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2020-04-09

Stability of MHD Shock Waves Stability of MHD Shock Waves

Bryn Nicole Barker Brigham Young University

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Stability of MHD Shock Waves

Bryn Nicole Barker

A thesis submitted to the faculty ofBrigham Young University

in partial fulfillment of the requirements for the degree of

Master of Science

Blake Barker, ChairEmily Evans

Jared Whitehead

Department of Mathematics

Brigham Young University

Copyright © 2020 Bryn Nicole Barker

All Rights Reserved

abstract

Stability of MHD Shock Waves

Bryn Nicole BarkerDepartment of Mathematics, BYU

Master of Science

This thesis focuses on the study of spectral stability of planar shock waves in 2-dimensionalmagnetohydrodynamics. We begin with a numerical approach, computing the Lopatinski de-terminant and Evans function with the goal of determining if there are parameters for whichviscous waves are unstable and the corresponding inviscid waves are stable. We also begindeveloping a method to obtain an explicit, analytical representation of the Evans function.We demonstrate the capabilities of this method with compressible Navier-Stokes and extendour results to 2-D MHD. Finally, using compressible Navier-Stokes again, we derive an energyestimate as a first step in improving the bound on possible roots of the Evans function.

Keywords: MHD, shocks, traveling waves, Lopatinski determinant, Evans function

Acknowledgements

A huge thanks to my advisor Dr. Blake Barker for his dedication to mentoring me

and preparing me for success in mathematical research. I am inspired by his passion for

mathematical discovery and his determination to make discoveries. I am also extremely

grateful for other math professors who have supported and mentored me throughout my

time at BYU, including Dr. Ben Webb, Dr. Emily Evans, and Dr. Wayne Barrett. Without

their encouragement, I would not have the appreciation and love for math that I do today.

In addition, I would like to thank my kids, George and Harris, for ensuring that finishing

this thesis was a real challenge and my family and friends for having confidence in my ability

to succeed. However, I owe the most thanks to my husband DJ for really making this happen.

Contents

Contents iv

List of Tables vi

List of Figures vii

1 Introduction 1

2 Mathematical Derivation 3

2.1 Magnetohydrodynamic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Lopatinski Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 The Evans Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Numerical Stability Results 14

3.1 STABLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Stability Bifurcation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 Convexity of Bifurcation Curve . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Navier-Stokes 21

4.1 Compressible Navier-Stokes . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.3 Bounding Real Valued λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.4 Bounding Complex Valued λ . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Analytic Stability 41

5.1 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Analytic Profile Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Analytic Evans System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

iv

5.4 Compound Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.5 Extending to MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendix A Analytic Evans System Solver 62

Bibliography 67

v

List of Tables

4.1 Bounds on Re(λ) + |Im(λ)|, where λ is an admissible eigenvalue, for various

values of Γ and ν, as given in [13]. . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2 Numerically approximated bounds on real valued λ for various values of Γ and

ν, computed based on Theorem 4.2. . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Approximated bounds on Re(λ) + |Im(λ)| for all λ with Re(λ) > 5.7. These

bounds were computed using Algorithm 1 which is based on Theorem 4.6. . . 41

vi

List of Figures

2.1 Diagrams showing the main idea behind evolving bases from each end of the

domain to determine existence of an eigenfunction for the Evans system. . . 12

3.1 Plotted contours for Lopatinski determinant and Evans function as computed

by STABLAB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Stability results comparing the inviscid and viscous models as h1 and u1+ vary. 17

3.3 Diagrams indicating possible relationships between roots of the inviscid system

and roots of the viscous system as parameterized by the Fourier coefficient ξ. 18

3.4 The real part of the roots of the viscous system as parameterized by h1 and

the Fourier coefficient ξ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 The real parts of the roots of the viscous system as parameterized by h1

for fixed values of ξ. The dotted line serves as a reference to highlight the

convexity of the curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 The real parts of the roots of the viscous system as parameterized by ξ for

fixed values of h1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 The convergence of the norms of the coefficient vectors corresponding to the

example system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 The series solution of e(σ) with 100 terms plotted against σ. . . . . . . . . . 46

5.3 The series solution of e(σ) with five terms plotted against σ. . . . . . . . . . 47

5.4 The absolute values of the coefficients |an| of e(σ). . . . . . . . . . . . . . . 47

5.5 Nullclines for the profile system (5.1) corresponding to v′ = 0 and e′ = 0,

which intersect at the fixed points of the system. . . . . . . . . . . . . . . . . 48

5.6 The fifteen-term series solution to the Evans system with λ = 10, scaled out

by the most negative eigenvalue of A+. . . . . . . . . . . . . . . . . . . . . 52

5.7 The trend in the norms of the coefficient vector for the series solution Z(σ). 53

vii

5.8 The fifteen-term series solution to the Evans system with λ = 10, scaled out

by the second most negative eigenvalue of A+. . . . . . . . . . . . . . . . . . 54

5.9 The fifteen-term series solution to the Evans system with λ = 10, scaled out

by the third most negative eigenvalue of A+. . . . . . . . . . . . . . . . . . . 54

5.10 The fifteen-term series solution to the lifted Evans system with λ = 10, scaled

by f ′(σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.11 The analytic solution to the Evans system evaluated on the unstable manifold

for various λ values satisfying Λ = 54.27, where λ is parameterized by θ

according to (5.17). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.12 Successive relative errors for finite series solution of the lifted Evans system

for various values of λ, satisfying Re(λ) > 0.001 and Λ = 54.27. Note that

λ is parameterized by θ according to (5.17). The dotted line, included for

reference, plots y = 1/n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.13 The fifteen-term series solution of the MHD profile e(σ) plotted against σ. . 59

5.14 The fifteen-term series solution to the Evans system for 2-D MHD with λ = 10,

scaled by f ′(σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.15 The five-term series solution to the lifted Evans system for 2-D MHD with

λ = 10, scaled by f ′(σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

viii

Chapter 1. Introduction

Magnetohydrodynamics (MHD) couples the Navier-Stokes equations for fluid dynamics with

Maxwell’s equation of electromagnetism to describe the behavior of conducting fluids, such

as plasmas. The main concept behind MHD is that magnetic fields can induce currents in

a moving conductive fluid, which in turn create forces on the fluid and affect the magnetic

field itself [22]. Research in MHD waves is a driving force in developing an experimental

tokamak nuclear fusion reactor which has huge potential impacts in power generation.

This thesis centers on the stability of traveling MHD waves. Traveling waves are consid-

ered nonlinearly or orbitally stable when after a small perturbation, the wave returns to its

original shape up to a spatial translate. Understanding traveling wave stability has notable

implications in model verification. When mathematically modeling fluid flow, many simpli-

fications are made that could cause solutions of the mathematical model to diverge from the

physical phenomena. As such, proving stability of traveling wave solutions is one important

step in ensuring the model will not diverge from the physical event it simulates. This has

immediate relevance in relation to MHD waves as the traveling waves solutions correspond

to MHD shocks.

When studying traveling wave stability, it is useful to consider both the inviscid and

viscous case. The relationship between inviscid and viscous traveling wave stability has been

studied previously [6]. It has been proven that in both 1-D and 2-D, inviscid instability

implies viscous instability; see [23, 27, 28]. Additionally, it has been shown that this rela-

tionship does not hold between dimensions, meaning that inviscid instability in 2-D does

not imply viscous instability in 1-D [10]. However, in both 1-D and 2-D, we do not know if

viscous instability implies inviscid instability [20].

So how do we determine stability of traveling waves? Linearizing the system about a

given profile (assumed stationary by Galilean invariance), and taking the Laplace transform

in time and the Fourier transform in the hyperplane orthogonal to the direction of propaga-

1

tion, allows the formulation of an eigenvalue equation for a differential operator with variable

coefficients [25].Traveling wave stability can then be shown by proving this linearized oper-

ator has no unstable spectra [26]. This follows from the fact that spectral stability implies

nonlinear stability [12, 18, 19].

In the inviscid and viscous cases, the unstable spectra of the linear operator correspond

to the roots of the Lopatinski determinant and the Evans function respectively. Detailed

descriptions of these functions can be found in Section 2.3 and Section 2.4. The Lopatinski

determinant measures the linear independence of the decaying eigenspaces of the unstable

and stable manifolds of the linear operator together with the jump condition corresponding

to the discontinuity. Due to the relative simplicity of the inviscid system, the Lopatinski

determinant can usually be calculated directly.

The Evans function, however, typically must be approximated numerically [25]. To com-

pute the Evans function, we write the eigenvalue problem as a first order system and find

decaying eigenspaces of the linear operator tangent to the unstable and stable manifolds. We

then evolve these bases and evaluate them at zero. The Wronskian of the resulting system

determines the linear independence of the evolved bases. A Wronskian of zero corresponds

to an intersection of the bases of the unstable and stable manifolds, implying the existence

of an eigenfunction for the particular eigenvalue.

One important thing to note is that the derivative of the profile is an eigenfunction

of the linear operator corresponding to the zero eigenvalue [21]. Because of this, when

evaluating the Wronskian at potential unstable spectra, we must be careful to avoid this

trivial eigenfunction. Alternatively, we can use what we call the integrated Evans system

in which we substitute each quantity, for example, u(x), with its integral,∫ x−∞ u(t)dt. This

removes the eigenvalue associated with translation invariance. We can achieve a similar

improvement for multi-D by using what are called pseudo-Lagrangian coordinates, as given

in [3]. Using pseudo-Lagrangian coordinates significantly reduces the amount of winding

2

in the Evans function computations. Note that in 1-D, pseudo-Lagrangian coordinates are

equivalent to Lagrangian coordinates.

As mentioned, in order to actually determine traveling wave stability, we need to deter-

mine if the Lopatinski determinant and Evans function are nonzero for all eigenvalues with

positive real part. In general, various techniques have been shown useful in bounding the

contour containing potential unstable spectra for a system. Once the contour is bounded,

we only require the Lopatinski determinant and the Evans function computations within the

bounded region, which greatly reduces the complexity of the stability problem. Additionally,

if we are careful to initialize the systems associated with the Lopatinski determinant and

Evans function in a way that preserves analyticity, we can apply the argument principle to

determine if there are unstable spectra within the contour [5]. When using the argument

principle, a resulting winding number of zero corresponds to traveling wave stability while

a nonzero winding number corresponds to instability. This bounded contour combined with

the use of the argument principle results in much simpler computations required to determine

stability.

As mentioned, MHD waves couple Navier-Stokes with Maxwell’s equations, making the-

oretical results extremely complicated and difficult to come by. Throughout this thesis, we

will introduce new theoretical methods and techniques that we are developing applied to

compressible Navier-Stokes instead of applied to full MHD. The compressible Navier-Stokes

equations have very similar structure to MHD so these results are relevant to the discovery

of similar results for MHD, some of which we will discuss later on.

In this thesis we will: (1) analyze the relationship between inviscid and viscous stability

for 2-D MHD, (2) derive energy estimates that may improve the bound on the contour

containing possible unstable spectra for compressible Navier-Stokes and (3) describe new

analytical tools for non-numerical Evans function computations for compressible Navier-

Stokes and 2-D MHD.

3

Chapter 2. Mathematical Derivation

In this section we introduce the mathematical background for the 2-D MHD system and

formulate the Lopatinski determinant and Evans function.

2.1 Magnetohydrodynamic Waves

In vector notation, the equations of 2-D MHD are given in Eulerian coordinates by

ρt + div(ρu) = 0, (2.1a)

(ρu)t + div(ρu⊗ u− h⊗ h) +∇q = µ∆u+ (η + µ)∇divu, (2.1b)

ht −∇× (u× h) = ν∆h, (2.1c)

(ρE +1

2h2)t + div(A) = div

(∑u)

+ κ∆T + ν div(B), (2.1d)

div(h) = 0, (2.1e)

where p = p(ρ, T ) is pressure, ρ is specific volume, T is temperature, u = (u1, u2, 0) is velocity,

h = (h1, h2, 0) is the magnetic field, E is the total energy given in terms of specific energy e

by E := e + u21/2 + u2

2/2, and q = p + |h|22

. The constants µ and η are viscosity coefficients

and κ is the heat conductivity coefficient. In addition,∑

:= ηdiv(u)I + µ(∇u + (∇u)t),

A = (ρE + p)u + h × (u × h), and B = h × (O × h); see [7, 8, 9, 15, 17]. Here, t is time

and (x1, x2, x3) is spatial location. Because we are only considering 2-D MHD, the solution

is independent of x3. We also note that × is the cross product and ⊗ is the outer product.

We consider an ideal gas, so e(ρ, T ) = cνT where cν > 0 is the specific heat coefficient at

constant volume, and the pressure function is given by p(ρ, e) = Γρe, where Γ = R/cν and

R is the universal gas constant; see [14].

In this thesis we will use the β-model of (2.1) [6]. We use this model because it has the

right form for the highly tested numerical package which we will use later; see Section 3.1.

More specifically, the β-model has consistent splitting which is necessary in order to use the

4

Lopatinski determinant to analyze stability. For a system to have consistent splitting means

that the end states are hyperbolic and the dimensions of the stable and unstable manifold

are constant and sum to the dimension of the full space [13]. The β-model is formed by

adding a multiple of div(h) to (2.1c), yielding the following updated system,

ρt + div(ρu) = 0, (2.2a)

(ρu)t + div(ρu⊗ u− h⊗ h) +∇q = µ∆u+ (η + µ)∇divu, (2.2b)

ht −∇× (u× h) + βdiv(h)e1 = ν∆h, (2.2c)

(ρE +1

2h2)t + div(A) = div

(∑u)

+ κ∆T + ν div(B), (2.2d)

div(h) = 0, (2.2e)

where β is a real valued parameter and e1 = (1 0)T . For full proofs of the mentioned

properties of the β-model see [6].

