Resonant leading term geometric optics expansions with boundary

layers for quasilinear hyperbolic boundary problems

Matthew Hernandez∗

Abstract

We construct and justify leading order weakly nonlinear geometric optics expansions for nonlinearhyperbolic initial value problems, including the compressible Euler equations. The technique of simul-taneous Picard iteration is employed to show approximate solutions tend to the exact solutions in thesmall wavelength limit. Recent work [2] by Coulombel, Gues, and Williams studied the case of reflectingwave trains whose expansions involve only real phases. We treat generic boundary frequencies by incor-porating into our expansions both real and nonreal phases. Nonreal phases introduce difficulties such asapproximately solving complex transport equations and result in the addition of boundary layers withexponential decay. This also prevents us from doing an error analysis based on almost periodic profilesas in [2].

Contents

1 Introduction 11.1 The statement of the main theorem and the form of the approximate solution . . . . . . . . . 21.2 The profile equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Assumptions and other preliminaries 13

3 Profile equations: formulation with periodic profiles 163.1 The solvability condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 The large system for individual profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Solution of the hyperbolic sub-system of the large system . . . . . . . . . . . . . . . . . . . . 263.4 Approximate solution of the equations for the elliptic profiles . . . . . . . . . . . . . . . . . . 27

4 Convergence of the approximate solution to the exact solution 314.1 The trigonometric expansion for the approximate solution . . . . . . . . . . . . . . . . . . . . 314.2 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1 Introduction

In this paper we consider quasilinear hyperbolic fixed boundary problems on the domain Rd+1

+ = {x =(x′, xd) = (t, y, xd) = (t, x′′) : xd ≥ 0} for a class of equations which includes, in particular, the compressibleEuler equations. The class consists of systems of the following form:

d∑j=0

Aj(vε)∂xjvε = f(vε),

b(vε)|xd=0 = g0 + εG

(x′,

x′ · βε

),

vε = u0 in t < 0.

(1.1)

∗Princeton University; mbh3@math.princeton.edu

1

Here, A0 = I, and we assume Aj ∈ C∞(RN ,RN2

), f ∈ C∞(RN ,RN ), and b ∈ C∞(RN ,Rp). For theboundary data, we take G(x′, θ0) ∈ C∞(Rd × T1,Rp) periodic in θ0 and supported in {x0 ≥ 0} and theboundary frequency β = (β0, . . . , βd−1) ∈ Rd \ 0. We assume that β belongs to the set of regular boundaryfrequencies1, a subset of Rd \ 0 whose complement has zero measure.

The goal of this paper is to obtain qualitative information about the exact solution to (1.1) by explicitlyconstructing an approximate solution with a leading order geometric optics expansion that exhibits thequalitative information, and then showing that this approximate solution tends to the exact solution asε → 0. In [2], this problem was solved under the stronger assumption that β belongs to the hyperbolicregion2. There, the authors constructed approximate solutions in the form of leading order expansions ofhighly oscillatory wavetrains with real phase functions. Our expansions include similar wavetrains with realphases, but in order to handle more general β, we are required to add to our expansions terms of a differentkind, which we call elliptic boundary layers. These are highly oscillatory with envelopes of rapid exponentialdecay, the result of incorporating nonreal phase functions into our construction. Indeed, working with nonrealphase functions is forced on us in the case that β lies outside of the hyperbolic region. It appears that thispaper is the first to rigorously justify leading order expansions involving multiple real and nonreal phasefunctions, and thus both hyperbolic and elliptic profiles, in quasilinear hyperbolic boundary problems.

1.1 The statement of the main theorem and the form of the approximate solu-tion

The precise statement of the main theorem is given in this section and is followed by a formal constructionof the approximate solution to aid in interpreting the theorem and discussing its consequences. Observingthe features of the approximate solution gives an idea of the qualitative information we obtain in the paperon the exact solution.

The system (1.1) is first reformulated in terms of a perturbation of a constant state solution, and thetheorem is stated in terms of the solution of the resulting system. We set vε = u0 + εuε, a perturbation ofa constant state u0 with f(u0) = 0 and b(u0) = g0, and we translate (1.1) to the following problem for uε(where the Aj have been altered accordingly:)

(a) P (εuε, ∂x)uε :=

d∑j=0

Aj(εuε)∂xjuε = F(εuε)uε,

(b) B(εuε)uε|xd=0 = G

(x′,

β · x′

ε

),

(c) uε = 0 in t < 0.

(1.2)

where B(v) and F(v) denote the p×N and N ×N real matrices, C∞ in v, defined by

B(v)v = b(u0 + v)− b(u0), F(v)v = f(u0 + v).(1.3)

The following theorem verifies that the approximate solution is in fact close to the exact solution uε(x)for small ε. Our proposed approximate solution, the leading order geometric optics expansion, is denoted byuaε (x).

Theorem 1.1. Suppose the assumptions hold that L(∂x) is hyperbolic with characteristics of constant mul-tiplicity, the boundary {xd = 0} is noncharacteristic, (L(∂x), B(0)) satisfies the uniform stability condition,and β is a regular boundary frequency (i.e. Assumptions 2.1, 2.2, 2.5, and 2.7 hold, resp.) Then given the

exact solution uε ∈ C1((−∞, T ] × Rd+) of (1.2), defined on the time interval (−∞, T ] independent of ε, foruaε as in (1.14), where v and the σm,k satisfy3 the profile equations, defined on a time interval (−∞, T ′]

1See Assumption 2.7. In particular, restricting β to the set of regular boundary frequencies prevents the case of glancing β.For a comprehensive discussion and treatment of the case of glancing β for the semilinear problem, we direct the reader to [12].

2It is worth noting that the hyperbolic region is not generally a set of full measure, typically having a cone (though possiblythe empty set) as its boundary. For a description, see Remark 2.9 (iii).

3The profile equations are not solved exactly. Here we refer to a kind of approximate solution whose error must satisfy thehypotheses of Proposition 4.6.

2

independent of ε, we have

|uε(x)− uaε (x)|L∞((−∞,T0]×R

d+)→ 0 as ε→ 0,(1.4)

where T0 is the minimum of T and T ′.

To begin with, this kind of result requires the existence of the exact solution uε on a time intervalindependent of ε, a result which is proven in [13] by M. Williams with the use of the singular system. Firstwe will say a few words about this to provide context, and then we will interpret specific consequences ofthe main theorem by discussing the form of uaε . It is the features of uaε that inform us on the behavior of uε,by the theorem.

The singular systemDespite the fact that we need to establish an existence time T of the exact solution uε to (1.2) which

holds uniformly for small ε, applying the standard theory to the problem (1.2) yields existence times Tεwhich shrink to zero since the Sobolev norms of the initial data blow up in the limit ε→ 0. To get estimatesuniform in ε, one may use a reformulation of the system (1.2) known as the singular system, which is alsoused in [2]. The singular system is obtained by recasting (1.2) in terms of an unknown Uε(x, θ0) periodic inθ0 such that the solution of (1.2) is formed by making the substitution

uε(x) = Uε

(x,β · x′

ε

).(1.5)

The singular system is written in the form

(a) ∂xdUε +

d−1∑j=0

Aj(εUε)

(∂xj +

βj∂θ0ε

)Uε = F (εUε)Uε,

(b) B(εUε)(Uε)|xd=0 = G(x′, θ0),

(c) Uε = 0 in t < 0,

(1.6)

where F = A−1d F . In [13], the solution is constructed and the necessary estimates uniform in ε for Uε areproven, estimates which are crucial to our analysis as well as the analysis of [2]. Under the hypotheses ofTheorem 4.7, the existence of the unique solution Uε is shown in [13] in the space4

EsT = C(xd, HsT (x′, θ0)) ∩ L2(xd, H

s+1T (x′, θ0)),(1.7)

implying the existence of the solution uε to (1.2) in the space C1((−∞, T ] × Rd+), on a fixed time intervalindependent of ε in particular, and so we are able to make sense of the limit which is claimed in the maintheorem. The singular system formulation is also used in our proof of the main theorem. We actually provea stronger result, Theorem 4.7, showing an approximate solution of the singular system tends to Uε in Es−1T

as ε→ 0, and the main theorem follows as a corollary.The form of the approximate solution

One key feature of the approximate solution is the manner in which it oscillates with linear phasefunctions. To explain this more precisely, we first point out that the solution restricted to the boundaryuε|xd=0 oscillates with phase φ0(x′), where

φ0(x′) := τt+ η · y = β · x′.(1.8)

Note the appearance of φ0(x′) in the boundary data (1.2)(b). This results in oscillations of the solution in

the whole domain Rd+1

+ . These oscillations are associated to linear phases φm(x) which have trace on theboundary equal to φ0(x′) and which are characteristic for the operator L(∂x), defined by

L(∂x) := P (0, ∂x) = ∂t +

d∑j=1

Aj(0)∂xj .(1.9)

4The subscript T is used to indicate the corresponding function space with time interval (−∞, T ] instead of R.

3

To find the characteristic phases, we consider the matrix

(1.10)1

iA(τ, η) = −A−1d (0)

τI +

d−1∑j=1

ηj Aj(0)

.

Let ωm, m = 1, . . . ,M denote the distinct eigenvalues of 1iA(τ , η) = 1

iA(β). We suppose that each ωm is asemisimple eigenvalue with multiplicity denoted by µm. Thus, the correct phase functions are

φm(x) := φ0(x′) + ωmxd = (β, ωm) · x, m = 1, . . . ,M.(1.11)

We remark that the statement that the solution oscillates with phases φm is a formal one. In particular,this is not precisely the correct statement for a phase which is nonreal, i.e. in the case that ωm is nonreal.However, we are still able to handle these cases. In fact, while in [2], restricting to hyperbolic β led to allthe ωm being real, by allowing for β outside the hyperbolic region in our study, we allow for nonreal ωm,and so to handle these cases is the main contribution of the paper.

Now we turn our attention to the natural basis in which to write down our approximate solution, beginningwith the following lemma, proved in [1]. Given Assumption 2.1, this provides a decomposition which is keyto the construction of our expansion and will serve us later in the construction of the projectors which areused in the derivation of the profile equations.

Lemma 1.2. The space CN admits the decomposition:

(1.12) CN = ⊕Mm=1 Ker L(dφm)

and for each m such that ωm is real, the associated vector space in (1.12) admits a basis of real vectors. Ifwe let P1, . . . , PM denote the projectors associated with the decomposition (1.12), then for all m = 1, . . . ,M ,there holds Im A−1d L(dφm) = Ker Pm.

For each m = 1, . . . ,M , we choose a basis of Ker L(dφm):

rm,k, k = 1, . . . , µm,(1.13)

where for real ωm, we take a basis of real rm,k. Now we are in a position to give the ansatz for our approximatesolution to (1.2).

uaε (x) = v(x) +

M∑m=1

µm∑k=1

σm,k

(x,φm(x)

ε

)rm,k,(1.14)

where v(x) and the σm,k(x, θm) are C1 functions defined for x ∈ Rd+ and, at least initially, θm on the real line.Later, some of these will be extended (in the domain for θm) to one of the half complex planes {Im z ≥ 0},{Im z ≤ 0}. Additionally, we require that each of the σm,k(x, θm) is periodic in θm with mean 0 (as θmvaries in R,) and we will refer to these as the periodic profiles. Equation (1.14) gives a decent picture of thedependence of our approximate solution on the linear phases, though there remain subtleties to be explainedin the case that some of the φm are nonreal.

To justify the form (1.14) for our approximate solution, consider the result of plugging (1.14) into (1.2)(a)and expanding the error in ascending powers of ε:

P (εuaε , ∂x)uaε −F(εuaε )uaε = ε−1

(M∑m=1

µm∑k=0

L(dφm)∂θmσm,k

(x,φm(x)

ε

)rm,k

)+ ε0 (. . .) + . . . .(1.15)

Observe that since each rm,k belongs to Ker L(dφm), the term of order 1ε in (1.15) is in fact zero. A higher

order geometric optics expansion can be formed by adding to uaε (x) higher order terms, setting the resultingcoefficients which appear in the right hand side equal to zero, and ensuring that those differential equationsare satisfied.

4

Expansions with elliptic boundary layers were constructed in [12], treating the semilinear problem forgeneric β,5 and in [8], where a semilinear dispersive problem with maximally dissipative boundary conditionswas considered. In both papers, higher order expansions in ascending powers of ε are used to justify theapproximate solution as well as construct the exact solution. High order expansions involving surface waves,which arise from a failure of the uniform Lopatinski condition at a frequency β in the elliptic region, wereconstructed and justified in [9] for quasilinear hyperbolic systems. Here, in the spirit of [2], we justify theleading term expansion uaε (x). In this situation it is impossible to construct a high order expansion without asmall divisor assumption, an assumption we do not make. While the small divisor assumption would excludeβ only from a set of measure zero, verifying it for a given β can be difficult if not impossible.

We have yet to specify the profiles v(x) and σm,k(x, θm), and a good choice of these is required to achievethe correct leading term. That choice is related to the fact that, while the order 1

ε error has been eliminated,these profiles still appear nontrivially in the remaining error in the right hand side of (1.15). With simplyuaε (x) plugged into (1.15), an O(1) error remains, in particular. This error arises as an obstacle in our proofof the main theorem, though we are able to produce terms which cancel out all but a negligible amount ofthe error as a consequence of the fact that v(x) and σm,k(x, θm) are approximate solutions of the profileequations. We explain this process in greater detail in sections 1.2 and 1.3.

Plugging complex phases into profiles and the result: boundary layers and almost periodicprofiles

Now we take some time to elaborate on what we mean in (1.14) when we plug the phases φm(x)/εinto the periodic arguments of the profiles. For each m such that ωm is real, we call the σm,k(x, θm),k = 1, . . . , µm, hyperbolic profiles, and we note that in order to make the substitution

θm =φm(x)

ε=β · x′

ε+ ωm

xdε,(1.16)

such as those made in (1.14), a hyperbolic profile needs only to be defined for real θm. On the other hand,when ωm is nonreal, we call the σm,k(x, θm) elliptic profiles, and making this substitution requires us toevaluate these at nonreal values. Let us introduce the place holders θ0 for β · x′/ε and ξd for xd/ε, so thatthe analogous substitution to (1.16) is

θm = θ0 + ωmξd(1.17)

Let us also introduce the functions

ψm,k(x, θ0, ξd) := σm,k(x, θ0 + ωmξd).(1.18)

Thus, in addition to the representation of uaε given in (1.14), we have

uaε (x) = v(x) +

M∑m=1

µm∑k=1

ψm,k

(x,φ0(x′)

ε,xdε

)rm,k.(1.19)

Observe that for each elliptic profile, as we vary the parameters θ0 ∈ R and ξd ≥ 0 (corresponding to

varying x = (x′, xd) ∈ Rd+1

+ ,) the value θm = θ0 + ωmξd varies throughout only one of the half complexplanes {Im θm ≥ 0}, {Im θm ≤ 0}. To make sense of the complex substitution in the periodic argumentof an elliptic profile σm,k, we define the profile first for real θm and then holomorphically extend the θm-dependence into the appropriate half complex plane. This is done by making use of the Fourier expansionsof the profiles.

Consider a profile σm,k(x, θm), periodic in θm with mean 0, and its expansion of the form

σm,k(x, θm) =∑j∈Z\0

aj(x)eijθm .(1.20)

For the moment we proceed formally, assuming that one can evaluate the above expression at θm = θ0+ωmξdby substituting θm = θ0 + ωmξd in the argument of each exponential in the expansion for σm,k(x, θm) and

5[12] handles more general β for the problem studied there than we do here, treating glancing modes of order two. There itis also shown that higher-order glancing cases can result in blow-up.

5

yield a convergent expansion for the profile ψm,k(x, θ0, ξd) as defined in (1.18). That is, given (1.20), we alsohave

ψm,k(x, θ0, ξd) =∑j∈Z\0

aj(x)eij(θ0+ωmξd).(1.21)

We remark that we have yet to make sense of this sum and that the space of convergence must be clarifiedeven when ωm ∈ R. The convergence is made rigorous for any ωm ∈ C with Proposition 4.5, given that theprofiles satisfy criteria to be explained in full later. Let us rewrite such an expansion in the following form:

ψm,k(x, θ0, ξd) =∑j∈Z\0

aj(x)eijθ0eijRe(ωm)ξde−jIm(ωm)ξd .(1.22)

Suppose σm,k(x, θm) is elliptic. We first consider the case that Im ωm > 0. Then, the terms in the sum in(1.22) with j < 0 grow exponentially with ξd. It is easy to check that such terms result in an unsatisfactorycandidate for our approximate solution uaε (x), as defined in (1.14), which is nonphysical in the sense that itblows up in L∞ as ε→ 0. Thus we will need to have σm,k(x, θm) such that aj(x) = 0 for j < 0, i.e.

