇢N + o(|(⇢,�)|� (↵� ↵

0

)�vI = 0,

where

(5.63) J :=

0

BB@

0 1 0 · · · 00 0 1 0 · · ·...

......

......

0 0 0 · · · 0

1

CCA ,

vI is the coordinate representation of v in the s-dimensional total eigenspace, andM and N are given by

(5.64) M := L(A1)�1R

and

(5.65) N := L(A1)�1(B⇠0⇠0 � ✓1

)R,

respectively, where R and L are the initializing (right) basis, and its corresponding(left) dual.

Now, we have only to recall that, as may be readily seen by the defining relation

L(A1)�1(i⌧0

+ iAe⇠0

)R = J,

150 KEVIN ZUMBRUN

or equivalently (A1)�1(i⌧0

+ iAe⇠0

)R = RJ and L(A1)�1(i⌧0

+ iAe⇠0

) = JL, the firstcolumn of R and the last row of L are genuine left and right eigenvectors r and lof (A1)�1(i⌧

0

+ iAe⇠0

), hence without loss of generality

r = r, l = plA1

as in the previous (simple eigenvalue) case, where p is an appropriate nonzero realconstant. Applying again Observation 5.15, we thus find that the crucial s, 1 entriesof the perturbations M , N , namely p and pl(B⇠0⇠0�✓

1

)r =: q, respectively, are real,nonzero and of the same sign. Recalling, by standard matrix perturbation theory,that this entry when nonzero is the only significant one, we have reduced finally(modulo o(|�|+ |⇢|)1/s) errors) to the computation of the eigenvalues/eigenvectorsof

(5.66) i

0

BB@

0 1 0 · · · 00 0 1 0 · · ·...

......

......

p� � iq⇢ 0 0 · · · 0

1

CCA ,

from which results (5.54)–(5.56) follow by an elementary calculation, for any fixed(e⇠

0

, ⌧0

), and some choice of ✏ > 0; as in the previous case, the correspondingglobal results then follow by compactness. Finally, bound (5.57) follows from (5.54)and (5.56) by direct calculation. (Note that the addition of further O(|�| + |⇢|)perturbation terms in entries other than the lower lefthand corner of (5.66) doesnot a↵ect the result.) This completes the proof in the strictly hyperbolic case.

Case (H2)(ii): We next turn to the more complicated symmetrizable, constant-multiplicity case; here, we make essential use of recent results of Metivier [Me.4]concerning the spectral structure of matrix (A1)�1(i⌧ + iA

e⇠). Without loss ofgenerality, take A⇠, B⇠,⇠ (but not necessarily A1) symmetric; this may be achievedby the change of coordinates A⇠ ! A1/2

0

A⇠A�1/2

0

, B⇠,⇠ ! A1/2

0

B⇠,⇠A�1/2

0

. From(H3), we find, additionally, that B⇠,⇠ � ✓|⇠|2 for all ⇠ 2 Rd.

With these assumptions, the kernel and co-kernel of (A⇠0 + ⌧0

) are of fixeddimension m, not necessarily equal to one, and are spanned by a common set of zero-eigenvectors r

1

, . . . , rm. Vectors r1

, . . . , rm are necessarily right zero-eigenvectors of(A1)�1(i⌧

0

+ iA⇠0) as well. Branch singularities correspond to the existence of oneor more Jordan chains of generalized zero-eigenvectors extending up from genuineeigenvectors in their span, which by the argument of Lemma 3.4 is equivalent to

(5.67) det (rtjA

1rk) = 0.

In fact, as pointed out by Metivier [Me.4], the assumption of constant multiplicityimplies considerable additional structure.


Observation 5.16 [Me.4]. Let (e⇠0

, ⌧0

) lie at a branch singularity involving root↵

0

= i⇠01 in (5.47), with ⌧

0

an m-fold eigenvalue of A⇠0 . Then, for (e⇠, ⌧) in thevicinity of (e⇠

0

, ⌧0

), the roots ↵ bifurcating from ↵0

in (5.47) consist of m copies ofs roots ↵

1

, . . . ,↵s, where s is some fixed positive integer.

Proof of Observation. Let a(e⇠,↵) denote the unique eigenvalue of A⇠ lying near�⌧

0

, where, as usual, �i⇠1

:= ↵; by the constant multiplicity assumption, a(·, ·) isan analytic function of its arguments. Observing that

det [(A1)�1(i⌧ + iAe⇠)� ↵] = det i(A1)�1det (⌧ + A⇠)

= e(e⇠, ⌧,↵)(⌧ + a(e⇠,↵))m,

where e(·, ·, ·) does not vanish for (e⇠, ⌧,↵) su�ciently close to (e⇠0

, ⌧0

,↵0

), we seethat the roots in question occur as m-fold copies of the roots of

(5.68) ⌧ + a(e⇠,↵) = 0.

But, the lefthand side of (5.68) is a family of analytic functions in ↵, continuouslyvarying in the parameters (e⇠, ⌧), whence the number of zeroes is constant near(e⇠

0

, ⌧0

). ⇤Observation 5.17 [Z.3]. The matrix (rt

jA1rk), j, k = 1, . . . ,m is a real multiple

of the identity,

(5.69) (rtjA

1rk) = (@a/@⇠1

)Im,

where a(⇠) denotes the (unique, analytic) m-fold eigenvalue of A⇠ perturbing from�⌧

0

.More generally, if

(5.70) (@a/@⇠1

) = · · · = (@s�1a/@⇠s�1

1

) = 0, (@sa/@⇠s1

) 6= 0

at ⇠0

, then, letting r1

(e⇠), . . . , rm(e⇠) denote an analytic choice of basis for theeigenspace corresponding to a(e⇠), orthonormal at (e⇠

0

, ⌧0

), we have the relations

(5.71) (A1)�1(⌧0

+ A⇠0)rj,p = rj,p�1

,

for 1 p s� 1, and

(5.72) (rtj,0A

1rk,p�1

) = p! (@pa/@⇠p1

)Im,

for 1 p s, whererj,p := (�1)pp(@prj/@⇠

p1

).

152 KEVIN ZUMBRUN

In particular,{rj,0, . . . , rj,s�1

}, j = 1, . . . ,m

is a right Jordan basis for the total zero eigenspace of (A1)�1(⌧0

+ A⇠0), for whichthe genuine zero-eigenvectors lj of the dual, left basis are given by

(5.73) lj = (1/s!(@sa/@⇠s1

))A1rj .

Proof of Observation. Considering A⇠ as a matrix perturbation in ⇠1

, we find bystandard spectral perturbation theory that the bifurcation of the m-fold eigenvalue⌧0

as ⇠1

is varied is governed to first order by the spectrum of (rtjA

1rk). Since theseeigenvalues in fact do not split, it follows that (rt

jA1rk) has a single eigenvalue.

But, also, (rtjA

1rk) is symmetric, hence diagonalizable, whence we obtain result(5.69).

Result (5.72) may be obtained by a more systematic version of the same argu-ment. Let R(⇠

1

) denote the matrix of right eigenvectors

R(⇠1

) := (r1

, . . . , rm)(⇠1

).

Denoting by

(5.74) a(⇠1

+ h) =: a0 + a1h + · · ·+ aphp + · · ·

and

(5.75) R(⇠1

+ h) =: R0 + R1h + · · ·+ Rphp + · · ·

the Taylor expansions of functions a(·) and R(·) around ⇠0

as ⇠1

is varied, andrecalling that

A⇠ = A⇠0 + hA1,

we obtain in the usual way, matching terms of common order in the expansion ofthe defining relation (A� a)R = 0, the hierarchy of relations:

(5.76)

(A⇠0 � a0)R0 = 0,

(A⇠0 � a0)R1 = �(A1 � a1)R0,

(A⇠0 � a0)R2 = �(A1 � a1)R1 + a2R0,

...

(A⇠0 � a0)Rp = �(A1 � a1)Rp�1 + a2Rp�2 + · · ·+ apR0.

Using a0 = · · · = as�1 = 0, we obtain (5.71) immediately, from equations p =1, . . . , s�1, and Rp = (1/p!)(r

1,p, . . . , rm,p). Likewise, (5.72), follows from equations


p = 1, . . . , s, upon left multiplication by L0 := (R0)�1 = (R0)t, using relationsL0(A⇠0 � a0) = 0 and ap = (@pa/@⇠p

1

)/p!.From (5.71), we have the claimed right Jordan basis. But, defining lj as in (5.73),

we can rewrite (5.72) as

(5.77) (ltjrk,p�1

) =⇢

0 1 p s� 1,

Im, p = s;

these ms criteria uniquely define lj (within the ms-dimensional total left eigenspace)as the genuine left eigenvectors dual to the right basis formed by vectors rj,p (seealso exercise just below). ⇤

Observation 5.17 implies in particular that Jordan chains extend from all ornone of the genuine eigenvectors r

1

, . . . , rm, with common height s. As suggestedby Observation 5.16 (but not directly shown here), this uniform structure in factpersists under variations in e⇠, ⌧ , see [Me.4]. Observation 5.17 is a slightly moreconcrete version of Lemma 2.5 in [Me.4]; note the close similarity between theargument used here, based on successive variations in basis rj , and the argumentof [Me.4], based on variations in the associated total projection.

With these preparations, the result goes through essentially as in the strictlyhyperbolic case. Set

(5.78) p := 1/(s!(@sa/@⇠s1

)).

In the result of Observation 5.17, we are free to choose any orthonormal basis rj at(e⇠

0

, ⌧0

); choosing a basis diagonalizing the symmetric matrix Be⇠0,e⇠0 , we define

(5.79) pRtBe⇠0,e⇠0R =: diag{q

1

, . . . , qm}.

Note, as claimed, that p 6= 0 by assumption (@sa/@⇠s1

) 6= 0 in Observation 5.17,and sgn qj = sgn p by positivity of B

e⇠0,e⇠0 (an easy consequence of (2.55)).Then, working in the Jordan basis defined in Observation 5.17, we find simi-

larly as in the strictly hyperbolic case that the matrix perturbation problem (5.50)reduces to an ms⇥ms system consisting of m s⇥ s equations

(5.80)�iJ + i�Mj + ⇢Nj + o(|(⇢,�)|)� (↵� ↵

0

)�vI

j

=X

k 6=j

(i�Mjk + ⇢Njk)vIk

(J denoting the standard Jordan block (5.63)), weakly coupled in the sense thatlower lefthand corner elements vanish in the coupling blocks Mjk, Njk, but in thediagonal blocks �iMj + ⇢Nj are p� � iqj⇢ ⇠ |�| + |⇢|. To order (|�| + |⇢|)1/s,

154 KEVIN ZUMBRUN

therefore, the problem decouples, reducing to the computation of eigenvectors andeigenvalues of the block diagonal matrix

(5.81) diag�i

0

BB@

0 1 0 · · · 00 0 1 0 · · ·...

......

......

p� � iqj⇢ 0 0 · · · 0

1

CCA ,

from which results (5.54)–(5.57) follow as before; see Section 2.2.4, Splitting of ablock-Jordan block. (Note that the simple eigenvalue case s = 1 follows as a specialcase of the Jordan block computation.) Finally, the uniform condition (5.58) is aconsequence of (5.55), orthonormality of bases {rj}, and continuity of the Taylorexpansions (5.75). ⇤

Remark 5.18[Z.3]. Fixing j, consider a single cycle of roots ⇡jk := "ki(p� �

iqj⇢)1/s in (5.56), j = 1, . . . , s, " := 11/s, sgn p = sgn qj , on the set ⇢ � 0, ⇢,� 2 R of interest. By direct evaluation at � = 0, ⇢ = 1/|qj |, for ⇢ > 0, the roots ⇡j

k

split into s+

unstable modes (Re ⇡ > 0) and s� stable modes (Re ⇡ < 0), where

(5.82) (s+

, s�) =

8><

>:

(r, r) for s = 2r,(r + 1, r) for s = 2r + 1 and p > 0,(r, r + 1) for s = 2r + 1 and p < 0.

(At � = 0, ⇢ = 1/|qj |, we have simply ⇡jk = "k(�is+1 sgn qk)1/s which in the

critical case s = 2r + 1 becomes "k(�1)r+2 sgn qk = "k(�1)r sgn p.)

Remark 5.19 [Z.3]. Defining wj 2 Rs by wj := (1,↵j ,↵2

j , . . . ,↵s�1

j )t, 1 j p s, with |↵j | C

1

, we find that |w1

^ · · ·^wp| ⇠ ⇧j<k|↵j �↵k|, with constantsdepending only on C

1

, or equivalently

(5.83)

��det

0

B@↵q1

1

· · · ↵q1p

......

...↵

qp

1

· · · ↵q

p

p

1

CA

�� C⇧j<k|↵j � ↵k|

for each minor 0 q1

< · · · < qp s � 1 of matrix (w1

, . . . , wp) 2 Rs⇥p, withequality for the Vandermonde minor qr := r�1. For, considering the determinantsin (5.83) as complex polynomials P (↵

1

, . . . ,↵p), and noting that P = 0 whenever↵j = ↵k, we may use successive applications of the Remainder Theorem to concludethat ⇧j<k(↵j � ↵k) divides P .

Remark 5.20. The direct calculations of Lemma 5.14 verifies Lemma 3.3 in thecase that (H5) holds.

Remark 5.21. With Lemma 5.14, Corollary 5.11 follows, also, by the Kreisssymmetrizer argument of [GMWZ.2].