We now write (2.2) in the general flux form; see [2],

f 0(U)t +2∑

k=1

fk(U)xk =2∑

j,k=1

(Bjk(U)Uxk

)xj, (2.3)

where in general x = (x1, x2) ∈ R2, t ∈ R, and U ∈ Rn with f j : Rn → Rn and it is assumed

that each Bjk has the block structure

Bjk =

0r×r 0r×(n−r)

0(n−r)×r bjk(U)

.

Notice that in the ideal gas case of 2-D MHD, n = 6, r = 1 (where r gives the number ofhyperbolic equations in the system), and

U =

ρ

u1

u2

h1

h2

T

, f

0(U)t =

ρ

ρu1

ρu2

h1

h2

12

(h21 + h22) + ρ(cνT + 1

2(u2

1 + u22))

,

5

f1(U) =

ρu1

RρT + 12

(h22 − h

21) + ρu2

1

ρu1u2 − h1h2

βh1

u1h2 − h1u2

(R + cν)ρu1T + h2(u1h2 − h1u2) +ρu12

(u21 + u2

2)

, f

2(U) =

ρu2

ρu1u2 − h1h2

RρT + 12

(h21 − h

22) + ρu2

2

βh2 + h1u2 − u1h2

0

(R + cν)ρu2T + h1(h1u2 − u1h2) +ρu22

(u21 + u2

2)

B11

(U) =

0 0 0 0 0 0

0 η + 2µ 0 0 0 0

0 0 µ 0 0 0

0 0 0 ν 0 0

0 0 0 0 ν 0

0 u1(η + 2µ) µu2 0 h2ν κ

, B

12(U) =

0 0 0 0 0 0

0 0 η + µ 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 µu2 ηu1 −h2ν 0 0

,

B21

(U) =

0 0 0 0 0 0

0 0 0 0 0 0

0 η + µ 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 ηu2 µu1 0 −h1ν 0

, B

22(U) =

0 0 0 0 0 0

0 µ 0 0 0 0

0 0 η + 2µ 0 0 0

0 0 0 ν 0 0

0 0 0 0 ν 0

0 µu1 u2(η + 2µ) h1ν 0 κ

.

We now compute the Jacobians of f 0, f 1, and f 2, given by

df0(U) =

1 0 0 0 0 0

u1 ρ 0 0 0 0

u2 0 ρ 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

cνT + 12

(u21 + u2

2) ρu1 ρu2 h1 h2 cνρ

,

df1(U) =

u1 ρ 0 0 0 0

RT + u21 2ρu1 0 −h1 h2 Rρ

u1u2 ρu2 ρu1 −h2 uh1 0

0 0 0 β 0 0

0 h2 −h1 −u2 u1 0

Rνu1T +u12

(u21 + u2

2) RνρT + h22 + ρ

2(3u2

1 + u22) −h1h2 + ρu1u2 −h2u2 2h2u1 − h1u2 Rνρu1

,

df2(U) =

u2 0 ρ 0 0 0

u1u2 ρu2 ρu1 −h2 −h1 0

RT + u22 0 2ρu2 h1 −h2 Rρ

0 −h2 h1 u2 β − u1 0

0 0 0 0 0 0

Rνu2T +u22

(u21 + u2

2) −h1h2 + ρu1u2 RνρT + h21 + ρ

2(u2

1 + 3u22) 2h1u2 − u1h2 −h1u1 Rνρu2

,

where Rν := R + cν .

It is worth noting that in this thesis we are considering the parallel case. In the parallel

case, multidimensional stability reduces to the two-dimensional case by rotational symmetry

about x1. For this reason, considering the parallel case in two dimensions does not lose

generality in the results [10].

6

2.2 Traveling Waves

A viscous shock profile of (2.2) is a traveling wave solution of the form

U(x, t) = U(x1 − st),

which is moving at speed s and has constant connecting states given by U±. Note that x1

is chosen without loss of generality due to the symmetry of (2.2) in the first and second

dimensions. Additionally, due to Galilean invariance, which holds for compressible MHD,

we can, again without loss of generality, choose s = 0 which corresponds to a standing shock

profile. With this choice of s we have that Ut = sU ′ = 0. Also notice that Ux2 = ∂∂x2U = 0.

As mentioned, in this thesis we are considering planar waves in the parallel case and thus h2

and u2 are equivalently zero in the profile.

Substituting our choice for U into our flux form (2.3) we get

f 1(U)x1 = (B11(U)Ux1)Ux1 ,

or equivalently (now incorporating h2 = u2 = 0 as well),

(ρu1)′ = 0,

(RρT )′ − 1

2(h2

1)′ + (ρu21)′ = (2µ+ η)u′′1,

βh′1 = νh′′1,

(Rνρu1T )′ + (1

2(ρu3

1)′ = (2µ+ η)(u1u′1)′ + κT ′′.

First notice that the only bounded solution of βh′1 = νh′′1 is constant and thus h1 is constant.

Similarly, (ρu1)′ = 0 implies that ρu1 = m for some constant m. Substituting a constant h1

and m into the remaining equations gives the following system,

R(ρT )′ +mu′1 = (2µ+ η)u′′1,

7

mRνT′ +

m

2(u2

1)′ = (2µ+ η)(u1u′1)′ + κT ′′.

Now we will integrate this system from −∞ to x1. Recall that at ±∞, U ′ = 0 by the nature

of the traveling wave. After integration we obtain,

m(u1 − u1−) +R(ρT − ρ−T−) = (2µ+ η)u′1,

mRν(T − T−) +m

2(u2

1 − u21−) = (2µ+ η)u1u

′1 + κT ′.

Rearranging this system and solving for u′1 and e′ yields,

u′1 =1

2µ+ η[m(u1 − u1−) +R(ρT − ρ−T−)] , (2.5)

e′ =cνκ

[m(e− e−)− m

2(u1 − u1−)2 +RcνT−ρ−(u1 − u1−)

]. (2.6)

It is useful to note that because u2 and h2 are zero in the profile and h1 is constant, the

profile for MHD is equivalent to the profile for the ideal gas case of compressible Navier-

Stokes.

We now rescale (2.5) to eliminate the arbitrary constant m using the fact that this system

is invariant under the following rescaling; see [14],

(x1, x2, t; ρ, u1, u2, T )→(mx1,mx2, εm

2t; ερ,u1

εm,u2

εm,T

ε2m2

).

We will now choose m and ε so that ρ− = u1− = 1, implying that m = 1 also. With this

particular rescaling, we obtain our final profile system,

u′1 =1

2µ+ η

[u1 − 1 + Γ(

e

u1

− e−)

], (2.7a)

e′ =cνκ

[e− e− −

(u1 − 1)2

2+ (u1 − 1)Γe−

], (2.7b)

8

with the corresponding Jacobian

J =

12µ+η

(1− Γeu2

1) Γ

(2µ+η)u1

cνκ

(Γe− − (u1 − 1)) cνκ

.

We denote the solutions to the profile system as ρ, u1, u2, h1, h2, and T . As a note, recall

that ρ = 1/u1, u2 = h2 = 0, h1 = c for some constant c, and T = e/cν .

2.3 The Lopatinski Determinant

The Lopatinski determinant is used to determine inviscid stability of a system. In order to

derive the Lopatinski determinant, we begin with the inviscid case of (2.2) which in flux

form is given by

f 0(U)t +2∑

k=1

fk(U)xk = 0. (2.8)

Notice this is equivalent to (2.3) with the right-hand side set to zero.

We will now linearize (2.8) about the stationary planar shock profile U = (ρ u1 u2 h1 h2 T )T .

We define U± = limx1→±∞ U(x1) and note that U± are constant, and set A±i = Ai(U±) =

Dfi(U±) for i ∈ {0, 1, 2}. The linearized system is then given by

A±0 Ut + A±1 Ux1 + A±2 Ux2 = 0. (2.9)

We can analyze (2.9) by taking the Laplace transform in t (with parameter λ) and the

Fourier transform in x2 (with parameter ξ). This yields

λA±0 V + A±1 Vx1 + iξA±2 V = 0, (2.10)

where V is the Laplace-Fourier transform of U . We write (2.10) as the following first order

system

Vx1 = −(A±1 )−1(λA±0 + iξA±2 )V. (2.11)

9

The Lopatinski determinant is then defined by

∆(ξ, λ) = det([R−1 , ...,R−p−1, λ[f 0(U)] + iξ[f 2(U)], R+p+1, ..., R

+n ]), (2.12)

where ξ ∈ R and λ ∈ C with Re(λ) > 0. Here {R+p+1, ...,R+

n } and {R−1 , ...,R−p−1} denote

respectively bases for the unstable/stable subspaces of A±(λ, ξ) = (A±1 )−1(λA±0 + iξA±2 ), and

λ[f 0(U)] + iξ[f 2(U)] is the jump condition given by the Rankine-Hugoniot condition of the

unperturbed shock; see [16, 27] for details.

When the Lopatinski determinant is zero for a particular value of λ, this implies that the

bases on each manifold together with the jump condition are linearly dependent and thus

there is an eigenfuction for the inviscid system with corresponding eigenvalue λ. Notice that

we can check the linear independence of the bases at the manifolds because the system (2.11)

is constant coefficient and thus the linear independence will not change as the solution bases

are evolved across the domain.

As a side, note that the Lopatinski determinant varies linearly in the Fourier coefficient

ξ. This will be relevant in Section 3.3.

2.4 The Evans Function

To create the Evans System we will linearize (2.3) about the steady state solution U(x, t) =

U(x1). By doing this we create an equation which describes the approximate evolution of

a perturbation. Recall that without loss of generality we chose the shock profile associated

with s = 0.

The resulting linearized equations are given by

A0(U)t +2∑j=1

(AjU)xj =2∑

j,k=1

(BjkUxk)xj , (2.13)

10

where

A0 = A0(U), AjU = Aj(U)U − dBj1(U)(U, Ux1), and Bjk = Bjk(U).

We now take the Laplace transform in time with variable λ, and the Fourier transform

in x2 with variable ξ, to obtain the generalized eigenvalue equation

λA0U + (A1U)′ + iξA2U = (B11U ′)′ + iξ(B12U)′ + iξB21U ′ − ξ2B22U. (2.14)

Note that (2.14) is equivalent to (2.10) but with the retention of the linearization of the

viscous terms.

We recall that to show stability of the viscous shock profile U , we must prove that the

eigenvalue problem (2.14) has no unstable spectra. As mentioned, this is equivalent to

showing that the Evans function D(λ, ξ) is nonzero for all λ such that Re(λ) ≥ 0 excluding

λ = 0. In order to simplify the computations required to check for zeros of the Evans

function, we aim to construct the Evans function in a way that ensures analyticity in λ

which in turn warrants use of the argument principle when searching for zeros.

Our approach is to first formulate (2.14) as the first order system,

W ′ = A(x1;λ, ξ)W, (2.15)

where based on the block structure of Bjk (as mentioned earlier), A is an N × N matrix

where N = 2n− r (this is because we have a second derivative in n− r equations and thus

when made into a first order system we will have 2(n − r) equations from these n − r and

then an additional r hyperbolic equations).

We then define A±(λ, ξ) = limx1→±∞A(x1;λ, ξ) and proceed by building subspaces of

solutions for (2.15). We let {W+1 , ...,W

+k } be an unstable basis for A+ and {W−

k+1, ...,W−N }

be a stable basis for A−. Notice that the basis {W+1 , ...,W

+k } decays as x1 → +∞ and

11

{W−k+1, ...,W

−N } decays as x1 → −∞.

With these bases defined, the Evans function is

D(λ, ξ) = det(W+1 , ...,W

+k ,W

−k+1, ...W

−N )|x1=0. (2.16)

If there exist λ′, ξ′ with Re(λ′) ≥ 0 such that D(λ′, ξ′) = 0 then our two bases are linearly

independent, implying existence of an eigenfunction of (2.15) which in turn implies the λ is

an eigenvalue of (2.15) and the underlying wave is unstable.

Notice that in this case, the system (2.15) is not constant coefficient so we must evolve

our bases to the center of the domain before determining linear independence. This idea is

displayed in Figure 2.1. We begin by determining a basis for each end of our domain, as

shown in Figure 2.1 (a). Then to determine existence of an eigenfunction that spans the

entire domain, we evolve the bases to the center and determine if they are linearly dependent

at their intersection, as shown in Figure 2.1 (b).

(a) (b)

Figure 2.1: Diagrams showing the main idea behind evolving bases from each end of thedomain to determine existence of an eigenfunction for the Evans system.

12

To create the Evans system for 2-D MHD, we start with (2.14), which we can rearrange

to

λA0U + iξA2U + ξ2B22U =(B11U ′ + iξB2U − A1U

)′,

where A2 = A2 + (B21)′ and B2 = B21 + B12.

From this formulation we are able to define our flux variable

f = B11U ′ + iξB2U − A1U,

and we will set Aξ = ξA2, Bξ = ξB2, and Bξξ = ξξB22. We are now ready to cast (2.14) asthe first order system (2.15) with

W =

f

u2

,A =

−(λA0

11 + iAxi11)(A111)−1 0 −λ(A0

11(A111)−1A1

12 − A012)− iAξ11(A1

11)−1A112 + iA

ξ12

−(λA021 + iAxi21)(A1

11)−1 0 −λ(A021(A1

11)−1A112 − A

022)− iAξ21(A1

11)−1A112 + iA

ξ22 + bξξ

−(b11)−1A121(A1

11)−1 (b11)−1 (b11)−1(A122 − A

121(A1

11)−1A112 − b

ξ)

. (2.17)

The full derivation of this system can be found in [2].