σm,k(x, θm) =∑

j∈Z+\0

aj(x)eijθm , for Im ωm > 0.(1.23)

Provided the above is a Fourier series in real θm converging in Hd2+3(x, θm), one can show that it holo-

morphically extends into {Im θm ≥ 0}. It follows that we can make sense of the corresponding profileψm,k(x, θ0, ξd) = σm,k(x, θ0 + ωmξd) for the values (x, θ0, ξd) ∈ Rd × T× R+, and Proposition 4.5 shows

ψm,k(x, θ0, ξd) =∑

j∈Z+\0

aj(x)eij(θ0+ωmξd) =∑

j∈Z+\0

aj(x)eijθ0eijRe(ωm)ξde−jIm(ωm)ξd , for Im ωm > 0.

(1.24)

As a result, the profile ψm,k(x, θ0, ξd) must decay exponentially in ξd.In the case that Im ωm < 0, similar considerations lead us to construct a corresponding elliptic profile of

the form

σm,k(x, θm) =∑

j∈Z−\0

aj(x)eijθm , for Im ωm < 0,(1.25)

a sum which holomorphically extends into {Im θm ≤ 0}. This kind of elliptic profile also results inψm,k(x, θ0, ξd) = σm,k(x, θ0 + ωmξd) which decays exponentially as ξd increases. For a hyperbolic profileσm,k(x, θm), which has Im ωm = 0, we do not make such restrictions on its coefficients aj(x), so we describeit with (1.20) and ψm,k(x, θ0, ξd) with the expansions (1.21) and (1.22). In general, any one of our profilesψm,k(x, θ0, ξd) is periodic in θ0, and either decays in ξd if it is elliptic or is almost periodic in ξd if it ishyperbolic. Since the elliptic profiles σm,k(x, θm) result in ψm,k(x, θ0, ξd) which decay in ξd, upon replacingthe placeholder ξd with xd

ε we see they contribute to a boundary layer with rapid exponential decay in xdfor small ε, i.e. the elliptic boundary layer part of the approximate solution described by (1.14).

Another important consequence of taking each elliptic profile either to have only positive or negativespectrum is that it considerably simplifies the couplings between the profiles that can occur in the profileequations in a way which is very useful for our solution.6 Ultimately, it is perfectly fine that we make suchrestrictions on the spectra since we are able to solve the profile equations for profiles of this form, whichallows us to satisfy a useful solvability condition.

With the following remark, we summarize the different forms of expansions for profiles.

6We stress that this restriction of couplings is not a kind of extra condition which we must impose. It is instead impliedwhen we write down the system of profile equations for profiles specifically of this form. Moreover, without this consideration itis not clear that the hyperbolic equations (together with the equation for v(x)) form their own sub-system, which is exploitedin a critical way in our solution.

6

Remark 1.3. For Zm = {n ∈ Z : nImωm ≥ 0}, i.e.

Zm =

Z if ωm ∈ R,Z+ if Im ωm > 0,Z− if Im ωm < 0,

(1.26)

each periodic profile has an expansion of the form

σm,k(x, θm) =∑

j∈Zm\0

aj(x)eijθm ,(1.27)

and each profile ψm,k(x, θ0, ξd) has

ψm,k(x, θ0, ξd) =∑

j∈Zm\0

aj(x)eij(θ0+ωmξd),(1.28)

where the sense of convergence is clarified with Proposition 4.5.

1.2 The profile equations

In order to have an approximate solution uaε (x), as in equation (1.14), such that the main theorem holds,our profiles v(x), σm,k(x, θm) must approximately satisfy the system of equations which we refer to as theprofile equations. To see where this additional criterion comes from, we recall that the idea that uaε (x) is theleading order term of a geometric optics expansion suggests we may add to uaε (x) higher order terms whichresult in solving (1.2)(a) to higher order. In particular, we should be able to find a function, say u1ε(x),such that when we replace uaε (x) with uaε (x) + εu1ε(x) in (1.15), both the O(1/ε) and O(1) error terms in theright hand side are annihilated. Ensuring the possibility of constructing such u1ε , which suggests we havethe right leading term uaε , requires solvability conditions to hold; these will be translated into conditions onthe profiles known as the profile equations.

DerivationWe may lift uaε (x) as defined in (1.14) under the substitution θ = (θ1, . . . , θM ) = (φ1/ε, . . . , φM/ε) =

φ/ε to get the following function

V0(x, θ1, . . . , θM ) := v(x) +

M∑m=1

µm∑k=0

σm,k(x, θm)rm,k.(1.29)

An additional useful representation of the leading term uaε can be found, with (1.19) in mind, if we analogouslylift the function uaε (x) under the substitution (θ0, ξd) = (φ0/ε, xd/ε) to get the function

U0(x, θ0, ξd) := v(x) +

M∑m=1

µm∑k=0

ψm,k(x, θ0, ξd)rm,k.(1.30)

We use only the formulation V0(x, θ) for the derivation and solution of the profile equations, though we needthe formulation U0(x, θ0, ξd) in addition to V0(x, θ) for the error analysis.

A calculation shows that taking

uaε (x) := U0(x, θ0, ξd)|θ0=φ0ε ,ξd=

xdε,(1.31)

or, equivalently,

uaε (x) := V0(x, θ(θ0, ξd))|θ0=φ0ε ,ξd=

xdε,(1.32)

where we use the notation

θ(θ0, ξd) = (θ0 + ω1ξd, . . . , θ0 + ωMξd),(1.33)

7

results in the vanishing of the terms of order 1ε when plugging uaε into P (εuε, ∂x)uε in (1.2)(a). We define

the operator

L(∂θ) :=

M∑m=1

L(dφm)∂θm(1.34)

where we have denoted L(dφm) = A−1d L(dφm). The O(1/ε) error term in (1.2)(a) can be written in termsof V0 as

(L(∂θ)V0)|θ=φ/ε.(1.35)

Thus the condition L(∂θ)V0 = 0 implies the cancellation of the O(1/ε) error, and this is satisfied by thedefinition of V0 regardless of the choice of our profiles v, σm,k.

Now let us consider plugging in a corrected approximate solution

ucε(x) :=(U0(x, θ0, ξd) + εU1(x, θ0, ξd)

)|θ0=

φ0ε ,ξd=

xdε

=(V0(x, θ(θ0, ξd)) + εV1(x, θ(θ0, ξd))

)|θ0=

φ0ε ,ξd=

xdε.

(1.36)

Here we do not necessarily have V1 of the form (1.29), but we do impose other constraints7 on its form (andanalogously for U1.) Let us define the operator

M(V)∂θ :=

M∑m=1

d−1∑j=0

βj∂uAj(0)V∂θm ,(1.37)

Then the cancellation of the O(1) error term in (1.15) (with uaε replaced by ucε) is satisfied if

L(∂θ)V1 + L(∂x)V0 +M(V0)∂θV0 = F (0)V0.(1.38)

Observe that the operator L(∂θ) is singular, so that one cannot simply invert it to solve for V1 in (1.38).One way to proceed is to ensure the existence of a solution V1 by imposing the following condition on V0:

L(∂x)V0 +M(V0)∂θV0 − F (0)V0 ∈ Im L(∂θ).(1.39)

This condition turns out to be equivalent to a differential equation in V0. To get an idea of how this works,recall from Lemma 1.2 that Im L(dφm) = Ker Pm, where Pm is the projection to the subspace spanned by

rm,k, k = 1, . . . , µm. Using the fact that L(∂θ) =∑Mm=1 L(dφm)∂θm and the decomposition of Lemma 1.2,

we are able to construct a projection operator E[ such that if a function of a certain type8 V(x, θ) satisfiesE[V = 0, then V ∈ Im L(∂θ). Such a projector E[ is also defined in the study [6]. Thus, a sufficient conditionfor (1.39) is that

E[(L(∂x)V0 +M(V0)∂θV0) = E[(F (0)V0),(1.40)

with the caveat that, in order for this to work, the functions to which E[ is applied must satisfy the hypothesisof Proposition 3.8, a consideration which we will discuss in detail later. This equation is closely related tothe profile equations, but it is not quite what we want. The current task is to solve for the correct choicesof v and the σm,k, but if we simply impose the condition (1.40) on V0 it is not clear how (or even whetheror not) that would determine v and the σm,k.

To get the profile equations, we define another projection operator, E, very closely related to E[,9 whichsatisfies

E[(V)|θ=φ/ε = E(V)|θ=φ/ε.(1.41)

7We will require that V1(x, θ) ∈ Hs;2(x, θ), defined in (3.4).8It is required that V(x, θ) ∈ Hs;2(x, θ) and that it is a trigonometric polynomial in θ. See Proposition 3.8.9The relation between the two projectors is clarified in Section 3 and will be used in the error analysis. Both projectors are

defined in Definition 3.5.

8

Thus we have that the following agrees with (1.40) after making the substitution θ = θ(θ0, ξd):

E(L(∂x)V0 +M(V0)∂θV0) = E(F (0)V0).(1.42)

Together with the appropriate boundary conditions, (1.42) forms the profile equations. The set of equationsfor V0, including the interior profile equations and the boundary conditions, are given in (3.31). This isindeed equivalent to a system of transport equations in terms of v and the σm,k which we will solve, at leastapproximately.

We digress to mention one interesting difference in our study from [2]: in that study (in which theboundary frequency is assumed to be in the hyperbolic region) they formulate and solve both a formulationof the profile equations in terms of V0, as we have with (3.31), and another formulation in terms of U0, referredto as the almost periodic formulation. For the more general setting we encountered serious difficulties10 tomaking a derivation in terms of U0, and we show it is possible to solve the general problem by only using aformulation of the profile equations in terms of V0. In fact, we found there were many instances where wewere able to do our analysis in one fell swoop in terms of V0, especially by using Proposition 4.5 in the erroranalysis, whereas in [2], the two settings are addressed separately.

Large system formulation and its solutionThe profile equations (1.42) can be expressed as a quasilinear differo-integral system of equations with

v and the profiles σm,k as the unknowns. This system is explicitly written out in Proposition 3.18 andis derived in the following way: the expression for V0 in the right hand side of (1.29) is regarded as anansatz, which we plug into (1.42). The projector E, which is designed to project to the characteristic modesfor the operator L(∂θ), decomposes into a sum of projectors E0, Em,k, where E0 generally projects to acomponent with nonzero mean, and each Em,k projects to a component with mean zero. The result is anN ×N system for v(x) (the component of (1.42) in the image of E0) coupled with a set of complex transportequations indexed by (m, k), m = 1, . . . ,M , k = 1, . . . , νkm , where the (m, k) equation (the component inthe image of Em,k) is the transport equation for σm,k(x, θm). In this equation we have the correspondingcharacteristic, complex vector field Xφm

11 applied to σm,k(x, θm). More precisely, for each hyperbolic profile,the corresponding vector field is real, and for each elliptic profile, the vector field is nonreal.

The nonlinear quadratic term M(V0)∂θV0 in (1.42) results in couplings between the profiles. Thesemanifest as integrals of two types in the large system. The first type is a mean of the product of two periodicfunctions, in lines (3.51) and (3.52)(b), and the second type is the interaction integrals, such as Jk,k

′

p,nq,nr ,defined in (3.47) and appearing in (3.53) and (3.55). The interaction integrals account for resonances, asdescribed in Proposition 3.16. Resonances occur in products such as σq,k(x, φq/ε)∂θrσr,k′(x, φr/ε) when wehave a relation among the phases of the form

nφp = nqφq + nrφr,(1.43)

where p ∈ {1, . . . ,M} \ {q, r}, nq and nr are in the spectra of σq,k and σr,k′ , respectively, and n is al-lowed to be any integer12. The implication of (1.43) is that oscillations with phases φq and φr interactto produce oscillations with phase φp. In the equation for σp,l(x, θp), in which we apply Ep,l to the prod-uct σq,k(x, θq)∂θrσr,k′(x, θr)M(rq,k)rr,k′ (a constituent of the term M(V0)∂θV0 from (1.42)), the projectoraccounts for these relations and outputs the appropriate function which depends only on (x, θp).

In [2], the authors derive and solve a large system for the profiles which is almost identical to the systemwe give in Proposition 3.18, except the restrictions on the boundary frequency result in the absence of anyelliptic profiles. As a result, the system in that case was a system of real transport equations. In that study,the authors develop an iteration scheme of the form (3.71).13 Equation (3.71)(a) implies that V0,n

h has adecomposition such as (1.29), but with other components, vn(x), and the profile iterates σnm,k(x, θm) in placeof v(x) and the σm,k(x, θ). Corresponding to (3.71)(b) is an equivalent large system in the profile iterates,

10Essentially, the difficulties arise from having nonreal ωm, which prevent functions U(x θ0, ξd) we use from being almostperiodic. Our attempts to construct a projector for forming profile equations in a space large enough to include such functionswhich were not almost periodic resulted in unbounded projectors, which were thus unusable.

11The definition of Xφm is given in Lemma 3.10.12It turns out a p-resonance can only be produced with n ∈ Zp, which is justified in Remark 3.12, though this cannot be

assumed a priori.13In the special case handled in [2], the set of equations (3.71) reduces to the iteration scheme from [2]. In that case, V0

h and

V0 are the same, as are V0,nh and V0,n, and the projectors E and Eh.

9

essentially the large system (3.51)-(3.52) with respective iterates in place. In [2] the authors prove an energyestimate for this system by combining a Kreiss-type estimate for the nonzero mean system (for vn) with anestimate for the zero mean system (for the profile iterates σnm,k.)

The latter is proven in [2] by taking the L2(dx dθm) pairing of σnm,k(x, θm) with its corresponding equationand integrating by parts to transfer derivatives off of the n-th iterates. The profiles (which are all hyperbolicin that case) are categorized as either incoming or outgoing according to their corresponding group veloc-ities. This determines whether the corresponding boundary terms appear with a good or a bad sign uponperforming the integration by parts on the equation for a given profile. In the result, the L2 norms of theoutgoing profiles σnm,k, m ∈ O (good sign) and their traces σnm,k|xd=0 appear with positive sign on the sameside (the lower bound,) bounded by the norms of the previous iterates, a source term, and the prescribedfunction on the boundary G, as desired. On the other hand, when writing the bound resulting from thesame calculation, but for the incoming profiles σnm,k, m ∈ I (bad sign), the norms of the traces fall on theupper bound side, and so they are forced to separately bound the traces of the incoming profiles in orderto proceed. Due to the uniform stability assumption14, the authors are able to estimate these terms for theincoming profiles by using the boundary condition to rewrite the trace of incoming σnm,k in terms of G andthe traces of the outgoing profiles, which had already been estimated, having occurred with good sign.

In contrast, for our study we must deal with complex transport equations, and while methods from[2] are used to a degree in handling the large system, since complex transport equations are generally notsolvable, a different approach is required. Dealing with the more general complex phases also forced us tohandle complex resonances, which determine the ways in which hyperbolic and elliptic profiles are coupledwith each other in the large system. Careful examination of these resonances and the possible interactionsrevealed the following about the system: the subset formed by the equations for v and the hyperbolic profilesis actually a sub-system of real transport equations not dependent on the elliptic profiles, which we refer toas the hyperbolic sub-system, given in (3.69). On the other hand, resonances between hyperbolic and ellipticphases do cause the complex transport equations to depend on the hyperbolic profiles. This does not causemuch additional difficulty since we are able to work at first strictly within the hyperbolic sub-system to solvefor v and the hyperbolic profiles. The procedure used to solve the full system in [2] can be straightforwardlyadapted to solve our real sub-system; our hyperbolic profiles are split into two groups, the incoming and theoutgoing, and essentially we use the same method sketched above with the iteration scheme (3.71).

As mentioned before, in contrast with the real transport equations, the complex transport equations forthe elliptic profiles are generally not solvable. In [12], a Taylor expansion in xd is developed to approximatelysolve the corresponding complex profile equations by satisfying the equations evaluated at xd = 0 along withhigh order transverse derivatives of the equations at xd = 0. One could use a similar approach for thisproblem, but using the error bounds of Taylor’s theorem to get an appropriate bound on the error in Hs

would require us also to take a large number of derivatives of the equations and require those to hold onthe boundary. Taking derivatives of the other profiles appearing in those equations means we would have torequire them to be in a higher order Sobolev space. Interestingly, however, in our proof of Theorem 4.7 wewere able to show that merely requiring the complex transport equations themselves to hold to at xd = 0,disregarding the higher order xd derivatives, is sufficient for our purposes. The error term is handled by anapplication of Proposition 4.6 in a process which we explain further in our discussion on the error analysis.

