Stability and dynamics of viscous shock waves

Kevin Zumbrun∗

July 16, 2009

IMA Summer School Lectures in Conservation Laws: preliminary version

Abstract

We examine from a classical dynamical systems point of view stability, dynamics,and bifurcation of viscous shock waves and related solutions of nonlinear pde. Thecentral object of our investigations is the Evans function: its meaning, numerical ap-proximation, and behavior in various asymptotic limits.

Contents

1 Introduction 51.1 Motivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Scalar reaction–diffusion equation . . . . . . . . . . . . . . . . . . . 51.1.2 Burgers equation: a scalar viscous conservation law . . . . . . . . . . 61.1.3 Isentropic gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.4 Isentropic MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Traveling and standing waves . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.1 Reaction–diffusion equation . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 Equations of gas dynamics . . . . . . . . . . . . . . . . . . . . . . . 91.2.4 Equations of MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.5 Numerical approximation of profiles . . . . . . . . . . . . . . . . . . 11

2 Spectral stability 112.1 Autonomous equations and linearized and spectral stability . . . . . . . . . 122.2 The eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Reaction–diffusion equation . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 Gas dynamics equations . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.4 MHD equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

∗Indiana University, Bloomington, IN 47405; kzumbrun@indiana.edu: Research of K.Z. was partiallysupported under NSF grants no. DMS-0300487 and DMS-0801745.

1

2.3 Expression as first-order system . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.1 Reaction–diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.2 Burgers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.4 MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 The Evans function 173.1 Limiting eigenprojections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Limiting eigenbases: Kato’s ODE . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Limiting behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Evans function construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 A fundamental example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Abstract stability theorem: the Evans condition 224.1 Profile facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2 The Evans function condition . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Basic stability theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.4 Meaning of the Evans condition . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Heuristic discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6 Verification for small amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 274.7 Verification for large amplitudes . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Numerical approximation of the Evans function 285.1 Shooting algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.1.1 Solution one: the centered exterior product method . . . . . . . . . 295.1.2 Solution two: the polar coordinate method . . . . . . . . . . . . . . 305.1.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Initialization of eigenbases at infinity . . . . . . . . . . . . . . . . . . . . . . 325.2.1 Numerical implementation of Kato’s ODE . . . . . . . . . . . . . . . 32

5.3 Initialization of Evans function ODE . . . . . . . . . . . . . . . . . . . . . . 335.3.1 Centered exterior product scheme . . . . . . . . . . . . . . . . . . . 335.3.2 Polar coordinate scheme . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.4 Error control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4.1 Shooting integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4.2 Kato integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6 Global stability analysis: a case study 356.1 The rescaled equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.1.1 Profile equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356.1.2 Eigenvalue equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

6.2 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 Evans function formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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6.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4.1 The strong shock limit . . . . . . . . . . . . . . . . . . . . . . . . . . 386.4.2 The weak shock limit . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4.3 Intermediate strength shocks . . . . . . . . . . . . . . . . . . . . . . 396.4.4 Numerical convergence . . . . . . . . . . . . . . . . . . . . . . . . . . 396.4.5 Global picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.5 Nonisentropic gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 416.6 MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.6.1 The case of infinite resistivity/permeability . . . . . . . . . . . . . . 426.6.2 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.7 Stability of detonations in the ZND limit . . . . . . . . . . . . . . . . . . . 43

7 Conditional stability of viscous shock waves 437.1 Existence of Center Stable Manifold . . . . . . . . . . . . . . . . . . . . . . 457.2 Conditional stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 45

7.2.1 Projector bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2.2 Linear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467.2.3 Reduced equations II . . . . . . . . . . . . . . . . . . . . . . . . . . . 497.2.4 Nonlinear damping estimate . . . . . . . . . . . . . . . . . . . . . . . 507.2.5 Proof of nonlinear stability . . . . . . . . . . . . . . . . . . . . . . . 51

8 Longitudinal bifurcation of viscous shock waves 538.0.6 Equations and assumptions . . . . . . . . . . . . . . . . . . . . . . . 548.0.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.1 Linearized estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.1.1 Short time estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 578.1.2 Pointwise Green function bounds . . . . . . . . . . . . . . . . . . . . 57

8.2 Nonlinear energy estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.2.1 Hs linearization error . . . . . . . . . . . . . . . . . . . . . . . . . . 638.2.2 Weighted variational bounds . . . . . . . . . . . . . . . . . . . . . . 65

8.3 Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 668.3.1 Construction of the period map . . . . . . . . . . . . . . . . . . . . . 668.3.2 Lyapunov–Schmidt reduction . . . . . . . . . . . . . . . . . . . . . . 688.3.3 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . . . . . 71

9 Cellular instability for flow in a duct 739.1 Equations and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 739.2 Stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 759.3 Bifurcation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769.4 Longitudinal vs. transverse bifurcation . . . . . . . . . . . . . . . . . . . . . 769.5 The refined stability condition and bifurcation . . . . . . . . . . . . . . . . 77

9.5.1 The inviscid stability condition . . . . . . . . . . . . . . . . . . . . . 789.5.2 The refined stability condition . . . . . . . . . . . . . . . . . . . . . 78

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9.5.3 Transverse bifurcation of flow in a duct . . . . . . . . . . . . . . . . 809.6 Discussion and open problems . . . . . . . . . . . . . . . . . . . . . . . . . . 829.7 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

9.7.1 Projector bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 839.7.2 Linear estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.7.3 Nonlinear damping estimate . . . . . . . . . . . . . . . . . . . . . . . 859.7.4 Proof of nonlinear stability . . . . . . . . . . . . . . . . . . . . . . . 869.7.5 Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A Traveling-wave equations for isentropic MHD 86A.1 The case σ = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

B Further example systems 87

C Asymptotic linear ODE theory 89C.1 The gap and conjugation lemmas . . . . . . . . . . . . . . . . . . . . . . . . 89C.2 The convergence lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

C.2.1 Regular perturbation of traveling waves . . . . . . . . . . . . . . . . 93C.3 The tracking lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

D Center Stable Manifold for ODE 96D.1 Frechet differentiability of substitution operators . . . . . . . . . . . . . . . 97D.2 Smooth dependence of fixed-point solutions . . . . . . . . . . . . . . . . . . 98D.3 Global Center Stable Manifold construction . . . . . . . . . . . . . . . . . . 100D.4 Local construction: proof of Proposition D.1 . . . . . . . . . . . . . . . . . . 102D.5 Addendum: The Stable (Unstable) Manifold Theorem . . . . . . . . . . . . 103