The system given by (2.17) is in Eulerian coordinates. It turns out that when computing

the Evans function in Eulerian coordinates the modulus and argument of the output changes

rapidly. This complicates calculations significantly when we use winding number computa-

tions on the values of the Evans function evaluated along a simple contour. We avoid this

issue by instead using pseudo-Lagrangian coordinates for which, as mentioned, the winding

of the Evans function is greatly reduced; for details see [3].

Using the work of [2], we define our 2-D Evans system under this formulation by first

setting r(λ, ξ) = |λ, ξ| and defining

f ] = F/r(λ, ξ), λ] = λ/r(λ, ξ), and ξ] = ξ/r(λ, ξ).

With these changes we arrive at the Evans system in pseudo-Lagrangian coordinates whichis given by W ′ = A](x1;λ, ξ)W , where

W =

f]u2

,A =

−r(λ]A0

11 + iAxi11)(A111)−1 0 −λ](A0

11(A111)−1A1

12 − A012)− iAξ

]

11(A111)−1A1

12 + iAξ]

12

−r(λ]A021 + iAxi21)(A1

11)−1 0 −λ](A021(A1

11)−1A112 − A

022)− iAξ

]

21(A111)−1A1

12 + iAξ]

22 + rbξ]ξ]

−r(b11)−1A121(A1

11)−1 r(b11)−1 (b11)−1(A122 − A

121(A1

11)−1A112 − rb

ξ] )

. (2.18)

13

Following the process explained above, we now build the integrated Evans function. We

define A]±(λ], ξ]) = limx1→∞A](x1;λ], ξ]), with bases {W+

1 , ...,W+k } and {W−

k+1, ...,W−N } as

described previously.

Taking the determinant of these bases evolved to x1 = 0 gives our integrated Evans

function

D(λ], ξ]) = det(W+1 , ...,W

+k ,W

−k+1, ...W

−N )|x1=0. (2.19)

Recall, we evolve the bases to x1 = 0 because due to the complexity of the system, in order

to determine linear independence of the two bases, we must evaluate the bases at the same x1

value. Whereas with the Lopatinski determinant we had a constant coefficient system which

allowed us to determine linear independence without evolving our bases. We also note that

we could alternatively evaluate the determinant at any value of x1. It is somewhat common

to instead evolve the solutions to x1 = −∞ or x1 = +∞.

Chapter 3. Numerical Stability Results

In this section we describe our study of the stability of traveling wave solutions of inviscid and

viscous models for 2-D MHD. More specifically, we seek to determine if there exist unstable

viscous waves with corresponding stable inviscid waves. Computationally, this occurs when

the Evans function has a real root while the Lopatinski determinant does not. We note that

the reverse relationship has already been proven, in other words, we know that instability in

the inviscid model implies instability in the viscous model.

Understanding this relationship between viscous and inviscid stability for 2-D MHD has

meaningful implications in mathematical modeling. At the present, it is significantly easier

to determine stability of an inviscid model than it is for a viscous model. As a result, there

is strong motivation to use the inviscid models rather than the viscous models, even though

we are not fully aware of how stability of these two models differ. In fact, there are some

14

applications that currently use an inviscid model to study a viscous system, justifying this

choice by the assumption that these two models have very similar stability properties and

that the inviscid model is much simpler computationally.

If we find that there exist unstable viscous waves with corresponding stable inviscid waves,

then when using an inviscid model to study a viscous system, in some cases the model will

assume stability where in reality the system in unstable which will lead to inaccurate results.

On the other hand, if unstable viscous waves always correspond to unstable inviscid waves,

then we can use inviscid models to study viscous systems more frequently and with more

confidence in the accuracy of results.

3.1 STABLAB

Throughout this chapter, we will use STABLAB to perform Lopatinski determinant and

Evans function computations. STABLAB is a numerical library available in Matlab and

Python. The library was originally designed for Evans function computation but has since

been expanded to provide additional computational tools for analyzing wave stability [4].

In our use of the code, we provide the profile system, Lopatinski system, and Evans system

as well as a contour bound r. STABLAB then numerically approximates the stability of the

inviscid and viscous models for the system. More specifically, using the contour radius bound

r, STABLAB computes the Lopatinski determinant and numerically approximates the Evans

function for values of λ along the computed contour.

As an example, the normalized contours for the Lopatinski determinant and Evans func-

tion associated with the system given by h1 = 0.75, u1+ = 1.3, Γ = 2/3, µ = 0.1, η = −2/3µ,

κ = 0.1 and cν = 1, are shown in Figure 3.1. In this example, both systems have a winding

number of zero and are thus stable. Note that the points along each contour indicate where

the functions were evaluated.

15

Figure 3.1: Plotted contours for Lopatinski determinant and Evans function as computedby STABLAB.

We strive to determine existence of unstable viscous waves with corresponding stable

inviscid waves in two ways. First, we search for roots of the Evans function and Lopatinski

determinant for a variety of parameter values. Second, we study the convexity of the curve

along which the roots of the Evans function bifurcate.

3.2 Stability Bifurcation Analysis

Recall that in our construction of the Evans system, we were careful to maintain analyticity

in our spectral parameter λ. As a result, we may use the argument principle to identify

unstable spectra in a bounded contour in the right half complex plane. This contour can

take a variety of shapes depending on the system but must be simple and connected.

As mentioned above, we define r as the radius of our bounded contour. We abuse the

word radius in this context as the contour is likely not circular. When we say a contour has

radius r this implies that the contour is contained inside the circle with radius r centered

at the origin. We compute r by curve fitting the Evans function to the known asymptotic

behavior iteratively, terminating when the curve fitting has relative error less than a certain

tolerance (0.2 in our case). Note that the asymptotic behavior is given by C1e√C2λ with

constants C1 and C2. With this method for choosing r, it is possible that our choice of r is

16

far larger than need be to locate roots. However, without a formal proof of a tighter bound

on r, we must use this method to ensure accuracy in our results.

We compute the stability of the inviscid and viscous systems for 50,000 different com-

binations of (h1, u1+, φ) satisfying h1 > 1, u1+ ∈ (0, 1), and φ > 0 where φ is a constant

indicating the viscosity level of the system. (In particular, we set ν = µ = φ.)

More specifically, we use STABLAB to compute the viscous and inviscid stability for

every combination (h1, u1+, φ) where h1 ranges from 1 to 4 incremented by 0.005, u1+ ranges

from 0 to 1 incremented by 0.005, and φ ranges from 0.01 to 0.2 incremented by 0.01. Some

of the results for φ = 0.1 are shown in Figure 3.2.

Figure 3.2: Stability results comparing the inviscid and viscous models as h1 and u1+ vary.

In Figure 3.2, the black line shows where the bifurcation occurs. Through all of our

experiments, we found that this line was exactly the same for the inviscid and viscous

systems for every choice of φ. Based on this result, it is likely that in 2-D MHD, viscous

instability does imply inviscid instability, though still not certainly.

3.3 Convexity of Bifurcation Curve

In the previous section we computed the winding number corresponding to the Evans function

for 50 different values of the Fourier coefficient ξ, ranging from ξ = 1e − 4 to ξ = 0.1, in

order to determine stability of a system characterized by a fixed h1, u1+, and φ. This was

17

necessary because the Evans function is not linear in ξ and thus we must do an exhaustive

search for the ξ value for which the Evans function has a root with positive real part. If we

do not find a root, we suspect the system is stable. As a result, in the previous section, it is

possible that we did not exhaust enough possible ξ values to definitely determine stability.

Note that the roots of the Lopatinski determinant are linear in the Fourier coefficient ξ

and thus we only need to preform computations for a single ξ value to determine stability

or instability, see [6]. In the previous section, we used ξ = 1.0.

Another way to study the relationship between inviscid and viscous stability, is to analyze

the roots of the system as parameterized by the Fourier coefficient ξ. Figure 3.3 gives an

example of what these parameterized curves of roots might look like. We know that initially,

the curve corresponding to the viscous roots will be tangent to that of the inviscid roots, as

is evident in all three figures; see [24].

(a) (b) (c)

Figure 3.3: Diagrams indicating possible relationships between roots of the inviscid systemand roots of the viscous system as parameterized by the Fourier coefficient ξ.

The situation depicted in Figure 3.3 (a) is what we typically expect to happen. In this

case, we see that as ξ increases, the roots of the viscous system stay negative. If we were

to consider a slightly different system where the line corresponding to the inviscid system

was located in the first quadrant, then we would see the roots of the viscous system starting

positive along with the inviscid, and then becoming negative for larger values of ξ.

18

The remaining two plots, (b) and (c) of Figure 3.3, display scenarios in which we might

see viscous instability and corresponding inviscid stability. In both cases, there are values

of ξ for which the viscous system has positive real roots while the inviscid system has only

negative real roots for all ξ. Another alternative is that we have a curve similar to the viscous

case in Figure 3.3 (c), but with the line curving back to the left half plane after crossing the

y-axis.

We seek to determine which of these cases is occurring in 2-D MHD. We do this by

calculating the roots of the Evans function for a fixed u1+ and φ, letting h1 and ξ vary. We

then study the convexity of the parameterized roots. The results are shown in Figure 3.4.

Figure 3.4: The real part of the roots of the viscous system as parameterized by h1 and theFourier coefficient ξ.

From Figure 3.4, it appears that Re(λ) varies nonlinearly in ξ and h1 though it is difficult

to see the exact relation. We can plot the cross sections of Figure 3.4 for fixed h1 and for

fixed ξ to better understand what is happening.

Figure 3.5 shows three different cross sections of Figure 3.4 for fixed values of ξ. By

plotting these cross sections we can better understand the relationship between h1 and

Re(λ). The dashed line plotted alongside the bifurcation curves serves to help determine

19

the convexity of the curves. In this case, we see that there is a quadratic relationship between

these two quantities. It is also relevant to note that the bifurcation curves are convex.

Figure 3.5: The real parts of the roots of the viscous system as parameterized by h1 for fixedvalues of ξ. The dotted line serves as a reference to highlight the convexity of the curves.

Similarly, Figure 3.6 shows three different cross sections of Figure 3.4 but this time for

fixed values of h1. This allows us to more closely examine the relationship between ξ and

Re(λ). We hope to use these results to identify which of the three cases from Figure 3.3 is

occurring in this system.

Figure 3.6: The real parts of the roots of the viscous system as parameterized by ξ for fixedvalues of h1.

As we can see, for each fixed h1, the bifurcation curve is convex and matches the far

left plot from Figure 3.3 This strengthens the possibility that viscous instability does in fact

imply inviscid instability.

20

It is also interesting to notice the linear black line plotted in Figure 3.4. This line

corresponds to where the system bifurcates and the roots transition to having nonzero real

part. The linearity of this line is surprising and suggests that there is an underlying linearity

when Re(λ) is parameterized by h1 and ξ simultaneously.

Chapter 4. Navier-Stokes

As mentioned, MHD waves consist of the Navier-Stokes equations and Maxwell’s equations

coupled together to describe the behavior of conducting fluids. As a result, analyzing the

Navier-Stokes equations can provide meaningful insight into the structure and behavior of

MHD waves. It should also be noted that there is significant relevance to the study of

Navier-Stokes outside of its applications to MHD waves.

In the remainder of this thesis, we describe the development of analytical and computa-

tional tools for determining the stability of traveling waves in the equations of compressible

Navier-Stokes. We do so because carrying out analysis of the compressible Navier-Stokes

equations is a first step to developing analysis for MHD. We now formally introduce the

system.

4.1 Compressible Navier-Stokes

The Navier-Stokes equations for a compressible gas in Lagrangian coordinates are given by,

vt − ux = 0,

ut + (p(v, T ))x =(µuxv

)x,

Et + (up(v, T ))x =(µuux

v

)x

+

(κTxv

)x

,

where E = e + u2/2 is the kinetic energy, e(v, T ) is the internal energy, u is velocity, v

is specific volume, p(v, T ) is the pressure law, and µ and κ are respectively coefficients of

21

viscosity and heat conductivity. The temperature, T , is proportional to internal energy,

e = cνT .

We are interested in analyzing the stability of viscous shock profiles v(x, t) = v(x− st),

u(x, t) = u(x− st), e(x, t) = e(x− st) of (4.1), which correspond to traveling waves moving

with constant speed s. Equivalently, the shock profiles are the stationary solutions of

vt − cvx − ux = 0,

ut − cux + (p(v, T ))x =(µuxv

)x, (4.1)

Et − cEx + (up(v, T ))x =(µuux

v

)x

+

(κTxv

)x

.

We use the scaling of [13], that is, we rescale by

(x, t, v, u, T )→ (−εcx, εc2t, v/ε,−u/(εc), T/(ε2c2))

and choose ε so that v− := limx→−∞ v(x) = 1, which yields

vt + vx − ux = 0

ut + ux + (p(v, T ))x =(µuxv

)x

Et + Ex + (up(v, T ))x =(µuux

v

)x

+

(κTxv

)x

,

where the new energy and pressure functions e and p relate to the old ones e0 and p0 by

e(v, T ) = e0(εv, ε2c2T )/(ε2c2), p(v, T ) = p0(εv, ε2c2T )/(εc2). (4.2)

Under this rescaling, the pressure and energy functions are unchanged up to a rescaling of

parameters for an ideal gas equation of state,

e(v, T ) = cνT, p(v, T ) =RT

v, R = gas constant. (4.3)

22

We solve for the profile system following the same procedure as described in Section 2.2.

The resulting profile solution satisfies the system,

v′ =1

µ[v(v − 1) + Γ(e− νe−)] , (4.4a)

e′ =v

ν

[−(v − 1)2

2+ e− e− + (v − 1)Γe−

]. (4.4b)

We denote the solutions to the profile system by u, v, and e. Then we linearize (4.1)

about the profile solutions to obtain,

λv + v′ − u′ = 0, (4.5a)

λu+ u′ +Γ

ve′ +

Γuxvu+

u

vv′ =

1

vu′′, (4.5b)

λe+ e′ +

[ux −

νuxxv

]u+

[Γe

v− (ν + 1)

uxv

]u′ +

[νexv2

]v′ =

ν

ve′′. (4.5c)

4.2 Energy Estimates

As mentioned, to determine stability, we evaluate the Lopatinski determinant and Evans

function over a contour containing the possible unstable spectra. Methods have been devel-

oped which can be used to bound the contour associated with the system, which reduces the

amount of required computation. Two of these methods are tracking estimates and energy

estimates.