Lemma 5.22[Z.3]. Under the hypotheses of Theorem 5.5, for � 2 �e⇠ (defined in(5.36)) and ⇢ := |(e⇠,�)|, ✓

1

> 0, and ✓ > 0 su�ciently small, there exists a choiceof bases {'±j }, {e'±j }, { ±j }, { e ±j } of solutions of the variable-coe�cient eigenvalueproblem such that, at z = 0, the wedge products

(5.84)✓�

1

�1

0

◆^ · · · ^

✓�n

�n0

◆

±(z)

and determinants

(5.85) det✓� �0 0

◆

±(z)

and

(5.86) det✓� �0 0

◆

±(z)

are bounded in modulus (measured, in the case of wedge products, with respect tocoordinates in some fixed basis) uniformly above and below, and, moreover, therehold bounds:

(5.87)✓'±j'±j

0

◆= �

21,'±j

eµ±

j

x

✓v±j

µjv±j

◆+O(e�✓|x|)

�, x

1

? 0,

(5.88)✓

e'±je'±j 0

◆= �

21,e'±j

e�µ±

j

x

✓v±j

�µjv±j

◆+O(e�✓|x|)

�, x

1

? 0,

and

(5.89)✓ ±j ±j

0

◆= �

21, ±j

e⌫

±j

x

✓v±j⌫jv±j

◆+O(e�✓|x|)

�, x

1

? 0,

(5.90)

e ±je ±j 0

!= �

21, e ±j

e�⌫

±j

x

✓v±j

�⌫jv±j

◆+O(e�✓|x|)

�, x

1

? 0,

where |v±j |, |v±j | are uniformly bounded above and below, and:

(i) For fast and intermediate-slow modes, or super-slow modes for which Im �

is bounded distance ✓1

✏ away from any associated branch singularities ⌘h(e⇠), v±j areuniformly transverse to all other v±k , while, for super-slow modes for which Im �

is within ✓1

✏ of an (necessarily unique) associated branch singularity ⌘j(e⇠), v±j are

156 KEVIN ZUMBRUN

as described in Lemma 5.14, with � := ⇢�1( Im � � ⌘h(e⇠)); moreover, the “groupvectors” associated with each cluster vj

k are uniformly transverse to each other andto all other modes. The dual vectors v±j satisfy identical bounds.

(ii) The decay/growth rates µ±j /⌫±j satisfy

(5.91) |Re µ±j |, |Re ⌫±j | ⇠ 1,

for fast modes,

(5.92) |Re µ±j |, |Re ⌫±j | ⇠ ⇢

for intermediate-slow modes, and

(5.93) |Re µ±j |, |Re ⌫±j | ⇠ ⇢2

for super-slow modes; moreover,

(5.94) |µ±j |, |⌫±j | = O(⇢)

for both intermediate- and super-slow modes.

(iii) The factors �21,'±

j

, �21,e'±

j

, �21, ±

j

, and �21, e ±

j

satisfy

(5.95) �21,� ⇠ 1, � = '±j , ±j , e'±j , e ±j ,

for fast and intermediate-slow modes, and for super-slow modes for which Im � isbounded distance ✓

1

✏ away from any associated branch singularities ⌘h(e⇠), and

(5.96) �21,� ⇠ (|�|+ |⇢|)�t

� , � = '±j , ±j , e'±j , e ±j ,

for super-slow modes for which Im � is within ✓1

✏ of an (necessarily unique) as-sociated branch singularity ⌘h(e⇠), with

(5.97)

(t'±j

,t ±j

, te'±j

, t e ±j

) :=8><

>:

�r�1

4r , 3r�1

4r , 3r�1

4r , r�1

4r

�for s = 2r,

�r

2(2r+1)

, 3r+1

2(2r+1)

, 3r2(2r+1)

, r�1

2(2r+1)

�for s = 2r + 1 and p ? 0,

�r�1

2(2r+1)

, 3r2(2r+1)

, 3r+1

2(2r+1)

, r2(2r+1)

�for s = 2r + 1 and p 7 0;

here, s := K±h denotes the order of the associated branch singularity ⌘±h (e⇠), and

� := ⇢�1( Im �� ⌘h(e⇠)), as above.

Proof. Rescaling v±j ! �21,v±

j

v±j in (5.44), for any choice of �21,v±

j

, we ob-tain immediately relations (5.87)–(5.93) and statements (i)–(ii) from the bounds


(2.20)–(2.21) guaranteed by the conjugation Lemma together with Lemma 5.14and discussion above; recall the relation µ ⇠ �⇢↵ in (5.48).

Using the results of Remarks 5.18–5.19, we readily obtain bounds (5.96)–(5.97)from the bounds on wedge product (5.84) and determinants (5.85)–(5.86), or equiv-alently (bounds (2.20)–(2.21); see also Exercise 4.20, [Z.3]) the limiting, constant-coe�cient versions

(5.98)

�

1

�1

0

!^ · · · ^

�n

�n0

!

±(z),

(5.99) det✓� �0 0

◆

±(z),

and

(5.100) det

� �0 0

!

±(z),

be bounded above and below at z = 0: more precisely, from the correspondingrequirements on the restriction to each s := K±

h -dimensional subspace of solu-tions corresponding to a single branch singularity ⌘±h (·) (clearly su�cient, by grouptransversality, statement (i)).

For example, in the representative case s = 2r, we find from Remark 5.18(c)

that there are r decaying modes �±j and r growing modes

±j , which at z = 0 are

given by rescalings �21,v±

j,k

v±j,k of the vectors v±j,k associated with the branch cycle

in question. Choosing common scalings �21,v±

j,k

:⌘ �' for all decaying modes '±j

and �21,v±

j,k

:⌘ � for all growing modes ±j , and applying the result of Remark

5.19, we find that ('1

^ · · ·^'n)±(0) ⇠ 1 and det (�, )±(0) ⇠ 1 are equivalent to

�r'

�(|�|+ |⇢|)1/s

�r(r�1)/2 ⇠ 1

and�r'�

r

�(|�|+ |⇢|)1/s

�2r(2r�1)/2 ⇠ 1,

respectively, from which we immediately obtain the stated bounds on t'±j

and te'±j

.The bounds on te'±

j

and t e ±j

then follow by duality,

(5.101)(�, ) = [(�, )�1S e⇠�1]⇤

= S e⇠�1⇤(�, )�1⇤,

158 KEVIN ZUMBRUN

S e⇠ ⇠ I as defined in (4.25), which, restricted to the s⇥ s cycle in question, yields�e' ⇠ ��1

' (|�| + |⇢|)(s�1)/s and � e ⇠ ��1

(|�| + |⇢|)(s�1)/s, where �e', � e denotecommon scalings for the dual growing, decaying modes associated with the branchcycle in question. The computations in other cases are analogous. ⇤

Example. In the “generic” case of a square-root singularity s = 2, calculation(5.101) becomes simply

(e', e ) := S�1⇤(', )�1⇤ = (1/2)S�1⇤✓

1 ⇡1/2

⇡�1/2 �1

◆,

where

(', ) :=✓

1 ⇡�1/2

⇡1/2 �1

◆

(recall: chosen so that det (', ) = 2 ⇠ 1) and S denotes the (undetermined) re-striction of S e⇠ to the two-dimensional total eigenspace associated with the branch-ing eigenvalue. Note that we do not require knowledge of S in order to determinethe orders of e' and e . This simple case is the one relevant to gas dynamics, forwhich glancing involves only the pair of strictly hyperbolic acoustic modes; seeRemark 4.14, [Z.3].

Remark. The requirements |�1

^ · · · ^ �n|±(0) ⇠ 1 are needed in order thatthe bases chosen in Lemma 5.14 be consistent with those defined in previous sec-tions, in particular that they induce the same Evans function. (Note: since wedo not require individual analyticity of our basis elements, we are free to chooseC1 rescalings �

21,� so as to exactly match the Evans function defined in Section3). The requirements det (�, )±(0), det (�, )±(0) ⇠ 1 are only a convenientnormalization, to simplify certain calculations later on.

From now on, we will assume that the bases {'±j }, {e'±j }, { ±j }, { e ±j } have beenchosen as described in Lemma 5.22. For later use, we record the useful relations:

(5.102)

(t'j

+ te'j

)±, (t j

+ t e j

)± ⌘ (1� 1/s),

(t j

� t'j

)±, (te'j

� t e j

)± ⌘ 1/2,

(t'j

+ t e j

)±, (t j

+ te'j

)± ⌘ 1/2� 1/s,

t'±j

, t e ±j

(1/4)(1� 1/s)

between exponents t� in (5.97); these may be verified case-by-case.

Refined derivative bounds. Similarly as in the one-dimensional case, for slow,dual modes the derivative bounds (5.88), (5.90) can be considerably sharpened,provided that we appropriately initialize our bases at ⇢ = 0, and this observationis significant in the Lax and overcompressive case.


Due to the special, conservative structure of the underlying evolution equations,the adjoint eigenvalue equation

(5.103) (B11⇤ ew0)0 = �A1⇤ ew0

at ⇢ = 0 admits an n-dimensional subspace of constant solutions; this is equivalentto the familiar fact that integral quantities are conserved under time evolution forsystems of conservation laws. Thus, at ⇢ = 0, we may choose, by appropriate changeof coordinates if necessary, that slow-decaying dual modes e'±j and slow-growingdual modes e ±j be identically constant, or equivalently that not only fast-decayingmodes '±j , but also fast-growing modes ±j be solutions of the linearized first-ordertraveling wave ODE

(5.104) B11w0 = A1w.

Note that this does not interfere with our previous choice in Lemma 3.4, nor doesit interfere with the specifications of Lemma 5.22, since these concern only thechoice of limiting solutions w

±j of the asymptotic, constant coe�cient equations at

x1

! ±1, and not the particular representatives w±j that approach them (which,

in the case of slow modes, are specified only up to the addition of an arbitraryfast-decaying mode).

Lemma 5.23[Z.3]. With the above choice of bases at ⇢ = 0, slow modes e'±k , e ±k ,

for � 2 �e⇠ and ⇢ := |(e⇠,�)| su�ciently small, satisfy

(5.105)|(@/@y1) e ±k | C⇢| e ±k |,|(@/@y1)e'±k | C⇢|e'±k |.

Proof. Recall that slow dual modes e'k or e'k, by our choice of basis, are iden-tically constant for ⇢ = 0, (i.e (@/@y1)e'k|

⇢=0, (@/@y1)e'k|

⇢=0⌘ 0, making (5.105)

plausible). Indeed, the manifold of all slow dual modes, both decaying and growing,is exactly the manifold of constant functions. Now, consider the evolution of thismanifold along the curve (e⇠,�)(e⇠

0

, ⌧0

, ⇢) defined in (5.43), �0

=: i⌧0

. The conglom-erate slow manifold is C1 in ⇢, even though the separate decaying and growingmanifolds may be only C0; for, it is spectrally separated from all fast modes. Thus,the subspace

(5.106) Span{(w,w0)t : w 2 slow manifold}

lies for each y1

within angle C⇢ of the space {(w, 0)t} of constant functions, where“0” denotes @/@y1 . But, this is exactly the statement (5.105). ⇤

160 KEVIN ZUMBRUN

Bounds (5.105) are to be compared with the bounds

|(@/@y1) e ±k | C⇢| e ±k |+ C�21, e ±

k

e�✓|y1|,

|(@/@y1)e'±k | C⇢|e'±k |+ C�21, e ±

k

e�✓|y1|,

resulting from (5.88), (5.90), and (5.94).

Low-frequency Evans function bounds. The results of Section 5.1 suggestthat the zeroes �⇤(e⇠) of the Evans function satisfy

(5.107) Re �⇤(e⇠) �✓|e⇠|, | Im �⇤(e⇠)| C|e⇠|; ✓, C > 0,

for uniformly inviscid stable shocks, i.e. they move linearly into the stable complexhalf-plane with respect to |e⇠|, and

(5.108) �⇤(e⇠) = �⇤j := i⌧j(e⇠)� �j(e⇠)|e⇠|2 + o(|e⇠|2), j = 1, . . . , `,

⌧j real, distinct, �j real, > 0, for weakly inviscid stable shocks satisfying the refineddynamical stability condition, where i⌧j = �j(e⇠) are the roots of �(e⇠j , ·) (recall,strong refined dynamical stability requires that they be simple, Definition 1.21).The following result quantifies these observations via appropriate polar coordinatecomputations centered around the refined dynamical stability condition.

Lemma 5.24[Z.3]. Under the hypotheses of Theorem 5.5, for � 2 @⇤e⇠ and ⇢ :=|(e⇠,�)| su�ciently small, there holds

(5.109) |D(e⇠,�)|�1 C⇢�`

for uniformly inviscid stable (U.I.S) shocks, and(5.110)|D(e⇠,�)|�1 C⇢�`+1[min

j(| Im (�)� i⌧j(e⇠)|+ |Re (�) + �j(e⇠)|e⇠|2|+ o(|e⇠|2))]�1

otherwise, where ⌧j, �j are as in (5.108), and ⇤e⇠ is as in (2.62).

Proof. Define the curves

(5.111) (e⇠,�)(⇢, e⇠0

,�0

) :=�⇢e⇠

0

, ⇢�0

� ✓(⇢|�0

|)2 � ✓(⇢|e⇠0

|)2�,

for e⇠0

2 Rd�1, Re �0

= 0, |e⇠0

|+ |�0

| = 1, and set

(5.112)=

D (⇢, e⇠0

,�0

) := D(e⇠(⇢, e⇠0

,�0

),�(⇢, e⇠0

,�0

)).

Inspection of the proof of Lemma 3.3 (either in [MeZ.2] or the later calculationsin Section 5.3.2) show that continuity of slow decaying modes holds not only along


rays, but also along the curved paths (e⇠,�)(⇢, e⇠0

,�0

) defined in (5.111), provided✓ > 0 is taken su�ciently small. Thus, we obtain by a calculation identical to thatin the proof of Theorem 3.5, for ✓ > 0 su�ciently small, that

(5.113)=

D (e⇠0

,�0

, 0) = ⇢`�(e⇠0

,�0

) + o(⇢`).