In the original boundary condition, the value we must prescribe for elliptic σm,k|xd=0 can be straightfor-wardly read off from linear relations with G and the hyperbolic, outgoing profiles at xd = 0, as a result ofthe uniform stability condition. The value we must prescribe for ∂xdσm,k|xd=0 is determined by taking thetransport equation for σm,k from the large system, evaluating this at xd = 0, and plugging in the (zerothorder in xd) traces of the other profiles, all of which have already been obtained. The problem reduces toextending elliptic σm,k from the data {σm,k|xd=0, ∂xdσm,k|xd=0} ∈ Hs×Hs−1 on the boundary to a functionin Hs in the interior, which is also supported in {t ≥ 0}. We accomplish this by solving a wave equation

∂2xdς −∆x′,θ0ς = 0,(1.44)

with slightly modified initial data, and then translating the solution away from the boundary with thesubstitution σm,k(x, θm) = ς(t− xd, x′′, θm),15 taking advantage of finite speed of propagation to ensure the

14We also assume this; see Assumption 2.5.15Strictly speaking, this is not precisely the definition of σm,k(x, θm), which actually includes multiplication by a smooth

cutoff in the variable xd.

10

support is contained in {t ≥ 0}. This process is described in detail with Lemma 3.28 and Proposition 3.29.

1.3 Proof of the main theorem

Recall the singular system (1.6) discussed in Section 1.1. As we stated there, in order to prove the maintheorem, we show a stronger result in terms of the solution of this system, Uε. The stronger result we showis the following:

Theorem 1.4. Define

U0ε (x, θ0) = V0

(x, θ0 + ω1

xdε, . . . , θ0 + ωM

xdε

)(1.45)

where V0 is as in (1.29). If G(x′, θ0) ∈ Hs+1T for sufficiently large s, then for the exact solution Uε of the

singular system (1.6), we have

|Uε(x, θ0)− U0ε (x, θ0)|L∞(xd,Hs−1(x′,θ0)) → 0 as ε→ 0.(1.46)

Recall that the exact solution of (1.2) is formed by making the substitution

uε(x) = Uε (x, θ0) |θ0=

φ0ε.(1.47)

Our approximate solution uaε (x) satisfies

uaε (x) = U0ε (x, θ0) |

θ0=φ0ε,(1.48)

and it follows that Theorem 1.1 is a direct consequence of Theorem 1.4.Before sketching the proof of Theorem 1.4, we define the following norm which is useful in the error

analysis:

EsT = {U(x, θ0, ξd) : supξd≥0|U(·, ·, ξd)|EsT <∞}.(1.49)

Typically, taking functions such as V0(x, θ) ∈ Hs+1 whose dependence on θ is extended to the productof complex half-spaces discussed in Remark 3.3,16 and then substituting in the complex valued functionθ = θ(θ0, ξd) yields a function which is bounded in the EsT norm. It is convenient at times to work in the

spaces for such functions, Es;kT , before evaluating at ξd = xd/ε as in (1.45).Proof of the main result

In the proof of Theorem 7.1 of [13], for some T0 > 0, the iteration scheme for the singular system,(4.41), is used to produce Uε(x, θ0) and iterates Unε (x, θ0) bounded in EsT0

uniformly with respect to n andε and which satisfy

limn→∞

Unε = Uε in Es−1T0uniformly with respect to ε ∈ (0, ε0],(1.50)

where Uε solves the singular system (1.6).As in [2], we use the strategy of simultaneous Picard iteration to prove the main theorem, an idea due

to the study [5], whereby we construct iterates U0,nε (x, θ0) converging to U0

ε uniformly as n→∞ and showthat U0,n

ε is close to Unε for each n. In Section 3.3, Proposition 3.29 of Section 3.4, and Definition 3.31 we

construct a function V0(x, θ) ∈ Hs(Rd+1

+ × TM ) which approximately satisfies the profile equations, anditerates V0,n(x, θ) approximately satisfying the equations of a corresponding iteration scheme. The iteratesV0,n are bounded in Hs+1

T0uniformly with respect to n and satisfy

limn→∞

V0,n = V0 in HsT0.(1.51)

16This is the product of half-spaces on which V0(x, θ) becomes defined once we have extended each of the elliptic profilesσm,k(x, θm) to its natural complex half-space, as discussed in the subsection on plugging in complex phases of Section 1.1.

11

Again, we plug in the function θ(θ0, ξd) in the placeholder argument θ to get

U0,n(x, θ0, ξd) := V0,n(x, θ0 + ω1ξd, . . . , θ0 + ωMξd),(1.52)

followed by ξd = xdε , yielding

U0,nε (x, θ0) := V0,n

(x, θ0 + ω1

xdε, . . . , θ0 + ωM

xdε

).(1.53)

Thus, by using (1.51) and applying the estimates given by Proposition 4.5 and Lemma 4.8, we get that

limn→∞

U0,nε = U0

ε in Es−1T0uniformly with respect to ε ∈ (0, ε0],(1.54)

where U0ε is as in Theorem 1.4. Therefore, to conclude limε→0 U0

ε (x, θ0) − Uε(x, θ0) = 0 in Es−1T0and finish

the proof of Theorem 1.4, it is sufficient to show

limε→0|U0,nε − Unε |Es−1

T0

= 0 for all n.(1.55)

Indeed, (1.55) is proved in Section 4.2 by induction on n. One might try to prove the statement in thisway by applying the estimate for the linearized singular system17 of Proposition 4.9 to

(U0,n+1ε − Un+1

ε

). The

problem with this is that if we take A(εUnε ), which we define to be the operator appearing in the left handside of the equation for Un+1

ε , i.e. (4.41)(a), and apply it to the difference(U0,n+1ε − Un+1

ε

), the resulting

quantity leaves an O(1) error, so we do not get convergence to zero as ε→ 0. This happens for the followingreason: while Un+1

ε solves this equation exactly, U0,n+1ε was only designed to eliminate the O(1/ε) order

error term. To be more precise, recalling our discussion on the formulations of the analogous error termsin (1.15) in terms of V0 in the section on the profile equations, one sees that an O(1) error is left in (1.15)when we form uaε = U0

ε |θ0=φ0/ε.We can amend this argument by using a similar approach to the O(1) error elimination discussed in

Section 1.2. The fix lies in constructing a corrector for V0,n+1, say V1,n+1, analogous to the function V1

which we discussed there. In the following we refer to the corresponding equations18 having the appropriateiterates in place of the original functions. Suppose for the moment that there exists V1,n+1 which results inthe elimination of the O(1) terms upon the replacement of V0,n+1 by V0,n+1 + εV1,n+1. Then one can obtaina corrector for U0,n+1

ε , say U1,n+1ε (x, θ0), by evaluating V1,n+1(x, θ) at θ = (θ0 + ω1xd/ε, . . . , θ0 + ωMxd/ε).

Applying A(εU0,nε ) to

(U0,n+1ε + εU1,n+1

ε − Un+1ε

)results in an error which converges to zero as ε→ 0. Then,

assuming for the moment that U1,n+1ε is bounded uniformly in ε, an application of the estimate from [13]

gives us

limε→0|U0,n+1ε − Un+1

ε |Es−1T0

= 0,(1.56)

which would allow us to conclude the induction step.However, there are two major obstacles to constructing such a corrector V1,n+1 for V0,n+1. To see the

first, recall that we mentioned in our discussion on solving the elliptic profile equations that in general theycannot be exactly solved. It is the same situation for the iterated profile equations. We form V0,n+1 so thatthe error Rn+1(x, θ) is zero at the boundary {xd = 0} and elliptically polarized19. This implies that whenwe make the substitution θ = θ(θ0, ξd) the error decays exponentially in ξd. We describe these conditionson Rn+1 precisely in Proposition 4.6 and prove that they imply Rn+1(x, θ(θ0, xd/ε)) goes to zero in Es−1T asε→ 0. Thus, the solution of the profile equations modulo this error is a sufficient solvability condition.

The second issue is that, since we do not make a small-divisor assumption, we can only guaranteesolvability if we are working with finite trigonometric polynomials20, as opposed to general elements of Hs

T0

which may have infinitely many nonzero Fourier coefficients. This is resolved with an approach similar to

17This linear estimate is taken from [13], where it is used to prove the existence of the solution to the singular system.18Here, we refer to equations (1.15), (1.38), etc.19This means EeRn+1 = Rn+1, where Ee is the elliptic projector defined in (3.13). The good decay properties of

Rn+1|θ=θ(θ0,ξd) basically come from the fact that Rn+1 then has zero mean and only oscillates nontrivially in variablesθj for j such that ωj is nonreal.

20For details on solvability, see Proposition 3.8.

12

that in [2], based on the idea due to [5] of using trigonometric polynomial approximations of the functions.We approximate each V0,n+1 by a trigonometric polynomial V0,n+1

p , construct a corresponding corrector V1p ,

and use Proposition 4.5 and Lemma 4.8 to work with U0,n+1p = V0,n+1

p |θ=θ(θ0,ξd) and U0,n+1p,ε = U0,n+1

p |ξd=xd/εapproximating U0,n+1 and U0,n+1

ε , respectively.We emphasize that for our study we are forced to handle the trigonometric polynomial approximations

differently from [2] as a result of dealing with nonreal ωm, since this means the U-type functions are nolonger almost periodic. Because we construct the corrector in the space for V-type functions, rather than inthe space for U , as done in [2], we must check that taking a trigonometric approximation for V(x, θ) in Hs+1,and then plugging in θ(θ0, ξd), yields a good approximation in the space for U(x, θ0, ξd), with respect to thenorm EsT . Establishing an estimate of the form |U|EsT ≤ C|V|Hs+1

Tis key, and having complex ωm introduces

new difficulties.A positive δ is fixed and each of these approximations has error bounded by O(δ) in the corresponding

norm. By carefully replacing the trigonometric polynomial approximations with respective quantities in theargument sketched above, we are able to solve away all but O(δ) of the O(1) error21. From this we canestablish

|U0,n+1ε − Un+1

ε |Es−1T0

≤ Cδ + c(ε) +K(δ)ε,(1.57)

for all δ, (where the error terms in the right hand side are given in the proof of Proposition 4.7,) so that theproof of the theorem can be concluded.

2 Assumptions and other preliminaries

We now take some time to go over the assumptions we make for the problem, some related definitions,and the example of the compressible Euler equations, to which our analysis applies.

We assume that L(∂x) is hyperbolic with characteristics of constant multiplicity, making the following:

Assumption 2.1. The matrix A0 = I. For an open neighborhood O of 0 ∈ RN , there exists an integerq ≥ 1, some real functions λ1, . . . , λq that are C∞ on O × Rd \ {0}, homogeneous of degree 1, and analyticin ξ, and there exist some positive integers ν1, . . . , νq such that:

(2.1) det[τ I +

d∑j=1

ξj Aj(u)]

=

q∏k=1

(τ + λk(u, ξ)

)νkfor u ∈ O, ξ = (ξ1, . . . , ξd) ∈ Rd \ {0}. Moreover the eigenvalues λ1(u, ξ), . . . , λq(u, ξ) are semi-simple (theiralgebraic multiplicity equals their geometric multiplicity) and satisfy λ1(u, ξ) < · · · < λq(u, ξ) for all u ∈ O,ξ ∈ Rd \ {0}.

Also, for our study we consider a noncharacteristic boundary {xd = 0}:

Assumption 2.2. For u ∈ O the matrix Ad(u) is invertible and the matrix B(u) has maximal rank, its rankp being equal to the number of positive eigenvalues of Ad(u) (counted with their multiplicity).

We now provide some notation regarding frequencies τ − iγ ∈ C and η ∈ Rd−1 dual to the variables tand y, respectively. We define the matrix

A(ζ) := −i A−1d (0)

(τ − iγ)I +

d−1∑j=1

ηj Aj(0)

, ζ := (τ − iγ, η) ∈ C× Rd−1,(2.2)

21To be more precise, the error we are referring to is in the right hand side of (4.60) (note the left hand side.) The first twoterms are small due to the approximate solution of the profile equations, and the last term is that which is solved away. By thiswe mean that after incorporating the corrector, which we refer to here as U1,n+1

p,ε , all that is left is the quantity (4.67) (wherethe corrector is instead denoted by U1

p,ε.)

13

and the sets of frequencies

Ξ :={

(τ − iγ, η) ∈ C× Rd−1 \ (0, 0) : γ ≥ 0}, Σ :=

{ζ ∈ Ξ : τ2 + γ2 + |η|2 = 1

},

Ξ0 :={

(τ, η) ∈ R× Rd−1 \ (0, 0)}

= Ξ ∩ {γ = 0} , Σ0 := Σ ∩ Ξ0 .

With these in mind, we define the symbol

L(τ, ξ) := τI +

d∑j=1

ξjAj(0).(2.3)

The following is a result due to Kreiss [7] in the case of strict hyperbolicity, i.e. when all the eigenvaluesin Assumption 2.1 have multiplicity νj = 1, and to Metivier [10] in our more general case.

Proposition 2.3 ([7, 10]). Let Assumptions 2.1 and 2.2 be satisfied. Then for all ζ ∈ Ξ \ Ξ0, the matrixA(ζ) has no purely imaginary eigenvalue and its stable subspace Es(ζ) has dimension p. Furthermore, Esdefines an analytic vector bundle over Ξ \ Ξ0 that can be extended as a continuous vector bundle over Ξ.

For (τ, η) ∈ Ξ0, we define Es(τ, η) to be the continuous extension obtained in Proposition 2.3 of Es to(τ, η).

Now we describe our assumption of uniform stability, defined as in [7, 3]:

Definition 2.4. The problem (1.2) is uniformly stable at u = 0 if the linearized operators (L(∂x), B(0)) atu = 0 are such that

B(0) : Es(τ − iγ, η)→ Cp is an isomorphism for all (τ − iγ, η) ∈ Σ.(2.4)

Assumption 2.5. The problem (1.2) is uniformly stable at u = 0.

The following is an important example, the Euler equations, which satisfies the above assumptions sothat our main results may be applied.

Example 2.6 (Euler equations). The following are the isentropic, compressible Euler equations in threespace dimensions on the half space {x3 ≥ 0}, in the unknowns density ρ and velocity u = (u1, u2, u3):

∂t

ρρu1ρu2ρu3

+ ∂x1

ρu1

ρu21 + p(ρ)ρu2u1ρu3u1

+ ∂x2

ρu2ρu1u2

ρu22 + p(ρ)ρu3u2

+ ∂x3

ρu3ρu1u3ρu2u3

ρu23 + p(ρ)

=

0000

,(2.5)

where p(ρ) is the pressure. The hyperbolicity assumption, Assumption 2.1, is satisfied in the region of statespace where ρ > 0, c2 = p′(ρ) > 0. In this case the eigenvalues λk(ρ, u, ξ) are

λ1 = u · ξ − c|ξ|, λ2 = u · ξ, λ3 = u · ξ + c|ξ|, with (ν1, ν2, ν3) = (1, 2, 1).(2.6)

For this problem, the boundary condition we impose is the natural “residual boundary condition,” which isobtained in the vanishing viscosity limit of the compressible Navier-Stokes equations with Dirichlet boundaryconditions. Fixing any constant state (ρ, u) with u3 /∈ {0,−c, c} about which to linearize the problem, so that(ρ, u) = u0 in (1.1), we have noncharacteristic boundary {x3 = 0} for the system. Consider in particularAssumption 2.5 in each of the following cases:

(a)Subsonic outflow: u3 < 0, |u3| < c. In this case there is exactly one positive eigenvalue ofA3(ρ, u) (p = 1), so we need one scalar boundary condition. Taking b(ρ, u) = u3 in (1.1), we have B(0) =[0 0 0 1

]and Assumption 2.5 is satisfied.