Tracking estimates are based on the idea of analytically bounding the solutions to the

system (4.5) in the large λ limit. This method uses the requirement that any potential

eigenfunction must be tangent to the stable and unstable manifolds as x → ±∞. Using

this fact, one can bound with a cone, the solutions leaving each end state tangent to the

respective stable and unstable manifold. We then evolve these cones from each end state and

require that they intersect, meaning that an evolved solution could potentially live in both

cones. As we increase λ, eventually the cones will no longer intersect, giving an effective

bound on the admissible eigenvalues of the system.

23

Energy estimates, on the other hand, are a method based on the assumption that λ

is an eigenvalue of the system with the corresponding eigenfunction given by the vector

of functions (e.g. [v u e]T for compressible Navier-Stokes). Energy estimates get their

name because they consider the overall energy of the system by taking its integral and then

carefully manipulating the remaining equations to produce a constraint on λ.

For the compressible Navier-Stokes system in the strong shock limit, tracking estimates

where developed in [13], providing an initial bound on the contour, given in Table 4.1.

Γν

0.2 0.5 1.0 2.0 5.0

0.2 398.6 388.8 385.3 733.8 1755.6

0.4 211.7 182.3 175.1 325.0 762.0

0.6 222.5 123.4 111.5 198.4 449.8

0.667 226.8 114.3 100.4 175.3 391.3

0.8 236.5 103.9 85.3 142.6 307.1

1.0 253.8 100.8 73.7 113.8 229.2

1.2 274.3 106.6 69.7 98.7 183.1

1.4 300.8 117.5 70.5 91.6 154.9

1.6 347.4 131.8 74.5 89.8 138.0

1.8 397.3 148.7 80.8 91.8 128.6

2.0 450.3 167.5 88.7 96.7 124.7

Table 4.1: Bounds on Re(λ) + |Im(λ)|, where λ is an admissible eigenvalue, for variousvalues of Γ and ν, as given in [13].

In the following sections, we work toward improving this bound on both the real and

complex parts of λ for compressible Navier-Stokes in Lagrangian coordinates by implement-

ing energy estimates. We implement energy estimates to improve this bound on both the

real and complex parts of λ for compressible Navier-Stokes in Lagrangian coordinates. We

24

use the linearized eigenvalue system for compressible Navier-Stokes as derived in [13] and

given in (4.5). For simplicity, we write (4.5c) as

λe+ e′ + c1u+ c2u′ + c3v

′ =ν

ve′′.

4.3 Bounding Real Valued λ

We will first assume that λ ∈ R.

Lemma 4.1. For v and u satisfying (4.5) with λ ∈ R,

∫|v′|2 ≤

∫|u′|2.

Proof. First multiply (4.5a) by v′ and integrate to obtain

λ

∫vv′ +

∫v′v′ =

∫u′v′. (4.6)

Now take the real part of (4.6) and apply Young’s inequality to the right-hand side, which

yields

∫|v′|2 = Re

(∫u′v′)≤∣∣∣∣∫ u′v′

∣∣∣∣ ≤ ∫ |u′||v′|≤ 1

2

∫|u′|2 +

1

2

∫|v′|2.

Subtracting 12

∫|v′|2 from both sides and multiplying by 2 produces the desired inequality.

This lemma becomes essential in proving the following bound on real-valued λ.

Theorem 4.2. If λ ∈ R and (u, v, e) are an eigenvalue-eigenfunction solution to (4.5), then

λ ≤ max(βu, βe),

25

where

βu =

∣∣∣∣∣∣∣∣α1|c1|+1

2

(1

v

)′′− Γux

v+

Γ

4θ1v− u

4θ2v

∣∣∣∣∣∣∣∣∞,

βe =

∣∣∣∣∣∣∣∣ν2(

1

v

)′′+|c1|4α1

+|c2|4α2

+|c3|4α3

∣∣∣∣∣∣∣∣∞,

and α1, α2, α3, θ1, and θ2 are real-valued, positive constants that satisfy

∫θ1Γ

v|e′|2 ≤

∫ν

v|e′|2 and

∫ [α2|c2|+ α3|c3| −

θ2u

v

]|u′|2 ≤

∫1

v|u′|2.

Proof. The proof can be broken up into three main steps: (1) bounding λ∫|u|2, (2) bounding

λ∫|e|2, and (3) combining the resulting two inequalities.

(1) Start by multiplying (4.5b) by u and integrating to get

λ

∫|u|2 +

∫u′u+

∫Γ

ve′u+

∫Γuxv|u|2 +

∫u

vv′u =

∫1

vu′′u. (4.7)

Take the real part of (4.7)

λ

∫|u|2 +

∫Γuxv|u|2 +Re

(∫Γ

ve′u+

∫u

vv′u

)=

1

2

∫ (1

v

)′′|u|2 −

∫1

v|u′|2,

and apply Young’s inequality twice, using Lemma 4.1 and the fact that −Re(z) ≤ |z|,

λ

∫|u|2 +

∫1

v|u′|2 =

1

2

∫ (1

v

)′′|u|2 −

∫Γuxv|u|2 −Re

(∫Γ

ve′u+

∫u

vv′u

)λ

∫|u|2 +

∫1

v|u′|2 ≤ 1

2

∫ (1

v

)′′|u|2 −

∫Γuxv|u|2 +

∣∣∣∣∫ Γ

ve′u+

∫u

vv′u

∣∣∣∣≤ 1

2

∫ (1

v

)′′|u|2 −

∫Γuxv|u|2 +

∫Γ

v|e′||u|+

∫−uv|v′||u|

≤ 1

2

∫ (1

v

)′′|u|2 −

∫Γuxv|u|2 + θ1

∫Γ

v|e′|2 +

1

4θ1

∫Γ

v|u|2

+ θ2

∫−uv|v′|2 +

1

4θ2

∫−uv|u|2

≤ 1

2

∫ (1

v

)′′|u|2 −

∫Γuxv|u|2 + θ1

∫Γ

v|e′|2 +

1

4θ1

∫Γ

v|u|2

26

+ θ2

∫−uv|u′|2 +

1

4θ2

∫−uv|u|2

= θ2

∫−uv|u′|2 + θ1

∫Γ

v|e′|2 +

∫ [1

2

(1

v

)′′− Γux

v+

Γ

4θ1v− u

4θ2v

]|u|2.

(4.8)

(2) Now multiply (4.5c) by e then integrate and take the real part which yields

λ

∫|e|2 +Re

(∫c1ue+

∫c2u′e+

∫c3v′e

)=ν

2

∫ (1

v

)′′|e|2 −

∫ν

v|e′|2.

Apply Lemma 4.1 and Young’s inequality to get the following,

λ

∫|e|2 +

∫ν

v|e′|2 ≤ν

2

∫ (1

v

)′′|e|2 +

∣∣∣∣∫ c1ue+

∫c2u′e+

∫c3v′e

∣∣∣∣≤ν

2

∫ (1

v

)′′|e|2 +

∫|c1||u||e|+

∫|c2||u′||e|+

∫|c3||v′||e|

≤ν2

∫ (1

v

)′′|e|2 + α1

∫|c1||u|2 +

1

4α1

∫|c1||e|2 + α2

∫|c2||u′|2

+1

4α2

∫|c2||e|2 + α3

∫|c3||v′|2 +

1

4α3

∫|c3||e|2

≤∫

[α2|c2|+ α3|c3|] |u′|2 + α1

∫|c1||u|2

+

∫ [ν

2

(1

v

)′′+|c1|4α1

+|c2|4α2

+|c3|4α3

]|e|2. (4.9)

(3) Combine (4.8) and (4.9) to obtain the resulting inequality:

λ

∫ (|u|2 + |e|2

)+

∫1

v|u′|2 +

∫ν

v|e′|2 ≤

∫ [α2|c2|+ α3|c3| −

θ2u

v

]|u′|2 +

∫θ1Γ

v|e′|2

+

∫ [α1|c1|+

1

2

(1

v

)′′− Γux

v+

Γ

4θ1v− u

4θ2v

]|u|2

+

∫ [ν

2

(1

v

)′′+|c1|4α1

+|c2|4α2

+|c3|4α3

]|e|2. (4.10)

27

We need to specify θ1, θ2, α2, and α3 so that

∫θ1Γ

v|e′|2 ≤

∫ν

v|e′|2 and

∫ [α2|c2|+ α3|c3| −

θ2u

v

]|u′|2 ≤

∫1

v|u′|2.

We first set θ1 = ν/Γ to make the first inequality true. And notice

∫[α2|c2|v + α3|c3|v − θ2u]

1

v|u′|2 ≤ [α2||c2v||∞ + α3||c3v||∞ + θ2||u||∞]

∫1

v|u′|2.

So we choose

α2 =1

3||c2v||∞, α3 =

1

3||c3v||∞, and θ2 =

1

3||u||∞

to satisfy the second inequality. Now let

βu =

∣∣∣∣∣∣∣∣α1|c1|+1

2

(1

v

)′′− Γux

v+

Γ

4θ1v− u

4θ2v

∣∣∣∣∣∣∣∣∞

and

βe =

∣∣∣∣∣∣∣∣ν2(

1

v

)′′+|c1|4α1

+|c2|4α2

+|c3|4α3

∣∣∣∣∣∣∣∣∞.

Substituting our chosen parameter values into (4.10) yields

λ

∫ (|u|2 + |e|2

)≤ max(βu, βe)

∫ (|u|2 + |e|2

),

and thus

λ ≤ max(βu, βe).

The bound on λ for various values of ν and Γ are given in Table 4.2. A comparison

between these bounds and the original bounds developed for this system (Table 4.1) shows

that these bounds are significantly tighter. Though it should be noted that these bounds

apply to a real valued λ where the previous allow λ to be complex valued.

28

Γν

0.2 0.5 1.0 2.0 5.0

0.2 7.3653 7.0340 6.9262 6.8732 6.8402

0.4 4.3233 3.6037 3.3641 3.2449 3.1730

0.6 3.8725 2.7025 2.3126 2.1179 2.0011

0.667 3.9091 2.5759 2.1314 1.9097 1.7764

0.8 4.1387 2.4587 1.8990 1.6192 1.4513

1.0 4.7494 2.4994 1.7497 1.2748 1.1500

1.2 5.5806 2.7010 1.7409 1.2610 0.9732

1.4 6.5796 3.0100 1.8201 1.2251 0.8683

1.6 7.7201 3.4005 1.9606 1.2407 0.8088

1.8 8.9878 3.8582 2.1484 1.2935 0.7806

2.0 10.3741 4.3746 2.3748 1.3749 0.7750

Table 4.2: Numerically approximated bounds on real valued λ for various values of Γ and ν,computed based on Theorem 4.2.

4.4 Bounding Complex Valued λ

We will now get a similar bound on Re(λ) + |Im(λ)|. For simplification, throughout this

section we will define

Λ = Re(λ) + |Im(λ)|.

We start by establishing three necessary lemmas.

Lemma 4.3. If (v, u, e) is an eigenfunction of (4.5), then∫v′v,

∫u′u, and

∫e′e are purely

imaginary.

Proof. Let (v, u, e) be an eigenfunction of (4.5). Without loss of generality we will show that∫v′v is purely imaginary. This proof does not rely on any specific properties of v and thus

29

can also be applied to u and e. We start by noting that we have

∫v′v =

1

2

∫v′v +

1

2

∫v′v

=1

2

∫v′v −

∫vv′

=1

2

(∫v′v −

∫v′v

),

which implies that∫v′v is purely imaginary, as desired.

Lemma 4.4. If (v, u, e) is an eigenfunction of (4.5), then

Re[λ

∫uv′ +

∫u′v′]

=

∫|u′|2 − 2Re(λ)2

∫|v|2. (4.11)

Proof. Let (v, u, e) be an eigenfunction of (4.5). Then we have

λ

∫uv′ +

∫u′v′ = (λ+ λ)

∫uv −

∫u(λv′ + v′′)

= −2Re(λ)

∫(u′v −

∫uu′′

= −2Re(λ)

∫(λv + v′)v +

∫|u′|2.

Taking the real part we get that

Re[λ

∫uv′ +

∫u′v′]

=

∫|u′|2 − 2Re(λ)2

∫|v|2.

Lemma 4.5. If λ ∈ C and (u, v, e) are an eigenvalue-eigenfunction solution to (4.5) and

Re(λ) > δ for δ > 0, with δ, β1, and β2 satisfying δ − β1(Γ + 2)− 2β2 > 0, then

∫|v′|2 ≤ 1

ε1

[∫|u′|2 +

Γ + 2

4β1

∫|e′|2 +

Γ + 2

2Γβ2

∫v|u|2

]. (4.12)

30

Proof. First we multiply (4.5b) by v′ and integrate,

λ

∫uv′ +

∫u′v′ +

∫Γ

ve′v′ +

∫Γuxvuv′ +

∫u

v|v′|2 =

∫1

vu′′v′

=

∫1

v(λv′ + v′′)v′

= λ

∫1

v|v′|2 +

∫1

vv′′v′.

Now take the real part and apply (4.11),

∫|u′|2 − 2Re(λ)2

∫|v|2 +Re

[∫Γ

ve′v′ +

∫Γuxvuv′]

+

∫u

v|v′|2

= Re(λ)

∫1

v|v′|2 +

∫vx2v2|v′|2.