In the U.I.S. case, �(e⇠0

,�0

) 6= 0, hence, for ⇢ su�ciently small,

(5.114)|

=

D (e⇠0

,�0

, ⇢)|�1 C|�(e⇠0

,�0

)|�1⇢�`

C2

(|e⇠|+ |�|)�`

as claimed.It remains to verify (5.110) for �

0

in the vicinity of a point i⌧j(e⇠0) such that�(e⇠

0

, i⌧j(e⇠0)) = 0, where, by assumption, � and D are analytic, with �� 6= 0.Introducing again the function

(5.115) ge⇠,⇢(�) := ⇢�`D(⇢e⇠, ⇢�)

used in Section 5.1, we thus have

(5.116)

=

D (e⇠0

,�0

, ⇢) = ⇢`ge⇠0,⇢(�

0

� ✓⇢)= ⇢`[g⇠0,0(i⌧j) + g⇠0,0

⇢ (i⌧j)⇢

+ g⇠0,0� (i⌧j)(�0

� ✓⇢0

� i⌧j) +O(⇢2 + |�0

� i⌧j |2 + ⇢2)].

Using ge⇠0,0(i⌧j) = �(e⇠

0

, i⌧j) = 0, ge⇠0,0� (i⌧j) = ��(e⇠0, i⌧j) 6= 0, and �� 6= 0,

together with definitions �j := (g⇢/g�)|e⇠0|�2 and �(e⇠0

,�0

, ⇢) := (�0

� ✓⇢)⇢, weobtain (factoring out the term g�)

(5.117)|

=

D (e⇠0

,�0

, ⇢)| � C�1|⇢`| |��| |�0

� ✓⇢� i⌧j + �j |e⇠0|2⇢+⇥|= C�1|⇢`�1| |��| |�� i⌧j(e⇠) + �j |e⇠|2 + ⇢⇥|,

where

(5.118) ⇥ = O(|�0

� i⌧j |2 + ⇢2).

The result then follows by the observation that |e⇠0

|/|⌧j(e⇠0)| is bounded from zero(recall, �(0, 1) 6= 0), whence ⇢ = O(e⇠). ⇤

Scattering coe�cients. It remains to estimate the behavior of coe�cientsM±

jk, d±jk defined in Corollary 2.26, as ⇢ ! 0, ⇢ := |(e⇠,�)|. Consider coe�cientsM±

jk. Expanding (2.101) using Cramer’s rule, and setting z = 0, we obtain

(5.119) M±jk = D�1C±

jk,

162 KEVIN ZUMBRUN

where

(5.120) C+ := (I, 0)✓�+ ��+0 ��0

◆adj

✓ � �0

◆

|z=0

and a symmetric formula holds for C�. Here, P adj denotes the adjugate matrix ofa matrix P , i.e. the transposed matrix of minors. As the adjugate is polynomialin the entries of the original matrix, it is evident that, at least away from branchsingularities ⌘±j (e⇠), |C±| is uniformly bounded and therefore

(5.121) |M±jk| C

1

|D�1| C2

⇢�`

8><

>:

1, U.I.S.[minj ⇢�1(| Im (�)� ⌧j(e⇠)|+ ⇢)]�1,

otherwise,

by Lemma 5.24, where C1

, C2

> 0 are uniform constants.However, the crude bound (5.121) hides considerable cancellation, a fact that

will be crucial in our analysis. Again focusing on the curve (e⇠,�)(e⇠0

,�0

, ⇢) definedin (5.43), let us relabel the {'±} so that, at ⇢ = 0,

(5.122) '+

j ⌘ '�j = @U�/@�j , j = 1, · · · , `.

(Note: as observed in Chapter 3, the fast decaying modes, among them {@U�/@�j},can be chosen independent of (e⇠

0

,�0

) at ⇢ = 0).

In the case that �(e⇠0

,�0

) = 0 or equivalently (@/@⇢)`=

D(⇠0

,�0

, ⇢) = 0, there isan additional dependency among

(@/@⇢)'±1

, · · · , (@/@⇢)'±` and '±1

, · · · ,'±n ,

so that we can arrange either

(5.123) '+

`+1

= '�`+1

or else (after C1 change in coordinates)

(5.124) (@/@⇢)'+

1

= (@/@⇢)'�1

modulo slow decaying modes. The corresponding functions '`+1

, (@/@⇢)'1

, respec-tively, have an interpretation in the one-dimensional case as “e↵ective eigenfunc-tions,” see [ZH], Section 5–6 for further discussion.

By assumption (1.42), in fact only case (5.124) can occur. For, review of the

calculation giving (@/@⇢)`=

D(e⇠0

,�, ⇢) = �(e⇠0

,�) reveals that a dependency of form(5.123) implies dependence of rows involving r±j in �. With convention (5.124), wehave the sharpened bounds:


Lemma 5.25[Z.3]. Under the hypotheses of Theorem 5.5, for � 2 �e⇠ and ⇢ :=|(e⇠,�)| su�ciently small, there holds

(5.125) |M±jk|, |d

±jk| C�

22,� �1

, � = M±jk, d±jk,

where(5.126)�22,M±

jk

�21,'±

j

�21, e ⌥

k

⇥1+

X

j

(|�+

j |+|⇢|)�12 (1�1/K+

j

)

⇤⇥1+

X

j

(|��j |+|⇢|)�12 (1�1/K�

j

)

⇤,

(5.127) (�22,d

jk

�21,'

j

�21, e

k

)± 1 +X

j

(|�±j |+ |⇢|)�(1�1/K±j

),

with �21,� as defined in (5.96)–(5.97), �±j := ⇢�1( Im � � ⌘±j (e⇠)), ⌘j(·) and K±

j

as in (H6), and, for uniformly inviscid stable (U.I.S) shocks,

(5.128) �1

:=⇢⇢�1 for j = 1, · · · , `,

1 otherwise,

while, for weakly inviscid stable (W.I.S) shocks,

(5.129) �1

:=

8><

>:

�minj |Im(�)� i⌧j(e⇠)|+ ⇢2

��1 for j = 1,

⇢�1 for j = 2, · · · , `,

1 otherwise,

⌧j(·) as in (5.108). (Here, as above, U.I.S. and W.I.S. denote “uniform inviscidstable” and “weak inviscid stable,” as defined, repectively, in (1.32), Section 1.4,and (1.28), Section 1.3; the classification of Lax, overcompressive, and undercom-pressive types is given in Section 1.2).

That is, apart from the factor induced by branch singularities, blowup in Mjk

occurs to order ⇢`�1|D�1| rather than |D�1|, and, more importantly only in fast-decaying modes.

Proof. Formula (5.120) may be rewritten as

(5.130) C+

jk = det

0

@'+

1

, · · · ,'+

j�1

, �k ,'+

j , · · · ,'+

n , ��

'+

1

0, · · · ,'+

j�1

0, �k0,'+

j0, · · · ,'+

n , ��0

1

A

|z=0

,

from which we easily obtain the desired cancellation in M+ = C+D�1. For ex-ample, in the U.I.S. case, we obtain for j > ` that, along path (e⇠,�)(e⇠

0

,�0

, ⇢), wehave, away from branch singularities ⌘±j (e⇠):

(5.131)C+

jk = det

0

@'+

1

+ ⇢'+

1

⇢

+ · · · , · · · , '+

n + ⇢'+

n⇢

+ · · ·

· · · , , · · · , '+

n0 + ⇢'+

n⇢

0 + · · ·

1

A

= O(⇢`) C|D|,

164 KEVIN ZUMBRUN

yielding |Mjk| = |Cjk||D|�1 C as claimed, by elimination of ` zero-order terms,using linear dependency among fast modes at ⇢ = 0. Note, similarly as in the proofof Lemma 5.24, that we require in this calculation only that fast modes be C1 in⇢, while slow modes may be only continuous. For this situation, �

22

⇠ 1 su�ces.In the vicinity of one or more branch singularities, we must also take account of

blowup in associated slow modes, as quantified by the factors �21,� in Lemma 5.22.

Branch singularities not involving �+

j or e �k clearly do not a↵ect the calculation,since we have chosen a normalization for which complete cycles of branching (super-slow) modes �±j have wedge products of order one; thus, �

22

⇠ 1 again su�ces forthis case. In the vicinity of a branch singularity involving either or both of the modes�+

j or e �k , however, there arises an additional blowup that must be accounted for.Since this blowup involves only slow modes, it enters (5.131) simply as an addi-

tional multiplicative factor corresponding to the modulus of their wedge productsat z = 0; as in the proof of Lemma 5.22, these can be estimated by the wedge prod-ucts of the corresponding modes for the limiting, constant coe�cient equations atx

1

! ±1, which at z = 0 are exactly the vectors vj , vjk described in Lemma 5.14.

Since complete cycles have order one wedge product, as noted above, we need onlycompute the blowup in branch the cycles directly involving �+

j / e �k , since these aremodified in expression (5.131), respectively by deleting one decaying mode, �+

j , andaugmenting one dual, growing mode, �k .

Using the result of Remark 5.19, and the scaling �21,� given in Lemma 5.22 for

the modulus of individual modes, we find that the deleted cycle has wedge productof order

(5.132) ⇡(r�1)(r�2)/2⇡�(r�1)(r�1)/2 = ⇡�(r�1)/2 = �21,�+

j

,

where r denotes the number of decaying modes in the original cycle: r = r fors := K+

h = 2r or s = 2r + 1 and p > 0, otherwise r = r + 1, and ⇡ := (|�|+ |⇢|)1/s

as in the notation of Lemma 5.22. Here, s := K+

h is the multiplicity of the nearbybranch singularity ⌘+

h (e⇠) associated with mode �+

j , and � = 21,�+

j is as defined in(5.97).

Similarly, the augmented cycle involving �k yields a wedge product of order

(5.133) ⇡(r+1)(r)/2⇡�(r)(r�1)/2⇡�(2s�r�2)/2 = ⇡�(3r�2s+1)/2 = �21, e �

k

,

where �21, e �

k

is as defined in (5.96)–(5.97). The asserted bound on �22

then followsby the observation (see (5.102)) that

t�±j

, t e ±j

(1/4)(1� 1/K±h )

in (5.97) (K±h =: s =: 2r in the notation of formulae (5.97)). This completes the

estimation of |Mjk| in the case x1

� y1

; the case x1

y1

may be treated by asymmetric argument.


The bounds on |d±jk| follow similarly, using (2.100) in place of (2.99). In thesecases, however, there is the new possibility that the augmented mode k and thedeleted mode 'j may belong to the same cycle, since both are associated with thesame spatial infinity. In this case, the total number of modes remains fixed, with agrowth mode replacing a decay mode, and the resulting blowup in the associatedwedge product is simply

(5.134) (�21,

j

/�21,'

j

)±.

The asserted bound on (�22

�21,'

j

�21, e

k

)± then follows from relation

(t j

� t'j

+ t'j

+ t e k

)± = (t j

� t'j

+ t'j

+ t e j

)±

⌘ 1� 1/K±h ,

again obtained by inspection of formulae (5.97) (see (5.102)), where K±h as usual

denotes the order of the associated branch singularity. If on the other hand theaugmented and deleted modes belong to di↵erent cycles, then the computationreduces essentially to that of the previous case, yielding

�22

= (�21,'

j

�21, e

k

)±,

from which the result follows as before (indeed, we obtain a slightly better bound,see (5.102)). ⇤

Remark 5.26. Note that frequency blowup in term Mjk'+

je �k splits evenly

between coe�cient Mjk and modes '+

je �k , see (5.132)–(5.133).

Lax and Overcompressive Case. For Lax and overcompressive shocks, wecan say a bit more. First, observe that duality relation(5.135)

det✓ e � e��e �0 e��0

◆⇤S✓�+ ��+0 ��0

◆= det

0

@

✓ e �e �0

◆⇤S✓�+

�+0

◆0

⇤ I

1

A

= det

0

@

✓ e �e �0

◆⇤S✓�+

�+0

◆⇤

0 I

1

A

= det✓ e � e +

e �0 e +0

◆⇤S✓�+ +

�+0 +0

◆

yields that D(e⇠,�) and the dual Evans function

(5.136) eD := det✓ e � e +

e �0 e +0

◆

|x1=0

166 KEVIN ZUMBRUN

vanish to the same order as (e⇠,�) ! (0, 0), since the determinants of✓ e � e��e �0 e��0

◆,

S, and✓�+ +

�+0 +0

◆are by construction bounded above and below. Indeed, sub-

spaces ker✓�+ ��+0 ��0

◆and ker

✓ e + e +

e �0 e +0

◆can be seen to have the same di-

mension, dim ker✓ e �e �0

◆⇤S✓�+

�+0

◆, which at ⇢ = 0 is equal to `. This compu-

tation partially recovers the results of [ZH], section 6, without the assumption ofsmoothness in {'±j }, { ±j }. The “contracted” Evans function

det✓ e �e �0

◆⇤S✓�+

�+0

◆

is discussed further in [BSZ.1].It follows that there is an `-fold intersection between

Span

e �je �j 0

!and Span

e +

j

e +

j0

!

We now recall the important observation of Lemma 4.28. Recall, Section 4.5.2,that we have fixed a choice of bases such that, at ⇢ = 0, both slow-decaying dualmodes e'±j and slow-growing dual modes e ±j are identically constant, or equivalentlythat not only fast-decaying modes '±j , but also fast-growing modes ±j are solutionsof the linearized traveling wave ODE (5.104). With this choice of basis, we have forLax and overcompressive shocks that that the only bounded solutions of the adjointeigenvalue equation at ⇢ = 0 are constant solutions, or, equivalently, fast-decayingmodes j at one infinity are fast-growing at the other, and fast-decaying j at oneinfinity are fast-growing at the other. Combining the above two results, we have:

Lemma 5.27. Under the hypotheses of Theorem 5.5, with |(e⇠,�)| su�ciently small.for Lax and overcompressive shocks, with appropriate basis at ⇢ = 0 (i.e. slow dualmodes taken identically constant), there holds

(5.137) |Mjk|, |djk| C�22

if e k is a fast mode, and

(5.138) |Mjk|, |djk| C�22

⇢

if, additionally, 'j is a slow mode, where �22

is as defined in (5.127)–(5.126).