E Center Stable Manifold for PDE 105E.1 Reduced equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106E.2 Preliminary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107E.3 Translation-invariant center stable manifold . . . . . . . . . . . . . . . . . . 109E.4 Application to viscous shock waves . . . . . . . . . . . . . . . . . . . . . . . 110E.5 A simple orbital stability theorem . . . . . . . . . . . . . . . . . . . . . . . 110

F Miscellaneous energy estimates 111F.1 Small-amplitude stability for isentropic gas dynamics . . . . . . . . . . . . . 111F.2 High-frequency bounds for isentropic gas dynamics . . . . . . . . . . . . . . 112F.3 Nonvanishing of D0 for isentropic gas dynamics . . . . . . . . . . . . . . . . 114

G Miscellaneous proofs 116

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1 Introduction

The object of this course is to study using Evans function and related techniques the stabilityand dynamics of traveling-wave solutions u(x, t) = u(x − st) of various systems of viscousconservation laws

(1.1) ut + f(u)x = (B(u)ux)x, u ∈ Rn

arising in continuum mechanics. Our emphasis is on concrete applications: in particular,numerical verification of stability/bifurcation conditions for viscous shock waves. However,the techniques we describe should have application also to other problems involving stabilityof traveling waves, particularly for large systems, or systems with no spectral gap betweendecaying and neutral modes of the linearized operator about the wave. We begin by intro-ducing some equations from Biology and Physics possessing traveling wave solutions.

1.1 Motivating examples

1.1.1 Scalar reaction–diffusion equation

Our first example (counterexample?) is not a conservation law. Let

(1.2) ut + g(u) = uxx,

u = u(x, t) ∈ R, x ∈ R, t ∈ R+, where

(1.3) g(u) = dG(u) = (1/2)(u2 − 1)u,

that is,

(1.4) G(u) = (1/8)(u2 − 1)2.

This equation is of “reaction–diffusion” type, combining the effects of spatial diffusion, asexemplified by the heat equation ut = uxx, with the effects of reaction, as exemplified bythe (spatially independent) ordinary differential reaction equation ut = −dG(u) (gradientflow) for which the function G represents a potential (chemical, for example) of the stateassociated with the value u (for example, densities of various chemical constituents).

This models for example the propagation of electro-chemical signals along an axon (nerveimpulse), pattern-formation in chemical systems, and population dynamics. It also servesas a model for phase-transitional phenomena in material science, where u represents an“order parameter” and G a free energy potential. The potential of (1.4) is the protoypicalexample of a “double-well”; other potentials (of similar topological configuration) lead toqualitatively similar behavior.

5

1.1.2 Burgers equation: a scalar viscous conservation law

Let

(1.5) ut + f(u)x = uxx,

u = u(x, t) ∈ R, x ∈ R, t ∈ R+, ou

(1.6) f(u) = u2/2.

This serves as a simple model for gas dynamics, traffic flow, or shallow-water waves, whereu represents the density of some conserved quantity and f its flux through a fixed point x.With the choice of flux (1.6), (1.5) becomes Burgers equation, the prototypical example of ascalar viscous conservation law. Behavior for other (convex) fluxes is qualitatively similar.

1.1.3 Isentropic gas dynamics

The equations of an isentropic compressible gas (also of the St. Venant equations for surfacewaves of an incompressible fluid) take the form, in Lagrangian coordinates1

(1.7)vt − ux = 0

ut + p(u)x =(ux

v

)x,

where u represents specific volume (one over density), v velocity, and p pressure. Thesemodel ideal gas dynamics when p′ < 0; in the case p′ ≷ 0, they give a model for phase-transitional gas dynamics or elasticity. Thus, these equations in a sense generalize both ofthe previous two examples. Here, we consider mainly a γ-law pressure

(1.8) p(v) = av−γ ,

modeling an ideal isentropic polytropic gas, where a > 0 and γ > 1 are constants thatcharacterize the gas. Similar pressures lead to qualitatively similar behavior.

1.1.4 Isentropic MHD

In Lagrangian coordinates, the equations for planar compressible isentropic magnetohydro-dynamics (MHD) take the form

(1.9)

vt − u1x = 0,u1t + (p+ (1/2µ0)(B2

2 +B23))x = ((1/v)u1x)x,

u2t − ((1/µ0)B∗1B2)x = ((µ/v)u2x)x,u3t − ((1/µ0)B∗1B3)x = ((µ/v)u3x)x,(vB2)t − (B∗1u2)x = ((1/σµ0v)B2x)x,(vB3)t − (B∗1u3)x = ((1/σµ0v)B3x)x,

1That is, coordinates in which x represents initial particle location, as opposed to Eulerian coordinates,in which x represents fixed location. The change from Lagrangian to Eulerian frame involves a change ofthe independent variable x, and typically of dependent variables as well.

6

where v denotes specific volume, u = (u1, u2, u3) velocity, p = p(v) pressure, B = (B∗1 , B2, B3)magnetic induction, B∗1 constant (by the divergence-free condition ∇x · B ≡ 0), µ = µ

2µ+ηwhere µ > 0 and η > 0 are the respective coefficients of Newtonian and dynamic viscosity,µ0 > 0 the magnetic permeability, and σ > 0 the electrical resistivity; see [A, C, J] forfurther discussion.

We take again a γ-law pressure (1.8). Though we do not specify η, we have in mindmainly the value η = −2µ/3 typically prescribed for (nonmagnetic) gas dynamics [Ba], or

µ = 3/4.

An interesting, somewhat simpler, subcase is that of infinite electrical resistivity σ = ∞, inwhich the last two equations of (1.9) are replaced by

(1.10) (vB2)t − (B∗1u2)x = 0, (vB3)t − (B∗1u3)x = 0.,

and only the velocity variables u = (u1, u2, u3) experience parabolic smoothing, throughviscosity. Here, we are assuming a planar multidimensional flow, i.e., one that depends onlyon a single space-direction x. In the simplest, parallel case u2 = u3 = B2 = B3 ≡ 0, theequations reduce to those of isentropic gas dynamics, (1.1.3).

1.2 Traveling and standing waves

We denote as traveling waves solutions

(1.11) u(x, t) = u(x− st),

s constant, of a general evolution equation (in the system case of Section 1.1.3, (v, u)(x, t) =(v, u)(x− st)). We call the function u(·) the profile of the wave. We consider here profilesof front or pulse type,

(1.12) limz→±∞

u(z) = u±.

Other types, for example, waves that are oscillatory at infinity, are also interesting butbehond the scope of our discussions. Such traveling-wave solutions move with constantspeed s without changing their form. If s = 0, they are equilibria, or stationary waves ofthe associated evolution equations. A traveling wave may always be converted to a standingwave by the change of coordinates x→ x− st to a frame moving with the same speed s.