Rearranging yields

∫|u′|2 +Re

[∫Γ

ve′v′ +

∫Γuxvuv′]

=

∫ [Re(λ) +

vx2v− u]

1

v|v′|2 + 2Re(λ)2

∫|v|2.

Notice

vx2v− u =

1

2u+

Γv

v− u =

Γe

v− u

2≥ 0 and

Γ + 2

Γ≥ 1

v≥ 1,

and thus

∫ [Re(λ) +

vx2v− u]

1

v|v′|2 + 2Re(λ)2

∫|v|2 ≥ Re(λ)

∫1

v|v′|2 ≥ Re(λ)

∫|v′|2.

Putting this together and applying Young’s inequality we get

Re(λ)

∫|v′|2 ≤

∫|u′|2 +Re

[∫Γ

ve′v′ +

∫Γuxvuv′]

≤∫|u′|2 +

∣∣∣∣∫ Γ

ve′v′ +

∫Γuxvuv′∣∣∣∣

31

≤∫|u′|2 +

∣∣∣∣∫ Γ

ve′v′∣∣∣∣+

∣∣∣∣∫ Γuxvuv′∣∣∣∣

≤∫|u′|2 +

∫Γ

v|e′||v′|+

∫|Γuxv||u||v′|

≤∫|u′|2 + β1

∫Γ

v|v′|2 +

1

4β1

∫Γ

v|e′|2 + β2

∫|Γuxv||v′|2 +

1

4β2

∫|Γuxv||u|2

≤∫|u′|2 + β1(Γ + 2)

∫|v′|2 +

Γ + 2

4β1

∫|e′|2 + 2β2

∫|v′|2 +

1

2β2

∫|u|2

=

∫|u′|2 + [β1(Γ + 2) + 2β2]

∫|v′|2 +

Γ + 2

4β1

∫|e′|2 +

1

2β2

∫|u|2

=

∫|u′|2 + [β1(Γ + 2) + 2β2]

∫|v′|2 +

Γ + 2

4β1

∫|e′|2 +

1

2β2

∫1

vv|u|2

≤∫|u′|2 + [β1(Γ + 2) + 2β2]

∫|v′|2 +

Γ + 2

4β1

∫|e′|2 +

1

2β2

Γ + 2

Γ

∫v|u|2

=

∫|u′|2 + [β1(Γ + 2) + 2β2]

∫|v′|2 +

Γ + 2

4β1

∫|e′|2 +

Γ + 2

2Γβ2

∫v|u|2.

Now we will require that Re(λ) ≥ δ for some δ > 0 and we will let ε1 = δ−β1(Γ+2)−2β2

and require that β1 and β2 are chosen so that ε1 > 0. With these requirements we get that

∫|v′|2 ≤ 1

ε1

[∫|u′|2 +

Γ + 2

4β1

∫|e′|2 +

Γ + 2

2Γβ2

∫v|u|2

]

With (4.11) and Lemma 4.5, we are ready for the main theorem of this section.

Theorem 4.6. If λ ∈ C and (u, v, e) are an eigenvalue-eigenfunction solution to (4.5) and

Re(λ) > δ for δ > 0, then

Λ ≤ max(uval, eval), (4.13)

where

uval =

∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

+√

2α1‖c1‖∞ +(Γ + 2)ε2

2Γβ2ε1

,

eval =1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

,

32

and δ, β1, β2, θ1, θ2, θ3, α2, α3, and α4 are positive valued constants satisfying

δ > β1(Γ + 2) + 2β2, θ3 +2√

2α2(ν + 1)

Γ + 2+ε2

ε1

≤ 1, and√

2θ1Γ + α4 +(Γ + 2)ε2

4β1ε1

≤ ν.

Proof. This proof can be broken up into three main steps: (1) bounding Λ∫|u|2, (2) bound-

ing Λ∫|e|2, and (3) combining the four resulting inequalities.

(1) First multiply (4.5b) by vu and integrate, applying integration by parts to the right-

hand side which yields

λ

∫v|u|2 +

∫vu′u+

∫Γe′u+

∫Γux|u|2 +

∫uv′u =

∫u′′u = −

∫|u′|2. (4.14)

Now take just the real part of (4.14) to obtain

Re(λ)

∫v|u|2 − 1

2

∫vx|u|2 +Re

[∫Γe′u

]+

∫Γux|u|2 +Re

[∫uv′u

]= −

∫|u′|2,

and simplify, using the fact that ux = vx which yields

Re(λ)

∫v|u|2 +

∫|u′|2 =

(1

2− Γ

)∫vx|u|2 −Re

[∫Γe′u

]−Re

[∫uv′u

]=

(1

2− Γ

)∫vxvv|u|2 −Re

[∫Γe′u

]−Re

[∫uv′u

].

Taking the absolute value of each side produces

∣∣∣∣Re(λ)

∫v|u|2 +

∫|u′|2

∣∣∣∣ =

∣∣∣∣(1

2− Γ

)∫vxvv|u|2 −Re

[∫Γe′u

]−Re

[∫uv′u

]∣∣∣∣Re(λ)

∫v|u|2 +

∫|u′|2 ≤

∣∣∣∣12 − Γ

∣∣∣∣ ∫ ∣∣∣∣ vxv∣∣∣∣ v|u|2 +

∣∣∣∣Re [∫ Γe′u

]∣∣∣∣+

∣∣∣∣Re [∫ uv′u

]∣∣∣∣ .Now we take the imaginary part of (4.14) to obtain

Im(λ)

∫v|u|2 + Im

[∫vu′u

]+ Im

[∫Γe′u

]+ Im

[∫uv′u

]= 0.

33

Again, rearrange and take the absolute value of both sides, producing

|Im(λ)|∫v|u|2 =

∣∣∣∣Im [∫ vu′u

]+ Im

[∫Γe′u

]+ Im

[∫uv′u

]∣∣∣∣≤∣∣∣∣Im [∫ vu′u

]∣∣∣∣+

∣∣∣∣Im [∫ Γe′u

]∣∣∣∣+

∣∣∣∣Im [∫ uv′u

]∣∣∣∣ .Now we can combine the two resulting inequalities and use the fact that for z ∈ C, |Re(z)|+

|Im(z)| ≤√

2|z|, which yields

Λ

∫v|u|2 +

∫|u′|2 ≤

∣∣∣∣12 − Γ

∣∣∣∣ ∫ ∣∣∣∣ vxv∣∣∣∣ v|u|2 +

√2

∣∣∣∣∫ Γe′u

∣∣∣∣+√

2

∣∣∣∣∫ uv′u

∣∣∣∣+

∣∣∣∣∫ vu′u

∣∣∣∣≤∣∣∣∣12 − Γ

∣∣∣∣ ∫ ∣∣∣∣ vxv∣∣∣∣ v|u|2 +

√2

∫Γ|e′||u|+

√2

∫|u||v′||u|+

∫v|u′||u|

Applying Young’s inequality three times gives us

Λ

∫v|u|2 +

∫|u′|2 ≤

∣∣∣∣12 − Γ

∣∣∣∣ ∫ ∣∣∣∣ vxv∣∣∣∣ v|u|2 +

√2θ1

∫Γ|e′|2 +

√2

4θ1

∫Γ|u|2

+√

2θ2

∫|u||v′|2 +

√2

4θ2

∫|u||u|2 + θ3

∫v|u′|2 +

1

4θ3

∫v|u|2

=

∣∣∣∣12 − Γ

∣∣∣∣ ∫ ∣∣∣∣ vxv∣∣∣∣ v|u|2 +

√2θ1

∫Γ|e′|2 +

√2

4θ1

∫Γ

vv|u|2

+√

2θ2

∫|u||v′|2 +

√2

4θ2

∫|u|vv|u|2 + θ3

∫v|u′|2 +

1

4θ3

∫v|u|2

≤∣∣∣∣12 − Γ

∣∣∣∣ 2

Γ + 2

∫v|u|2 +

√2θ1Γ

∫|e′|2 +

√2Γ

4θ1

Γ + 2

Γ

∫v|u|2

+√

2θ22

Γ + 2

∫|v′|2 +

√2

4θ2

2

Γ

∫v|u|2 + θ3

∫|u′|2 +

1

4θ3

∫v|u|2

=

[∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

]∫v|u|2

+√

2θ1Γ

∫|e′|2 +

2√

2θ2

Γ + 2

∫|v′|2 + θ3

∫|u′|2.

34

(2) Now we multiply (4.5c) by ve and integrate, applying integration by parts to the

right-hand side as before, to obtain

λ

∫v|e|2 +

∫ve′e+

∫c1vue+

∫c2vu

′e+

∫c3vv

′e =

∫νe′′e = −

∫ν|e′|2. (4.15)

Now we will take just the real part of (4.15), which yields

Re(λ)

∫v|e|2− 1

2

∫vx|e|2 +Re

[∫c1vue

]+Re

[∫c2vu

′e

]+Re

[∫c3vv

′e

]= −

∫ν|e′|2,

which we can rearrange and take the absolute value, resulting in

Re(λ)

∫v|e|2 +

∫ν|e′|2 =

1

2

∫vx|e|2 −Re

[∫c1vue

]−Re

[∫c2vu

′e

]−Re

[∫c3vv

′e

]Re(λ)

∫v|e|2 +

∫ν|e′|2 =

∣∣∣∣12∫vx|e|2 −Re

[∫c1vue

]−Re

[∫c2vu

′e

]−Re

[∫c3vv

′e

]∣∣∣∣≤1

2

∫|vx||e|2 +

∣∣∣∣Re [∫ c1vue

]∣∣∣∣+

∣∣∣∣Re [∫ c2vu′e

]∣∣∣∣+

∣∣∣∣Re [∫ c3vv′e

]∣∣∣∣ .

Take the imaginary part of (4.15) provides

Im(λ)

∫v|e|2 + Im

[∫ve′e

]+ Im

[∫c1vue

]+ Im

[∫c2vu

′e

]+ Im

[∫c3vv

′e

]= 0,

which again we can rearrange and take the absolute value of both sides to get

|Im(λ)|∫v|e|2 =

∣∣∣∣Im [∫ ve′e

]+ Im

[∫c1vue

]+ Im

[∫c2vu

′e

]+ Im

[∫c3vv

′e

]∣∣∣∣≤∣∣∣∣Im [∫ ve′e

]∣∣∣∣+

∣∣∣∣Im [∫ c1vue

]∣∣∣∣+

∣∣∣∣Im [∫ c2vu′e

]∣∣∣∣+

∣∣∣∣Im [∫ c3vv′e

]∣∣∣∣ .

35

Now we add the inequalities corresponding to the real and imaginary parts of (4.15) to

obtain

Λ

∫v|e|2 +

∫ν|e′|2 ≤1

2

∫|vx||e|2 +

√2

∣∣∣∣∫ c1vue

∣∣∣∣+√

2

∣∣∣∣∫ c2vu′e

∣∣∣∣+√

2

∣∣∣∣∫ c3vv′e

∣∣∣∣+

∣∣∣∣∫ ve′e

∣∣∣∣≤1

2

∫|vx|vv|e|2 +

√2

∫|c1|v|u||e|+

√2

∫|c2|v|u′||e|

+√

2

∫|c3|v|v′||e|+

∫v|e′||e|.

Notice

c2v = Γe− (ν + 1)ux = −(ν + 1)uv − νΓe ≤ −(ν + 1)uv ≤ 2(ν + 1)

Γ + 2,

c2 ≤2(ν + 1)

Γ + 2

1

v≤ 2(ν + 1)

Γ,

c3v =νexv≤ ν

2

ν(Γ + 2)2

Γ + 2

Γ=

2

Γ(Γ + 2),

c3 ≤2

Γ(Γ + 2)

1

v≤ 2

Γ(Γ + 2)

Γ + 2

Γ=

2

Γ2.

Now apply the previous bounds on c2 and c3 and Young’s inequality four times,

Λ

∫v|e|2 +

∫ν|e′|2 ≤1

2

∫|vx|vv|e|2 +

√2α1

∫|c1|v|u|2 +

√2

4α1

∫|c1|v|e|2

+√

2α2

∫|c2|v|u′|2 +

√2

4α2

∫|c2|v|e|2 +

√2α3

∫|c3|v|v′|2

+

√2

4α3

∫|c3|v|e|2 + α4

∫v|e′|2 +

1

4α4

∫v|e|2

≤1

2

2

Γ + 2

∫v|e|2 +

√2α1‖c1‖∞

∫v|u|2 +

√2‖c1‖∞4α1

∫v|e|2

+√

2α22(ν + 1)

Γ + 2

∫|u′|2 +

√2

4α2

2(ν + 1)

Γ

∫v|e|2

+√

2α32

Γ(Γ + 2)

∫|v′|2 +

√2

4α3

2

Γ2

∫v|e|2 + α4

∫|e′|2

+1

4α4

∫v|e|2

=1

Γ + 2

∫v|e|2 +

√2α1‖c1‖∞

∫v|u|2 +

√2‖c1‖∞4α1

∫v|e|2

36

+2√

2α2(ν + 1)

Γ + 2

∫|u′|2 +

√2(ν + 1)

2α2Γ

∫v|e|2

+2√

2α3

Γ(Γ + 2)

∫|v′|2 +

√2

2α3Γ2

∫v|e|2 + α4

∫|e′|2

+1

4α4

∫v|e|2

=

[1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

]∫v|e|2

+√

2α1‖c1‖∞∫v|u|2 +

2√

2α2(ν + 1)

Γ + 2

∫|u′|2

+2√

2α3

Γ(Γ + 2)

∫|v′|2 + α4

∫|e′|2.