Proof. Transversality, � 6= 0, follows from (D2), so that the results of Lemma4.28 hold. Consider first, the more di�cult case of uniform inviscid stability,�(e⇠

0

,�) 6= 0.


We first establish the bound (5.137). By Lemma 5.25, we need only considerj = 1, . . . , `, for which ��j = �+

j . By Lemma 4.28, all fast-growing modes +

k , �klie in the fast-decaying manifolds Span('�

1

, · · · ,'�i�), Span('+

1

, · · · ,'+

i+) at �1,

+1 respectively. It follows that in the right hand side of (5.130), there is at ⇢ = 0a linear dependency between columns

(5.139) '+

1

, · · · ,'+

j�1

, �k ,'+

j+1

,'+

i+and '�j = '+

j

i.e. an `-fold dependency among columns

(5.140) '+

1

, · · · , �k , · · · ,'+

i+ and '�1

, · · · ,'�` ,

all of which, as fast modes, are C1 in ⇢. It follows as in the proof of Lemma 5.25that |Cjk| C�

22

⇢` for ⇢ near zero, giving bound (5.137) for Mjk. If '+

j is a slowmode, on the other hand, then the same argument shows that there is a lineardependency in columns

(5.141) '+

1

, · · · ,'+

j�1

, �k ,'+

j+1

,'+

i+

and an (` + 1)-fold dependency in (5.140), since the omitted slow mode '+

j playsno role in either linear dependence; thus, we obtain the bound (5.138), instead.Analogous calculations yield the result for d±jk as well.

In the case of weak inviscid stable shocks, the above argument su�ces near (e⇠,�)such that �(e⇠,�) 6= 0 (again, computing as in the proof of Lemma 5.25 along thecurves (e⇠, ⇢)(e⇠

0

,�, ⇢)). Near points such that �(e⇠0

,�) = 0, all modes are C1 in ⇢,and the result can be obtained more simply by a dual version of the calculation in theproof of Lemma 5.25, together with the observation that the “dual eigenfunctions”e ±k lying in Span(e �)\ Span(e +) are by Lemma 4.28 restricted to the slow modes.⇤

Proof of Proposition 5.10. The proof of Proposition 5.10 is now straight-forward. Collecting the results of Lemmas 5.22–5.23, we have the spatial decaybounds

(5.142)|'j(x1

)| C�21,'

j

e�✓|x1|,

|'j0(x

1

)| C�21,'

j

e�✓|x1|,

for fast forward modes,

(5.143)|'j(x1

)| C�21,'

j

e�✓⇢2|x1|,

|'j0(x

1

)| C�21,'

j

(⇢e�✓⇢2|x1| + e�✓|x1|),

168 KEVIN ZUMBRUN

for slow forward modes

(5.144)| e k(y

1

)| C�21, e

k

e�✓|y1|,

| e k0(y

1

)| C�21, e

k

e�✓|y1|,

for fast dual modes, and

(5.145)| e k(y

1

)| C�21, e

k

e�⇢2✓|y1|,

| e k0(y

1

)| C�21, e

k

(⇢e�✓⇢2|y1| + ↵e�✓|y1|),

for slow dual modes, where �21,� are as defined in (5.96)–(5.97) and

↵ =⇢

0 for Lax and overcompressive shocks,1 for undercompressive shocks.

Likewise, the results of Lemmas 5.25 and 5.27 give the frequency growth bounds

(5.146) |M±jk|, |d

±jk| C�

22

for slow modes '±j , and

(5.147) |M±jk|, |d

±jk| C⇢�1�

22

�1

for fast modes '±j , where �1

is as defined in (5.37) and �22

as in (5.127)–(5.126).In the critical case that '±j is slow and e ±k is fast, (5.146) can be sharpened to

(5.148) |M±jk|, |d

±jk| C�

22

(⇢+ ↵),

↵ as above.Combining these bounds, we may estimate the various terms arising in the de-

compositions of Corollary 2.26 as:

|�jMjke k|, |�jdjk

e k| C�21,'

j

�21, e

k

�22

�1

⇢�1e�✓|x1|e�✓⇢2|y1|

and

|(@/@y1

)�jMjke k|, |(@/@y

1

)�jdjke k|

C�21,'

j

�21, e

k

�22

�1

⇢�1e�✓|x1|(⇢e�✓⇢2|y1| + ↵e�✓|y1|)

for fast modes 'j ;

|�jMjke k|, |�jdjk

e k| C�21,'

j

�21, e

k

�22

e�✓⇢2|x1|e�✓⇢

2|y1|


and

|(@/@y1

)�jMjke k|, |(@/@y

1

)�jdjke k|

C�21,'

j

�21, e

k

�22

e�✓⇢2|x1|(⇢e�✓⇢

2|y1| + ↵e�✓|y1|)

for slow modes 'j ;

|'j e'j |± C(�21,'

j

�21,e'

j

)±e�✓|x1�y1|

and|(@/@y

1

)'j e'j |± C(�21,'

j

�21,e'

j

)±e�✓|x1�y1|

for fast modes 'j ;

|'j e'j |± C(�21,'

j

�21,e'

j

)±e�✓⇢2|x1�y1|

and|(@/@y

1

)'j e'j |± C(�21,'

j

�21,e'

j

)±⇢e�✓⇢2|x1�y1|

for slow modes 'j ;

| je j |± C(�

21, j

�21, e

j

)±e�✓|x1�y1|

and|(@/@y

1

) je j |± C(�

21, j

�21, e

j

)±e�✓|x1�y1|

for fast modes j ; and

| je j |± C(�

21, j

�21, e

j

)±e�✓⇢2|x1�y1|

and|(@/@y

1

) je j |± C(�

21, j

�21, e

j

)±⇢e�✓⇢2|x1�y1|

for slow modes j . Using bounds (5.127)–(5.126), and recalling (see (5.102)) that

(t'j

+ te'j

)±, (t j

+ t e j

)± ⌘ 1� 1/K±j ,

we thus obtain the result by checking term by term. ⇤

170 KEVIN ZUMBRUN

5.3.2. High-frequency bounds.

Modifying the auxiliary energy estimate and one-dimensional resolvent estimatesalready derived, we obtain in straightforward fashion the following high-frequencyresolvent bounds.

Proposition 5.28. Under the hypotheses of Theorem 5.5, for |e⇠| su�ciently large,and some C, ✓ > 0,

(5.149) |eLe⇠

tf |ˆH1

(x1,e⇠) Ce�✓t|f |ˆH1

(x1,e⇠)

for |f |ˆH1

(x1,e⇠) := (1 + |e⇠|)|f |H1(x1).

Proof. Carrying out a Fourier-transformed version of the auxiliary energy esti-mate of Propositions 4.15 and 5.9 in the simpler, linearized setting, we obtain

(d/dt)E(W (t)) �✓E(t) + C|W |2L2(x1)

,

✓ > 0, where(5.150)E(W ) := M(1 + |e⇠|2)hW,↵A0W i+ h↵K(@x

1

, ie⇠)W,W i+ Mh@x1W,↵A0@x1W i,M , C > 0 su�ciently large constants, K(@x) a first-order di↵erential operator asdefined in (5.19)–(5.20), or, more generally, an analogous first-order pseudodi↵eren-tial operator as discussed in the proof of Proposition 5.9 (see also [Z.4]). Observingthat |W |2L2 C|e⇠|�2E(W ) can be absorbed in ✓E(W ), we obtain

(d/dt)E(W (t)) �(✓/2)E(t),

from which the result follows by the fact that E(W )1/2 is equivalent to the H1(x1

, e⇠)norm of W . ⇤Proposition 5.29. Under the hypotheses of Theorem 5.5, for |e⇠| R and |�|su�ciently large, Re � � �⌘ is contained in the resolvent set ⇢(Le⇠) for some uni-form ⌘ > 0 su�ciently small, with resolvent kernel Ge⇠,�(x, y) satisfying the samedecomposition (4.175)–(4.187) as in the one-dimensional case, but with dissipa-tion coe�cients ⌘⇤j replaced by multi-dimensional versions ⌘j(e⇠, x)⇤, exponentiallydecaying to asymptotic values

(5.151) Re �⌘j(e⇠,±1)⇤ > 0.

Proof. For |e⇠| bounded and |�| su�ciently large, we may treat e⇠ terms as first-order perturbations in the one-dimensional analysis in the proof of Proposition 4.33,order |�|�1 after the rescaling of (4.189)–(4.193). These a↵ect only the bounds onthe hyperbolic block, modifying the form of ⌘⇤j in (4.202). We do not require theprecise dependence of the resulting ⌘j(e⇠, x)⇤ on e⇠, but only the bound (5.151),which follows as in the one-dimensional case by (2.55) together with the fact that�Re �⌘j(e⇠,±1) corresponds to the zero-order term in the expansion of hyperbolicmodes of symbol �A

e⇠ �Be⇠,e⇠. ⇤


5.3.3 Bounds on the solution operator.

We are now ready to establish the claimed bounds on the linearized solution op-erator. Define “low-” and “high-frequency” parts of the linearized solution operatorS(t)

(5.152) S1

(t) :=1

(2⇡)di

Z

|e⇠|r

I

�

e⇠

e�t+ie⇠·x(Le⇠ � �)�1 d�de⇠

and

(5.153) S2

(t) := eLt � S1

(t),

�e⇠ as in (5.36) and r > 0 su�ciently small.

Proof of Proposition 5.7. We estimate S1

and S2

in turn, for simplicityrestricting to the case of a uniformly stable Lax or overcompressive shock, �

1

= 1.

S1

bounds. Let u(x1

, e⇠,�) denote the solution of (Le⇠ ��)u = f , where f(x1

, e⇠)denotes Fourier transform of f , and

u(x, t) := S1

(t)f =1

(2⇡)di

Z

|e⇠|r

I

�

e⇠

e�t+ie⇠·x(Le⇠ � �)�1f(x1

, e⇠) d�de⇠.

Bounding |f |L1(

e⇠,L1(x1))

|f |L1(x1,x)

= |f |1

using Hausdor↵–Young’s inequality,and appealing to the L1 ! Lp resolvent estimates of Corollary 5.11, we may thusbound

|u(x1

, e⇠,�)|Lp

(x1) |f |1

b(e⇠,�),

where b := C�2

⇢�1.L2 bounds. Using in turn Parseval’s identity, Fubini’s Theorem, the triangle

inequality, and our L1 ! L2 resolvent bounds, we may estimate

(5.154)

|u|L2(x1,x)

(t) =⇣ 1

(2⇡)d

Z

x1

Z

e⇠2Rd�1

��I

�2˜

�(

e⇠)e�tu(x

1

, e⇠,�)d��2

de⇠ dx1

⌘1/2

=⇣ 1

(2⇡)d

Z

e⇠2Rd�1

��I

�2˜

�(

e⇠)e�tu(x

1

, e⇠,�)d��2

L2(x1)

de⇠⌘

1/2

⇣ 1

(2⇡)d

Z

e⇠2Rd�1

��I

�2˜

�(

e⇠)|e�t||u(x

1

, e⇠,�)|L2(x1)d�

��2

de⇠⌘

1/2

|f |1

⇣ 1(2⇡)d

Z

e⇠2Rd�1

��I

�2˜

�(

e⇠)eRe �tb(e⇠,�)d�

��2

de⇠⌘

1/2

,

from which we readily obtain the claimed bound on |S1

(t)f |L2(x)

using the boundson b. Specifically, parametrizing �(e⇠) by

�(e⇠, k) = ik � ✓1

(k2 + |e⇠|2), k 2 R,

172 KEVIN ZUMBRUN

and observing that in nonpolar coordinates

(5.155)

⇢�1�2

h(|k|+ |e⇠|)�1(1 +

X

j�1

⇣ |k � ⌧j(e⇠)|⇢

⌘ 1s

j

�1

i

h|k|+ |e⇠|)�1(1 +

X

j�1

⇣ |k � ⌧j(e⇠)|⇢

⌘"�1

i,

where " := 1

max

j

sj

(0 < " < 1 chosen arbitrarily if there are no singularities), weobtain a contribution bounded by(5.156)

C|f |1

⇣Z

e⇠2Rd�1

��Z

+1

�1e�✓(k

2+|e⇠|2)t(⇢)�1�

2

dk��2

de⇠⌘

1/2

C|f |1

Z

e⇠2Rd�1

⇣e�2✓|e⇠|2t|e⇠|�2"

��Z

+1

�1e�✓|k|

2t|k|"�1dk��2

de⇠⌘

1/2

+CX

j�1

|f |1

Z

e⇠2Rd�1

⇣e�2✓|e⇠|2t|e⇠|�2"

��Z

+1

�1e�✓|k|

2t|k � ⌧j(e⇠)|"�1dk��2

de⇠⌘

1/2

C|f |1

Z

e⇠2Rd�1

⇣e�2✓|e⇠|2t|e⇠|�2"

��Z

+1

�1e�✓|k|

2t|k|"�1dk��2

de⇠⌘

1/2

C|f |1

t�(d�1)/4

as claimed. Derivative bounds follow similarly.L1 bounds. Similarly, using Hausdor↵–Young’s inequality, we may estimate

(5.157)

|u|L1(x1,x)

(t) supx1

1(2⇡)d

Z

e⇠2Rd�1

��I

�2˜

�(

e⇠)e�tu(x

1

, e⇠,�)d��de⇠

1(2⇡)d

Z

e⇠2Rd�1

I

�2˜

�(

e⇠)|e�t||u(x

1

e⇠,�)|L1(x1)d�de⇠

|f |1

1(2⇡)d

Z

e⇠2Rd�1

I

�2˜

�(

e⇠)eRe �tb(e⇠,�)d�de⇠,

to obtain the claimed bound on |S1

(t)f |1. Parametrizing �(e⇠) again by

�(e⇠, k) = ik � ✓1

(k2 + |e⇠|2), k 2 R,

we obtain a contribution bounded by

(5.158)

C|f |1

Z

e⇠2Rd�1

Z+1

�1e�✓(k

2+|e⇠|2)t⇢�1�

2

dkde⇠

C|f |1

Z

e⇠2Rd�1e�✓|e⇠|

2t|e⇠|�"Z

+1

�1e�✓|k|

2t|k|"�1dkde⇠

C|f |1

t�(d�1)/2


as claimed. Derivative bounds follow similarly.General 2 p 1. Finally, the general case follows by interpolation between

L2 and L1 norms.