Traveling waves are quite important in practice, as the simplest solutions that maintaintheir form while still reflecting the spatial dynamics of the problem; in particular, except inthe trivial case u ≡ constant, they depend nontrivially on the spatial variable x. They oftenserve to carry biological or mechanical signals. We derive below nontrivial traveling-wavesolutions for each of the models 1–4 just described.

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1.2.1 Reaction–diffusion equation

Observing that

(1.13) ∂tu(x− st) = −su′, ∂xu(x− st) = u′, ∂2xu(x− st) = u′′,

we obtain for a solution (1.11) the profile equation

(1.14) −su′ + dG(u) = u′′,

an ordinary differential equation of second order. Taking for simplicity

s = 0,

we find that (1.14) becomes a nonlinear oscillator equation

(1.15) dG(u) = u′′,

a Hamiltonian equation that may be reduced to first order.Specifically, multiplying by u′ on both sides, we obtain

dG(u)u′ = u′u′′,

where G(u)′ = [(u′)2/2]′. Integrating from −∞ to x then yields

(u′)2 = 2[G(u)−G(u−)].

Thanks to (1.12), we have u′(±∞) = 0, hence, recalling (1.15), we have dG(u±) = 0, whereu± = ±1, 0. As u = 0 is a nonlinear center which admits no orbital connections, we have

(1.16) u± = ±1, G(u±) = 0,

hence

(1.17) u′ = ±√

2G(u) = ±√

(1/4)(u2 − 1)2 = ±(1/2)(u2 − 1).

Choosing the sign +, we have finally, by separation of variables or simply by inspection, theprofile solution

(1.18) u(x) = − tanh(x/2).

(Of course, there exists also a symmetric solution u(x) = tanh(x/2) by reflection invarianceof (1.15), which corresponds to the other choice of sign − in (1.17).) This is the uniquesolution up to translation in x.

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1.2.2 Burgers equation

Using again (1.13), we obtain as the equation for a Burgers profile

(1.19) −su′ + f(u)′ = u′′.

Integrating from −∞ to x reduces (1.19) to a first-order equation

u′ = (f(u)− su)− (f(u−)− su−).

For definiteness taking again s = 0, u− = 1, we obtain therefore u′ = (1/2)(1 − u2) asin the reaction–diffusion case, and thus the same tanh solution (1.18), again unique up totranslation in x.

1.2.3 Equations of gas dynamics

Similarly, we find for gas dynamics the profile equations

(1.20)−sv′ − u′ = 0

−su′ + p(v)′ = (v−1u′)′,

or, substituting the first equation into the second, s2v′ + p(v)′ = −s(v−1v′)′. Integratingfrom −∞ to x, we obtain

(1.21) v′ = (−s)−1v[(p(v) + s2v)− (p(v−) + s2v−)].

Chooosing s = −1, v− = 1, a = v+, taking a barotropic, or constant-temperature,pressure p(v) = av−1,2 and setting vm := v−+v+

2 , δ := v− − v+, and v := vm + (δ/2)θ, weobtain, finally, θ′ = (δ/2)(θ2 − 1), hence θ(x) = − tanh((δ/2)x), giving a solution

(1.22) v := vm − (δ/2) tanh((δ/2)x)

similar to the Burgers profile (1.18), with u = v + C, where C is an arbitrary constant.(Observe that equations (1.7) are evidently invariant under translation in u, since u appearsonly with a derivative.) With C (or, equivalently, u±) fixed, solutions are unique up totranslation in x.

Remark 1.1. For p(v) = av−γ , arbitrary connections may be reduced by invariances ofthe equations to the case 1 = v− > v+ > 0, s = −1 as above; see Section 6.

Exercise 1.2. More generally, for a convex pressure function p, show that there exists atraveling-wave connection between any two zeros v± of the righthand side of (1.20) suchthat p(v−) > p(v+), which decays exponentially to its endstates as x→ ±∞.

2That is, holding temperature T ≡ constant in the ideal gas law p = RT/v, R the gas constant. Thespecial solution (1.22) is described by G.I. Taylor in [Tay] (thanks to C.M. Dafermos for this reference).

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1.2.4 Equations of MHD

In the parallel case u2 = u3 = B2 = B3 ≡ 0, the equations of isentropic MHD reduce tothose of isentropic gas dynamics, and so we have the same family of traveling waves alreadystudied. However, from the standpoint of behavior, there is a major difference that we wishto point out. Namely, taking v− = 1, u1,− = 0, s = −1, and for simplicity taking σ = ∞,but without making the parallel assumptions, we obtain the more general profile equation

(1.23)v′ = v(v − 1) + v(p− p−) +

12µ0v

((B− +B∗1w)2 − v2B2−),

µw′ = vw − B∗1µ0

(B−(1− v) +B∗1w

),

where w := (u2, u3) denotes transverse velocity components, and B ≡ v−1(B−+B∗1w). (SeeAppendix A for the derivation of these equations.)

The point we wish to emphasize is that even in the case of a parallel traveling-waveprofile, for which B2,± = B3,± = u2,± = u3,± = 0, there may exist other traveling-waveprofiles connecting the same endstates, but which are not of parallel type. Indeed, thisdepends precisely on the type of the rest points at ±∞, i.e., the number of stable (negativereal part) and unstable (positive real part) eigenvalues of the coefficient of the linearizedequations about these equilibria, which are readily determined by the observation (Ex. 1.3)that v and w equations decouple in the parallel endstate case.

Specifically, we have from existence of a gas-dynamical profile that the v equation con-tributes on unstable eigenvalue at −∞ and a stable eigenvalue at +∞, corresponding toa connecting solution decaying exponentially at ±∞. Meanwhile, the diagonal part of thelinearized w-equation is

µw′ =(v± −

(B∗1)2

µ0

)w,

so that the signs of the eigenvalues depends on the signs of 1 − (B∗1 )2

µ0v±, i.e., on where |B∗1 |

falls relative to √µ0v±. Thus, there are three generic possibilities:(i) ( |B∗1 | <

√µ0v±) The state at −∞ is a repellor, with three positive real eigenvalues,

and the state at +∞ is a saddle with one negative and two positive real eigenvalues.(ii) ( √µ0v− < |B∗1 | <

√µ0v+) The state at −∞ is a repellor, with three positive real

eigenvalues, and the state at +∞ is an attractor, with three negative real eigenvalues.(iii) ( √µ0v± < |B∗1 |) The state at −∞ is a saddle, with one positive and two negative

real eigenvalues, and the state at +∞ is an attractor, with three negative real eigenvalues.In cases (i) and (iii), for prescribed fixed endstates at ±∞, the profile is unique up to

translation in x as before, and is the one given by gas dynamics. However, in case (ii), thereexists a three-parameter family of connecting profiles for each pair of endstates at ±∞, ofwhich the unique parallel profile, up to translation in x, is that given by gas dynamics.