(b)Subsonic inflow: 0 < u3 < c. For this we get p = 3 and boundary condition b(ρ, u) = (ρu3, u1, u2)and linearized operator

B(0)(ρ, u) = (ρu3 + ρu3, u1, u2),(2.7)

14

which satisfies Assumption 2.5.(c)Supersonic inflow: 0 < c < u3. This case is trivial, with p = 4 and B(0) the 4× 4 identity matrix,

and so Assumption 2.5 holds.(d)Supersonic outflow: u3 < 0, |u3| > c. This is another trivial case, with p = 0, where B(0) is

absent, meaning Assumption 2.5 holds vacuously.With clear modifications of the above statements, the same holds for the 2D Euler equations, which are

in fact strictly hyperbolic. For complete proofs and discussion verifying these cases, we refer the reader to[4], Section 5.

Regular boundary frequenciesWe now discuss an assumption regarding the boundary frequency β = (τ , η) ∈ R × Rd−1 \ (0, 0) which

is relevant to the construction of the approximate solution, allowing us to obtain the phases φm and severalother objects used in the construction.

Assumption 2.7. For ζ = (τ − iγ, η) ∈ C× Rd−1 consider the matrix

(2.8)1

iA(τ − iγ, η) = −A−1d (0)

(τ − iγ) I +

d−1∑j=1

ηj Aj(0)

.

Let ωm, m = 1, . . . ,M denote the distinct eigenvalues of 1iA(τ , η). We suppose that each ωm is a semisimple

eigenvalue with multiplicity denoted µm. Moreover, we assume there is a conic neighborhood O of β inC×Rd−1\{0} on which the eigenvalues of 1

iA(ζ) are semisimple and given by functions ωm(ζ), m = 1, . . . ,Manalytic in ζ0 = τ − iγ and smooth in η, where ωm = ωm(β) and ωm(ζ) is of constant multiplicity µm.

We call β a regular boundary frequency provided Assumption 2.7 holds. We also have the complexcharacteristic vector field associated to φm:

Xφm := ∂xd +

d−1∑j=0

−∂ξjωm(β)∂xj ,(2.9)

which is real for real ωm. Moreover, for each ωm which is real, Assumption 2.1 implies there is a uniquekm ∈ {1, . . . , q} such that τ + λkm(η, ωm) = 0, and in this case we have µm = νkm .22 With this in mind, wemake the following definition:

Definition 2.8. (i) For m such that ωm is real, we call (β, ωm) a hyperbolic mode if

∂ξdλkm(η, ωm) 6= 0.(2.10)

(ii) For m with nonreal ωm, we call (β, ωm) an elliptic mode.

Remark 2.9. (i) For each real ωm, Assumption 2.7 guarantees that (β, ωm) is a hyperbolic mode. That(2.10) holds is a consequence of the semisimplicity of ωm.23 Also, the condition (2.10) and the implicitfunction theorem imply that ωm is real and of multiplicity νkm in a neighborhood of β. Thus, there is aslight redundancy in Assumption 2.7 when ωm is real.

(ii) When ωm is nonreal, the fact that the Aj are real implies that 1iA(β) also has the complex conjugate

ωm as an eigenvalue, and thus ωm = ωm′ for some m′ 6= m. Furthermore, if the vector r ∈ CN is aneigenvector of 1

iA(β) associated to ωm, then r is an eigenvector associated to ωm = ωm′ .(iii) The boundary frequency β lies in the hyperbolic region (as defined in [2]) if and only if Assumption

2.7 holds with all ωm real (with smoothness in real ζ0 replacing the analyticity condition.) The elliptic regionconsists of all β such that Assumption 2.7 holds with all ωm nonreal.

22Assumption 2.1 has no immediate implication for the multiplicity of nonreal ωm.23If ωm is real and ∂ξdλkm (η, ωm) = 0, we refer to (β, ωm) as a glancing mode. An explanation of how (2.10) follows from

semisimplicity can be found in the proof of Lemma 2.7 of [10].

15

Example 2.10. We return to the 3D Euler equations, linearizing about (ρ, u) = (ρ, u1, u2, u3), as in Example2.6.

For |u3| > c the set of regular boundary frequencies is all of R3 \ 0. In the case |u3| < c, one finds thatthe set of regular boundary frequencies is{

(τ, η) ∈ R3 : |τ + u1η1 + u2η2| 6=√c2 − u23 |η|

}.(2.11)

We remark that in the former case, the hyperbolic region is also H = R3 \ 0, and in the latter case

H ={

(τ, η) ∈ R3 : |τ + u1η1 + u2η2| >√c2 − u23 |η|

}.

Remark 2.11. A key feature related to the assumptions which distinguishes our study from that in [2] isthat in [2] it is assumed that β belongs to the hyperbolic region. This is equivalent to requiring that both(i) β is a regular boundary frequency (i.e. β satisfies Assumption 2.7) and (ii) all the eigenvalues ωm of thematrix 1

iA(β) are real. In our study, we allow for any regular boundary frequency β thus also handling casesin which any number of the ωm are nonreal.

The partition of the phases into incoming and outgoing hyperbolic phases and elliptic phaseswith positive and negative imaginary parts

For real ωm we have the associated real group velocity:

vm := ∇λkm(η, ωm),(2.12)

which has the following relationship with the characteristic vector field (2.9):

∂ξ0ωm(β) = − 1

∂ξdλkm(η, ωm), ∂ξjωm(β) = −

∂ξjλkm(η, ωm)

∂ξdλkm(η, ωm), j = 1, . . . , d− 1.(2.13)

Observe that each group velocity vm can be thought of as either incoming or outgoing with respect to theinterior of the domain Rd+, in particular since the last coordinate of vm is nonzero, by (2.10). With this inmind, we classify the phases in the following way:

Definition 2.12. For real ωm, the phase φm is incoming if the group velocity vm is incoming (that is,∂ξdλkm(η, ωm) > 0), and it is outgoing if the group velocity vm is outgoing (∂ξdλkm(η, ωm) < 0).

With this classification of the real phases, we let I denote the set of indices m ∈ {1, . . . ,M} such thatφm is incoming and O the set of m such that φm is outgoing. We classify the remaining complex phases φm,which correspond to nonreal ωm, by the distinction that P is the set of m such that Im ωm > 0 and N isthe set of m such that Im ωm < 0. We thus form the partition

{1, . . . ,M} = I ∪ O ∪ P ∪N .(2.14)

Recall the ωm(τ, η) are eigenvalues of 1iA(τ, η), meaning the iωm(τ, η) are the eigenvalues of A(τ, η).

Consider also the fact that Im ωm(τ, η) > 0 if and only if Re iωm(τ, η) < 0. From these observations, wesee that the stable subspace Es(τ, η) of A(τ, η) must contain each of the eigenspaces corresponding to someωm(τ, η) with Re iωm < 0, so that Es(τ , η) contains the subspaces Ker L(dφm) for m ∈ P. The incomingphases, corresponding to m ∈ I, also play a role in setting up a decomposition for the stable subspace Es atthe boundary, which is established in the following lemma.

Lemma 2.13. The stable subspace Es(τ , η) admits the decomposition:

Es(τ , η) = ⊕m∈I∪PKer L(dφm),(2.15)

where, in the decomposition, the vector spaces Ker L(dφm) for m ∈ I admit a basis of real vectors.

Proof. It is easy to show that the subspaces Ker L(dφm) for m ∈ P are in Es(τ , η). We will show that thisis also the case for the subspaces Ker L(dφm) with m ∈ I, from which the result will follow. Since Es(τ , η)

16

is close to Es(τ − iγ, η) for small γ > 0, in accordance with Proposition 2.3, it will suffice to show for m ∈ Ithat iωm(τ − iγ, η), close to iωm, satisfies

Re iωm(τ − iγ, η) < 0.(2.16)

Recalling (2.13), we see that since φm is incoming, i.e. m ∈ I, we have

∂ζ0ωm(τ , η) < 0.(2.17)

Using analyticity of ωm(ζ) with (2.17), since ωm(τ , η) has imaginary part equal to zero, it follows thatωm(τ − iγ, η) has positive imaginary part for small positive γ. Therefore (2.16) holds. The statement thateach of the vector spaces Ker L(dφm) for m ∈ I admits a basis of real vectors follows from Lemma 1.2,which was proved in [1].

3 Profile equations: formulation with periodic profiles

Definition 3.1. (i) We define ZM ⊂ ZM by

ZM := {α = (αi)Mi=1 : αi ∈ Zi},(3.1)

where Zi is defined by

Zi :=

Z for i ∈ I ∪ O,Z+ for i ∈ P,Z− for i ∈ N ,

(3.2)

where I, O, P, and N are as defined in the comments following Definition 2.12. We also have the equivalentformulation Zi = {n ∈ Z : nImωi ≥ 0}.(ii) We also define

ZM ;k := {α ∈ ZM : at most k components of α are nonzero}.(3.3)

Definition 3.2. For k = 1, 2, we define the following spaces:

Hs;k(Rd+1

+ × TM ) =

V(x, θ) ∈ Hs(Rd+1

+ × TM ) : V(x, θ) =∑

α∈ZM;k

Vα(x)eiα·θ

.(3.4)

For s > (d+ 1 + 2)/2, it is clear that multiplication defines a continuous map

Hs;1(Rd+1

+ × TM )×Hs;1(Rd+1

+ × TM )→ Hs;2(Rd+1

+ × TM ).(3.5)

Remark 3.3. We define

CM := C1 ×C2 × · · · ×CM ,(3.6)

where Ci is defined by

Ci :=

R for i ∈ I ∪ O,{Im z ≥ 0} for i ∈ P,{Im z ≤ 0} for i ∈ N .

(3.7)

For V ∈ Hs;2(Rd+1

+ × TM ) where s > d2 + 3, one can show that spec V ⊂ ZM ;2 implies V extends holomor-

phically in θ to the interior of CM . In particular, this uses the fact that then Im(αiθi) ≥ 0 for i ∈ P ∪ N .This allows us to make sense of

U(x, θ0, ξd) := V(x, θ0 + ω1ξd, . . . , θ0 + ωMξd)(3.8)

for θ0 ∈ T, ξd ≥ 0.

17

3.1 The solvability condition

Our periodic ansatz has the form

V0(x, θ) = v(x) +

M∑m=1

µm∑k=1

σm,k(x, θm)rm,k,(3.9)

a function in Hs;1(Rd+1

+ × TM ), where s is to be specified, and which is holomorphic in θ ∈ CM ⊂ CM(in each single variable θm,) in particular. To construct V0 (we still need to solve for the profiles) and itscorrector V1, we need to define the projection operators E and E[, which are used to get the profile equationsto be solved for the profiles.

Definition 3.4. Setting φ := (φ1, . . . , φM ), we call α ∈ ZM ;2 a characteristic mode provided detL(d(α·φ)) =0 and write α ∈ C. We decompose

C = ∪Mm=1Cm, where Cm = {α ∈ ZM ;2 : α · φ = nαφm for some nα ∈ Z}.(3.10)

Now we are ready to define the projectors which will give us the profile equations and the differentialequation for the condition (1.39).

Definition 3.5. Let V ∈ Hs+1;2T . The action of E on V is defined by

E = E0 +

M∑m=1

Em, where E0V = V0 and EmV =∑

α∈Cm\0

PmVα(x)einαθm ,(3.11)

where Pm denotes the projection onto Ker L(dφm), the action of E[ on V is defined by

E[ = E[0 +

M∑m=1

E[m, where E[0V = V0 and E[mV =∑

α∈Cm\0

PmVα(x)eiα·θ,(3.12)

and we use the notation

Eh = E0 +∑

m∈I∪OEm, Ee =

∑m∈P∪N

Em.(3.13)

Remark 3.6. (i) It is shown in [2] that E is a continuous map in the Hs norm. Here, to complete thestatement E : Hs;2 → Hs;1, we must also have for each α ∈ Cm that nα ∈ Zm. This is easily verified fromthe definitions of Zm, Cm, and the nα. A similar calculation is done in Remark 3.12. (ii) It is not hard toshow the continuity of E[ : Hs;2 → Hs;2.

Remark 3.7. Denoting evaluation at θ = θ(θ0, ξd) by Φ, observe Φ ◦ E = Φ ◦ E[. The projector E servesas the main tool for solving for v(x) and the periodic profiles σm,k(x, θm) of our ansatz. The projector E[ iskey to describing solvability (such as the condition (1.39)) and is thus used in the error analysis in solvingaway an O(1) error output of Φ ◦ (I − E) written in the form Φ ◦ (I − E[), so that we can prove Theorem4.7. That is the purpose which the following proposition serves.

Proposition 3.8. Given H ∈ Hs;2T (x, θ) with finitely many nonzero Fourier coefficients, suppose

E[H = 0,(3.14)

Then there exists V ∈ Hs;2T (x, θ) such that

L(∂θ)V = H.(3.15)

18

Proof. We write out the series of H as

H(x, θ) =∑

α∈ZM;2

Hα(x)eiα·θ.(3.16)

Now we search for Vα(x) such that

V(x, θ) :=∑

α∈ZM;2

Vα(x)eiα·θ,(3.17)

satisfies

L(∂θ)V(x, θ) =

M∑m=1

∑α∈ZM;2

iαmL(dφm)Vα(x)eiα·θ =∑

α∈ZM;2

Hα(x)eiα·θ.(3.18)

This holds if and only if for all α ∈ ZM ;2

iL

(∑m

αmβ, α · ω

)Vα(x) = Hα(x),(3.19)

where ω = (ω1, . . . , ωM ). We handle first the case where α ∈ Cj \ 0 for some j = 1, 2, . . . ,M . Note that

0 = E[jH(x, θ) =∑

α∈Cj\0

PjHα(x)eiα·θ,(3.20)

which implies for each α ∈ Cj

0 = PjHα(x).(3.21)

Recalling from Lemma 1.2 that Im A−1d L(dφj) = Ker Pj , we see there exists Wα(x) ∈ HsT (x) such that

Hα(x) = iL(dφj)Wα(x).(3.22)

Observe then

Hα(x) = iL(nαβ, nαωj)1

nαWα(x),(3.23)

= iL

(M∑m=1

αmβ, α · ω

)1

nαWα(x).(3.24)

So we may satisfy (3.19) by defining Vα(x) = 1nαWα(x).

For the case α = 0, where Hα = E[0H = 0, or for any other α such that Hα = 0, we may satisfy (3.19) bytaking Vα = 0.For α /∈ C with Hα 6= 0, we have det L(

∑m αmβ, α · ω) 6= 0 by definition of C, so then (3.19) can be solved

directly for Vα(x). Since there are only finitely many nonzero Hα, we have no issues with convergence. Thesame holds for the Vα, and so the sum in (3.17) clearly describes an element of Hs;2

T (x, θ).

Remark 3.9. (i) In view of Proposition 3.8, in order to construct a corrector V1 appropriate for our periodicansatz V0 to satisfy (1.38), it is tempting to require something like

E[(L(∂x)V0 +M(V0)∂θV0) = E[(F (0)V0).(3.25)

However, note that to use Proposition 3.8, if the V0 we seek has infinitely many nonzero Fourier coefficients,(3.25) alone is insufficient; we must use finite trigonometric polynomial approximations. Furthermore, evenif V0 is replaced with a finite trigonometric polynomial approximation, it is not clear how one might solve(3.25) for V0 of the form (3.9). For an explanation, consider the components E[m of the projector E[. The

19

goal is to obtain a profile, say, σm,k(x, θm) in (3.9) by examining the mth component of some projector.However, E[m maps into Hs;2(x, θ), meaning its output generally varies with more than one component of θas opposed to varying with only the θm component, and so we have little hope of getting from it an equationfor σm,k(x, θm). Meanwhile, E has components Em mapping into Hs(x, θm), spaces suited to σm,k(x, θm),and these will give us the equations for the σm,k(x, θm). In fact, V0 having the form (3.9) is equivalent tohaving

EV0 = V0.(3.26)

(ii) With the considerations made in Remark 3.7, it is natural to require instead of (3.25) that

E(L(∂x)V0 +M(V0)∂θV0) = E(F (0)V0).(3.27)

Observe that if we satisfy (3.27), we have

(I − E)(L(∂x)V0 +M(V0)∂θV0 − F (0)V0) = L(∂x)V0 +M(V0)∂θV0 − F (0)V0.(3.28)

Once V0 is determined by imposing (3.27), with Proposition 3.8 one can solve for a corrector V1 in

L(∂θ)V1 = −(I − E[)(L(∂x)V0 +M(V0)∂θV0 − F (0)V0),(3.29)

(if we assume for the moment that we are working with finite trigonometric polynomials.) Although such V1

does not necessarily satisfy (1.38), after evaluating at θ = θ(θ0, ξd), which, recall, is denoted by Φ, it followsfrom (3.28) and (3.29) that

Φ(L(∂θ)V1

)= −Φ

(L(∂x)V0 +M(V0)∂θV0 − F (0)V0

),(3.30)

where we have used the property Φ ◦ (I − E[) = Φ ◦ (I − E). Thus (3.30) implies that we have a correctedapproximate solution solving (1.2)(a) to order O(1) (disregarding for now the error introduced by usingtrigonometric approximations.)