S2

bounds. Using

S(t) =1

(2⇡)d�1

Ze

e⇠·xeLe⇠

t de⇠,

we may split S2

(t) into two parts,

SI2

(t) :=1

(2⇡)d�1

Z

|e⇠|�R

ee⇠·xeLe

⇠

t de⇠,

R > 0 su�ciently large, and

SII2

(t) :=1

(2⇡)d�1

Z

|e⇠|R

ee⇠·xeLe

⇠

t de⇠ � S1

(t)

=1

(2⇡)d�1

Z

|e⇠|R

P.V.Z �0+i1

�0�i1eie⇠·x+�t(Le⇠ � �)�1 d�de⇠

� 1(2⇡)d�1

Z

|e⇠|r

I

�

e⇠

eie⇠·x+�t(Le⇠ � �)�1 d�de⇠.

The first part, SI2

(t) satisfies the claimed bounds by Parseval’s identity togetherwith (5.149), together with the observation that @x commutes with Le⇠ and thus allSk

j , by invariance of coe�cients with respect to x. The second part, SII2

(t), satisfiesthe claimed bounds by Parseval’s identity together with calculations like those ofthe corresponding one-dimensional high- and intermediate-frequency estimates inthe proof of Proposition 4.39 and the H1(x

1

) ! H1(x1

) estimates of Lemma 4.14,together with the same observation that @x commutes all Sk

j .This completes the proof, and the article. ⇤

APPENDICES A. AUXILIARY CALCULATIONS.

A.1. Applications to example systems.

In this appendix, we present simple conditions for (A1)–(A3) and (H1)–(H2),and use them to verify the hypotheses for various interesting examples from gasdynamics and MHD. Consider a system (1.1) satisfying conditions (1.5) and (1.6).(Note: (1.6) is implied by (A3) under the class of transformations considered, sinceRe b > 0 ) Re �(A0)�1b = (@W/@U)b(@U/@W ) > 0.)

Definition 6.1. Following [Kaw], we define functions ⌘(G), qj(G) : Rn ! R1 tobe a convex entropy, entropy flux ensemble for (1.1), (1.6) if: (i) d2⌘ > 0 (⌘ convex),(ii) d⌘dF j = dqj , and (iii) d2⌘Bjk is symmetric (hence positive semidefinite, by(1.6)).

174 KEVIN ZUMBRUN

Proposition 6.2[KSh]. A necessary and su�cient condition that a system (1.1),(1.6) can be put in symmetric form (1.7) by a change of coordinates U ! W (U)for U in a convex set U is existence of a convex entropy, entropy flux ensemble ⌘,qj defined on U , with W = d⌘(U) (known as an “entropy variable”).

Proof. (() The change of coordinates U ! W := d⌘(U) is invertible on anyconvex domain, since its Jacobian d2⌘ is symmetric positive definite, by Defini-tion 6.1 (i), and thus we can rewrite (1.1) as (1.7), as claimed, where F j(W ) :=F j(U(W )) and Bjk(W ) := Bjk(U(W ))(@U/@W ) = Bjkd2⌘�1. Symmetry ofdF j = dF jd2⌘�1 and Bjk = Bjkd2⌘�1 follow from symmetry of d2⌘dF j andd2⌘Bjk, the first a well-known consequence of Definition 6.1 (ii) [G, Bo, Se.3], thesecond just Definition 6.1(iii). Finally, block structure and uniform ellipticity ofBjk follow by symmetry of Bjk, the fact that the wI rows must vanish, by the cor-responding property of left factor Bjk, and the fact that Re �

Pjk bjk⇠j⇠k � ✓|⇠|2,

since Bjk is similar to Bjk.()) Symmetry of A0 = dW (U) implies that d⌘(U) := W (u) is exact on any sim-

ply connected domain, while positive definiteness implies that ⌘ is convex. Likewise,symmetry of d(d⌘dF j) (following by symmetry of d2⌘dF j , a consequence of sym-metry of dF jd2⌘�1, plus the reverse calculation to that cited above) implies thatdqj(U) := d⌘dF j is exact. Finally, symmetry of d2⌘Bjk follows by symmetry ofBjkd2⌘�1. ⇤Proposition 6.3 [GMWZ.4]. Necessary and su�cient conditions that there ex-ist coordinate change W (U) and matrix multiplier S(W ) taking (1.1) to form(1.11) satisfying (A1)–(A3) are: (i) there exist symmetric, positive definite (right)symmetrizers Q(U) such that (AjQ)± and (AjQ)

11

are symmetric, and BjkQ =block-diag {0,�jk} with �jk satisfying uniform ellipticity condition (1.6), and (ii)there exists wII : U ! Rr such that Bjk(U)Ux

k

= (0, bjk(U)wII(U)xk

).If there exist such transformations, then S must be lower block-triangular and

Bjk(@U/@W ) block-diagonal, with S(U) = (@U/@W )trQ�1, and

W (U) := (wI(uI), wII(U))

for wI invertible and and wII satisfying (ii). Moreover, we may always chooseW with (@W/@U) lower block-triangular, in which case A0 = S(@U/@W ) is block-diagonal as well. (Indeed, this structure follows already from block structure of Bjk

and positive symmetric definiteness of A0 alone, without any further considera-tions; conversely, W (U) := (wI(uI), wII(U)) and S = A0(@U/@W ) for symmetricpositive definite block-diagonal A0 is su�cient to imply block structure of Bjk.)

Proof. ()) Define W (U) = (wI , wII)(U), where wI , wII are as described inthe hypotheses. Then, @W/@U is lower block triangular, of form

✓dwI 0

@wII/@uI @wII/@uII

◆,


with Bjk22

= bjk(@wII/@uII) by assumption (ii), hence (taking, e.g., j = k = 1)is invertible, by assumption (1.6). Moreover, Bjk := Bjk(@U/@W ) by assumption(ii) is block-diagonal of form block-diag {0, bjk}. Thus, choosing a lower block-triangular multiplier

S(W ) := P (W )(@W/@U), P = block-diag {P1

, P2

},

for any symmetric positive definite Pj , we obtain form (1.11), with A0 = P symmet-ric positive definite and Bjk = block-diag {0, bjk} in the required block-diagonalform. Indeed, only multipliers of this form will su�ce, since (@W/@U)Bjk(@U/@W )is also of block-diagonal form, by lower triangularity of (@W/@U), so that P mustbe lower triangular as well to preserve this property, but also P = A0 is requiredto be symmetric. Writing the block-diagonal Bjk as BjkQ

⇣Q�1(@U/@W )

⌘, where

BjkQ is block-diagonal, we find similarly that Q�1(@U/@W ) must be upper block-triangular to preserve block structure. Therefore, S(U) := (@U/@W )trQ�1 is lowerblock-triangular, and so is its product with the lower block-triangular @U/@W , sothat A0 = S(@U/@W ) = (@U/@W )trQ�1(@U/@W ) is both lower block-triangularand symmetric positive definite, hence block-diagonal as well. Further, expanding

S(U)Bjk(@U/@W ) = (@U/@W )trQ�1(BjkQ)Q�1(@U/@W )

= S(BjkQ)Str = block-diag {0, S22

(BjkQ)22

Str22

},

we find that Bjk satisfies ellipticity condition (1.6) as well. Likewise, lower block-triangularity of S yields SAj(@U/@W )

11

= S11

(AjQ)11

Str11

, verifying symmetryof SAj(@U/@W )

11

, and SAj(@U/@W )± =�S(AjQ)Str

�± and SBjk(@U/@W )± =�

S(BjkQ)Str�±, verifying symmetry of SAj(@U/@W )± and SBjk(@U/@W )±.

(() By nonsingularity of (@U/@W ), and vanishing of the first block-row ofSBjk(@U/@W ), we find that the first block-row of SBjk must vanish, hence Smust be lower block-triangular by vanishing of the first block-row of Bjk togetherwith nonsingularity of Bjj

22

. It follows by nonsingularity of S that S11

and S22

are nonsingular, whence the product of lower block-triangular S with vanishingfirst block-row Bjk(@U/@W ) can be block-diagonal only if Bjk(@U/@W ) is alreadyblock-diagonal, implying condition (ii). From the assumption that A0 = S(@U/@W )is symmetric positive definite, we find that

Q := S�1(@U/@W )tr = (@U/@W )(A0)�1(@U/@W )tr = (@U/@W )S�1,tr

is symmetric positive definite as well. Factoring AjQ = S�1

⇣SAj(@U/@W )

⌘S�1,tr,

BjkQ = S�1

⇣SBjk(@U/@W )

⌘S�1,tr, we find, again by lower block-triangularity of

S, that right symmetrizer Q preserves all properties of Aj and Bjk, thus verifying(i). Once existence of such Q is verified, it follows from the analysis in the firstpart that we can take W of form (wI(uI), wII(U)) as claimed. ⇤

176 KEVIN ZUMBRUN

Remark 6.4. The above construction may be recognized as a generalizationof the Kawashima normal form (N), described, e.g., in [GMWZ.4]; indeed, at theendstates (more generally, wherever symmetry of Aj , Bjk are enforced), the twoconstructions coincide. From the proof of Proposition 6.3, we find, more generally,that, assuming any subset of the properties asserted on AjQ, BjkQ, these propertieswill be inherited by the system (1.11), independent of the others.

Corollary 6.5. Suppose it is known that system (1.1) admits right symmetrizersQ± at U± as described above, for example, if there exist local convex entropies⌘±(U) near these states. Then, necessary and su�cient conditions for existence ofa transformed system (1.11) satisfying (A1)–(A3) are (ii) of Proposition 6.3, to-gether with the existence of symmetric positive definite matrices P

1

and P2

such thatP

1

simultaneously symmetrizes ↵j⇤ := (@wI/@uI)Aj

⇤(@wI/@uI)�1, P1

↵j⇤ symmet-

ric, and P2

simultaneously stabilizes �jk := (@wII/@uII)bjk = (@wII/@uII)Bjk22

=(@wII/@uII)�1, W as defined in (ii) above, in the sense that �jk satisfy a uniformellipticity condition (1.6).

Proof. Necessity follows from Proposition 6.3, block-diag {P1

, P2

} = A0. Su�-ciency follows by a partition of unity argument, interpolating the symmetric positivedefinite matrices Pj guaranteed at each point U , taking care at the endstates U± tochoose the special Pj (guaranteed by the existence of symmetrizers Q±) that alsosymmetrize Aj . ⇤

Corollary 6.5 suggests a strategy for verifying (A1)–(A3), namely to make thecomplete coordinate change U ! W (U), S(U) = (@W/@U), with W = (uI , wII(U))for simplicity, then look directly for simultaneous symmetrizer/stabilizers P

1

/P2

.This works extremely well for the Navier–Stokes equations of compressible gasdynamics, which appear as

(6.1)

⇢t + div(⇢u) =0,

(⇢u)t + div(⇢u⌦ u) +rp =

div⌧z }| {µ�u + (�+ µ)rdivu,

(⇢(e +12u2))t + div(⇢(e +

12u2)u + pu) = div(⌧ · u) + �T,

where ⇢ > 0 denotes density, u 2 Rd fluid velocity, T > 0 temperature, e = e(⇢, T )internal energy, and p = p(⇢, T ) pressure. Here, ⌧ := �div(u)I + 2µDu, whereDujk = 1

2

(ujx

j

+ ujx

k

) is the deformation tensor, �(⇢, T ) > 0 and µ(⇢, T ) > 0 areviscosity coe�ents, and (⇢, T ) > 0 is the coe�cient of thermal conductivity.

For these equations, U = (⇢, ⇢u,E = e + |u|2/2) and W = (⇢, u, T ), and ↵j , �jk

are symmetric, withP

j ↵j⇢x

j

= u ·r⇢ (scalar convection) and

X

jk

(�jk(u, T )xk

)xj

= ⇢�1(µ�u + (µ + �)rdivu,Te�T )


(see, e.g., [GMWZ.4]), whence suitable Pj exist if and only if Te > 0, in which casethey may be simply taken as the identity. Here, E represents total energy density.Likewise, MHD may be treated similarly, with W = (⇢, u, T,B), B the magneticfield. Note that this indeed allows the case of van der Waals equation of state.

Alternatively, we may start with the Kawashima form (N) at the endstates,and try to deduce directly an appropriate modification preserving block structurealong the profile. For example, examining form (N) for Navier–Stokes equations,for which (see, e.g., [KS], or equation (2.13) of [GMWZ.4])

A0 =

0

@p⇢/⇢ 0 0

0 ⇢I3

00 0 ⇢eT /T

1

A ,

X

j

⇠jAj =

0

@(p⇢/⇢)(u · ⇠) p⇢⇠ 0

p⇢⇠tr ⇢(u · ⇠)I3

pT ⇠tr

0 p⇢⇠ (⇢eT /T )(u · ⇠)

1

A ,

andX

j,k

Bjk =

0

@0 0 00 µ|⇠|2I

3

+ (µ + �)⇠tr⇠ 00 0 T�1|⇠|2

1

A ,

we find that we may simply multiply the first row by �(U)p�1

⇢ +(1��(U)) to obtainform (1.11) satisfying (A1)–(A3), where � is a smooth cuto↵ function supportedon the set where the Navier–Stokes equations support a convex entropy and equalto unity at U±. A similar procedure applies to MHD.