Exercise 1.3. Determine by explicit computation the type of the rest points at x = ±∞by linearizing (1.23) about each equilibrium and noting that the v and w equations decoupleinto triangular form for parallel endstate B2,± = B3,± = u2,± = u3,± = 0.

10

Remark 1.4. A straightforward computation shows that the characteristics of the first-order hyperbolic system obtained by neglecting second-derivative terms in (1.9) at theendstates v±, in frame x = x− st, s = −1, have values

(1.24) (1± c(v±)), 1, 1,(1± B∗1√

µ0v±

),

where c(v) :=√−p′(v) =

√γav−γ−1 is the gas-dynamical sound speed, satisfying c+ >

1 > c−. Thus, the shock is a Lax 1-shock for 0 ≤ B∗1 <√µ0v+, meaning that it has six

positive characteristics at v− and one at v+; an intermediate doubly overcompressive shockfor√µ0v+ < B∗1 <

√µ0v−, meaning that it has six positive characteristics at v− and three at

v+; and a Lax 3-shock for √µ0v− < B∗1 , meaning that it has 4 positive characteristics at v−and three at v+. This illustrates the general principle that the dimension ` of the family ofviscous shock connections is generically equal to the number of incoming characteristics (i.e.,the number of positive characteristics relative to the shock speed s at −∞ plus the numberof negative characteristics at +∞) minus the dimension n of the system: in the Lax case,` = 7−6 = 1, in the overcompressive case ` = 9−6 = 3. For further discussion of hyperbolicshock type and its relation to existence of viscous profiles, see [MP, LZ, ZH, Z1, Z2, MaZ3].See also Proposition 4.4, below.

1.2.5 Numerical approximation of profiles

Above, we have chosen parameters carefully in order to have explicit solutions of thetraveling-wave equations. Generally, one must approximate profiles numerically, since forgeneral choices of f , g, p, the profile equations are not explicitly soluble. But, the propertiesof the solutions remain similar; in particular, profiles decay exponentially as x→ ±∞, whichis the principle property we will need in what follows. We shall take the form of profiles asgiven and focus on other aspects. Numerical approximation of profiles is another interestingproblem, but one that has already been well studied. See, for example, the works of Beynand colleagues on boundary-value methods (e.g., [Be]), of which a satisfactory version issupported in MATLAB and easy to use.

2 Spectral stability

A fundamental issue in physical applications is nonlinear (time-evolutionary) stability ofsuch traveling-wave solutions: whether or not a perturbed solution

u = u+ u

of a traveling wave with perturbation u sufficiently small at t = 0 approaches as t → +∞to a translation u(x− x0) of the original wave u(x). Observe that a perturbation

u(x, t) = u(x− x0 − st)− u(x− st)

11

is a traveling wave itself, hence does not decay. Thus, approach to a translate– “orbitalstability”– is all that we can expect, and not approach to the original wave. Unstable solu-tions are impermanent, since they disappear under small perturbations, and typically playa secondary role if any in behavior: for example, as transition states lying on separatricesbetween basins of attraction of neighboring stable solutions, or as background organizingcenters from which bifurcate more complicated (stable) solutions.

2.1 Autonomous equations and linearized and spectral stability

Let us briefly recall the study of asymptotic stability of an equilibrium U ∈ Rn of anautonomous ordinary differential equation

(2.1) Ut = F(U)

in Rn. We first linearize about U , to obtain the linearized equations

Ut = LU := dF(U)U,

where L, since U is constant and F is autonomous, is constant. The formal justification forthis step is that if perturbation U = U − U is small, then terms of second order are muchsmaller than terms of first order in the perturbation, and so can be ignored. If the solutionsof (2.1) decay as t → ∞, we say that the equilibrium U is linearly stable. Linear stabilityis nearly but not completely necessary for nonlinear stability, defined as decay as t→∞ ofperturbations U = U − U that are sufficiently small at t = 0, for U = U + U a solution ofthe nonlinear equation (2.1).

A necessary condition for linear stability is spectral stability, or nonexistence of eigen-values λ of L with <λ ≥ 0. For, if there are W ∈ Rn, λ ∈ C such that LW = λW with<λ ≥ 0, then U(t) := eλtW is a growing mode of (2.1), growing as |eλt| = e<λt: a solutionof (2.1) that does not decay.

These three notions of stability are linked in Rn by Lyapunov’s Theorem, which assertsthat linearized and spectral stability are equivalent, and imply nonlinear exponential sta-bility. One generally studies the simplest condition of spectral stability which correspondsto nonexistence of solutions W , λ with <λ ≥ 0 of the eigenvalue equation

(2.2) LW = λW,

that is, determination of the eigenvalues of the linear operator L, in this case, an n × nmatrix. The latter problem may be studied by Nyquist diagram, Routh–Hurwitz stabilitycriterion [Hol], or other methods based on the characteristic polynomial p(λ) := det(λ−L)associated with L.

We will pursue a similar program in the case of interest of partial differential equations,linearizing and studying the associated eigenvalue equations with the aid of the Evansfunction, an infinite-dimensional generalization of the characteristic polynomial. As a firststep, we derive here the linearized and eigenvalue equations.

12

2.2 The eigenvalue equations

2.2.1 Reaction–diffusion equation

Let ut + g(u) = uxx be a solution of (1.2) different from u, also satisfying ut + g(u) = uxx.Subtracting u from u, we obtain for the perturbation variable

u := u− u

the nonlinear perturbation equation

(2.3) ut + (g(u+ u)− g(u)) = uxx.

Taylor expanding(g(u+ u)− g(u)) = dg(u)u+O(u2)

and dropping nonlinear terms O(u2), we obtain the linearized equation

(2.4) ut + dg(u)u = uxx,

where

(2.5) dg(u) = d2G(u) = (1/2)(3u2 − 1), u(x) = − tanh(x/2),

in particular,

(2.6) dg(u±) = 1.

Finally, seeking normal mode solutions u(x, t) = eλtw(x), we obtain the eigenvalue equation

(2.7) λw + dg(u)w = w′′,

where ′ denotes ∂x.

Stationary modes. A complication is that the equation (2.3), by translational-invarianceof the original equation (1.2), admits stationary solutions

u(x, t) ≡ u(x+ c)− u(x)

corresponding with translations u(x, t) = u(x − c) of u, which do not decay. Likewise, thelinearized equation (2.4) admits the stationary solutions

u(x, t) ≡ cu′(x).