It is from the components of (3.27) that we obtain the components v, σm,k, m = 1, . . . ,M , k = 1, . . . , µm,

of V0. Now we collect the set of equations which determines V0(x, θ) ∈ Hs;1(Rd+1

+ × TM ) for s sufficientlylarge (specified later):

a) EV0 = V0

b) E(L(∂x)V0 +M(V0)∂θV0

)= E(F (0)V0) in xd ≥ 0

c) B(0)V0(x′, 0, θ0, . . . , θ0) = G(x′, θ0)

d) V0 = 0 in t < 0.

(3.31)

To summarize, (3.31)(a) implies L(∂θ)V0 = 0 so that (1.2)(a) is solved to order 1ε . Equation (3.31)(b)

represents the solvability conditions for the existence of a corrected approximate solution, and equations(3.31)(c) and (3.31)(d) straightforwardly correspond to (1.2)(b) and (1.2)(c).

3.2 The large system for individual profiles.

The next major task is to rephrase (3.31)(b) in terms of v and the σm,k to get the interior profile equations.Equation (3.31)(b) will be broken down into components with E0 and Em,k, m = 1, . . . ,M , k = 1, . . . , µm,replacing E, yielding equations for v and the σm,k.

Recall the decomposition of the projector

E = E0 +

M∑m=1

Em(3.32)

20

For m = 1, . . . ,M , we enumerate by {`m,k, k = 1, . . . , νm} a basis of vectors for the left eigenspace of

iA(β) = A−1d (0)(τI +

d−1∑j=1

Aj(0)ηj)(3.33)

associated to the eigenvalue −ωm, and with the following property:

`m,k · rm′,k′ =

{1, if m = m′ and k = k′

0, otherwise.(3.34)

For v ∈ CN set

Pm,kv = (`m,k · v)rm,k (no complex conjugation here) .(3.35)

So now we may write

E = E0 +M∑m=1

µm∑k=1

Em,k,(3.36)

where

Em,k(Vαeiα·θ) :=

{(Pm,kVα)einαθm , α ∈ Cm \ 0

0, otherwise; i.e., Em,k = Pm,kEm.(3.37)

Having defined the Em,k, we can proceed with the explicit derivation of the large system in terms ofv(x) and the σm,k(x, θm) by applying E0 and Em,k to (3.31)(b). We begin with the identity of the followinglemma, whose proof we omit here, as it is almost identical to that of Lemma 2.11 of [2].

Lemma 3.10. Suppose EV0 = V0. Then

Em,k(L(∂x)V0) = (Xφmσm,k)rm,k(3.38)

where Xφm is the characteristic vector field associated to φm:

Xφm := ∂xd +

d−1∑j=0

−∂ζjωm(β)∂xj .(3.39)

ResonancesLet us fix p ∈ {1, . . . ,M} and consider the projector Ep in an instance where there exists a relation of

the form

npφp = nqφq + nrφr where q, r 6= p and (np, nq, nr) ∈ Z× Zq × Zr.(3.40)

It takes some effort to evaluate certain terms arising from the quantity Ep(M(V0)∂θV0) which then dependon the profiles σq,k and σr,k′ , in particular. As discussed in Section 1.2, we must account for the ways inwhich the φq oscillations interact with φr oscillations to produce φp oscillations. This is the purpose of theinteraction integrals, (3.47), appearing in the profile equations. To handle these difficulties we introducesome new definitions regarding resonances and interaction integrals.

Definition 3.11. We say (φp, φq, φr) is an ordered triple of resonant phases and that (φq, φr) forms ap-resonance if

npφp = nqφq + nrφr(3.41)

for some (np, nq, nr) ∈ Z × Zq × Zr, each entry nonzero. The triple (φp, φq, φr) is called normalized if wehave both (i) gcd(np, nq, nr) = 1 and (ii) if p ∈ I ∪ O, then np > 0.

21

We explicitly treat the case that the only normalized triples of resonant phases are permutations of(φp, φq, φr) corresponding to one of the six rearrangements of (3.41). It follows from the next remark thatthe sign of np is determined, so this normalization uniquely determines np, nq, nr.

Remark 3.12. For a triple of resonant phases (φp, φq, φr) with (3.41), one has (np, nq, nr) ∈ Zp×Zq ×Zr.To see this, taking the imaginary part of npφp = nqφq + nrφr, observe

npIm ωp = nqIm ωq + nrIm ωr.(3.42)

Thus, if (φq, φr) forms a p-resonance, since nqIm ωq and nrIm ωr are nonnegative, so is npIm ωp, so itfollows that np ∈ Zp.

We now make a related observation with the following proposition which will greatly simplify our solutionof the interior equations.

Proposition 3.13. (i) A hyperbolic resonance (i.e. a p-resonance, where p ∈ I ∪O) can only be formed bya pair of hyperbolic phases, and (ii) an elliptic resonance can only be formed by a pair including at least oneelliptic phase.

Proof. (i) Suppose for some p ∈ I∪O that npφp = nqφq+nrφr with (np, nq, nr) ∈ Zp×Zq×Zr. Consideringthe imaginary part gives

0 = nqIm ωq + nrIm ωr,(3.43)

so recalling ni ∈ Zi, niIm ωi ≥ 0, we see Im ωq = Im ωr = 0. Hence q, r ∈ I ∪ O. The proof of (ii) issimilar.

We proceed by defining the interaction integrals which will appear in the interior equations.

Definition 3.14. Suppose (φq, φr) forms a normalized p-resonance, with

npφp = nqφq + nrφr.(3.44)

For any f ∈ HsT (x, θq) define fnq ∈ Hs

T (x, θq) to be the image of f under the preparation map

f(x, θq) =∑k∈Z

fk(x)eikθq →∑k∈Z

fknq (x)eiknqθq .(3.45)

Suppose s > d+32 + 1 and that σq,k, σr,k′ ∈ Hs

T (Rd+1

+ × T). We get exactly two normalized p-resonanceformations from the arrangements of (3.44):

npφp = nqφq + nrφr, npφp = nrφr + nqφq.(3.46)

To these equations we associate, respectively, the two families of prepared integrals:

Jk,k′

p,nq,nr (x, θp) :=

1

2π

∫ 2π

0

(σq,k)nq

(x,npnqθp −

nrnqθr

)∂θrσr,k′(x, θr)dθr, k ∈ {1, . . . , µq}, k′ ∈ {1, . . . , µr},

Jk,k′

p,nr,nq (x, θp) :=

1

2π

∫ 2π

0

(σr,k)nr

(x,npnrθp −

nqnrθq

)∂θqσq,k′(x, θq)dθq, k ∈ {1, . . . , µr}, k′ ∈ {1, . . . , µq}.

(3.47)

Remark 3.15. Strictly speaking, Jk,k′

p,nq,nr and Jk,k′

p,nr,nq are symbolically Ik,k′

nq,np,n′rand Ik,k

′

n′r,−np,nq(as defined

in [2], with respect to the triple (φq, φp, φr) and nqφq = npφp+n′rφr) where n′r = −nr. The difference is thatwe do not require nq > 0, q < p < r – recall, instead, our main requirement is that (np, nq, nr) ∈ Zp×Zq×Zr.

22

The following proposition shows that the prepared integrals ‘pick out’ the p-resonances.

Proposition 3.16. Suppose s > d+32 + 1 and that σq,k, σr,k′ ∈ Hs

T (Rd+1

+ × T) have Fourier series

σq,k(x, θq) =∑

j∈Zq\0

aj(x)eijθq and σr,k′(x, θr) =∑

j∈Zr\0

bj(x)eijθr .(3.48)

Given (φq, φr) forms a normalized p-resonance, the prepared integral Jk,k′

p,nq,nr (x, θp) belongs to Hs−1T (x, θp)

and has Fourier series

Jk,k′

p,nq,nr (x, θp) =∑j∈Z

ajnq (x)bjnr (x)i · (jnr)eijnpθp .(3.49)

For the other integral in (3.47), one switches nq and nr above. Moreover, the functions Jk,k′

p,nq,nr (x, θp) canbe described for θp ∈ Cp through analytic extension of the expansions into the complex half plane Cp in

the same way that we extend V0(x, θ) to CM and each of the profiles σm,k(x, θm) into Cm in the mannerdiscussed in Remark 3.3.

Remark 3.17. We note that for p ∈ P ∪ N , in (3.49), we can sum over just j ∈ Z+ \ 0. To see this,observe that then (φq, φr) forms an elliptic resonance, so Proposition 3.13 guarantees one of φq, φr is elliptic– without loss of generality, say φq. Thus, Zq \ 0 consists only of negative integers or positive integers, andthe only nonzero Fourier coefficients of σq,k are those aj(x) appearing in (3.48), with j ∈ Zq \ 0; we haveaj(x) = 0 for all other j. For example, if q ∈ P, then Zq = Z+, and so, noting nq > 0, we have the ajnq (x)appearing in (3.49) are only nonzero for j > 0. Indeed, q ∈ N implies the same of the ajnq (x), so we maysum over just j ∈ Z+ \ 0 in either case. We add that, for such j, the eijnpθp factors are bounded as θp rangesover Cp.

Proof of Proposition 3.16. The main part of the proof is showing that the expansion (3.49) converges to thedesired result in Hs−1

T (x, θp), where we have restricted our attention to real θp. The proof of this is verysimilar to the proof of Proposition 2.13 from [2]. Analytic extension into the complex half plane Cp is donein the same way that the expansions for the profiles are extended into their corresponding complex halfplanes, as in Remark 3.3.

Interior equationsBy applying E0 and the Em,k to (3.31)(b), we obtain the system for v(x) and the σm,k(x, θm) with the

following proposition.

Proposition 3.18. Suppose (φq, φr) forms a normalized p-resonance, with

npφp = nqφq + nrφr.(3.50)

and that all other normalized triples of resonant phases are permutations of (φp, φq, φr) each corresponding

to one of the six rearrangements of (3.50).(i) Given V0 as in (3.26) and represented by (3.9), the equation (3.31)(b) is equivalent to the followingsystem:

L(∂x)v +

d−1∑j=0

M∑m=1

µm∑k,k′=1

1

2π

(∫ 2π

0

σm,k(x, θm)∂θmσm,k′(x, θm)dθm

)Rk,k

′

j,m = F (0)v ;(3.51)

23

(a) Xφpσp,l(x, θp) +

d−1∑j=0

µp∑k′=1

ak′

p,l,j(v)∂θpσp,k′(x, θp)+

(b)

d−1∑j=0

µp∑k=1

µp∑k′=1

bk,k′

p,l,jσp,k(x, θp)∂θpσp,k′(x, θp)−d−1∑j=0

µp∑k=1

µp∑k′=1

bk,k′

p,l,j

1

2π

∫ 2π

0

σp,k(x, θp)∂θpσp,k′(x, θp)dθp+

(c)

d−1∑j=0

Jp,l,j(x, θp) =

µp∑k=1

ekp,lσp,k(x, θp)

for p ∈ I ∪ O, l ∈ {1, . . . , µp},

(3.52)

where

Jp,l,j(x, θp) =

µq∑k=1

µr∑k′=1

ck,k′

p,l,j Jk,k′

p,nq,nr (x, θp) +

µr∑k=1

µq∑k′=1

dk,k′

p,l,j Jk,k′

p,nr,nq (x, θp), p = p,

µp∑k=1

µr∑k′=1

ck,k′

p,l,j Jk,k′

q,−np,nr (x, θp) +

µr∑k=1

µp∑k′=1

dk,k′

p,l,j Jk,k′

q,nr,−np(x, θp), p = q,

µq∑k=1

µp∑k′=1

ck,k′

p,l,j Jk,k′

r,nq,−np(x, θp) +

µp∑k=1

µq∑k′=1

dk,k′

p,l,j Jk,k′

r,−np,nq (x, θp), p = r,

0, otherwise;

(3.53)

(a) Xφpσp,l(x, θp) +

d−1∑j=0

µp∑k′=1

ak′

p,l,j(v)∂θpσp,k′(x, θp)+

(b)

d−1∑j=0

µp∑k=1

µp∑k′=1

bk,k′

p,l,jσp,k(x, θp)∂θpσp,k′(x, θp)+

(c)

d−1∑j=0

Jp,l,j(x, θp) =

µp∑k=1

ekp,lσp,k(x, θp)

for p ∈ P ∪N , l ∈ {1, . . . , µp},

(3.54)

where

Jp,l,j(x, θp) =

µq∑k=1

µr∑k′=1

ck,k′

p,l,j Jk,k′

p,nq,nr (x, θp) +

µr∑k=1

µq∑k′=1

dk,k′

p,l,j Jk,k′

p,nr,nq (x, θp), p = p,

0, otherwise.

(3.55)

The constant vectors Rk,k′

j,m appearing in (3.51) are given by

Rk,k′

j,m = βj(∂uAj(0) · rm,k)rm,k′ .(3.56)

The constant scalars bk,k′

p,l,j, ck,k′

p,l,j, dk,k′

p,l,j, ekp,l, and the coefficients of the scalar linear function of v, akp,l,j(v),

in (3.52)-(3.55) are given by similar formulas, but now involving dot products with the vector `p,l.(ii) Equations (3.51), (3.52), form a hyperbolic sub-system, that is, a system in v and σp,l for p ∈ I ∪ Oindependent of σq,k for all q ∈ P ∪N .

Remark 3.19. While the hyperbolic interior equations are independent of the elliptic profiles, the ellipticinterior equations (3.54) are not in general independent of the hyperbolic profiles, since these appear in theelliptic interaction integrals if some phase paired with a hyperbolic phase forms an elliptic resonance.

24

Proof of Proposition 3.18. Regarding (i), (3.51) follows from application of E0 to (3.31)(b), and (3.52) fromapplication of Ep,l to (3.31)(b). One similarly arrives at (3.54): in doing so, one finds that the second triplesum appearing in (3.52)(b) is zero for p ∈ P ∪ N , since then spec σp,k, spec σp,k′ ⊂ Zp \ 0 which is eitherZ+ \0 or Z− \0, resulting in (3.54)(b). The differences between (3.53) and (3.55) merely account for the factthat for p ∈ I ∪ O, we have np ∈ Zp \ 0 = Z \ 0 implies −np ∈ Zp \ 0 = Z \ 0, while for p ∈ P ∪N , we havenp ∈ Zp \ 0 implies −np /∈ Zp \ 0. The reverse implication of (i) follows upon recalling the decomposition(3.36).For proof of (ii), first we show the elliptic profiles drop out of (3.51). This follows from the argument madein proving (i) that terms such as those in the triple sum of (3.51) with m ∈ P ∪ N are zero, since thenspec σm,k, spec σm,k′ ⊂ Zp \ 0. Finally, to see that no σq,k with q ∈ P ∪ N appears in (3.52), we have onlyto check the interaction integral terms Jp,l,j of (3.52)(c). However, the terms of (3.53) defining Jp,l,j involveonly profiles corresponding to p-resonant phases; Jp,l,j is only nonzero for some p ∈ I ∪ O if p = p, q, or rand, by Proposition 3.13, p, q, r ∈ I ∪ O. Thus, in this case, no elliptic profiles appear in Jp,l,j .

From the above, we find the hyperbolic sub-system has the same form as the system in [2] and we willsolve it similarly. On the other hand, we will not solve the elliptic interior equations exactly.

Boundary equationsNow that we have the interior equations for the profiles, we formulate the boundary conditions to be

imposed on the profiles. From (3.31)(c), we get

(a) B(0)v = G(x′)

(b) B(0)V0∗(x′, 0, θ0, . . . , θ0) =

B(0)

( ∑m∈I∪P∪N

νkm∑k=1

σm,k(x′, 0, θ0)rm,k +∑m∈O

νkm∑k=1

σm,k(x′, 0, θ0)rm,k

)= G∗(x′, θ0),

(3.57)

where V0∗ denotes the mean zero part of the periodic function V0, and we have similar for G∗. While (3.57)(a)gives boundary data for v, it remains to establish satisfactory boundary conditions on the individual profilesσm,k from (3.57)(b). This is done with Proposition 3.21.