By direct calculation, we find that the eigenvalues of (A0)�1

Pj Aj⇠j =

Pj Aj⇠j

for gas dynamics are of constant multiplicity with respect to both U and ⇠; seeAppendix C. For MHD on the other hand, they are of constant multiplicity (underappropriate nondegeneracy conditions) only with respect to U (see [Je], pp. 122-127), and this limits the applications for the moment to one dimension. In bothcases, A⇤ = (u · ⇠) is scalar, hence constant multiplicity. Thus, to complete theverification of (A1)–(A3), (H0)–(H2), it remains only to check that det (A1 � s)does not vanish at the endstates U±, and that A1

⇤ � s = u1 � s does not vanishalong the profile, both purely one-dimensional consideration associated with thetraveling-wave ODE.

We conclude by examining further various example systems in one dimension,expressed for simplicity in Lagrangian coordinates. For ease of notation, we dropthe superscripts j and jk from Aj , Aj

⇤ and Bjk.

Example 6.6. (standard gas dynamics) The general Navier–Stokes equationsof compressible gas dynamics, written in Lagrangian coordinates, appear as

(6.2)

vt � ux = 0,

ut + px = ((⌫/v)ux)x,

(e + u2/2)t + (pu)x = ((/v)Tx + (µ/v)uux)x,

178 KEVIN ZUMBRUN

where v > 0 denotes specific volume, u velocity, e internal energy, T = T (v, e)temperature, p = p(v, e) pressure, and µ > 0 and > 0 are coe�cients of viscosityand heat conduction, respectively.

Note, as described in [MeP], that the “incomplete equation of state” p = p(v, e)is su�cient to close the compressible Euler equations, consisting of the associatedhyperbolic system (1.4), whereas the compressible Navier–Stokes equations requirealso the relation of temperature to v and e. Accordingly, following [MeP], considerthe “complete equation of state”

(6.3) e = e(v, s)

expressing internal energy in terms of v and entropy s, defining temperature andpressure through the fundamental thermodynamic relation (combining first andsecond laws)

(6.4) de = Tds� pdv,

under the standard constraints of positive temperature,

(6.5) T = es > 0,

and global thermodynamic stability (precluding van der Waals type equation ofstate),

(6.6) e(v, s) convex.

Note that we do not require positivity of p = ev or pe, which may fail under certaininteresting circumstances (for example, in water, near the transition from liquid toice).

Under assumption (6.5), we may invert relation (6.3) to obtain s = s(v, e) as afunction of the primary variables v and e. A straightforward calculation then gives

Lemma 6.7. For T = es positive, e(v, s) is convex if and only if �s(v, e) is convex,if and only if �s(v,E�u2/2) is convex with respect to (v, u, E), where E := e+u2/2.

An immediate consequence of Lemma 6.7 is that

(6.7) (@/@E)T (v,E � u2/2) = Te = (es)e = (1/se)e = �see/s2

e > 0,

whence (6.2) has the structural form (1.1)–(1.5), where the lower triangular matrix

b2

=✓

1/v 0⇤ (@/@E)T

◆

evidently has real, positive spectrum 1/v, (@/@E)T . Likewise, it is easily checkedthat genuine coupling, (A2), holds.


Appealing to relation (6.4), we find that

(6.8) ⌘(v, u, E) := �s(v, u, E � u2/2), q(v, u, E) := 0

satisfy conditions (ii)-(iii) of Definition 6.1. (Here, it is helpful to take u = 0 byGalillean invariance; for details, see, e.g., Section 9 of [LZe].) By Lemma 6.7,condition (i) is satisfied as well, whence ⌘, q is a convex entropy, entropy flux pairfor (6.2). Applying Proposition 6.2, we thus find that (A1)–(A3) are satisfied for(6.2) written in terms of the entropy variable

(6.9) d⌘ = (�sv, use,�se) = (ev/es, u/es,�1/es) = T�1(�p, u,�1).

Straightforward calculation then gives A⇤ = dF11

= dF12

b�1

2

b1

⌘ 0 as in theisentropic case, verifying (H1) and completing the verification of the hypotheseseunder assumptions (6.5) and (6.6). Note: zero-speed profiles, s = 0 do not exist.For, stationary solutions of (6.2) are easily seen to be constant in u, p, and T . But,(T,�p) = (es, ev) is the (necessarily invertible) Legendre transform of (s, v) withrespect to the convex function e(s, v), whence s, v, and therefore e and E must beconstant as well.

As discussed above, the assumption of global thermodynamic stability may berelaxed to Te > 0 together with thermodynamic stability at endstates U±, thusaccomodating models for van der Waals gas dynamics. Again, stationary profilescannot occur, since U = (v, u, e) must remain constant so long as it stays su�cientlynear U±, and thus remains so forever. This completes the verification of (A1)–(A3),(H0)–(H2) in the general case; condition (H3) holds for extreme, Lax-type profiles,but not necessarily in general.

Example 6.8. (MHD) Next, consider the equations of MHD:(6.10)

vt � u1x = 0,

u1t + (p + (1/2µ

0

)(B2

2

+ B2

3

))x = ((⌫/v)u1x)x,

u2t � ((1/µ

0

)B⇤1

B2

)x = ((⌫/v)u2x)x,

u3t � ((1/µ

0

)B⇤1

B3

)x = ((⌫/v)u3x)x,

(vB2

)t � (B⇤1

u2

)x = ((1/�µ0

v)B2x)x,

(vB3

)t � (B⇤1

u3

)x = ((1/�µ0

v)B3x)x,

(e + (1/2)(u2

1

+ u2

2

+ u2

3

) + (1/2µ0

)v(B2

2

+ B2

3

))t

+ [(p + (1/2µ0

)(B2

2

+ B2

3

))u1

� (1/µ0

)B⇤1

(B2

u2

+ B3

u3

)]x= [(⌫/v)u

1

u1x + (µ/v)(u

2

u2x + u

3

u3x)+

(/v)Tx + (1/�µ2

0

v)(B2

B2x + B

3

B3x)]x,

where v denotes specific volume, u = (u1

, u2

, u3

) velocity, p = P (v, e) pressure,B = (B⇤

1

, B2

, B3

) magnetic induction, B⇤1

constant, e internal energy, T = T (v, e)

180 KEVIN ZUMBRUN

temperature, and µ > 0 and ⌫ > 0 the two coe�cients of viscosity, > 0 thecoe�cient of heat conduction, µ

0

> 0 the magnetic permeability, and � > 0 theelectrical resistivity.

Calculating similarly as in the Navier–Stokes case, under the same assumptions(6.5) and (6.6), we find again that ⌘ := �s, q := 0 is an entropy, entropy flux pairfor system (6.10), ⌘, q now considered as functions of the conserved quantities

(v, u1

, u2

, u3

, vB2

, vB3

, E),

where E := e + (1/2)(u2

1

+ u2

2

+ u2

3

) + (1/2µ0

)v(B2

2

+ B2

3

) denotes total energy;for details, see again Section 9 of [LZe]. Likewise, we obtain by straighforwardcalculation that A⇤ = dF

11

= dF12

b�1

2

b1

⌘ 0, verifying (H1) and completing theverification of the hypothesese under assumptions (6.5) and (6.6), plus the addi-tional assumption (possibly superflous, as in the Navier–Stokes case) that speed sbe nonzero.

Example 6.9. (MHD with infinite resistivity/permeability) An interesting vari-ation of (6.10) that is of interest in certain astrophysical parameter regimes is thelimit in which either electrical resistivity �, magnetic permeability µ

0

, or both, goto infinity, in which case the righthand sides of the fifth and sixth equations of (6.10)go to zero and there is a three-dimensional set of hyperbolic modes (v, vB

2

, vB3

)instead of the usual one. Nonetheless, A⇤ = dF

11

= dF12

b�1

2

b1

⌘ 0 in this caseas well; that is, though now vectorial, hyperbolic modes still experience passive,scalar convection, and so (H1) remains valid whenever s 6= 0. (Note: in Lagrangiancoordinates, zero speed corresponds to “particle”, or fluid velocity.) Reduction tosymmetric form may be achieved by the same entropy, entropy flux pair as in thestandard case of Example 6.8, under assumptions (6.5) and (6.6).

Example 6.10. (multi-species gas dynamics or MHD) Another simple examplefor which the hyperbolic modes are vectorial is the case of miscible, multi-speciesflow, neglecting species di↵usion, in either gas- or magnetohydrodynamics. In thiscase, the hyperbolic modes consist of k copies of the hyperbolic modes for a singlespecies, where k is the number of total species, with a single, scalar convection rateA⇤ ⌘ 0, corresponding to passive convection by the common fluid velocity.

A.2. Structure of viscous profiles.

In this appendix, we prove the results cited in the introduction concerning struc-ture of viscous profiles.

Proof of Lemma 1.6. Di↵erentiating (1.16) and rearranging, we may write(1.16)–(1.17) in the alternative form

(6.11)✓

uI

uII

◆0=✓

dF 1

11

dF 1

12

b11

1

b11

2

◆�1

✓0

f II � f II�

◆.


Linearizing (6.11) about U±, we obtain

(6.12)✓

uI

uII

◆0=✓

dF 1

11

dF 1

12

b11

1

b11

2

◆�1

✓0 0

dF 1

21

dF 1

22

◆

|(U±)

✓uI

uII

◆,

or, setting ✓z1

z2

◆:=

✓dF 1

11

dF 1

12

b11

1

b11

2

◆

|(U±)

✓uI

uII

◆,

the pair of equationsz01

= 0

and

(6.13) z02

= ( dF 1

21

dF 1

22

)✓

dF 1

11

dF 1

12

b11

1

b11

2

◆�1

✓0Ir

◆

|(U±)

z2

,

the latter of which evidently describes the linearized ODE on manifold (1.16).Observing that

det ( dF 1

21

dF 1

22

)✓

dF 1

11

dF 1

12

b11

1

b11

2

◆�1

✓0Ir

◆

|(U±)

=

det✓

dF 1

11

dF 1

12

dF 1

21

dF 1

22

◆✓dF 1

11

dF 1

12

b11

1

b11

2

◆�1

|(U±)

6= 0

by (H2) and (H1)(i), we find that the coe�cient matrix of (6.13) has no zero eigen-values. On the other hand, it can have no nonzero purely imaginarly eigenvaluesi⇠, since otherwise

✓F 1

11

F 1

12

dF 1

21

dF 1

22

◆✓F 1

11

F 1

12

b11

1

b11

2

◆�1

✓0

uII

◆= i⇠

✓0

uII

◆,

and thush� i⇠

✓dF 1

11

dF 1

12

dF 1

21

dF 1

22

◆� ⇠2

✓0 0

b11

1

b11

2

◆ih✓ dF 1

11

dF 1

12

b11

1

b11

2

◆�1

✓0

uII

◆i=✓

00

◆

for ⇠ 6= 0 2 R, in violation of (2.55), Corollary 2.19. Thus, we find that U± arehyperbolic rest points, from which (2.18) follows. ⇤

Proof of Lemma 1.7. We follow the notation of Sections 1.2 and 3.1. Equat-ing by consistent splitting/continuous extension to � = 0 the dimensions of thestable subspace S+ of the limiting coe�cient matrix A

+

(�) appearing in eigenvalueequation (2.17) at � = 0 and at �! +1, we obtain

dim U(A+

) + d+

= dim U(A⇤+) + r,

182 KEVIN ZUMBRUN

or(n� i

+

) + d+

= dim U(A⇤+) + r,

and similarly at x ! �1, where the lefthand side follows from Lemma 4.20 andthe righthand side from the proof of Lemma 2.21. Rearranging, we obtain (1.19).If there exists a connecting profile, then we have by (H1)(i) that dimS(A⇤) anddimU(A⇤) are constant along the profile and sum to n � r. Thus, dimS(A⇤�) +dimU(A⇤+) = n� r, and we obtain n� i = r� d by summing relations (1.19). ⇤

A.3. Asymptotic ODE estimates.

For completeness, we prove in this appendix the asymptotic ODE estimates citedin Section 2.2 We first recall the gap lemma of [GZ,KS,ZH].

Lemma 6.11 (the gap lemma). Consider (2.17), under assumption (h0), with✓ < ✓. If V �(�) is an eigenvector of A� with eigenvalue µ(�), both analytic in �,then there exists a solution of (2.17) of form

W (�, x) = V (x,�)eµ(�)x,

where V is C1 in x and locally analytic in � and for each j = 0, 1, . . . satisfies

(6.14) V (x,�) = V �(�) + O(e�¯✓|x||V �(�)|), x < 0,

for all ✓ < ✓. Moreover, if Re µ(�) > Re eµ(�)� ✓ for all (other) eigenvalues eµ ofA�, then also W is uniquely determined by (6.14), and (6.14) holds for ✓ = ✓.