(3) From (1) and (2) we have

Λ

∫v|u|2 +

∫|u′|2 ≤

[∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

]∫v|u|2

+√

2θ1Γ

∫|e′|2 +

2√

2θ2

Γ + 2

∫|v′|2 + θ3

∫|u′|2

and

Λ

∫v|e|2 +

∫ν|e′|2 ≤

[1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

]∫v|e|2

+√

2α1‖c1‖∞∫v|u|2 +

2√

2α2(ν + 1)

Γ + 2

∫|u′|2

+2√

2α3

Γ(Γ + 2)

∫|v′|2 + α4

∫|e′|2.

Summing these two inequalities yields

Λ

∫v(|u|2 + |e|2) +

∫|u′|2 +

∫ν|e′|2

≤

[∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

+√

2α1‖c1‖∞

]∫v|u|2

+

[1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

]∫v|e|2

37

+

[θ3 +

2√

2α2(ν + 1)

Γ + 2

] ∫|u′|2 +

[√2θ1Γ + α4

] ∫|e′|2

+

[2√

2θ2

Γ + 2+

2√

2α3

Γ(Γ + 2)

]∫|v′|2.

Finally, set ε2 = 2√

2θ2Γ+2

+ 2√

2α3

Γ(Γ+2)and apply (4.12) to obtain

Λ

∫v(|u|2 + |e|2) +

∫|u′|2 +

∫ν|e′|2

≤

[∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

+√

2α1‖c1‖∞

]∫v|u|2

+

[1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

]∫v|e|2

+

[θ3 +

2√

2α2(ν + 1)

Γ + 2

] ∫|u′|2 +

[√2θ1Γ + α4

] ∫|e′|2

+ε2

ε1

[∫|u′|2 +

Γ + 2

4β1

∫|e′|2 +

Γ + 2

2Γβ2

∫v|u|2

]=

[∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

+√

2α1‖c1‖∞ +(Γ + 2)ε2

2Γβ2ε1

]∫v|u|2

+

[1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

]∫v|e|2

+

[θ3 +

2√

2α2(ν + 1)

Γ + 2+ε2

ε1

] ∫|u′|2 +

[√2θ1Γ + α4 +

(Γ + 2)ε2

4β1ε1

] ∫|e′|2.

(4.16)

We assume that δ, β1, β2, θ1, θ2, θ3, α2, α3, and α4 are chosen to be positive constants

satisfying

δ > β1(Γ + 2) + 2β2, θ3 +2√

2α2(ν + 1)

Γ + 2+ε2

ε1

≤ 1, and√

2θ1Γ + α4 +(Γ + 2)ε2

4β1ε1

≤ ν.

38

A method for sufficiently defining these parameters is given in Algorithm 1 on page 40. With

correctly chosen parameter values, we define

uval =

∣∣∣∣1− 2Γ

Γ + 2

∣∣∣∣+

√2(Γ + 2)

4θ1

+2√

2

4Γθ2

+1

4θ3

+√

2α1‖c1‖∞ +(Γ + 2)ε2

2Γβ2ε1

and

eval =1

Γ + 2+

√2‖c1‖∞4α1

+

√2(ν + 1)

2α2Γ+

√2

2α3Γ2+

1

4α4

.

This reduces (4.16) to

Λ

∫v(|u|2 + |e|2) ≤ max(uval, eval)

∫v(|u|2 + |e|2),

or equivalently

Λ ≤ max(uval, eval).

To find somewhat optimal values for the parameters of Young’s inequality, we use a

Monte Carlo type method given in Algorithm 1. Notice, we have chosen δ = 5.7. With

this choice of δ it remains to find a bound for |Im(λ)| when Re(λ) < δ. This value of δ

was chosen based on experimental results in an attempt to minimize δ and the bound on λ

simultaneously. The algorithm randomly selects 1000 sets of parameter values that satisfy

the necessary inequalities and saves the set that results in the best bound. The computed

bounds for various choices of Γ and ν are given in Table 4.3.

39

Algorithm 1 Monte Carlo Method for Determining bound on Λ

δ ← 5.7λmin ←∞for i ≤ 1000 doβ1 ← rand(0, δ

Γ+2)

β2 ← rand(0, 12(δ − β1(Γ + 2)))

ε1 ← δ − β1(Γ + 2)− 2β2

m← min(ε1,4νβ1ε1

Γ+2)

ε2 ← rand(0,m)

θ2 ← rand(0, (Γ+2)ε22√

2)

α3 = Γ(Γ+2)

2√

2(ε2 − 2

√2θ2

Γ+2)

θ3 ← rand(0, 1− ε2ε1

)

α2 = Γ+22√

2(ν+1)(1− ε2

ε1− θ3)

α4 ← rand(0, ν − (Γ+2)ε24β1ε1

)

θ1 = 1√2Γ

(ν − (Γ+2)ε24β1ε1

− α4)

α1 ← rand(0, 1)

uval ← |1−2ΓΓ+2|+

√2(Γ+2)4θ1

+ 2√

24∗Γθ2 + 1

4θ3+√

2α1||c1||∞ + (Γ+2)ε22Γβ2ε1

eval = 1Γ+2

+√

2||c1||∞4α1

+ sqrt2(ν+1)2α2Γ

+√

22α3Γ2 + 1

4α4

bound← max(uval, eval)if bound < λmin thenλmin ← bound

end ifend for

40

Γν

0.2 0.5 1.0 2.0 5.0

0.2 151.0930 100.6399 104.3654 139.9998 336.7644

0.4 39.7031 26.8213 27.8682 40.8144 117.9493

0.6 26.6082 15.3777 14.8138 22.3819 64.9350

0.667 25.5299 13.4558 12.9988 18.6730 55.2181

0.8 22.8556 11.5034 10.0113 14.3735 42.6807

1.0 22.0219 10.6719 8.7641 10.7626 31.3285

1.2 23.4886 10.7742 7.6080 8.1634 23.4666

1.4 24.9883 11.7047 7.4325 7.3801 18.6681

1.6 26.9214 11.9224 7.5844 6.3488 15.7451

1.8 30.5783 13.1597 7.9733 6.2060 12.8241

2.0 34.8105 15.1891 8.5906 5.8791 11.3767

Table 4.3: Approximated bounds on Re(λ) + |Im(λ)| for all λ with Re(λ) > 5.7. Thesebounds were computed using Algorithm 1 which is based on Theorem 4.6.

Chapter 5. Analytic Stability

In this section we introduce work on a method to analytically show the stability of traveling

wave solutions using the Evans function. We first motivate this method with a rudimentary

example, then describe the method and its results for compressible Navier-Stokes in the

strong shock limit given in Lagrangian coordinates. We end by applying the method to 2-D

MHD.

41

5.1 Motivating Example

In order to introduce they key ideas for our method, we start with a simple toy problem that

displays the potential success of this method.

Consider an Evans ODE system given by

W ′(x;λ) = A(x;λ)W (x;λ),

where ′ = d/dx, and

A(x;λ) =

3v λ 1

0 −v λ

0 0 −1− v

.

We consider v to be the traveling wave solution for our example system defined by v(x) =

tanh(x) and we define v± = limx→±∞ v(x). We make the coordinate change W (x;λ) =

e−µxV (x;λ) where µ is an eigenvalue of A± := limx→±∞A(x;λ). The resulting system is

given by

V ′(x;λ) = (A(x;λ)− µI)V (x;λ).

We make another coordinate change by defining σ = v − v+ = v − 1. This gives rise to the

new system

(−2σ − σ2)dZ(σ)

dσ= B(σ;λ)Z(σ),

where

B(σ;λ) =

3 + 3σ − µ λ 1

0 −σ − 1− µ λ

0 0 −2− σ − µ

.

We seek to solve this new system by finding a series solution for Z(σ). This is done by

letting Z(σ) =∑∞

n=0Bnσn and then solving the system to find each coefficient vector Bn.

42

After solving this system we get that for a given eigenpair (α, vα), of the matrix A+,

B0 = vα,

B1 =

5− α λ 1

0 1− α λ

0 0 −α

−1−3 0 0

0 1 0

0 0 1

B0,

Bn =

3− α + 2n λ 1

0 2n− 1− α λ

0 0 2n− 2− α

−1−2− n 0 0

0 2− n 0

0 0 2− n

Bn−1,

for n ≥ 2. It is important to mention that we can choose any eigenvector of A+ to define

B0.

To check the relevance of our solution, we compare the norms of each successive coefficient

vector Bn. Figure 5.1 shows the trend of these norms for various values of λ, with the second

most negative eigenvalue of A+ being chosen for α.

Figure 5.1: The convergence of the norms of the coefficient vectors corresponding to theexample system.

As seen in the graph, in all cases the norms converge exponentially as n increases. This

43

is a very positive result because it gives reason to believe that our solution will not blow up

when |σ| ≥ 1.

5.2 Analytic Profile Solution

Notice that in the motivating example, the entire profile system (v(x), e(x)) was expressed

in terms of our new coordinate σ. In order to analytically compute the Evans function for

compressible Navier-Stokes, we will similarly need the profile solutions to be expressed as

functions of σ.

Recall that the profile for the ideal gas case of compressible Navier-Stokes is equivalent

to the profile for MHD (as mentioned in Section 2.2), though it should be noted that the

two systems have different parameters. Additionally, we multiply the profile system by v to

transform to Lagrangian coordinates and further simplify by substituting e− = 0. With this,

our profile system is given by

v′ =1

µ[v(v − 1) + Γe] (5.1a)

e′ =v

ν

[−(v − 1)2

2+ e

], (5.1b)

where v and e are both functions of x. The Jacobian of this system is given by

J =

1µ(2v − 1) Γ

1ν

(e− (v−1)2

2

)− v

ν(v − 1) v

ν

.

This system has two fixed points corresponding to x going to negative and positive infinity.

These are given by (v−, e−) = (1, 0) and (v+, e+) =(

ΓΓ+2

, 2(Γ+2)2

).

Observe the stability of the solutions around the fixed point (v−, e−), found using the

Jacobian,

J(v−, e−) =

1µ

Γ

0 1ν

.

44

This system has eigenvalues λ = 1/µ and λ = 1/ν from which we know this fixed point

is unstable because µ, ν > 0. As a result, we will use the fixed point (v+, e+) to solve the

system.

We seek to find a series solution by making the coordinate change given by

σ = v(x)− v+.

We choose this to be our coordinate change because σ = 0 initializes our solutions at the

stable fixed point. We hope that by starting at this fixed point, we will be able to evolve the

solution in the stable eigendirection across our entire domain to the unstable fixed point.

Notice that with this coordinate change we have

∂e

∂x=∂e

∂σ

∂σ

∂x,

where

∂σ

∂x=∂v

∂x.

This gives us the following

v

ν

[−(v − 1)2

2+ e

]=∂e

∂σ

1

µ[v(v − 1) + Γe] . (5.2)

We can now rewrite v in terms of σ as

v = σ + v+ (5.3)

and we can write e as a series expansion given by

e =∞∑n=0

anσn. (5.4)

45

Substituting (5.3) and (5.4) into (5.2), results in

(σ + v+)ν

[−(σ + v+ − 1)2

2+∞∑n=0

anσn

]=

∞∑n=1

nanσn−1 1

µ

[(σ + v+)(σ + v+ − 1) + Γ

∞∑n=0

anσn

]. (5.5)

We solve (5.5) to find the values of the coefficients an,

a0 = e+

a1 =1

2A(−B −

√B2 − 4AC)

a2 =a1µ− a1ν − 3

2µv+ + µ

3Γa1ν − µv+ + 4νv+ − 2ν(5.6)

a3 =−2Γa2

2ν + a2µ− 2a2ν − 12µ

4Γa1ν − µv+ + 6νv+ − 3ν

an =(µ− (n− 1)ν)an−1 − νΓ

(∑n−1r=2 raran−r+1

)nν(Γ−2)−µΓ

Γ+2+ (n+ 1)νΓa1

,

where A = νΓ, B = 2νv+ − ν − µv+, and C = −e+µ+ 32µv2

+ − 2µv+ + 12µ.

This gives us a series solution for e(σ). We check our accuracy by comparing this plot to

the numerical solution found using STABLAB. The results are shown below.

Figure 5.2: The series solution of e(σ) with 100 terms plotted against σ.

46

Figure 5.3: The series solution of e(σ) with five terms plotted against σ.

Notice that even with just five terms in the series, this approximation does very well

against the numerical solution. We can verify that the solution converges by creating a plot

of the series coefficients analogous to Figure 5.1. Figure 5.4 shows the trend in coefficients for

the analytic profile solution. Unlike the example system, these coefficients do not converge

to zero. Fortunately σ ∈ [0, 1) on our domain and the coefficients grow slower than σn for

all σ ∈ [0, 1). As a result, our series solution does not diverge and evolves to the unstable

fixed point.

Figure 5.4: The absolute values of the coefficients |an| of e(σ).

.

It is important to address why this coordinate change does not affect the behavior of the

profile system. This is justified in the following theorem.

47

Theorem 5.1. [11] Let v(x) and e(x) be the solutions to the profile system (2.7). The

function v(x) is monotone in x.

Proof. Recall our profile system (5.1) is given by

v′ =1

µ[v(v − 1) + Γe]

e′ =v

ν

[−(v − 1)2

2+ e

],

with corresponding nullclines e = −v(v−1)Γ

(corresponding to v′ = 0) and e = 12(v − 1)2

(corresponding to e′ = 0). We have already found that these nullclines intersect at the fixed

points (v+, e+) =(

ΓΓ+2

, 2(Γ+2)2

)and (v−, e−) = (1, 0). Now to show that v(x) is monotone,

it suffices to show that the solution v(x) remains strictly below the nullcline given by v′ = 0

from x = −∞ to x = +∞. This can be done by showing that the upper nullcline creates

a trapping region for solutions which implies that solutions that start below the nullcline

cannot cross the nullcline; see [11] for details.

Figure 5.5: Nullclines for the profile system (5.1) corresponding to v′ = 0 and e′ = 0, whichintersect at the fixed points of the system.