Observe that u(x+c)− u(x) = cu′(x)+O(c2), hence the two (nonlinear vs. linear) solutionsapproach each other as c→ 0.

Also, as remarked above, we cannot expect in terms of stability more than approach toa translation for (2.3) and a multiple cu′ of u′ for (2.4). As spectral stability, therefore, werequire that there exist no solution w ∈ L2, λ ∈ C of the eigenvalue equation (2.7) with<λ ≥ 0 except for the “translational solution” w = cu′, λ = 0: that is, orbital spectralstability.

13

2.2.2 Burgers equation

Reasoning in the same way, we obtain for Burgers equation the nonlinear perturbationequation

ut + (f(u+ u)− f(u))x = uxx.

Expandingf(u+ u)− f(u) = df(u)u+O(u2)

and discarding the nonlinear part, we obtain the linearized equation

ut + (df(u)u)x = uxx

and the eigenvalue equation

(2.8) λw + (df(u)w)′ = w′′,

where ′ denotes ∂x and

(2.9) df(u) = u, u(x) = − tanh(x/2).

Integrated equation. A standard trick is to integrate the equation (2.8) to obtain theintegrated eigenvalue equation

(2.10) λw + df(u)w′ = w′′,

where w(x) :=∫ x−∞w(z) dz.

Lemma 2.1 ([ZH]). For <λ ≥ 0, the eigenvalues (L2) of (2.8) and (2.10) coincide, exceptfor λ = 0.

Proof. Integrating (2.8) from −∞ to +∞, we obtain λ∫ +∞−∞ w(z)dz = 0, since all other

terms are exact derivatives, hence for λ 6= 0∫ +∞

−∞w(z)dz = 0.

Thus, w(±∞) = 0, and w ∈ L2, if w ∈ L2, and vice versa. To make rigorous this intuitiveproof, it suffices to observe that w ∈ L2 and <λ ≥ 0, λ 6= 0 implies that w and all of itsderivatives approach exponentially to zero as x→ ±∞; see Section (3.3) below.

The advantage of (2.10) vs. (2.8) is that (2.10) does not admit any bounded solution wfor λ = 0, that is, we have removed the eigenvalue λ = 0 by integration. As spectral stabilityfor Burgers equation, therefore, we require simply that there exist no solution w ∈ L2, λ ∈ Cof the integrated eigenvalue equation (2.10) with <λ ≥ 0.

14

2.2.3 Gas dynamics equations

Reasoning as before, we obtain in the frame x = x− st the linearized equations

(2.11)vt − svx − ux = 0

ut − sux +((dp(v) + v−2ux)v

)x

= ((1/v)ux)x,

the eigenvalue equations

(2.12)λv − sv′ − u′ = 0

λu− su′ +((dp(v) + v−2u′)v

)′ = ((1/v)u′)′,

where ′ denotes ∂x. Integrating, we obtain finally the integrated eigenvalue equations

(2.13)λv + v′ − u′ = 0

λu+ u′ + (dp(v) + v−2u′)v′ = (1/v)u′′,

As for the Burgers equation, we require as spectral stability that there exist no solutionw ∈ L2, λ ∈ C of the integrated eigenvalue equation (2.13) with <λ ≥ 0.

2.2.4 MHD equations

Linearizing (1.9) about a parallel shock profile (v, u1, 0, 0, B∗1 , 0, 0), we obtain a decoupledsystem

(2.14)

vt + vx − u1x = 0

u1t + u1x − aγ(v−γ−1v

)x

=(u1x

v+u1x

v2v)

x

u2t + u2x −1µ0

(B∗1B2)x = µ(u2x

v

)x

u3t + u3x −1µ0

(B∗1B3)x = µ(u3x

v

)x

(vB2)t + (vB2)x − (B∗1u2)x =((

1σµ0

)B2x

v

)x

(vB3)t + (vB3)x − (B∗1u3)x =((

1σµ0

)B3x

v

)x

,

consisting of the linearized isentropic gas dynamic equations in (v, u1) about profile (v, u1),and two copies of an equation in variables (uj , vBj), j = 2, 3.

Introducing integrated variables v :=∫v, u :=

∫u1 and wj :=

∫uj , αj :=

∫vBj ,

j = 2, 3, we find that the integrated linearized eigenvalue equations decouple into theintegrated linearized eigenvalue equations for gas dynamics (2.13) and two copies of

(2.15)

λw + w′ − B∗1α

′

µ0v= µ

w′′

v

λα+ α′ −B∗1w′ =

1σµ0v

(α′

v

)′15

in variables (wj , αj), j = 2, 3. As spectral stability, therefore, we require the pair of con-ditions that there exist no solution w ∈ L2, λ ∈ C with <λ ≥ 0 of either the integratedgas-dynamical eigenvalue equation (2.13) or the integrated transverse eigenvalue equation(2.15).

Remark 2.2. In the overcompressive case, the change to integrated coordinates removes twoadditional zeros of the Evans function for the transverse equations (2.15) that would other-wise occur at λ = 0, making possible a unified study across different parameter values/shocktypes.

2.3 Expression as first-order system

The various eigenvalue equations may be put in a common framework as (complex) systemsof first-order ODE

(2.16) W ′ = A(λ, x)W, W ∈ Cn, A ∈ Cn×n.

2.3.1 Reaction–diffusion

Adjoining first derivatives, we obtain

(2.17)(ww′

)′=(

0 1λ+ dg(u) 0

)(ww′

), dg(u) = (1/2)(3u2 − 1), u = − tanh(x/2).

2.3.2 Burgers

Likewise, we obtain

(2.18)(ww′

)′=(

0 1λ df(u)

)(ww′

), df(u) = u = − tanh(x/2).

2.3.3 Gas dynamics

Following [HLZ], we obtain after a brief calculation

(2.19)

uvv′

′

=

0 λ 10 0 1λv λv f(v)− λ

uvv′

,

where

(2.20) f(v) = 2v − (γ − 1)(1− v+

1− vγ+

)(v+v

)γ−(1− v+

1− vγ+

)vγ+ − 1.

16

2.3.4 MHD

The transverse eigenvalue equations (2.15) may be written as a first-order system

(2.21)

wµw′

αα′

σµ0v

′

=

0 1/µ 0 0λv v/µ 0 −σB∗1 v0 0 0 σµ0v0 −B∗1 v/µ λv σµ0v

2

wµw′

αα′

σµ0v

,

indexed by the three parameters B∗1 , σ, and µ0 := σµ0, augmenting (6.8),

2.4 Discussion

For four models of physical interest, we have found traveling-wave solutions and character-ized spectral stability in terms of existence or nonexistence of solutions decaying as x→ ±∞of an associated eigenvalue equation of linear ordinary differential type. However, the so-lution of a linear ODE with variable coefficients is itself nontrivial, and in general cannotbe carried out explicitly. Following, we will study the question of existence numericallyby an efficient “shooting” method based on the Evans function, and analytically by theexamination of various asymptotic limits.