Proof. Setting W(x) = eµxV (x), we may rewrite W0 = AW as

(6.15) V 0 = (A� � µI)V + ✓V, ✓ := (A� A�) = O(e�✓|x|),

and seek a solution V (x,�) ! V �(x) as x ! 1. Choose ✓ < ✓1

< ✓ suchthat there is a spectral gap |Re

��A� � (µ + ✓

1

)�| > 0 between �A� and µ + ✓

1

.Then, fixing a base point �

0

, we can define on some neighborhood of �0

to thecomplementary A�-invariant projections P (�) and Q(�) where P projects ontothe direct sum of all eigenspaces of A� with eigenvalues eµ satisfying Re (eµ) <Re (µ)+ ✓

1

, and Q projects onto the direct sum of the remaining eigenspaces, witheigenvalues satisfying Re (eµ) > Re (µ) + ✓

1

. By basic matrix perturbation theory(eg. [Kat]) it follows that P and Q are analytic in a neighborhood of �

0

, with

(6.16)��e(A��µI)xP

�� C(e✓1x), x > 0,��e(A��µI)xQ

�� C(e✓1x), x < 0.

It follows that, for M > 0 su�ciently large, the map T defined by

(6.17)T V (x) = V � +

Z x

�1e(A��µI)(x�y)P✓(y)V (y)dy

�Z �M

x

e(A��µI)(x�y)Q✓(y)V (y)dy


is a contraction on L1(�1,�M ]. For, applying (6.16), we have(6.18)

|T V1

� T V2

|(x)

C|V1

� V2

|1✓Z x

�1e✓1(x�y)e✓ydy +

Z �M

x

e✓1(x�y)e✓ydy

◆

C1

|V1

� V2

|1✓

e✓1xe(✓�✓1)y|x�1 + e✓1xe(✓�✓1)y|�Mx

◆

C2

|V1

� V2

|1e�¯✓M <12|V

1

� V2

|1.

By iteration, we thus obtain a solution V 2 L1(�1,�M ] of V = T V withV C

3

|V �|; since T clearly preserves analyticity V (�, x) is analytic in � as theuniform limit of analytic iterates (starting with V

0

= 0). Di↵erentiation shows thatV is a bounded solution of V = T V if and only if it is a bounded solution of (6.15)(exercise). Further, taking V

1

= V , V2

= 0 in (6.18), we obtain from the second tolast inequality that

(6.19) |V � V �| = |T (V )� T (0)| C2

e¯✓x|V | C

4

e¯✓x|V �|,

giving (6.14). Analyticity, and the bounds (6.14), extend to x < 0 by stan-dard analytic dependence for the initial value problem at x = �M . Finally, ifRe (µ(�)) > Re (eµ(�)) � ✓

2

for all other eigenvalues, then P = I, Q = 0, andV = T V must hold for any V satisfying (6.14), by Duhamel’s principle. Further,the only term appearing in (6.18) is the first integral, giving (6.19) with ✓ = ✓. ⇤

Proof of Lemma 2.5. Substituting W = P+

Z into (2.17), equating to (2.19),and rearranging, we obtain the defining equation

(6.20) P 0+

= A+

P+

� P+

A, P+

! I as x ! +1.

Viewed as a vector equation, this has the form P 0+

= AP+

, where A approachesexponentially as x ! +1 to its limit A

+

, defined by A+

P := A+

P � PA+

. Thelimiting operator A

+

evidently has analytic eigenvalue, eigenvector pair µ ⌘ 0,P

+

⌘ I, whence the result follows by Lemma 6.11 for j = k = 0. The x-derivativebounds 0 < k K + 1 then follow from the ODE and its first K derivatives, andthe �-derivative bounds from standard interior estimates for analytic functions. Asymmetric argument gives the result for P�. ⇤

Proof of Proposition 2.13. Setting �2

:= 1

�1

2

, 1

2 C(N�k)⇥k, 2

2 Ck⇥k,where ( t

1

, t2

)t 2 CN⇥k satisfies (2.29), we find after a brief calculation that �2

satisfies

(6.21) �02

= (M1

�2

� �2

M2

) + �Q(�2

),

where Q is the quadratic matrix polynomial Q(�) := ⇥12

+⇥11

��⇥22

+�⇥21

�.Viewed as a vector equation, this has the form

(6.22) �02

= M�2

+ �Q(�2

),

184 KEVIN ZUMBRUN

with linear operator M� := M1

� � �M2

. Note that a basis of solutions of thedecoupled equation �0 = M� may be obtained as the tensor product � = ��⇤ ofbases of solutions of �0 = M

1

� and �0 = �M⇤2

�, whence we obtain from (4.257)the bound

(6.23) |Fy!x| |Fy!x1

||Fy!x2

| = |Fy!x1

||(Fy!x2

)�1| Ce�⌘|x�y|

for x > y, where F denotes the flow of the full decoupled matrix-valued equation, Fj

the flow of �0 = Mj� and Fj the adjoint flow associated with equation �0 = �M⇤j �.

(Recall the standard duality relation Fj = F�1

j ).That is, F is uniformly exponentially decaying in the forward direction. Thus,

assuming only that �2

is bounded at �1, we obtain by Duhamel’s principle theintegral fixed-point equation

(6.24) �2

(x) = T �2

(x) := �

Z x

�1Fy!xQ(�

2

)(y) dy.

Using (6.23), we find that T is a contraction of order O(�/⌘), hence (6.24) deter-mines a unique solution for �/⌘ su�ciently small, which, moreover, is order �/⌘ asclaimed. The bound on @x�✏j then follows by di↵erentiation of (6.24), using thefact that @xFy!x CFy!x, since the coe�cients of the decoupled (linear) floware bounded. A symmetric argument establishes existence of �

1

. ⇤Remarks 6.12. 1. Note that the above stable/unstable manifold construc-

tion requires only boundedness of coe�cients at infinity, and not any special (e.g.,asymptotically constant or periodic) structure. This reflects the local nature oflarge frequency/short time estimates in the applications.

2. Equation (6.21) could alternatively be derived from the point of view ofLemma 2.5, as defining a change of coordinates

P :=✓

I �2

�1

I

◆

conjugating (2.29) to diagonal form. The result of Proposition 2.13 shows thatthere is a unique such transformation that is bounded on the whole line.

3. If desired, one can obtain a full, and explicit Neumann series expansion inpowers of �/⌘ from the fixed-point equation (6.24), using our explicit descriptionof the flow F as the tensor product � = ��⇤ of bases of solutions of �0 = M

1

� and�0 = �M⇤

2

�. For our purposes, it is only the existence and not the precise form ofthe expansion that is important.

Proof of Corollary 2.14. Similarly as in the proof of Proposition 2.13 justabove, we may obtain a fixed-point representation

Fy!xj = T Fy!x

j := Fy!x1

+ �

Z x

y

Fy!z�⇥

11

+⇥12

�2

�(z, ✏)Fz!x

j dz,

where T is a contraction of order O(�/⌘), yielding the result. ⇤


Lemma 6.13. Let d1

2 Cm1⇥m1 and d2

2 Cm2⇥m2 have norm bounded by C1

andrespective spectra separated by 1/C

2

> 0. Then, the matrix commutator equation

(6.25) d1

X �Xd2

= F,

X 2 Cm1⇥m2 , is soluble for all F 2 Cm1⇥m2 , with |X| C(C1

, C2

)|F |.

Proof. Consider (6.25) as a matrix equation DX = F where X and F arevectorial representations of the m

1

⇥m2

dimensional quantities X and F , and Dis the matrix representation of the linear operator (commutator) corresponding tothe lefthand side. It is readily seen that �(D) is just the di↵erence �(d

1

) � �(d2

)between the spectra of d

1

and d2

, with associated eigenvectors of form r1

l⇤2

, wherer1

are right eigenvectors associated with d1

and l2

are left eigenvectors associatedwith d

2

. (Here as elsewhere, ⇤ denotes adjoint, or conjugate transpose of a matrixor vector). By assumption, therefore, the spectrum of D has modulus uniformlybounded below by 1/C

2

, whence the result follows. ⇤Proof of Proposition 2.15. Substituting W = TW into (2.43) and rearrang-

ing, we obtainW 0 = (T�1AT � T�1T 0)W

= (T�1AT � "T�1Ty)W ,

yielding the defining relation

(6.26) (T�1AT � "T�1Ty) = D

for T .By (h2), there exists a uniformly well-conditioned family of matrices T

0

(x) suchthat T�1

0

A0

T0

= D0

, D0

as in (2.42); moreover, these may be chosen with the sameregularity in y as A

0

(i.e., the full regularity of A). Expanding

(6.27) T�1(y, ") =

0

@I �� pX

j=2

"jT�1

0

Tj

�+� pX

j=2

"jT�1

0

Tj

�2 � . . .

1

AT�1

0

by Neumann series, and matching terms of like order "j , we obtain a hierarchy ofsystems of linear equations of form:

(6.28) D0

T�1

0

Tj � T�1

0

TjD0

� Fj = Dj ,

where Fj depends only on Ak for 0 k j and Tk, (d/dy)Tk for 0 k j � 1.On o↵-diagonal blocks (k, l), (6.28) reduces to

(6.29) dl[T�1

0

Tj ](k,l) � [T�1

0

Tj ](k,l)dk = F (k,l)j ,

186 KEVIN ZUMBRUN

uniquely determining [T�1

0

Tj ](k,l), by Lemma 6.13. On diagonal blocks (k, k), weare free to set [T�1

0

Tj ](k,k) = 0, whereupon (6.28) reduces to

(6.30) [Dj ](k,k) = �[Fj ](k,k),

determining the remaining unknown [Dj ](k,k).Thus, we may solve for Dj , Tj at each successive stage in a well-conditioned way.

Moreover, it is clear that the regularity of Dj , Tj is as claimed, since we lose oneorder of regularity at each stage, through the dependence of Fj on derivatives oflower order Tk, k < j. ⇤

A.4. Expansion of the one-dimensional Fourier symbol

In this appendix, we record the Taylor expansions about ⇠ = 0 and ⇠ = 1 ofthe Fourier symbol P (i⇠) = �i⇠A1(x) � ⇠2B11(x) of the one-dimensional frozen,constant-coe�cient operator at a given x. Henceforth, we suppress the superscriptsfor A and B.

Low frequency expansion. Under assumption (P0), straightforward matrixperturbation theory [Kat] yields that the second-order expansion at i⇠ = 0 of P (i⇠)with respect to i⇠ is just

P = �i⇠A� ⇠2R diag {�1

, . . . ,�n}L,

diag {�1

, . . . ,�n} := LBR,

where L, R are composed of left, right eigenvectors of A, as described in the intro-duction.

High frequency expansion. We next consider the expansion of P at infinitywith respect to (i⇠)�1. Recall that

(6.31) B :=✓

0 0b1

b2

◆, A :=

✓A

11

A12

A21

A22

◆, A⇤ = A

11

�A12

b1

b�1

2

.

Claim. To zeroth order, ”hyperbolic” dispersion relations

(6.32) �(i⇠) 2 �(�⇠2B � i⇠A), � ⇠ ⇠,

are given by

(6.33) �j(⇠) = �i⇠a⇤j � ⌘⇤j + . . . , j = 1, · · ·n� r,

with corresponding right and left eigenvectors

(6.34) R⇤j (i⇠) =✓

r⇤j�b�1

2

b1

r⇤j

◆+ . . . and L⇤j (i⇠) =

✓l⇤j0

◆+ . . . ,


where

(6.35) ⌘⇤j := l⇤j D⇤r⇤j , D⇤ := A12

b�1

2

hA

21

�A22

b2

b1

+ b�1

2

b1

A⇤i,

and a⇤j , r⇤j , l⇤j are eigenvalues and associated right and left eigenvectors of A⇤. Theremaining r “parabolic” relations � ⇠ ⇠2 are given to second order by

(6.36) �n�r+j(⇠) = �⇠2�j + . . . , j = 1, · · · r,

with corresponding right and left eigenvectors

(6.37) Rn�r+j(i⇠) =✓

0sj

◆+ . . . and Ln�r+j(i⇠) =

✓��1

j btr1

tjtj

◆,

where �j , sj , tj are the eigenvalues and right and left eigenvectors of b2

.

Proof. Expressing P (i⇠) = �⇠2(B + ✏A), ✏ := (i⇠)�1 and expanding around✏ = 0 by standard matrix perturbation theory, we obtain the results after rearrange-ment; for details, see the dual calculation carried out in Appendix A2, [MaZ.1] inthe relaxation case. ⇤

Remark 6.14. Note that we have without calculating that Re �⌘⇤j > 0 asx ! ±1, by the spectral bounds of Lemma 2.18.

APPENDIX B. A WEAK VERSION OF METIVIER’S THEOREM.

In this appendix, we present a simplified version of the results of Metivier [Me.1,FM, Me.5] on inviscid stability of small-amplitude shock waves, establishing sharpuniform lower bounds on the Lopatinski determinant but not the associated uniformL2 bounds on the solution operator. This gives the flavor of Metivier’s analysis,while avoiding many of the technical di�culties; at the same time, it illustrates inthe simpler, inviscid setting some of the issues arising in the verification of strongspectral stability in the viscous small-amplitude case [PZ, FreS.2]. This material,presented first in [Z.2], was developed in part together with H. Freistuhler; for anextension to the overcompressive case, see [FreZ.2]. Thanks also to K. Jenssen forhelpful comments and corrections.

Consider an inviscid system

(7.1) Ut +X

j

F j(U)xj

,

and a one-parameter family of stationary Lax p-shocks (defined in (1.23)) (U"+

, U"�)

with normal e1

and amplitude |U"+

� U"�| =: ", satisfying the Rankine–Hugoniot

condition[F 1(U")] = 0,

188 KEVIN ZUMBRUN

with U± converging in the small-amplitude limit " ! 0 to some base point U0

.Setting Aj := dF j(U), and denoting by aj , rj , lj the eigenvalues and associatedright and left eigenvectors of A1, we make the assumptions:

(h0) F j 2 C2 (regularity).(h1) There exists A0(·), symmetric positive definite and smoothly depending on

U , such that A0Aj(U) is symmetric for all 1 j d in a neighborhood of U0

(simultaneous symmetrizability, ) hyperbolicity).(h2) ap = 0 is a simple eigenvalue of dF 1(U

0

) with rap · rp(U0

) 6= 0 (stricthyperbolicity and genuine nonlinearity of the principal characteristic field ap in thenormal spatial direction x

1

).