To do this, we will look at e′ along the line e = −v(v−1)Γ

. So we have

e′ =v

ν

(−(v − 1)2

2− v(v − 1)

Γ

).

48

Or equivalently

e′ =−v(v − 1)

ν

(v − 1

2+v

Γ

).

Notice that for v ∈ (v+, v−) we have v < 1 and v > ΓΓ+2

. So

v <Γ

Γ + 2

=⇒ Γv + 2v < Γ

=⇒ Γv + 2v − Γ < 0

=⇒ Γv + 2v − Γ

2Γ< 0

=⇒ Γ(v − 1) + 2v

2Γ< 0

=⇒ v − 1

2+v

Γ< 0

Thus v > 0, v − 1 < 0 and v−12

+ vΓ< 0. Which implies that along this curve, e′ < 0 as

desired. Thus our solution lies strictly below this nullcline and thus v′ < 0 along our solution

which implies that v(x) is monotone decreasing.

5.3 Analytic Evans System

We can express (4.5) as the first order system

W ′(x;λ) = A(x;λ)W (x;λ), (5.7)

with

A(x;λ) =

0 1 0 0 0

λν−1v ν−1v ν−1vux − uxx λg(U) g(U)− h(U)

0 0 0 λ 1

0 0 0 0 1

0 Γ λv + Γux λv f(U)− λ

, (5.8)

49

W = (e, e′, u, v, v′)T , ′ = d/dx. (5.9)

We note that u = v − 1 and vx = ux, where vx and ex are given by (5.1) and we define

g(U) = ν−1(Γe− (ν + 1)ux),

f(U) =ux − Γe

v+ v = 2v − 1− Γe−,

h(U) = − exv

= −ν−1

(−(v − 1)2

2+ (e− e−) + (v − 1)Γe−

).

The quantities v and u are the traveling wave solutions of the Navier-Stokes equations for

compressible gas. As before, we now define v± := limx→±∞ v(x) and e± := limx→±∞ e(x). In

this case we have

v+ =Γ

Γ + 2, e+ =

2

(Γ + 2)2, v− = 1, and e− = 0.

Similarly we compute A± := limx→±∞A(x;λ).

A+(λ) =

0 1 0 0 0

λν−1v+ ν−1v+ 0 λν−1Γe+ ν−1Γe+

0 0 0 λ 1

0 0 0 0 1

0 Γ λv+ λv+ 2v+ − 1− λ

, (5.10)

A−(λ) =

0 1 0 0 0

λν−1 ν−1 0 0 0

0 0 0 λ 1

0 0 0 0 1

0 Γ λ λ 1− λ

. (5.11)

50

Following the same procedure as in the example, we make the coordinate changeW (x;λ) =

eαxV (x;λ) where α is an eigenvalue of A± := limx→±∞A(x;λ).

V ′(x;λ) = (A(x;λ)− αI)V (x;λ). (5.12)

We also make the coordinate change σ = v − v+. This gives rise to the new system

1

µ((σ + v+)(σ + v+ − 1) + Γe(σ))

dZ(σ)

dσ= B(σ;λ)Z(σ), (5.13)

with

B(σ;λ) =

−α 1 0 0 0

λ(σ+v+)ν

(σ+v+)ν− α (σ+v+)uσ

ν− uσσ λg(σ) g(σ)− h(σ)

0 0 −α λ 1

0 0 0 −α 1

0 Γ λ(σ + v+) + Γuσ λ(σ + v+) f(σ)− λ− α

, (5.14)

where

e(σ) =∞∑n=0

anσn as defined in Section 5.2,

uσ =1

µ((σ + v+)(σ + v+ + 1) + Γe(σ)),

eσ =σ + v+

ν

(−(σ + v+ − 1)2

2+ e(σ)

),

uσσ =1

µ((2(σ + v+)− 1)uσ + Γeσ) ,

g(σ) =1

ν(Γe(σ)− (ν + 1)uσ),

f(σ) = 2(σ + v+)− 1,

h(σ) =−1

ν

(−(σ + v+ − 1)2

2+ e(σ)

).

51

We seek to solve this new system to find a series solution to Z(σ). This is done by letting

Z(σ) =∑∞

n=0 Bnσn and then solving the system to find each coefficient vector Bn. Using

SymPy, we have developed a generalized linear solver which computes the coefficient vectors

Bn, the code for this solver can be found in Appendix A.

After using the solver to compute the coefficient vectors, we can look at our results for

various values of λ. We chose λ = 10. With this choice, we find the eigenvalues of A+ to

be 9.692,−10.565,−5.119, 2.2208,−1.775. For any of the negative eigenvalues, α, of A+, we

can initialize our series solution with the eigenpair (α, vα) where vα is the eigenvalue of A+

associated with α. Initializing the solution with an eigenpair is equivalent to setting B0 = vα

and using α as our scale when defining B(σ;λ) as in (5.14).

When solving this system with α = −10.565, we get a stable solution all the way from

x = −∞ to x = +∞ or equivalently from σ = v− − v+ to σ = v+ − v+ = 0. The solution is

shown in Figure 5.6, where the asterisks correspond to the eigenvector vα.

Figure 5.6: The fifteen-term series solution to the Evans system with λ = 10, scaled out bythe most negative eigenvalue of A+.

Our fifteen-term series solution with this eigenpair is very similar to the numerical solution

generated by STABLAB, which is quite promising. However, if we consider the relevance by

examining the norms of each successive coefficient vector Bn, as was done in Figure 5.1, we

see that the coefficients grow quite quickly, shown in Figure 5.7.

52

Figure 5.7: The trend in the norms of the coefficient vector for the series solution Z(σ).

From Figure 5.7, it is not surprising that as more terms are added to the series, the

solution diverges from the actual Evans system for x in the unstable manifold. As a result,

the solution can only be used from x = +∞ to x = 0.

We tried the same approach to solve the system scaled out by the other two negative

eigenvalues with less positive results, as shown below. In both cases, the solutions blew up

before reaching the fixed point at x = −∞. But we were able to get a system that stayed

reasonably bounded for part of the domain from x = +∞ to x = 0 for both additional

negative eigenvalues. Figures 5.8 and 5.9 show the evolved solutions from x = −∞ to x = 0

scaling the system by the second and third most negative eigenvalues respectively. Notice

for the third most negative eigenvalue, shown in Figure 5.9, the solution is unstable before

evolving to x = 0.

53

Figure 5.8: The fifteen-term series solution to the Evans system with λ = 10, scaled out bythe second most negative eigenvalue of A+.

Figure 5.9: The fifteen-term series solution to the Evans system with λ = 10, scaled out bythe third most negative eigenvalue of A+.

Recall that in order to compute the Evans function, we must take the determinant of the

stable eigenvectors from both the unstable and stable manifolds, evaluated at x = 0. With

the results above, we will not be able to evolve the entire basis to the center of our domain.

In the next section, we introduce an alternative method that will only require one solution

be evolved across the domain.

5.4 Compound Matrix Method

Because our previous approach was unsuccessful in evolving the entire basis from the stable

manifold to the center of our domain, we try a new approach. Instead of finding five basis

54

elements for the Evans function, we lift the system to ten dimensions and find one stable

solution from the stable manifold to the unstable manifold. Recall that our goal is to

determine if there is an eigenfunction for a particular potentially unstable eigenvalue. When

we lift our system using the complex matrix method, one single exterior product encodes

the information for the entire stable basis for the non-lifted stable manifold. As a result, if

we are able to evolve one exterior product across our entire domain, we can then evaluate

the Wronskian at x = −∞ to determine linear independence of our bases which in turn

determines the existence of an eigenfunction; see [1] for details.

We denote the matrix A(x;λ) lifted into∧2(C4) by A(2)(x;λ). Using properties of the

exterior product, we can find the leading eigenvalues and eigenvectors for this matrix at

x = +∞ and x = −∞. Let α1, α2, α3 be the three most negative eigenvalues of A+ with

corresponding eigenvectors v1, v2, v3. Then α1 + α2 + α3 is the most negative eigenvalue for

A(2)+ with corresponding eigenvector v1

∧v2

∧v3. We can use this same technique to find the

most negative eigenvalue and eigenvector for the lifted system at negative infinity A(2)− .

Now we will introduce a different coordinate change to solve for solution using the lifted

system A(2). The hope is that we can find a solution that is stable all the way from x = +∞

to x = −∞. If we are able to make it all the way from the stable manifold to the unstable

manifold, we avoid the issue of not being able to get a stable solution starting at x = −∞.

We make the coordinate change W (x;λ) = eg(x)V (x;λ) for some function g(x). This

gives rise to the new system.

V ′(x;λ) = (A(2) − g′(x))V (x;λ) (5.15)

Next, we make the familiar coordinate change σ = v − v+, which gives us

1

µ((σ + v+)(σ + v+ − 1) + Γe(σ))

dZ(σ)

dσ= B(2)(σ;λ)Z(σ), (5.16)

where B(2)(σ;λ) = A(2)(σ;λ)− f ′(σ) and f ′(σ) = g′(x).

55

To control our solution we want to choose f ′(σ) so that at both x = +∞ and x = −∞

(or equivalently at σ = 0 and σ = v− − v+) the solution stays stable. One way to do this is

by choosing f ′(σ) so that at x = ±∞, f ′(σ) scales the solutions out by the corresponding

dominating eigenvalues. Because we are going from positive infinity to negative infinity, the

dominating eigenvalue will be the most negative in both cases. Let α+ be the most negative

eigenvalue for A(2)+ and α− be the most negative eigenvalue for A

(2)− . Then set

f ′(σ) = α+ + (α− − α+)σ

v− − v+

.

Notice f ′(0) = α+ and f ′(v− − v+) = α− as desired.

Using this choice of f ′(σ) we define Z(σ) =∑∞

n=0Bnσn and solve for each coefficient

vector Bn with positive results. Figure 5.10 shows the evolved fifteen-term lifted system

solution corresponding to λ = 10. As you can see, the solution stays stable across the entire

domain and thus can be used to compute the Evans function. Note that we initialize our

solution with the eigenvector of A(2)+ corresponding the most negative eigenvector. That is,

we set B0 = vα, where vα is the eigenvector of A(2)+ corresponding to α which is the most

negative eigenvalue of A(2)+ .

Figure 5.10: The fifteen-term series solution to the lifted Evans system with λ = 10, scaledby f ′(σ).

56

We now want to test if this method is stable for all relevant values of λ on the contour

bounding unstable spectra. Using Algorithm 1, we can bound Λ for this system. With

δ = 0.001, meaning for all λ with Re(λ) > 0.001, we compute that Λ ≤ 54.27.

Figure 5.11 demonstrates the stability of the system for λ satisfying Λ = 54.27 and

Re(λ) ≥ 0.001. The figure on the left shows the complex values of λ for which the system is

evaluated. The figure on the right shows the value of the evolved eigenvector at the unstable

manifold for values of λ along the contour. Notice that the contour shown on the left is

parameterized by θ such that

tan(θ) =Re(λ)

Im(λ). (5.17)

Figure 5.11: The analytic solution to the Evans system evaluated on the unstable manifoldfor various λ values satisfying Λ = 54.27, where λ is parameterized by θ according to (5.17).

From Figure 5.11, we can conclude that the series solution is stable across the entire

domain for all λ values in the bounded unstable contour. With this result, we can now

use the series solution to evaluate the Evans function. We further confirm this by ensuring

the series solution converges all along the specified contour. This is done by comparing the

relative error between consecutive series solutions evaluated at the unstable manifold. More

precisely, comparing

||Zm(v− − v+)− Zm−1(v− − v+)||2||Zm(v− − v+)||2

,

as m → ∞ where Zm(v− − v+) =∑m

n=0Bn(v− − v+)n. The resulting norms for various λ

57

values satisfying Λ = 54.27 are shown in Figure 5.12.As you can see, the relative error is

order O(1/n), where n is the number of terms in the series.

Figure 5.12: Successive relative errors for finite series solution of the lifted Evans system forvarious values of λ, satisfying Re(λ) > 0.001 and Λ = 54.27. Note that λ is parameterizedby θ according to (5.17). The dotted line, included for reference, plots y = 1/n.

With the convergence of relative error and stability for λ all along our contour bounded

by Theorem 4.6, we have shown that this method for analytically computing the Evans

function has significant potential, especially when applied to the Navier-Stokes equations.

In the next section, we study this method for 2-D MHD.

5.5 Extending to MHD

We now extend our method to 2-D MHD. Again we start by introducing the coordinate

change σ = u1 − u1+, with u1 = σ + u1+ and e =∑∞

n=0 anσn. Note that due to the

equivalence of the profile systems of Navier-Stokes and MHD, we have the same fixed points,

(u1−, e−) = (1, 0) and (u1+, e+) = ( ΓΓ+2

, 2(Γ+2)2 ). And again, from the Jacobian we determine

that (u1−, e−) is unstable and (u1+, e+) is stable, which is why we center our coordinate

change around the stable fixed point.

Using the relation 5.2 we again seek to solve

cν(σ + u1+)

κ

[∞∑n=0

anσn − e− −

(σ + u1+ − 1)2

2+ (σ + u1+ − 1)Γe−

]=

58

∞∑n=1

nanσn−1 1

2µ+ η

[(σ + u1+)(σ + u1+ − 1) + Γ

(∞∑n=0

anσn − e−

)](5.18)

for the coefficients an. Notice that we have transformed our system into pseudo-Lagrangian-

coordinates by multiplying by u1. Solving for the coefficients results in a recurrence relation

nearly identical to (5.6). It is no surprise that this series solution for the MHD profile

converges and agrees well with the numerical solution with very few terms. Figure 5.13

shows the analytic series compared with the numerical solution for the MHD profile.

Figure 5.13: The fifteen-term series solution of the MHD profile e(σ) plotted against σ.