Lemma 3.20. Each of the sets of vectors

{B(0)rm,k : m ∈ I ∪ P, k = 1, . . . , µm},(3.58)

{B(0)rm,k : m ∈ I ∪ N , k = 1, . . . , µm},(3.59)

is a basis of Cp.

Proof. First we prove (3.58) is a basis of Cp. Recall our assumption of uniform stability, Assumption (2.5),which tells us that B(0) maps a basis for the stable subspace Es(τ , η) to a basis for Cp. Thus, according toLemma 2.13, which states that

Es(τ , η) = ⊕m∈I∪PKer L(dφm),(3.60)

we have that the set (3.58) is a basis for Cp. To see the same holds for (3.59), recall from Remark 2.9(ii)the fact that for each nonreal ωm, say, without loss of generality, m ∈ P, there is another eigenvalue ωm′satisfying ωm′ = ωm, thus with m′ ∈ N , and that similar holds for the associated eigenvectors. It followsthat

{B(0)rm,k : m ∈ I ∪ N , k = 1, . . . , µm} = {B(0)rm,k : m ∈ I ∪ P, k = 1, . . . , µm}(3.61)

= {B(0)rm,k : m ∈ I ∪ P, k = 1, . . . , µm}.(3.62)

Clearly, (3.62) spans the same subspace as (3.58), i.e. Cp, and so (3.59) is also a basis of Cp.

25

Proposition 3.21. The data (σm,k(x′, 0, θ0);m ∈ I ∪ P ∪ N , k ∈ {1, . . . , µm}) are determined by the data(σm,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}) in that there exist constant matrices M± such that the zero-meanboundary condition,

B(0)V0∗(x′, 0, θ0, . . . , θ0) = B(0)

(M∑m=1

µm∑k=1

σm,k(x′, 0, θ0)rm,k

)= G∗(x′, θ0),(3.63)

is equivalent to the condition

(σ±m,k(x′, 0, θ0);m ∈ I ∪ P ∪N , k ∈ {1, . . . , µm}) = M±(G∗,±, σ±m,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}),(3.64)

(using + (−) to denote a part with positive (negative) spectrum.)

Proof. We rewrite (3.63) as

B(0)V0∗(x′, 0, θ0, . . . , θ0) =

B(0)

( ∑m∈I∪P∪N

µm∑k=1

σm,k(x′, 0, θ0)rm,k +∑m∈O

µm∑k=1

σm,k(x′, 0, θ0)rm,k

)= G∗(x′, θ0).

(3.65)

Recall P-profiles have positive spectra and N -profiles have negative spectra. Thus, using the subscript n todenote the nth Fourier coefficient, we get for n > 0,

B(0)

( ∑m∈I∪P

µm∑k=1

σm,k,n(x′, 0)rm,k

)= G∗n(x′)−B(0)

(∑m∈O

µm∑k=1

σm,k,n(x′, 0)rm,k

),(3.66)

and for n < 0,

B(0)

( ∑m∈I∪N

µm∑k=1

σm,k,n(x′, 0)rm,k

)= G∗n(x′)−B(0)

(∑m∈O

µm∑k=1

σm,k,n(x′, 0)rm,k

).(3.67)

By Lemma 3.20, both sets {B(0)rm,k : k ∈ {1, . . . , µm},m ∈ I∪P}, {B(0)rm,k : k ∈ {1, . . . , µm},m ∈ I∪N}are bases for Cp, and the desired result follows from this.

So we now have a sub-system ((3.51), (3.52)) in just the hyperbolic profiles σm,k, m ∈ I∪O, and the meanv with boundary data for v and determination of I-boundary data from O-boundary data as in (3.57)(b),where (3.57)(b) is equivalent to (3.64).

Remark 3.22. We will seek v, σp,l such that, in addition to the equations above, we satisfy the initialconditions

v = 0 and σp,l = 0 in t ≤ 0 for all p, l.(3.68)

The large system consists of the interior equations (3.51)-(3.55) with boundary equations (3.57) and theinitial conditions (3.68).

3.3 Solution of the hyperbolic sub-system of the large system

First we will obtain the ‘hyperbolic part’ of our solution, V0h, which will satisfy, in place of (3.31),

a) EhV0h = V0

h

b) Eh(L(∂x)V0

h +M(V0h)∂θV0

h

)= Eh(F (0)V0

h) in xd ≥ 0

c) B(0)v = G(x′),

(σm,k(x′, 0, θ0);m ∈ I, k ∈ {1, . . . , µm}) = Lh(G∗, σm,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}),d) V0

h = 0 in t < 0.

(3.69)

26

Here, in the zero-mean boundary data in (3.69)(c), we denote by Lh the appropriate linear function oneobtains from the condition (3.64). Equation (3.69)(a) serves the same purpose as (3.31)(a), but adds therestriction that V0

h only consists of the mean and the hyperbolic profiles. Thus, recalling the form for V0,we see V0

h takes the form

V0h(x, θ1, . . . , θM ) = v(x) +

M∑m∈I∪O

µm∑k=1

σm,k(x, θm)rm,k.(3.70)

The condition (3.69)(b) is exactly the hyperbolic sub-system ((3.51),(3.52)) found in Proposition 3.18. Thisfollows directly from Proposition 3.18 and the definition of Eh.

To solve this, we employ the same approach as that which is used in [2] to solve the full system of profileequations. This is done by solving the system with an iteration scheme such as the following

a) EhV0,nh = V0,n

h

b) Eh(L(∂x)V0,n

h +M(V0,n−1h )∂θV0,n

h

)= Eh(F (0)V0,n−1

h ) in xd ≥ 0

c) B(0)vn = G(x′),

(σnm,k(x′, 0, θ0);m ∈ I, k ∈ {1, . . . , µm}) = Lh(G∗, σnm,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}),

d) V0,nh = 0 in t < 0,

(3.71)

where we note that (3.71)(a) means V0,nh is of the form

V0,nh (x, θ1, . . . , θM ) = vn(x) +

M∑m∈I∪O

µm∑k=1

σnm,k(x, θm)rm,k,(3.72)

where iterates vn(x), σnm,k(x, θm) for m ∈ I ∪ O satisfy an iterated version of the hyperbolic sub-system ofthe profile equations, ((3.51),(3.52)) which is encapsulated by (3.71)(b).

The following proposition gives the solution to the iterated system as well as the solution to the hyperbolicsub-system itself. We omit the proofs of these as they are almost identical to the proof of Proposition 2.19together with that of Proposition 2.21 from [2].

Proposition 3.23. Let T > 0, m > d+32 + 1 and suppose that G(x′, θ0) ∈ Hm

T .

(i) Setting V0,0h = 0, there exist unique iterates V0,n

h ∈ HmT , n ≥ 1, solving the system (3.71).(ii) For some 0 < T0 ≤ T the system (3.69) has a unique solution V0

h ∈ HmT0. Furthermore,

limn→∞

V0,nh = V0

h in Hm−1T0,(3.73)

and the traces V0h|xd=0, v|xd=0, and σp,l|xd=0, p ∈ I ∪ O, all lie in Hm

T0.

3.4 Approximate solution of the equations for the elliptic profiles

Recall from Proposition 3.18 the elliptic interior equations:

(a) Xφpσp,l(x, θp) +

d−1∑j=0

µp∑k′=1

ak′

p,l,j(v)∂θpσp,k′(x, θp)+

(b)

d−1∑j=0

µp∑k=1

µp∑k′=1

bk,k′

p,l,jσp,k(x, θp)∂θpσp,k′(x, θp)+

(c)

d−1∑j=0

Jp,l,j(x, θp) =

µp∑k=1

ekp,lσp,k(x, θp)

for p ∈ P ∪N , l ∈ {1, . . . , µp}.

(3.74)

27

These complex transport equations may not generally have exact solutions. Instead, our elliptic profiles σp,lwill approximately solve these equations in the sense that they will simply hold at the boundary xd = 0. Itis sufficient to simply evaluate the above expression at xd = 0, isolate ∂xdσp,l(x

′, 0, θp), and require that σp,lhas the appropriate xd-derivative at the boundary. Of course, we will also have to adhere to the boundaryconditions, so that the trace σp,l(x

′, 0, θ0) at the boundary satisfies (3.57)(b), and we must satisfy initialconditions (3.68). With Proposition 3.29, we construct such elliptic profiles σp,l(x, θp) by solving some waveequations in which xd plays the role of the time variable and with appropriate boundary data correspondingto σp,l(x

′, 0, θp) and ∂xdσp,l(x′, 0, θp).

In addition to the exact elliptic interior equations, we consider semilinear equations in iterates σnp,l forp ∈ P ∪ N , l ∈ {1, . . . , µp}, almost identical to the iteration scheme which is used to obtain the hyperbolicprofiles. While we do not need elliptic iterates σnp,l to get the elliptic profiles σp,l, which are instead obtainedby the process described above, they allow us to handle hyperbolic parts and elliptic parts uniformly in theerror analysis. The semilinear equations in the elliptic iterates σnp,l are

Xφpσnp,l(x, θp)+(3.75)

d−1∑j=0

µp∑k′=1

ak′

p,l,j(vn−1)∂θpσ

np,k′(x, θp) +

d−1∑j=0

µp∑k=1

µp∑k′=1

bk,k′

p,l,jσn−1p,k (x, θp)∂θpσ

np,k′(x, θp)+(3.76)

d−1∑j=0

µp∑k=1

µp∑k′=1

ck,k′

p,l,jJk,k′,np,nq,nr (x, θp) +

d−1∑j=0

µp∑k=1

µp∑k′=1

dk,k′

p,l,jJk,k′,np,nr,nq (x, θp)(3.77)

=

µp∑k=1

ekp,lσn−1p,k (x, θp), for p ∈ P ∪N , l ∈ {1, . . . , µp},(3.78)

where we define

Jk,k′,n

p,nq,nr (x, θp) =1

2π

∫ 2π

0

(σn−1q,k )nq

(x,npnqθp −

nrnqθr

)∂θrσ

nr,k′(x, θr)dθr,(3.79)

and similar for Jk,k′,n

p,nr,nq . Imposed on the iterates σnp,l are boundary conditions identical to those for theprofiles σp,l, i.e. (3.64) with σnm,k in place of each σm,k, and initial conditions

σnp,l(x) = 0 for t ≤ 0.(3.80)

Similar to the elliptic profiles, the elliptic iterates σnp,l will only approximately solve (3.75)-(3.78) in the sensethat these equations will hold at xd = 0. In fact, the elliptic iterates σnp,l are obtained with the same kindof construction used for the elliptic σp,l. In addition to the elliptic profiles, their corresponding iterates areconstructed in Proposition 3.29.

For now, let us fix p ∈ {1, . . . ,M}, l ∈ {1, . . . , µp} and an integer n and begin the task of findingapproximate solutions σp,l and σnp,l of (3.74) and the system (3.75)-(3.78), respectively. First we rearrange(3.74), isolating the vector field applied to σp,l in the left hand side, and denoting the resulting right handside by fp,l, getting something of the form

Xφpσp,l(x, θp) = fp,l(x, θp).(3.81)

Similarly, we isolate Xφpσnp,l in the left hand side of (3.75)-(3.78), and define fnp,l(x, θp) to be what remains

on the right hand side, so (3.74) becomes

Xφpσnp,l(x, θp) = fnp,l(x, θp).(3.82)

Now we isolate the quantities this prescribes for the traces of the σp,l and the σnp,l at the boundary xd = 0.

28

Definition 3.24. (i) The coefficient of ∂xdσp,l in the left hand side of (3.81) is one, so we may isolate it inthis expression. We will require this equation to hold at xd = 0, and define bp,l so that doing so is representedby the condition

∂xdσp,l|xd=0 = bp,l,(3.83)

noting that the right hand side depends only on the boundary data σm,k|xd=0, m = 1, . . . ,M , k = 1, . . . , µm.(ii) Observe that (3.64) determines a linear function Le giving us the boundary data for the elliptic

profiles:

(σm,k(x′, 0, θ0);m ∈ P ∪N , k ∈ {1, . . . , µm}) = Le(G∗, σm,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}).(3.84)

Note that the right hand side has already been determined by our solution of the hyperbolic profiles. Wedefine ap,l such that the boundary condition on σp,l in (3.64) is equivalent to

σp,l|xd=0 = ap,l,(3.85)

This thus determines the right hand side of (3.83).(iii) We isolate ∂xdσ

np,l on the left hand side of (3.75)-(3.78), and define bnp,l such that requiring this to

hold at xd = 0 is equivalent to

∂xdσnp,l|xd=0 = bnp,l.(3.86)

(iv) We impose the condition

(σnm,k(x′, 0, θ0);m ∈ P ∪N , k ∈ {1, . . . , µm}) = Le(G∗, σnm,k(x′, 0, θ0);m ∈ O, k ∈ {1, . . . , µm}).(3.87)

Let us define anp,l such that the condition on σnp,l in (3.87) is equivalent to

σnp,l|xd=0 = anp,l.(3.88)

Lemma 3.25. Suppose G ∈ Hs+1T0

. Let V0h ∈ Hs+1

T0be the solution of the hyperbolic sub-system constructed

in Proposition 3.23. For anp,l, bnp,l, ap,l, and bp,l as defined in Definition 3.24, we have anp,l ∈ H

s+1T0

(x′, θp),bnp,l ∈ Hs

T0(x′, θp), and

limn→∞

anp,l = ap,l in Hs+1T0

(x′, θp),(3.89)

limn→∞

bnp,l = bp,l in HsT0

(x′, θp).(3.90)

Proof. Checking anp,l ∈ Hs+1T0

(x′, θp) is straightforward, since G(x′, θp), σnm,k(x′, 0, θp), m ∈ O, are in

Hs+1T0

(x′, θp).Now we show bnp,l ∈ Hs

T0(x′, θp). Observe bnp,l consists of the terms evaluated at xd = 0 (3.76), (3.77), (3.78),

and

(∂xd −Xφp)σnp,l(x′, 0, θp).(3.91)

Since the vnp,k, σnp,k(x′, 0, θp), p ∈ I ∪ O, are in Hs+1T0

(x′, θp), and HsT0

(x′, θp) is a Banach algebra, it followsthat (3.91) and both (3.76) and (3.78) at xd = 0 are in Hs

T0(x′, θp). It remains to show this for (3.77) at

xd = 0. This follows from Proposition 3.16 with x′ in place of x. Taking into account that, for p ∈ I ∪ O,

limn→∞

σnp,k|xd=0 = σp,k|xd=0 in Hs+1T0

(x′, θp),(3.92)

limn→∞

∂xdσnp,k|xd=0 = ∂xdσp,k|xd=0 in Hs

T0(x′, θp),(3.93)

the proof that (3.89) and (3.90) hold is similar.

29

Remark 3.26. To simplify obtaining our approximate solutions of (3.75)-(3.78), it will be useful to extendfunctions anp,l, ap,l ∈ H

s+1T0

= Hs+1((−∞, T0) × Rd−1 × T), bnp,l, bp,l ∈ HsT0

= Hs((−∞, T0) × Rd−1 × T) on

the half-space to elements of Hs+1(Rd × T) and Hs(Rd × T), respectively.

Lemma 3.27. For s ≥ 0 there is a continuous extension map

E : Hs((−∞, T0)× Rd−1 × T)→ Hs(Rd × T).(3.94)

Proof. It is shown in 4.4 of [11] that there is a continuous extension map from Hs(Rd+) to Hs(Rd).

From now on, in place of anp,l, ap,l, bnp,l, bp,l we refer to their extensions to the respective spaces mentioned

above unless we explicitly state otherwise.The following lemma is a standard kind of result in the theory of hyperbolic initial/boundary value problems.

Lemma 3.28. (i) For a ∈ Hs+1(Rd × T) and b ∈ Hs(Rd × T), there exists unique ς ∈ Hs+1D = Hs+1(Rd ×

[0, D]× T) solving the wave equation

∂2xdς −∆x′,θ0ς = 0,(3.95)

and initial-xd conditions

ς|xd=0 = a,(3.96)

∂xdς|xd=0 = b.(3.97)

(ii) Furthermore, ς satisfies

|ς|Hs+1D≤ C (|a|Hs+1 + |b|Hs) ,(3.98)

for some constant C independent of the choice of initial data {a, b}.

Justification of Lemma 3.28 can be found in Chapter 6 (see 6.18 and Remark 6.21) of [3].