Remark 2.3. In the scalar second-order cases 1 and 2, it is known by the adaption ofSturm–Liouville theory to the case of the whole line that nonzero eigenvalues with <λ ≥ 0exist if and only if the linear stationary mode u′ does not vanish [He, Sa], that is, the profileu is monotone (as it is for the example − tanh(x/2)). Nonetheless, they serve as usefulpractice cases in setting up numerics, etc.

3 The Evans function

We now present in a general framework the Evans function introduced in [E1]–[E4] andelaborated and generalized in [AGJ, PW, GZ, HuZ2]. Let L be a linear differential opera-tor with asymptotically constant coefficients along some preferred spatial direction x, andsuppose that the eigenvalue equation

(3.1) (L− λ)w = 0

may be expressed as a first-order ODE in an appropriate phase space:

(3.2) W ′ = A(x, λ)W, limx→±∞

A(x, λ) = A±(λ),

with A analytic in λ as a function from C to C1(R,Cn×n).Suppose that:(h0) for <λ > 0, the dimension k of the stable subspace S+ of A+ and dimension n− k

of the unstable subspace U− of A− summing to the dimension n of the entire phase space

17

(“consistent splitting” [AGJ]); moreover, the associated eigenprojections extend analyticallyto <λ = 0.

(h1) In a neighborhood of any λ0 ∈ <λ ≥ 0, for fixed C, θ > 0,

|A−A±|(x, λ) ≤ Ce−θ|x| for x ≷ 0.

Remark 3.1. The first part of condition (h0) corresponds to the absence of spectra <λ > 0of the limiting constant coefficient operators of L as x → ±∞. This in turn correspondsto the absence of essential spectrum of L on <λ > 0, by a standard connection betweenessential spectrum of asymptotically constant coefficient operators and the spectrum of theirlimiting constant-coefficient operators as x→ ±∞ [He, Sa].

Condition (h1) holds in the traveling-wave context whenever the underlying travelingwave is a connection between hyperbolic rest points of the traveling-wave ODE, definedas equilibria whose linearized equations have no center subspace, by the Stable (Unstable)Manifold Theorem (Theorem D.9, Appendix D.5). Both conditions are satisfied for theexamples considered here.

3.1 Limiting eigenprojections

Lemma 3.2. Denote by Π+ and Π− the eigenprojections of A+ onto its stable subspaceand A− onto its unstable subspace, with A± defined as above. Then, condition (h0) impliesthat Π± are analytic on <λ > 0 and extend analytically to <λ = 0.

Proof. The first assertion follows by standard matrix perturbation theory, and the assump-tion that stable and unstable subspaces remain spectrally separated [K], the second byassumption.

3.2 Limiting eigenbases: Kato’s ODE

Denote by Π+ and Π− the eigenprojections of A+ onto its stable subspace and A− ontoits unstable subspace, with A± defined as above. Assume that these are analytic on somedesired region of investigation λ ∈ Λ (in this case Λ = <λ ≥ 0. Introduce the complexODE

(3.3) R′ = Π′R, R(λ∗) = R∗,

where ′ denotes d/dλ, λ∗ ∈ Λ is fixed, Π = Π±, and R = R± with R+ and R− n × k andn × (n − k) complex matrices, and R∗ is full rank and satisfies Π(λ∗)R∗ = R∗: that is, itscolumns are a basis for the stable (resp. unstable) subspace of A+ (resp. A−).

Lemma 3.3 ([K, Z2]). For Λ simply connected, there exists a unique global analyticsolution R of (3.3) on Λ such that (i) rankR ≡ rankR∗, (ii) ΠR ≡ R, and (iii) ΠR′ ≡ 0.

Proof. As a linear ODE with analytic coefficients, (3.3) possesses a unique analytic solutionin a neighborhood of λ∗, that may be extended globally along any curve, whence, by the

18

principle of analytic continuation, it possesses a global analytic solution on any simplyconnected domain containing λ∗ [K]. Property (i) follows likewise by the fact that R satisfiesa linear ODE. Differentiating the identity Π2 = Π following [K] yields ΠΠ′ + Π′Π = Π′,whence, multiplying on the right by Π, we find the key property

(3.4) ΠΠ′Π = 0.

From (3.4), we obtain

(ΠR−R)′ = (Π′R+ ΠR′ −R′) = Π′R+ (Π− I)Π′R = ΠΠ′R,

which, by ΠΠ′Π = 0 and Π2 = Π gives

(ΠR−R)′ = −ΠΠ′(ΠR−R), (ΠR−R)(λ∗) = 0,

from which (ii) follows by uniqueness of solutions of linear ODE. Expanding ΠR′ = ΠΠ′Rand using ΠR = R and ΠΠ′Π = 0, we obtain ΠR′ = ΠΠ′ΠR = 0, verifying (iii).

Remark 3.4. Property (iii) indicates that the Kato basis is an optimal choice in the sensethat it involves minimal variation in R. It is also useful as a direct characterization of theKato basis independent of (3.3); see [HSZ, BDG] or Section 3.5 below. From the propertiesΠR′ = 0 and ΠR = R, we obtain the theoretically useful alternative formulation

(3.5) R′ = [Π′,Π]R,

where [P,Q] denotes the commutator PQ−QP . (See Exercises 3.10 below.)

3.3 Limiting behavior

Lemma 3.5. Assuming (h1), there exist coordinate changes W = P±Z±, P± = I + Θ±,defined and uniformly invertible on x ≷ 0, with

(3.6) |(∂/∂λ)jΘp±| ≤ C(j)e−θ|x| for x ≷ 0, 0 ≤ j,

for any 0 < θ < θ, converting (3.1) to the constant-coefficient limiting systems Z ′ = A±Z.

Proof. See the conjugation lemma, Lemma C.4.

Remark 3.6. Equivalently, solutions of (3.1) may be factored as

(3.7) W = (I + Θ±)Z±,

where Z± satisfy the limiting, constant-coefficient equations and Θ± satisfy (3.6).

Corollary 3.7. On <λ > 0, the manifolds of solutions of (3.1) decaying as x → ±∞ arespanned by (PRj)±, where R±j are the columns of R± defined in Lemma 3.3.