(h3) rp is not an eigenvector of A˜⇠ :=

Pj 6=1

⇠jAj for ⇠ 2 Rd�1 6= 0 (su�cient forextreme shocks, p = 1 or n); more generally,

(7.2) hrp, A0A

˜⇠⇧(A1 � ap)�1⇧A˜⇠ rpi 6= 0

for all ⇠ 2 Rd�1 6= 0, where ⇧ denotes the eigenprojection complementary tothe eigenprojection of A1 onto Range (rp) (genuine hyperbolic coupling [Me.1–4,FreZ.2]).

Assumptions (h0)–(h2) are standard conditions satisfied by many physical sys-tems. Assumption (h3) is a technical condition that will emerge through the anal-ysis. It ensures that the system does not decouple in the pth field, in particularexcluding the (non-uniformly stable) scalar case.

Propositions 1.10 and 1.12 and Remark 1.14.1, show that the described sce-nario of a converging family of Lax p-shocks is typical under (h0)–(h2). As pointedout by Metivier, condition (h3) has the following easily checkable characterization(proof deferred to the end of this section), which shows also that (h3) is gener-ically satisfied in two dimensions or for extreme shocks in higher dimensions, inboth cases corresponding to nonsingularity of the associated Hessian (1 ⇥ 1 fortwo dimensions; semidefinite for extreme shocks, as a consequence of hyperbolic-ity). Conditions (h0)–(h3) are generically satisfied for extreme shocks of gas- andmagnetohydrodynamics near points U

0

of thermodynamic stability.

Lemma 7.1 [Me.1]. Under assumptions (h0)–(h2), condition (h3) is equivalentto strict concavity (resp. convexity) of eigenvalue ap(U, ⇠) of A⇠ :=

Pj ⇠jA

j withrespect to e⇠ at the base point U = U

0

, ⇠ = (⇠1

, e⇠) = (1, 0).

Then, we have the following two stability results, respectively concerning theone- and multi-dimensional case.

Proposition 7.2 [M.4]. (One-dimensional stability) In dimension d = 1, hypothe-ses (h0)–(h2) are su�cient to give uniform stability for " su�ciently small; more-over, we have

(7.3) |�(0,�)| � C�1"|�|,


for some C > 0, uniformly as "! 0.

Proof. Recalling (1.29), we have

(7.4) |�(0,�)| :=��det (r�

1

, . . . , r�p�1

,�[U ], r+

p+1

, . . . , r+

n )�� .

Without loss of generality choosing rj(U) continuously in U , |rj | ⌘ 1, we have

r±j = rj(U0

) + o(1)

as "! 0. At the same time,

(7.5) [U ] = "(rp(U0

) + o(1)),

by the Lax structure theorem, Theorem 1.10. Combining, we have

|�(0,�)| = "|�| |det (r1

, . . . , rn)(U0

) + o(1)| ,

giving the result. ⇤Proposition 7.3 [Me.1]. (Multidimensional stability). Assuming (h0)–(h3), wehave uniform stability for " := |U

+

� U�| su�ciently small; moreover

(7.6) |�(⇠,�)| � C�1"3/2|(⇠,�)|,

for some uniform C > 0.

Proof. Without loss of generality take d = 2 and (by the change of coordinatesAj ! (A0)1/2Aj(A0)�1/2) A1, A2 symmetric, A0 = I. For simplicity, restrict tothe extreme shock case p = 1; the intermediate case goes similarly [Me.1]. Then,up to a well-conditioned rescaling, we may rewrite (1.29) as

(7.7) � = l+1

· (�[U ] + i⇠2

[F 2(U)]),

where l+1

(e⇠,�) denotes the unique stable left eigenvector for Re � > 0 of

A+

(⇠,�) := (�+ i⇠2

A2

+

)(A1

+

)�1,

extended by continuity to Re � = 0. We use form (7.7) to simplify the computa-tions.

Observation 1. By appropriate choice of coordinates, we may arrange that

(7.8) A1

+

=✓�" 00 A1

◆,

190 KEVIN ZUMBRUN

where A1 2 R(n�1)⇥(n�1) is symmetric, positive definite and bounded uniformlyabove and below, " ⇠ " by the Lax structure theorem and genuine nonlinearity,condition (h2);

(7.9) A2

+

=

0

B@

0 1 01

A2

0

1

CA ,

where A2 2 R(n�1)⇥(n�1) is symmetric; and

(7.10) �[U ] + i⇠2

[F 2(U)] = "

0

BBB@

�i⇠

2

0· · ·0

1

CCCA(c +O(")) ,

c 6= 0. (Here, O(") is matrix-valued, for brevity of exposition.)

Proof of Observation. First, take coordinates such that A1 is block-diagonaland A2 symmetric. Changing to a moving coordinate frame x0

2

= x2

� ct in thetransverse direction, we may arrange that (A2

+

)11 = 0, i.e., hr+

1

, A2

+

r1

+i = 0, henceby a change of dependent variables in the second ((n�1)⇥(n�1)) block of A1 thatA2

+

r+

1

lies in the second coordinate direction. Since A2

+

r+

1

6= 0, by (h3), we mayarrange by a rescaling of x

2

that |A2

+

r+

1

| = 1. This verifies (7.8)–(7.9). Relation(7.10) then follows by the Lax structure theorem and Taylor expansion about U

+

of F 2(U). ⇤From here on, we drop the hat, and write " for ".

Observation 2. The matrix A+

(⇠,�) := (�+ i⇠2

A2

+

)(A1

+

)�1 then has the form

(7.11)

✓��/" i⇠

2

/a�i⇠

2

/" 0

◆0

0 0

!+ (|⇠

2

|+ |�|)

0

@0 O(") O(1)0 O(1) O(1)0 O(1) O(1)

1

A ,

a > 0, or, “balancing” using transformation

T :=

0

@"�1/2 0 0

0 1 00 0 In�2

1

A ,

(7.12)

TA+

T�1 =

✓��/" i⇠

2

/a"1/2

�i⇠2

/"1/2 0

◆0

0 0

!+ (|⇠

2

|+ |�|)

0

@0 O("1/2) O("�1/2)0 O(1) O(1)0 O(1) O(1)

1

A

= "�1/2

0

@�� i⇠

2

/a wTr

�i⇠2

0 00 0 0

1

A+⇣|⇠

2

|+ "1/2|�|⌘0

@0 O("1/2) 00 O(1) O(1)0 O(1) O(1)

1

A ,


where � := �"�1/2, |w| = O(1).

Proof of Observation 2. We have from (7.8) that

(7.13) (A1

+

)�1 =

0

@�"�1 0

0 (A1)�1

1

A ,

where

(7.14) (A1)�1 =✓

a�1 ⇤⇤ ⇤

◆

is symmetric positive definite, whence the diagonal entry a�1 is symmetric positivedefinite as well. The result then follows by direct computations, combining (7.9)with (7.14). ⇤

Without loss of generality fixing ⇠2

⌘ 1 (the case ⇠2

= 0 may be obtained bycontinuity, and has anyway been treated already in the one-dimensional case), weconsider the resulting matrix perturbation problem (7.12).

Case (i) (|�|� "12 ). First we treat the straightforward case |�|� 1. In this case,

�� dominates all entries in the rescaled perturbation problem (7.12), from whichwe find that the associated left eigendirection of TA

+

T�1 is (1, 0, . . . , 0)Tr + o(1),and thus the left eigenvector l+

1

of A+

is (rescaled to order one)

(7.15) "1/2

⇣(1, 0, . . . , 0)Tr + o(1)

�T ⇠ (1, 0, . . . , 0)Tr + o("1/2).

Computing � = l+1

· (�[U ] + i⇠2

[F 2(U)]) following (7.7), we obtain from (7.10) and(7.15) that |�| ⇠ "� � C�1"3/2, as asserted.

Case (ii) (|�| C"1/2). Next, we treat the critical case that |�| is uniformlybounded. Noting that the upper lefthand corner

(7.16) ↵ :=✓�� i⇠

2

/a�i⇠

2

0

◆=✓�� i/a�i 0

◆

of block-upper triangular matrix

A =

0

@�� i⇠

2

/a wTr

�i⇠2

0 00 0 0

1

A =

0

@�� i/a wTr

�i 0 00 0 0

1

A

is uniformly nonsingular, with determinant �⇠22

/a = �1/a < 0, we have by stan-dard matrix perturbation theory that the stable left eigenvector L+

1

of

TA+

T�1 = "�1/2A+O(1)

192 KEVIN ZUMBRUN

is an O("1/2) perturbation of the left stable eigenvector L = (`Tr , wTr )Tr of A,where

` := (1, i/aµ)Tr

is the left stable eigenvector of ↵ and

µ :=��

q�2 + 4/a

2

is the associated stable eigenvalue. Scaling to order one length, we thus have

(7.17) l+1

= "1/2TL+

1

.

Computing � = l+1

· (�[U ] + i⇠2

[F 2(U)]) following (7.7), we obtain from (7.10),(7.17), and the above developments that

� = "1/2(L+

1

+O("1/2)) · T"

0

BBB@

�i⇠

2

0· · ·0

1

CCCA(c +O("))

= "3/2` ·✓�i

◆(1 +O("1/2))

= "3/2µ(1 +O("1/2)),

yielding |�| ⇠ "3/2 as claimed.This completes the proof. ⇤Remarks. 1. The upper bound in case (ii) shows that (7.6) is sharp.2. [Me.1] The matrix perturbation problem (7.11) may be recognized as essen-

tially that arising from the canonical 2⇥ 2 example

F 1(u, v) :=✓

u2/2v

◆, F 2(u, v) :=

✓vu

◆,

U± = (⌥", 0), for which the system at U+

is the 2⇥ 2 wave-type equation

A1 =✓�" 00 1

◆, A2 =

✓0 11 0

◆,

and the resulting matrix perturbation problem agrees with the determining 2 ⇥ 2upper-lefthand block of (7.11). That is, whereas one-dimensional behavior is (atleast at small amplitudes) essentially scalar, Proposition 7.2, multi-dimensional


behavior involves in an essential way the coupling between di↵erent hyperbolicmodes (precisely two for each fixed transverse direction e⇠).

3. In the intermediate shock case 1 < p < n, the strengthened version (7.2) ofcondition (h3) may be seen to correspond (see development of (7.14)) to nonvan-ishing of a in the 2 ⇥ 2 blocks analogous to ↵ in (7.16) (one block arising at +1and one at �1).

Proof of Lemma 7.1. Without loss of generality take d = 2 and (by the changeof coordinates Aj ! (A0)1/2Aj(A0)�1/2) A1, A2 symmetric, A0 = I. Let

(7.18) ap(⇠) = a0

p + a1

p⇠ + a2

p⇠2

and

(7.19) rp(⇠) = r0

p + r1

p⇠ + . . . ,

denote the eigenvalues and eigenvectors of

(7.20) A(⇠) := (A1 + ⇠A2),

analytic by simplicity of ap, (h2), without loss of generality |r0

p| = 1. Expandingthe defining relations

(7.21) (A(⇠)� ap(⇠))rp(⇠) = 0,

and matching terms of like order in ⇠, we obtain

(7.22)(0th order) :

(A1 � a0

p)r0

p = 0,

(7.23)(1st order) :

(A1 � a0

p)r1

p + (A2 � a1

p)r0

p = 0,

(7.24)(2nd order) :

(A1 � a0

p)r2

p + (A2 � a1

p)r1

p + (�a2

p)r0

j = 0.

Taking the inner product of r0

p with (7.23), we obtain

(7.25) 0 = hr0

p, (A2 � a1

p)r0

pi,

and thus

(7.26) a1

p = hr0

p, A2r0

pi.

194 KEVIN ZUMBRUN

Next, we may solve modulo span (r0

p) for r1

p in (7.23) by taking the inner productwith r0

k, k 6= p, to obtain

(7.27) hr0

k, r1

pi = �hr0

k, (A2 � a1

p)r0

pia0

k � a0

p

,

hence

(7.28) r1

p = �X

k 6=p

hr0

k, (A2 � a1

p)r0

pia0

k � a0

p

r0

k

is a solution. Finally, taking the inner product of r0

p with (7.24), and using (7.28)plus symmetry of Aj , we obtain expansion

(7.29)

a2

p = �X

k 6=p

hr0

k, (A2 � a1

p)r0

pihr0

p, (A2 � a1

p)r0

kia0

k � a0

p

= �X

k 6=p

hr0

k, (A2 � a1

p)r0

pi2a0

k � a0

p

.

Observing that r0

p, by (7.25) is an eigenvector of A2 if and only if

(7.30) (A2 � a1

1

)r0

p = 0 mod (spank 6=p(r0

k)),

we have the result in the extreme case p = 1 or n, for which a0

k�a0

p for k 6= p has adefinite sign. More generally, observing that (A2� a1

p)r0

p = ⇧A2r0

p, we may rewrite(7.28) as

r1

p = (A1 � a0

p)�1⇧A2r0

p

to obtaina2

p = hr0

p, (A2 � a1

p)r1

pi= h⇧A2r0

p, r1

pi= h⇧A2r0

p, (A1 � ap)�1⇧A2r0

pi

in place of (7.29). Thus, a2

p 6= 0 is equivalent to

hr0

p, A2⇧(A1 � ap)�1⇧A2r0

pi 6= 0,

as claimed. ⇤


APPENDIX C. EVALUATION OF THE LOPATINSKICONDITION FOR GAS DYNAMICS.31

(Previously submitted.)

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STABILITY OF LARGE-AMPLITUDE SHOCK WAVES OF … › ... › Zumbrun › handbook.pdf · of large-amplitude and or strongly nonlinear waves such as arise in the regime where physical