Proposition 3.29. For anp,l, bnp,l, ap,l, and bp,l as defined in Definition 3.24, where p ∈ P ∪ N there exist

σnp,l(x, θp), σp,l(x, θp) ∈ Hs+1T0

(x, θp) with compact xd-support in [0, D] satisfying

σnp,l = σp,l = 0, for t ≤ 0,(3.99)

boundary conditions

σnp,l|xd=0 = anp,l, σp,l|xd=0 = ap,l,(3.100)

∂xdσnp,l|xd=0 = bnp,l, ∂xdσp′l|xd=0 = bp,l,

and

limn→∞

σnp,l = σp,l in Hs+1T0

(x, θp).(3.101)

Proof. We apply Lemma 3.28 to initial data {ap,l, bp,l+∂tap,l} to obtain the corresponding solution of (3.95),denoted ςp,l ∈ Hs+1

D , and subsequently to each {anp,l, bnp,l + ∂tanp,l}, obtaining solutions ςnp,l ∈ H

s+1D . Now we

fix a smooth cutoff function χ(xd) supported in [0, D), and define

σp,l(t, x′′, θp) = χ(xd)ςp,l(t− xd, x′′, θp),(3.102)

σnp,l(t, x′′, θp) = χ(xd)ς

np,l(t− xd, x′′, θp).(3.103)

It is easy to check that then σnp,l, σp,l also belong to Hs+1D . Since ςp,l|xd=0, ςnp,l|xd=0 are supported in t > 0, by

finite speed of propagation for solutions to the wave equation (3.95), for any fixed r > 0, the (t, y, θp)-supportsof ςp,l|xd=r, ςnp,l|xd=r are contained in the union of balls of radius r about points in {(t, y, θp) : t > 0}. Thus,the support is contained in {(t, y, θp) : t > −r}. It follows that σp,l(t, y, xd, θp), σ

np,l(t, y, xd, θp) are zero for

all t ≤ 0, since the supports of σp,l, σnp,l consist only of points satisfying t− xd > −xd, i.e. t > 0. It is easy

30

to check from (3.102) that the σp,l, σnp,l have the desired traces at xd = 0, satisfying (3.100). To establish

(3.101), first we note we may regard the σp,l, σnp,l as elements of Hs+1

T0(x, θp) by defining them to be equal

to zero for xd > D and restricting to t < T0. One easily verifies the bound

|σnp,l − σp,l|Hs+1T0

≤ |σnp,l − σp,l|Hs+1D≤ C1|ςnp,l − ςp,l|Hs+1

D.(3.104)

Applying Lemma 3.28 to the solutions ςnp,l − ςp,l, from the above and the estimate (3.98) we get

|σnp,l − σp,l|Hs+1T0

≤ C2

(|anp,l − ap,l|Hs+1 + |bnp,l − bp,l|Hs

).(3.105)

Now we conclude (3.101) from Lemma 3.25 and the continuity of the extension map from Lemma 3.27.

Remark 3.30. The elliptic profiles obtained in Proposition 3.29 satisfy spec σp,l ⊂ Zp, spec σnp,l ⊂ Zp sincethe prescribed traces at xd = 0 satisfy this, which can be checked by examining the elliptic profile equations(3.54), their iterated versions (3.75)-(3.78), and the boundary data (3.84) and (3.87) coming from (3.64),and thus (3.66) and (3.67).

Now that we have obtained the elliptic profiles, we have determined the ansatz V0.

Definition 3.31. Let G ∈ Hs+1T0

and let V0h ∈ Hs+1

T0be the solution of the hyperbolic sub-system constructed

in Proposition 3.23, and let the V0,nh ∈ Hs+1

T0be the corresponding iterates. Additionally, let σp,l, σ

np,l ∈ H

s+1T0

,where p ∈ P ∪N , be the elliptic profiles and iterates obtained in Proposition 3.29.

(i) We define the elliptic part of our ansatz

V0e (x, θ1, . . . , θM ) :=

M∑m∈P∪N

µm∑k=1

σm,k(x, θm)rm,k,(3.106)

and the corresponding nth iterate

V0,ne (x, θ1, . . . , θM ) :=

M∑m∈P∪N

µm∑k=1

σnm,k(x, θm)rm,k.(3.107)

Our ansatz is defined to be

V0 := V0h + V0

e ,(3.108)

and we also define V0,n := V0,nh + V0,n

e .(ii) We isolate the error in the iterated interior profile equations, defining Rn(x, θ) by

Rn := E(L(∂x)V0,n +M(V0,n−1)∂θV0,n − F (0)V0,n−1).(3.109)

Proposition 3.32. Suppose the hypotheses of Definition 3.31 are satisfied. Then V0 ∈ Hs+1T0

, and

limn→∞

V0,n = V0 in HsT0.(3.110)

Proof. The claims follow directly from Proposition 3.23 and Proposition 3.29.

Remark 3.33. Recalling (3.71), note that

Eh(L(∂x)V0,n +M(V0,n−1)∂θV0,n − F (0)V0,n−1) = 0,(3.111)

and so Rn of (3.109) is purely elliptic in the sense that EeRn = Rn. Moreover,

Rn =∑

p∈P∪N

µp∑l=1

(Xφpσnp,l − fnp,l)rp,l,(3.112)

31

which is zero at the boundary xd = 0, since we have satisfied (3.81) for xd = 0. Summarizing the results ofProposition 3.23 and Proposition 3.29, we conclude

a) EV0,n = V0,n

b) E(L(∂x)V0,n +M(V0,n−1)∂θV0,n

)= E(F (0)V0,n−1) +Rn in xd ≥ 0

c) B(0)V0,n(x′, 0, θ0, . . . , θ0) = G(x′, θ0)

d) V0,n = 0 in t < 0.

(3.113)

The sense in which an error such as Rn is small is clarified by Proposition 4.6.

Remark 3.34. Recall the discussion in Remark 2.9(ii) which gives the bijection between the eigenvaluesωm with m ∈ P and ωm′ = ωm, m′ ∈ N and between the corresponding eigenvectors. A careful look atthe profile equations shows that the equations for the elliptic profiles come in conjugate pairs. That is,the equation for σm,k, some m ∈ P, is the conjugate of the equation for σm′,k′ with the correspondingm′ ∈ N and k′. As a result the solutions we have obtained, the elliptic profiles, also come in conjugate pairs,satisfying σm′,k′ = σm,k. Meanwhile, the hyperbolic part of the solution V0

h is real. It follows from theseobservations and the definition of our ansatz that when we plug in θ = θ(θ0, ξd), getting

U0(x, θ0, ξd) := V0(x, θ0 + ω1ξd, . . . , θ0 + ωMξd),(3.114)

we have a function U0(x, θ0, ξd) which is in fact real on Rd+1

+ × T× R+.

4 Convergence of the approximate solution to the exact solution

4.1 The trigonometric expansion for the approximate solution

For this study, we define the spaces EsT and EsT as in [2].

EsT = C(xd, HsT (x′, θ0)) ∩ L2(xd, H

s+1T (x′, θ0)),(4.1)

where by C(xd, HsT (x′, θ0)) we actually refer to functions in C(xd, H

sT (x′, θ0)) with xd-support in [0, D] for

some large enough D, and for C(xd, HsT (x′, θ0)) we use the L∞(xd, H

sT (x′, θ0)) norm where the supremum

is taken over xd ≥ 0. These spaces are algebras and are contained in L∞ for s > d+12 . Theorem 7.1 of

[13], as discussed in Section 1.1, regarding existence of solutions of the singular system, tells us that for slarge enough, one has existence of solutions to (1.6) in the space EsT on a time interval [0, T ] independent ofε ∈ (0, ε0]. For these reasons, the space EsT and related estimates are key to our analysis, in particular forour main theorem (Theorem 4.7) which relies on Proposition 4.9, also proved in [13]. Additionally we usethe spaces

EsT = {U(x, θ0, ξd) : supξd≥0|U(·, ·, ξd)|EsT <∞},(4.2)

which also play a role in Theorem 4.7. The proof of Theorem 4.7 uses estimates regarding functions suchas V in Hs+1

T and the corresponding U in EsT which results from the substitution θ = θ(θ0, ξd). While thismoves us out of the space of periodic profiles Hs+1

T (x, θ), we get functions in EsT (x, θ0, ξd) which we canapproximate with trigonometric polynomials (truncated expansions24) in θ = θ(θ0, ξd).

Definition 4.1. For k = 1, 2 we define

Es;kT := {U(x, θ0, ξd) = V(x, θ)|θ=θ(θ0,ξd) : V ∈ Hs+1;kT },(4.3)

with the norm | · |EsT . (Note: The subscript T has the same indication as it did for the Hs+1;kT used in [2],

where the subscript T is at first ignored.)

24These expansions are described in Proposition 4.5.

32

It is verified in Proposition 4.5 that elements of Es;2T are bounded in the EsT norm.Convergence of expansions in EsTThe following notation gives us a way to sort the spectra α ∈ ZM ;2 of elements V in Hs+1;2

T , which willaid in showing U(x, θ0, ξd) = V(x, θ)|θ=θ(θ0,ξd) has an expansion converging in EsT , with Proposition 4.5.

Definition 4.2. Let {zk}∞k=1 enumerate the set {α · ω : α ∈ ZM ;2}. For each j, k, we define

Cj,k := {α ∈ ZM ;2 :

M∑i=1

αi = j, α · ω = zk}(4.4)

and pick out one element α(j,k) ∈ Cj,k.

Remark 4.3. A nice consequence of the fact that we work with ZM ;2 instead of general ZM ;k is that wecan easily show the Cj,k are finite25. To see this, first fix j ∈ Z, k ∈ {1, 2, . . .}, and p, q ∈ {1, . . . ,M} andtake arbitrary α ∈ Cj,k such that all but the pth and qth components are zero (either of the pth and qthcomponents may be zero, as well.) We claim this is the only such element of Cj,k. This is because ωp 6= ωqimplies (

1 1ωp ωq

)(αpαq

)=

(jzk

)(4.5)

has a unique solution (αp, αq). Thus, the number of such α in Cj,k is bounded by the number of pairs ofcomponents, so |Cj,k| ≤M(M − 1)/2.

A function V(x, θ) ∈ Hs+1;2T (x, θ) of (x, θ) ∈ Rd+1 ×CM has a series

V(x, θ) =∑

α∈ZM;2

Vα(x)eiα·θ(4.6)

and, for fixed xd, squared HsT (x′, θ) norm

|V(x, θ)|2HsT (x′,θ) =∑

α∈ZM;2

∑|β|≤s

|∂βx′Vα(x)|2L2(x′)(1 + |α|)2(s−|β|).(4.7)

From Sobolev embedding and the fact that

Hs+1T (x, θ) ⊂ L2(xd, H

s+1T (x′, θ)) ∩H1(xd, H

sT (x′, θ)),(4.8)

we find V(x, θ) ∈ L2(xd, Hs+1T (x′, θ)) ∩ C(xd, H

sT (x′, θ)), implying the partial sums of the series (4.6) are

bounded and converge in HsT (x′, θ) uniformly with respect to xd ≥ 0. We will prove Proposition 4.5 by using

these facts with the following lemma, which shows for finite truncations of expansions (4.6) that the norm| · |HsT (x′,θ) dominates

∣∣·|θ=θ(θ0,ξd)∣∣HsT (x′,θ0) independent of (xd, ξd).

Lemma 4.4. Suppose V ∈ Hs+1;2T (x, θ) with series given by (4.6). For θ(θ0, ξd) as in (1.33) and integers

M1 ≤M2, 0 < N1 ≤ N2, we have the following inequality:

|M2∑j=M1

N2∑k=N1

∑α∈Cj,k

Vα(x)eiα·θ(θ0,ξd)|2HsT (x′,θ0) ≤ |M2∑j=M1

N2∑k=N1

∑α∈Cj,k

Vα(x)eiα·θ|2HsT (x′,θ).(4.9)

Proof. We estimate

|M2∑j=M1

N2∑k=N1

∑α∈Cj,k

Vα(x)eiα·θ(θ0,ξd)|2HsT (x′,θ0) = |M2∑j=M1

N2∑k=N1

(∑

α∈Cj,k

Vα(x)eiα·ωξd)eijθ0 |2HsT (x′,θ0),

=

M2∑j=M1

∑|β|≤s

|(N2∑k=N1

∑α∈Cj,k

∂βx′Vα(x)eiα·ωξd)|2L2(x′)(1 + |j|)2(s−|β|),(4.10)

≤M2∑j=M1

N2∑k=N1

∑α∈Cj,k

∑|β|≤s

|∂βx′Vα(x)eiα·ωξd |2L2(x′)(1 + |j|)2(s−|β|).(4.11)

25We work with ZM ;2 due to the fact that we only have to consider quadratic interactions. Further discussion on interactionsand resonances can be found in Section 3.2.

33

For (4.10), we used the formula which that in (4.7) generalizes. For α ∈ Cj,k, we have |j| = |∑i αi| ≤ |α|

and Im(α · ω) ≥ 0, so for all ξd ≥ 0, the sum in (4.11) is bounded by

|M2∑j=M1

N2∑k=N1

∑α∈Cj,k

Vα(x)eiα·θ|2HsT (x′,θ) =

M2∑j=M1

N2∑k=N1

∑α∈Cj,k

∑|β|≤s

|∂βx′Vα(x)|2L2(x′)(1 + |α|)2(s−|β|),(4.12)

where the equality (4.12) follows from the formula in (4.7).

With the estimate of Lemma 4.4, we are ready to prove that elements of Hs+1;2T yield elements with

expansions converging in EsT upon the substitution θ = θ(θ0, ξd).

Proposition 4.5. Let V ∈ Hs+1;2T (x, θ) with expansion (4.6) and set U = V|θ=θ(θ0,ξd) for θ(θ0, ξd) as in

(1.33). Then we have U ∈ EsT :

|U|EsT ≤ C|V|Hs+1T

,(4.13)

and we have convergence in EsT to U of finite partial sums, independent of arrangement, of the series

U(x, θ0, ξd) =∑

α∈ZM;2

Vα(x)eiα·θ(θ0,ξd).(4.14)

Proof of Proposition 4.5. We explicitly show the convergence of the finite partial sums∑−n≤j≤n1≤k≤n

∑α∈Cj,k

Vα(x)eiα·θ(θ0,ξd).(4.15)

For integers n1, n2 with n1 ≤ n2, an application of Lemma 4.4 gives

|∑

n1≤|j|≤n2

n1≤k≤n2

∑α∈Cj,k

Vα(x)eiα·θ(θ0,ξd)|2HsT (x′,θ0) ≤ |∑

n1≤|j|≤n2

n1≤k≤n2

∑α∈Cj,k

Vα(x)eiα·θ|2HsT (x′,θ).(4.16)

It follows from (4.16) that since the sequence of the partial sums∑−n≤j≤n1≤k≤n

∑α∈Cj,k

Vα(x)eiα·θ(4.17)

is Cauchy in HsT (x′, θ) uniformly with respect to (xd, ξd), so is the sequence (4.15) in Hs

T (x′, θ0), and thuswe have convergence. We similarly get convergence of the (4.15) in L2(xd, H

s+1T (x′, θ0)) after integrating the

inequality (4.16) (with s+1 in place of s) with respect to xd and noting the right hand side is independent ofξd. Hence, the (4.15) converge in EsT . It is not hard to show the limit is in fact V|θ=θ(θ0,ξd), and the estimate(4.13) easily follows. The same proof works for arbitrarily ordered sums, where one instead considers finiteBn ↗ ZM ;2 and similarly gets that the sequence of the∑

α∈Bn

Vα(x)eiα·θ(θ0,ξd)(4.18)

is Cauchy in the desired space because the sequence of the∑α∈Bn Vα(x)eiα·θ is Cauchy.

4.2 Error analysis

The following proposition is used to make precise the notion that σp,l(x, θp) and σnp,l(x, θp) are approximatesolutions of (3.74) and (3.75)-(3.78), respectively.