19

3.4 Evans function construction

Definition 3.1. We define the Evans function on Λ as

(3.8)D(λ) := det(P+R+

1 , · · · , P+R+

k , P−R−k+1, P

−R−n )|x=0,

= det(W+

1 · · · W+k W−

k+1 · · · W−n

)|x=0

,

where W+1 , . . . ,W

+k and W−

k+1, . . . ,W−n are analytically-chosen (in λ) bases of the manifolds

of solutions decaying as x→ +∞ and −∞.

See also [AGJ, PW, KS, GZ, Z1, HuZ2, HSZ] and references therein.

Proposition 3.8 ([GJ1, GJ2, MaZ3]). The Evans function is analytic in λ on <λ ≥ 0.On <λ > 0, its zeros agree in location and multiplicity with eigenvalues of L.

Proof. Local analyticity follows by construction, as does agreement in location (recallingthat eigenvalues are λ such that the manifolds of solutions decaying at ±∞ have nontrivialintersection). Agreement in multiplicity was shown in [GJ1, GJ2]. See also [MaZ3]. Globalanalyticity follows by the observation that the Evans function is uniquely determined by thebehavior at ±∞ of W±

j , which is independent of the choice of transformation P± → I.3

Remark 3.9. Note that this is indeed “the” Evans function, and not “an” Evans function,since we have uniquely specified a choice of asymptotic bases R±j .

Analogous to the characteristic polynomial for a finite-dimensional operator, D(·) isanalytic in λ with zeroes corresponding in both location and multiplicity to the eigenvaluesof the linear operator L [GJ1, GJ2]. Taking the winding number around a contour Γ =∂Λ ⊂ <λ ≥ 0, where Λ is a set outside which eigenvalues may be excluded by othermethods (e.g. energy estimates or asymptotic ODE theory), counts the number of unstableeigenvalues in Λ of the linearized operator about the wave, with zero winding numbercorresponding to stability.

In general, the Evans function, like the background profile, is only numerically evaluable.(Nonetheless, as we will see, it can yield valuable information through various asymptoticlimits.) In the exercises below, we describe some simple situations in which it is exactlyevaluable.

Exercises 3.10. In the following exercises, we use the formulation (3.5) of Kato’s ODE.1. Let Π = I − Π be the projector complementary to Π. Then, the associated Kato

basis S, determined by S′ = [Π′, Π]S, satisfies the same equation S′ = [Π′,Π]S as does R,since [Π′, Π] = [(I −Π)′, (I −Π] = [(Π− I)′, (Π− I)] = [Π′,Π].

3For example, one may observe as in [GZ, ZS] that the minor W+1 ∧ · · · ∧W+

k is uniquely determined upto a constant complex scalar factor as the strong stable manifold as x → +∞ of the asymptotically constantlinearized evolution induced on k-forms by the linear flow (3.2). Alternatively, noting that P+

1 R+ = P+2 R+α,

α ∈ Ck×k, for two different transformations P+1 , P+

2 , we may deduce from P+j → I as x →∞ that det α = 1.

20

2. Using the result of Exercise 1 together with Abel’s Lemma (in λ), show that theEvans function associated with a constant-coefficient operator is nonzero and identicallyconstant in λ.

3. Show that Kato’s equation (3.3) is invariant under a constant (with respect to λ, butnot necessarily with respect to x...) change of coordinates R = TR, that is, the solution ofR′ = [Π′, Π]R with Π := TΠT−1 is R ≡ TR, where R′ = [Π′,Π]R.

4. Assuming the existence of a linearized Hopf–Cole transformation taking the inte-grated eigenvalue equation for Burgers equation to that for the heat equation, reprove thefact that the Evans function of the integrated Burgers eigenvalue equation is identicallyconstant.

3.5 A fundamental example

Consider Burgers’ equation, ut + (u2)x = uxx, and the family of stationary viscous shocksolutions

(3.9) uε(x) := −ε tanh(εx/2), limz→±∞

uε(z) = ∓ε

of amplitude |u+ − u−| = 2ε going to zero as ε→ 0. Burgers’ equation models behavior inthe weak shock limit of general systems; see, e.g., [PZ].

The linearized eigenvalue equation u′′ = (uεu)′+λu about uε, expressed in the integratedvariable w(x) :=

∫ x−∞ u(y)dy, appears as

(3.10) w′′ = uεw′ + λw.

This reduces by the linearized Hopf–Cole transformation w = sech(εx/2)z to the constant-coefficient linear oscillator equation z′′ = (λ+ ε2/4)z, yielding exact solutions

(3.11) w±(x, λ) = sech(εx/2)e∓√

ε2/4+λx

decaying, respectively, as x→ ±∞, with asymptotic behavior

(3.12) W±(x, λ) ∼ eµ±(λ)xV±(λ),

where µ±(λ) := ∓(ε/2+√ε2/4 + λ and V± := (1, µ±(λ))T are the eigenvalues and eigenvec-

tors of the limiting constant-coefficient equations at plus and minus spatial infinity writtenas a first-order system, W± := (w,w′)T

±.Defining an Evans function D(λ) := det(W−,W+)|x=0, we may compute explicitly

(3.13) D(λ) = −2√ε2/4 + λ.

However, this is not “the” Evans function

D(λ) := det(W−,W+)|x=0

21

specified in the previous sections, which is constructed, rather, from a special basis

W± = c±(λ)W± ∼ eµ±xV ±,

where V± = c±(λ)V± are “Kato” eigenvectors determined uniquely (up to a constant factorindependent of λ) by the property that there exist corresponding left eigenvectors V ± suchthat

(3.14) (V · V )± ≡ constant, (V · V )± ≡ 0,

where “ ˙ ” denotes d/dλ; see [K, GZ, HSZ] for further discussion.Computing dual eigenvectors V± = (λ+µ2)−1(λ, µ±) satisfying (V ·V)± ≡ 1, and setting

V ± = c±V±, V ± = V±/c±, we find after a brief calculation that (3.14) is equivalent to thecomplex ODE

(3.15) c± = −( V · VV · V

)±c± = −

( µ

2µ− ε

)±c±,

which may be solved by exponentiation, yielding the general solution

(3.16) c±(λ) = C(ε2/4 + λ)−1/4.

Initializing at a fixed nonzero point, without loss of generality c±(1) = 1,4 and noting thatDε(λ) = c−c+Dε(λ), we thus obtain the remarkable formula

(3.17) Dε(λ) ≡ −2√ε2/4 + 1.

That is, with the “Kato” normalization, the Evans function associated with a Burgers shockis not only stable (nonvanishing), but identically constant.