Using this profile, we are now prepared to build our Evans system. We will use the Evans

system in pseudo-Lagrangian coordinates given in Section 2.4 and repeated below

W ′ = A](x1;λ, ξ)W ,

where

A]

=

−r(λ]A0

11 + iAxi11)(A111)−1 0 −λ](A0

11(A111)−1A1

12 − A012)− iAξ

]

11(A111)−1A1

12 + iAξ]

12

−r(λ]A021 + iAxi21)(A1

11)−1 0 −λ](A021(A1

11)−1A112 − A

022)− iAξ

]

21(A111)−1A1

12 + iAξ]

22 + rbξ]ξ]

−r(b11)−1A121(A1

11)−1 r(b11)−1 (b11)−1(A122 − A

121(A1

11)−1A112 − rb

ξ] )

, (5.19)

W = (f ], u2)T . (5.20)

Following the same procedure as before and using the special scaling function introduced

in Section 5.4, we make the coordinate change W (x;λ) = eg(x)V (x;λ) for some function g(x),

59

resulting in,

V ′(x;λ) = (A](x;λ)− g′(x)I)V (x;λ). (5.21)

We also make the coordinate change σ = u1 − u1+. This gives rise to the new system

1

2µ+ η((σ + u1+)(σ + u1+ − 1) + Γe(σ))

dZ(σ)

dσ= B](σ, λ)Z(σ), (5.22)

where B](σ;λ) = A](σ;λ)− f ′(σ)I and f ′(σ) = g′(x).

Again we want to choose f ′(σ) that will appropriately scale A] on the stable and unstable

manifold as we evolve toward the unstable manifold. As before, we choose

f ′(σ) = α+ + (α− − α+)σ

u1− − u1+

,

where α± is the most negative eigenvalue of A]± = limx1→±∞A].

If we are able to get solutions across the entire domain, corresponding to the six stable

eigenvectors of the system, then we will be able to use this method to compute the Evans

function. Unfortunately, this is not the case. None of the solutions remain stable across the

entire domain and only the solution corresponding to the first eigenvector remains stable

halfway across the domain before diverging. Figure 5.14 shows the solution corresponding

the first eigenvector on the stable half of the domain.

Figure 5.14: The fifteen-term series solution to the Evans system for 2-D MHD with λ = 10,scaled by f ′(σ).

60

The stability of this method for the non-lifted 2-D MHD system gives reason to believe

that we might have success using the lifted system as we did with compressible Navier-Stokes.

In this case, we start with an 11×11 system and so the resulting lifted system will be 55×55.

Our resulting Evans system is given by

1

2µ+ η((σ + u1+)(σ + u1+ − 1) + Γe(σ))

dZ(σ)

dσ= (B])(2)(σ, λ)Z(σ), (5.23)

where (B])(2)(σ;λ) = (A])(2)(σ;λ) − f ′(σ)I, (A])(2)(σ;λ) is the lifted system in pseudo-

Lagrangian coordinates, and f ′(σ) = g′(x).

We use the same choice for f ′(σ) as before, scaling by the dominating eigenvalues of

(A])(2) at both the stable and unstable manifold. Our results are shown in Figure 5.15.

Figure 5.15: The five-term series solution to the lifted Evans system for 2-D MHD withλ = 10, scaled by f ′(σ).

While these results seem promising with just five terms in the series, unfortunately when

we add more terms, our solutions diverge on the right. From this we see that we will

need further modifications to our method developed for compressible Navier-Stokes. These

modifications could involve adjusting f ′(σ) to have a stronger scale on the unstable manifold,

or even increasing the order of the function from linear to quadratic or cubic.

61

Appendix A. Analytic Evans System Solver

This appendix contains the generalized Python class developed for solving for the coefficient

vectors of the analytic solution to the Evans system. This code was used in Sections 5.1,

5.3, 5.4, and 5.5.

import numpy as np

import sympy as sp

class SystemSolve():

"""

This class analytically solves for the solution to the Evans

system £Z(\sigma)£ using the coordinate change £\sigma = v_+£.

Input:

dim (int) : dimension of the system

terms (int) : number of terms in the series solution £Z(\sigma)£

A (array) : the matrix dictating the unscaled Evans system

deriv (func) : the derivative £d\sigma/dx£ associated with the coordinate change

scale (func) : scaling function (typically called £f'(\sigma)£), can be a float

e_vec (vector): the eigenvector with which to initialize the system

display (bool): determines whether output is printed as system is solved

copmlex (bool): determines whether or not the system is complex valued

"""

def __init__(self,dim,terms,A,deriv,e_val, e_vec,\

display=True, complex=False):

""" Initialize the System """

self.dim = dim

self.terms = terms

self.A = A

self.deriv = deriv

self._createB()

self.p = display

self.comp = comp

self._setEigs(scale, e_vec)

return

def _createB(self):

""" Create symbolic coefficient vectors B_n """

self.B = []

for i in range(self.terms):

62

b = []

for j in range(1,self.dim+1):

b.append(sp.sympify('B{}_{}'.format(i,j)))

self.B.append(b)

return

def _setEigs(self, scale, e_vec):

""" Set scale and initial coefficient vector B_0 """

self.scale = scale

self.B_tups = []

b_tups = []

for i in range(self.dim):

b_tups.append((self.B[0][i],e_vec[i]))

self.B_tups = b_tups

return

def createSys(self):

""" Create system and solve for each vector B_n """

Z = []

Z_deriv = []

# Create symbolic Z

for j in range(self.dim):

z = sp.sympify(0)

for i in range(self.terms):

z += self.B[i][j]*sigma**i

z_deriv = sp.diff(z, sigma)

Z.append(z)

Z_deriv.append(z_deriv)

# Store Z and Z' as numpy arrays

self.Z = np.array(Z)

self.Z_deriv = np.array(Z_deriv)

# Replace matrix A by scaled A-scale*I

Anew = self.A - np.eye(self.dim)*self.scale

# Define system of equations from each row of A

eqs = []

for i in range(self.dim):

rhs = np.dot(Anew[i],self.Z)

lhs = self.Z_deriv[i]*self.deriv

eq = rhs-lhs

63

eqs.append(eq)

# Isolate coefficients for each power of sigma

eq_der = [eq.copy() for eq in eqs]

heqs = []

for i in range(self.terms):

if i==0: prod = 1

else:

prod *= i

if self.p: print('\n\npower = {}'.format(i))

res = [eqd.subs(sigma,0)/prod for eqd in eq_der]

if self.p:

for r in res:

print('',r,sep='\n')

heqs.append(res)

eq_der = [diff(eqd,sigma) for eqd in eq_der]

# Solve for the coefficients of Z

start = time()

for j in range(len(heqs)):

# Display progress (optional)

if self.p:

p1 = '-'*21+'\nsystem when sigma = '

p2 = str(j)+'\n'+'-'*21+'\n'

print(p1+p2)

# Put system of equations in matrix vector form

mat, vec = sp.linear_eq_to_matrix(heqs[j],self.B[j])

# Accomodate complex system

if self.comp:

vec = [complex(vec[i].subs(self.B_tups)) for i in range(self.dim)]

mat = [[complex(mat[i,j].subs(self.B_tups)) for j in range(self.dim)] \

for i in range(self.dim)]

vec = np.array(vec)

mat = np.array(mat)

# Solve the system

if self.comp:

soln = np.linalg.solve(mat,vec)

64

else:

soln = mat.LUsolve(vec)

# Iteratively solve for B_n

for i in range(self.dim):

# Determine value of B_i

if self.comp:

val = soln[i]

val = complex(val)

else:

val = soln[i].subs(self.B_tups)

# Store value

if j != 0:

self.B_tups.append((self.B[j][i],val))

# Display progress (optional)

if self.p:

val = str(self.B[j][i]) +' = '+str(val)

print(val,'\n\n')

def evalZ(self, sig_val):

""" Evaluate £Z(\sigma)£ """

to_sub = self.B_tups + [(sigma,sig_val)]

Z_vals = []

# Substitute sigma into each dimension of solution

for i in range(self.dim):

val = self.Z[i].subs(to_sub)

if self.comp: val = complex(val)

Z_vals.append(val)

# Return £Z(\sigma)£

return Z_vals

def checkBNorms(self):

""" Compare ||B_n|| for each n """

norms = []

# Iterate through each coefficient vector B_n

for b in self.B:

# Compute ||B_n}}

65

norm = sp.sympify(0)

for i in b:

norm = norm + sp.Abs(i)

norm = (norm)**(1/2)

norm = norm.subs(self.B_tups)

# Store resulting norm

norms.append(norm)

return norms

66

Bibliography

[1] Blake Barker. Evans function computation, 2009. Master’s Thesis.

[2] Blake Barker, Jeffrey Humpherys, Gregory Lyng, and Kevin Zumbrun. Balancedflux formulations for multidimensional Evans-function computations for viscous shocks.Quarterly of Applied Mathematics, 76(3):531–545, Oct 2017.

[3] Blake Barker, Jeffrey Humpherys, Gregory Lyng, and Kevin Zumbrun. Euler versusLagrange: The role of coordinates in practical Evans-function computations. SIAMJournal on Applied Dynamical Systems, 17(2):1766–1785, Jan 2018.

[4] Blake Barker, Jeffrey Humpherys, Joshua Lytle, and Kevin Zumbrun. STA-BLAB: A MATLAB-based numerical library for Evans function computation.https://github.com/nonlinear-waves/stablab.git.

[5] Blake Barker, Jeffrey Humpherys, Keith Rudd, and Kevin Zumbrun. Stability of viscousshocks in isentropic gas dynamics. Comm. Math. Phys., 281(1):231–249, 2008.

[6] Blake Barker, Rafael Monteiro, and Kevin Zumbrun. Transverse bifurcation of viscousslow MHD shocks. http://arxiv.org/pdf/1901.09153.pdf.

[7] G. K. Batchelor. An introduction to fluid dynamics. Cambridge Mathematical Library.Cambridge University Press, Cambridge, paperback edition, 1999.

[8] H. Cabannes. Theoretical magnetofluiddynamics. Academic Press, New York, 1970.

[9] Constantine M Dafermos. Hyperbolic conservation laws in continuum physics, vol-ume 325 of Grundlehren der mathematischen wissenschaften [Fundamental principlesof mathematical sciences], 2010.

[10] Heinrich Freistuhler and Yuri Trakhinin. On the viscous and inviscid stability of magne-tohydrodynamic shock waves;. Physica D: Nonlinear Phenomena, 237(23):3030–3037,2008.

[11] David Gilbarg. The existence and limit behavior of the one-dimensional shock layer.Amer. J. Math., 73:256–274, 1951.

[12] Peter Howard and Kevin Zumbrun. The Evans function and stability criteria for de-generate viscous shock waves. Discrete Contin. Dyn. Syst., 10(4):837–855, 2004.

[13] J. Humpherys, G. Lyng, and K. Zumbrun. Spectral stability of ideal-gas shock layers.Arch. Ration. Mech. Anal., 2009.

[14] Jeffrey Humpherys, Gregory Lyng, and Kevin Zumbrun. Multidimensional stability oflarge-amplitude Navier–Stokes shocks. Archive for Rational Mechanics and Analysis,226(3):923–973, Jul 2017.

[15] Alan Jeffrey. Magnetohydrodynamics. University Mathematical Texts, No. 33. Oliver &Boyd, Edinburgh, 1966.

67

[16] Helge Kristian Jenssen and Gregory Lyng. Evaluation of the Lopatinski determinantfor gas dynamics. arXiv:0904.0268. Appendix to Z2.

[17] Shuichi Kawashima. Systems of a hyperbolic–parabolic composite type, with applicationsto the equations of magnetohydrodynamics. PhD thesis, Kyoto University, 1983.

[18] Corrado Mascia and Kevin Zumbrun. Pointwise Green function bounds for shock profilesof systems with real viscosity. Arch. Ration. Mech. Anal., 169(3):177–263, 2003.

[19] Corrado Mascia and Kevin Zumbrun. Stability of large-amplitude viscous shock profilesof hyperbolic-parabolic systems. Arch. Ration. Mech. Anal., 172(1):93–131, 2004.

[20] Guy Metivier and Kevin Zumbrun. Large viscous boundary layers for noncharacteristicnonlinear hyperbolic problems. Mem. Amer. Math. Soc., 175(826):vi+107, 2005.

[21] D. H. Sattinger. On the stability of waves of nonlinear parabolic systems. Advances inMath., 22(3):312–355, 1976.

[22] Andreas Schafelner. Introduction to magnetohydrodynamics., 2016. Bachelor’s Thesis.

[23] Benjamin Texier and Kevin Zumbrun. Hopf bifurcation of viscous shock waves in com-pressible gas dynamics and MHD. Arch. Ration. Mech. Anal., 190(1):107–140, 2008.

[24] Kevin Zumbrun. Multidimensional stability of planar viscous shock waves. In Advancesin the theory of shock waves, volume 47 of Progr. Nonlinear Differential Equations Appl.,pages 307–516. Birkhauser Boston, Boston, MA, 2001.

[25] Kevin Zumbrun. Planar stability criteria for viscous shock waves of systems with realviscosity. In Hyperbolic systems of balance laws, volume 1911 of Lecture Notes in Math.,pages 229–326. Springer, Berlin, 2007.

[26] Kevin Zumbrun. Numerical error analysis for Evans function computations: a numer-ical gap lemma, centered-coordinate methods, and the unreasonable effectiveness ofcontinuous orthogonalization. arXiv.org:0904.0268, 2009.

[27] Kevin Zumbrun. The refined inviscid stability condition and cellular instability of vis-cous shock waves. Phys. D, 239(13):1180–1187, 2010.

[28] Kevin Zumbrun and D. Serre. Viscous and inviscid stability of multidimensional planarshock fronts. Indiana Univ. Math. J., 48(3):937–992, 1999.

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