34

Proposition 4.6. Let R(x, θ) ∈ Hs;1T (x, θ) have the property that it is polarized by the elliptic projector Ee

defined in (3.13), i.e. that Ee(R) = R, and suppose

R|xd=0 = 0.(4.19)

Then

limε→0|R(x, θ0 + ω1ξd, . . . , θ0 + ωMξd)|ξd= xd

ε|Es−1T (x,θ0)

= 0.(4.20)

Proof. It suffices to consider the case R(x, θp) ∈ HsT (x, θp), Ep(R) = R, for some p ∈ P ∪N , and show

limε→0|R(x, θ0 + ωpξd)|ξd= xd

ε|Es−1T (x,θ0)

= 0.(4.21)

First, we show

limε→0

supxd≥0

|R(x, θ0 + ωpξd)|ξd= xdε|Hs−1

T (x′,θ0)= 0.(4.22)

Note R has an expansion of the form

R(x, θp) =∑

j∈Zp\0

aj(x)eijθp .(4.23)

Thus, noting R(x, θ0) ∈ HsT (x, θ0) ⊂ C(xd, H

s−1T (x′, θ0)), fixing xd, we get the norm

|R(x, θp)|2Hs−1T (x′,θp)

=∑

j∈Zp\0

∑|β|≤s−1

|∂βx′aj(x)|2L2(x′)(1 + |j|)2(s−1−|β|).(4.24)

We let ξd = xdε and find

R(x, θ0 + ωpξd) =∑

j∈Zp\0

(e−Im(jωp)ξdeiRe(jωp)ξdaj(x))eijθ0 .(4.25)

Then it follows

|R(x, θ0 + ωpξd)|2Hs−1T (x′,θ0)

=∑

j∈Zp\0

∑|β|≤s−1

(e−Im(jωp)ξd)2|∂βx′aj(x)|2L2(x′)(1 + |j|)2(s−1−|β|)

≤ e−2|Imωp|ξd∑

j∈Zp\0

∑|β|≤s−1

|∂βx′aj(x)|2L2(x′)(1 + |j|)2(s−1−|β|)(4.26)

= e−2|Imωp|ξd |R(x, θ0)|2Hs−1T (x′,θ0)

(4.27)

Now we show that, as ε tends to zero,

supxd∈[0,

√ε]

|R(x, θ0 + ωpxdε

)|Hs−1T (x′,θ0)

→ 0,(4.28)

and

supxd≥√ε

|R(x, θ0 + ωpxdε

)|Hs−1T (x′,θ0)

→ 0.(4.29)

Set h(xd) = R(x′, xd, θ0) ∈ Hs−1T (x′, θ0). To see (4.28), observe that the term is bounded by

supxd∈[0,

√ε]

e−2|Imωp|xdε |h(xd)|2Hs−1

T (x′,θ0)≤ supxd∈[0,

√ε]

|h(xd)|2Hs−1T (x′,θ0)

,(4.30)

35

which converges to 0 as ε tends to zero, since h ∈ C(xd, Hs−1T (x′, θ0)) with h(0) = 0. That (4.29) holds

follows from the fact that this term is bounded by

supxd≥√ε

e−2|Imωp|xdε |h(xd)|2Hs−1

T (x′,θ0)≤ supxd≥0

|h(xd)|2Hs−1T (x′,θ0)

e−2|Imωp|

1√ε ,(4.31)

which also converges to 0 as ε tends to zero. Now we check that

limε→0|R(x, θ0 + ωpξd)|ξd= xd

ε|L2(xd,HsT (x

′,θ0)) = 0.(4.32)

We split the integral into the two pieces∫ √ε0

|R(x, θ0 + ωpxdε

)|2HsT (x′,θ0)dxd,(4.33)

and ∫ ∞√ε

|R(x, θ0 + ωpxdε

)|2HsT (x′,θ0)dxd.(4.34)

Using the inequality (4.27) with s in place of s− 1, we get that the first integral is bounded above by∫ √ε0

e−2|Imωp|xdε dxd|R(x, θ0)|2L2(xd,HsT (x

′,θ0))≤√ε|R(x, θ0)|2L2(xd,HsT (x

′,θ0)),(4.35)

and the second is bounded by

e−2|Imωp|

1√ε |R(x, θ0)|2L2(xd,HsT (x

′,θ0)),(4.36)

both of which tend to zero as ε→ 0, and so the proof is finished.

Now we are ready to show that, as ε → 0, the approximate solution uaε converges to the exact solutionuε in L∞. This is a corollary of the following theorem.

Theorem 4.7. For M0 = 2(d+ 2) + 1 and s ≥ 1 + [M0 + d+12 ] let G(x′, θ0) ∈ Hs+1

T have compact support inx′ and vanish in t ≤ 0. Let Uε(x, θ0) ∈ EsT0

be the exact solution to the singular system for 0 < ε ≤ ε0 given

by Theorem 7.1 of [13], let V0 ∈ Hs+1T0

be the profile given by (1.29), and let U0 ∈ EsT0be defined by

U0(x, θ0, ξd) = V0(x, θ0 + ω1ξd, . . . , θ0 + ωMξd).(4.37)

Here 0 < T0 ≤ T is the minimum of the existence times for the quasilinear problems (1.6) and (3.31). Define

U0ε (x, θ0) := U0(x, θ0,

xdε

).(4.38)

The family U0ε is uniformly bounded in EsT0

for 0 < ε ≤ ε0 and satisfies

|Uε − U0ε |Es−1

T0

→ 0 as ε→ 0.(4.39)

Upon evaluating our placeholder ξd = xdε in the argument of U0 ∈ EsT , we get the function U0

ε in EsT , thesame space containing the solutions of the singular system. Indeed, the EsT norm is that of the estimate ofProposition 4.9, key to our proof of Theorem 4.7, and the norm in which we show our approximate solutionU0ε is close to the exact solution of the singular system.

Lemma 4.8. For m ≥ 0 suppose V(x, θ) ∈ Hm+1T , EV = V, and set U(x, θ0, ξd) = V(x, θ0 + ω1ξd, . . . , θ0 +

ωMξd). Then, setting Uε(x, θ0) = U(x, θ0,xdε ),

|Uε|EmT ≤ |U|EmT .(4.40)

36

Proof. The proof is almost identical to the argument used in [2] to prove Lemma 2.25 (b) with Lemma2.7.

Consider again the singular system, whose solution is constructed in Theorem 7.1 of [13]. This is donewith the use of the following iteration scheme:

a) ∂xdUn+1ε +

d−1∑j=0

Aj(εUnε )

(∂xj +

βj∂θ0ε

)Un+1ε = F (εUnε )Unε ,

b) B(εUnε )(Un+1ε )|xd=0 = G(x′, θ0),

c) Un+1ε = 0 in t < 0.

(4.41)

Once the Unε are obtained, Uε is found by taking the limit as n→∞ of the Unε .In order to prove Theorem 4.7, we will use the technique of simultaneous Picard iteration. For this, we

take the iterates U0,nε , which are formed by evaluation of our iterates V0,n(θ) at θ = (θ0+ω1ξd, . . . , θ0+ωMξd),

and which converge to U0ε in the limit n → ∞ uniformly with respect to ε. We then show the U0,n

ε tend tothe iterates Unε for the singular system as ε→ 0. The proof uses the following linear estimate

Proposition 4.9 ([13], Cor. 7.2). Let s ≥ [M0 + d+12 ] and consider the problem (4.41), where G ∈ Hs+1

T

has compact support and vanishes in t ≤ 0, and where the right side of (4.41)(a) is replaced by F ∈ EsT withsupp F ⊂ {t ≥ 0, 0 ≤ xd ≤ E}. Suppose Unε ∈ EsT has compact x−support and that for some K > 0, ε1 > 0we have

|Unε |EsT + |ε∂xdUnε |L∞ ≤ K for ε ∈ (0, ε1].(4.42)

Then there exist constants T0(K) and ε0(K) ≤ ε1 such that for 0 < ε ≤ ε0 and T ≤ T0 we have

|Un+1ε |EsT +

√T 〈Un+1

ε 〉s+1,T ≤ C(K,E)√T(|F|EsT + 〈G〉s+1,T

).(4.43)

Our proof of Theorem 1.4 relies on an induction argument, and the above linear estimate plays a crucialrole in the inductive step, allowing us to show the closeness between an iterate U0,n+1

ε approximating ourleading order expansion and the iterate Un+1

ε .

Proof of Theorem 4.7. It suffices to prove boundedness of the family U0ε in EsT0

along with the followingthree statements:

(a) limn→∞

Unε = Uε in Es−1T0uniformly with respect to ε ∈ (0, ε0]

(b) limn→∞

U0,nε = U0

ε in Es−1T0uniformly with respect to ε ∈ (0, ε0]

(c) For each n limε→0|Unε − U0,n

ε |Es−1T0

= 0.

(4.44)

The uniform boundedness of Uε and the Unε in EsT0and that (4.44)(a) holds are proved in [13], Theorem

7.1, with the use of the iteration scheme (4.41) and the linear estimate Proposition [[13], Cor. 7.2]4.9.The desired boundedness of U0

ε and the U0,nε in EsT0

follows from the boundedness of V0 and the uniform

boundedness of the V0,n in Hs+1T0

, considered with Proposition 4.5 and Lemma 4.8. Similarly, from the factthat

limn→∞

V0,n = V0 in HsT0,(4.45)

we get (4.44)(b).Now we proceed by proving (4.44)(c). Fix δ > 0. Noting EV0,n = V0,n and EV0,n+1 = V0,n+1, choose

finite partial trigonometric polynomial sums V0,np and V0,n+1

p of (resp.) V0,n and V0,n+1 such that

(a) EV0,np = V0,n

p and EV0,n+1p = V0,n+1

p ,

(b) |V0,n − V0,np |Hs+1

T0

< δ, |V0,n+1 − V0,n+1p |Hs+1

T0

< δ,(4.46)

37

and note that, as a result of (4.46)(b), we also have

|∂xdV0,n+1 − ∂xdV0,n+1p |HsT0 < δ.(4.47)

Observe then, as a consequence of Proposition 4.5, we get

|U0,n − U0,np |EsT0 < Cδ, |U0,n+1 − U0,n+1

p |EsT0 < Cδ, and |∂xdU0,n+1 − ∂xdU0,n+1p |Es−1

T0

< Cδ,(4.48)

where U0,np and U0,n+1

p are obtained from (resp.) V0,np and V0,n+1

p via the substitution θ = θ(θ0, ξd). Theinduction assumption is

limε→0|Unε − U0,n

ε |Es−1T0

= 0.(4.49)

With the boundedness of the Unε in EsT0, it follows

limε→0|F (εUnε )(Unε )− F (0)U0,n

ε |Es−1T0

= 0.(4.50)

Observe that since V0,n and V0,np are invariant under E, Lemma 4.8 applies. Thus, using the estimate (4.40),

from (4.50) and (4.48) we obtain

|F (εUnε )(Unε )− F (0)U0,np,ε |Es−1

T0

≤ Cδ + c(ε),(4.51)

where c(ε)→ 0 as ε→ 0.Now we define

Gp = L(∂x)V0,n+1p +M(V0,n

p )∂θV0,n+1p .(4.52)

Then

E(Gp − F (0)V0,np ) =E(L(∂x)(V0,n+1

p − V0,n+1))+

E(M(V0,np )∂θV0,n+1

p −M(V0,n)∂θV0,n+1)+

E(F (0)(V0,n − V0,np )) +Rn+1.

(4.53)

Thus using (4.46)(b) and continuity of both E : Hs;2 → Hs;1 and multiplication from Hs;1 ×Hs;1 → Hs;2,we get

|EGp − E(F (0)V0,np )−Rn+1|HsT0 = O(δ).(4.54)

We define the operator

L0 = L(∂x) +1

εL(dφ0)∂θ0 +M′(U0,n

p,ε )∂θ0 .(4.55)

We claim

|L0Un+1ε − F (εUnε )Unε |Es−1

T0

≤ Cδ + c(ε),(4.56)

where c(ε)→ 0 as ε→ 0. This follows from (4.41)(a) and

|Aj(εUnε )∂xjUn+1ε − Aj(0)∂xjU

n+1ε |Es−1

T0

= O(ε)∣∣∣∣1ε Aj(εUnε )βj∂θ0Un+1ε −

(1

εAj(0)βj∂θ0U

n+1ε + ∂uAj(0)Unε βj∂θ0U

n+1ε

)∣∣∣∣Es−1T0

= O(ε)∣∣∣∂uAj(0)(Unε − U0,np,ε )βj∂θ0U

n+1ε

∣∣∣Es−1T0

≤ C|Unε − U0,np,ε |Es−1

T0

≤ c(ε) +O(δ).

(4.57)

38

Setting G′p = L(∂x)U0,n+1p +M′(U0,n

p )∂θ0U0,n+1p , since L′(∂θ0 , ∂ξd)U0,n+1

p = 0, we get

L0U0,n+1p,ε = G′p,ε.(4.58)

From now on we use |(θ0,ξd) to denote evaluation at θ = θ(θ0, ξd), and |(θ0,ξd),ε to indicate |(θ0,ξd) followedby evaluation at ξd = xd

ε . It is easy to check G′p = Gp|(θ0,ξd), particularly because V0,np , V0,n+1

p are finitepolynomials. Thus

L0U0,n+1p,ε − F (0)U0,n

p,ε = G′p,ε − F (0)U0,np,ε

= [Gp − F (0)V0,np ]|(θ0,ξd),ε(4.59)

= [(E(Gp − F (0)V0,np )−Rn+1) +Rn+1 + (I − E)(Gp − F (0)V0,n

p )]|(θ0,ξd),ε(4.60)

We claim:(i)

|(E(Gp − F (0)V0,np )−Rn+1)|(θ0,ξd),ε|Es−1

T0

= O(δ),(4.61)

(ii)

limε→0

∣∣Rn+1|(θ0,ξd),ε∣∣Es−1T0

(x,θ0)= 0,(4.62)

and (iii) there exists V1p ∈ H

s;2T0

such that for U1p = V1

p |θ(θ0,ξd),

L′(∂θ0 , ∂ξd)U1p = −[(I − E)(Gp − F (0)V0,n

p )]|θ(θ0,ξd).(4.63)

(i) follows from (4.54) followed by application of Proposition 4.5 and then Lemma 4.8.To see (ii), observe it is clear from our definition of Rn+1 that the continuity of E implies Rn+1(x, θ) ∈Hs;1T0

(x, θ), and recall from Remark 3.33 that we have EeRn+1 = Rn+1 and (3.112). By requiring (3.86), and

thus (3.82), to hold at xd = 0, we have obtained Rn+1|xd=0 = 0. So Proposition 4.6 grants us (4.62).Regarding (iii), note that, by Remark 3.7, (4.63) is equivalent to

L′(∂θ0 , ∂ξd)U1p = −[(I − E[)(Gp − F (0)V0,n

p )]|θ(θ0,ξd),(4.64)

for which a sufficient condition is that, where U1p = V1

p |θ(θ0,ξd),

L(∂θ)V1p = −(I − E[)(Gp − F (0)V0,n

p ).(4.65)

Proposition 3.8 guarantees the existence of such V1p , so proof of (iii) is complete.

Noting

L0(εU1p,ε) = (L′(∂θ0 , ∂ξd)U1

p )ε + (L(∂x)εU1p )ε +M′(U0,n

p,ε )∂θ0(εU1p,ε),(4.66)

it follows from (4.60)-(4.63) that

|L0(U0,n+1p,ε + εU1

p,ε)− F (0)U0,np,ε |Es−1

T0

≤ Cδ + c(ε) +K(δ)ε,(4.67)

where c(ε) has been altered, but still tends to zero as ε → 0. It follows from equations (4.51), (4.56), and(4.67) that

|L0(Un+1ε − (U0,n+1

p,ε + εU1p,ε))|Es−1

T0

≤ Cδ + c(ε) +K(δ)ε.(4.68)

Now we claim that we have the following estimates:

(a)

∣∣∣∣(∂xd + A(εU0,np,ε , ∂x′ +

β · ∂θ0ε

)

)(Un+1ε − (U0,n+1

p,ε + εU1p,ε))∣∣∣∣Es−1T0

≤ Cδ + c(ε) +K(δ)ε

(b)∣∣B(εU0,n

p,ε )(Un+1ε − (U0,n+1

p,ε + εU1p,ε))∣∣HsT0≤ Cδ + c(ε) +K(δ)ε.

(4.69)

39

That (4.69)(a) holds follows from (4.68) with the use of estimates similar to (4.57), and (4.69)(b) followsfrom the boundary conditions (1.6)(b) and (3.113)(c) together with Proposition 4.5 and Lemma 4.8. Afterapplying the estimate of Proposition 4.9, we get that

|Un+1ε − (U0,n+1

p,ε + εU1p,ε)|Es−1

T0

≤ Cδ + c(ε) +K(δ)ε.(4.70)

from which we conclude

|Un+1ε − U0,n+1

ε |Es−1T0

≤ Cδ + c(ε) +K(δ)ε.(4.71)

This finishes the induction step, completing the proof of the theorem.

40

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