4 Abstract stability theorem: the Evans condition

Consider a viscous shock solution

(4.1) U(x, t) = U(x− st), limz→±∞

U(z) = U±,

of a hyperbolic–parabolic system of conservation laws

(4.2) Ut + F (U)x = (B(U)Ux)x,

x ∈ R, U , F ∈ Rn, B ∈ Rn×n. Profile U satisfies the traveling-wave ODE

(4.3) B(U)U ′ = F (U)− F (U−)− s(U − U−).4In the numerics of Section 6.4.4, we initialize always at λ = 10.

22

In particular, the condition that U± be rest points implies the Rankine–Hugoniot condtions

(4.4) F (U+)− F (U−) = s(U+ − U−) = 0

of inviscid shock theory. Following this analogy, define the degree of compressivity d asthe total number of incoming hyperbolic characteristics toward the shock profile minus thedimension of the system, or d := dimU(A−)+dimS(A+)−n, where d = 1 corresponds to astandard Lax shock, d ≤ 0 to an undercompressive shock, and d ≥ 2 to an overcompressiveshock.

Denote A(U) = FU (U),

(4.5) B± := limz→±∞

B(z) = B(U±), A± := limz→±∞

A(z) = FU (U±).

Following [TZ3, Z2, Z3], we make the structural assumptions:

(A1) U =(U1

U2

), F =

(F1

F2

), B =

(0 00 b

), b nonsingular, where U ∈ Rn,

U1 ∈ Rn−r, U2 ∈ Rr, and b ∈ Rr×r; moreover, the U1-coordinate F1(U) of F is linear in U(strong block structure).

(A2) There exists a smooth, positive definite matrix A0(U), without loss of generalityblock-diagonal, such that A0

11A11 is symmetric, A022b is positive definite but not necessarily

symmetric, and (A0A)± is symmetric.(A3) No eigenvector of A± lies in KerB± (genuine coupling [Kaw, KSh]).

To (A1)–(A3), we add the following more detailed hypotheses. Here and elsewhere,σ(M) denotes the spectrum of a matrix or linear operator M .

(H0) F,B ∈ Ck, k ≥ 4.(H1) σ(A11) real, constant multiplicity, with σ(A11) < s or σ(A11) > s.(H2) σ(A±) real, semisimple, and different from s.(H3) Considered as connecting orbits of (4.3), U lies in an `-dimensional manifold,

` ≥ 1, of solutions (4.1) connecting the fixed pair of endstates U±.

Conditions (A1)–(A3) are a slightly strengthened version of the corresponding hypothe-ses of [MaZ4, Z2, Z3] for general systems with “real”, or partially parabolic viscosity, the dif-ference lying in the strengthened block structure condition (A1): in particular, the assumedlinearity of the U1 equation. Conditions (H0)–(H3) are the same as those in [MaZ4, Z2, Z3].The class of equations satisfying our assumptions, though not complete, is sufficiently broadto include many models of physical interest, in particular compressible Navier–Stokes equa-tions and the equations of compressible magnetohydrodynamics (MHD), expressed in La-grangian coordinates, with either ideal or “real” van der Waals-type equation of state asdescribed in Appendix B. See [TZ3, Z9] for further discussion/generalization.

Remark 4.1. Much more general assumptions (in particular, allowing Eulerian coordi-nates) are sufficient for the stability results reported here; see [MaZ4, Z1]. Lagrangiancoordinates are important for later material on conditional stability and bifurcation, how-ever, and so we make the stronger assumptions here for the sake of a common presentation.

23

Remark 4.2. In all of the examples discussed, A11 is just a scalar multiple αI of the iden-tity, corresponding with simple transport along fluid particle paths, with genuine couplingequivalent to A12 full rank.

By the change of coordinates x → x − st, we may assume without loss of generalitys = 0 as we shall do from now on. Linearizing (4.2) about the (now stationary) solutionU ≡ U yields linearized equations

(4.6) Ut = LU := −(AU)x + (BUx)x,

(4.7) B(x) := B(U(x)), A(x)V := dF (U(x))V − (dB(U(x))V )Ux,

for which the generator L possesses [He, Sa, ZH] both a translational zero-eigenvalue andessential spectrum tangent at zero to the imaginary axis.

4.1 Profile facts

Remark 4.3. Conditions (H1)–(H2) imply that U± are nonhyperbolic rest points of ODE(4.3) expressed in terms of the U2-coordinate, whence, by standard ODE theory,

(4.8) |∂rx(U − U±)(x)| ≤ Ce−η|x|, 0 ≤ r ≤ k + 1,

for x ≷ 0, some η, C > 0; in particular, |U ′(x)| ≤ Ce−η|x|.

Define d2 to be the dimension of the unstable manifold at −∞ of the traveling-waveODE expressed in the U2 coordinate plus the dimension of the stable manifold at +∞,minus the dimension r of the ODE.

Proposition 4.4 (MaZ3). Under (A1)–(A3), (H0)–(H2), d2 is equal to the degree of com-pressivity d, hence, if the traveling-wave connection is transversal, the set of all traveling-wave connections between U± form an `-dimensional manifold Uα, α ∈ R`, in the vicinityof U , with ` = d. In any case, for Lax or overcompressive shocks (d ≥ 1), the linearizedequations have a d-dimensional set of stationary solutions decaying as x → ±∞, given inthe transversal case by Span∂αj U

α. Undercompressive connections are never transversal.

Proof. Solving the U1 equation of the traveling-wave ODE, we obtain U1 = A−111 A12U2,

hence, subsituting into the U2 equation of the linearized equations about U±, we obtain

u′2 = b−1± (A22 −A21A

−111 A12)±U2.

The first result then follows by a homotopy argument to the case b = I, A12 = 0, notingthat det b, det(A22 − A21A

−111 A12)± = detA±/detA±,11, and detA±,11 do not vanish by

assumption. See [MaZ3] for extensions and further details. The second result follows bythe Rank Theorem for finite-dimensional linear operators.

24

4.2 The Evans function condition

Lemma 4.5. Under (A1)–(A3), (H0)–(H2),

(4.9) <σ(−iξA− ξ2B)± ≤ − θ|ξ|2

1 + |ξ|2

θ > 0, for all ξ ∈ R, where (here and elsewhere) σ(M) denotes the spectrum of a matrix orlinear operator.

Proof. A standard consequence of (A1)–(A3) is existence of a skew-symmetric matrix Ksuch that <(A0B+KA) ≥ θ > 0, where <M := (1/2)(M +M∗) denotes symmetric part ofa matrix or linear operator. In the case described in Remark 4.2 that A11 is a nonvanishing

scalar multiple of the identity, if also A0 = I, we may take K := η

(0 A12

−A21 0

)for

η > 0 sufficiently small (exercise); the general case A0 6= I may be reduced to this one bycoordinate transformation. Assuming

(−iξA− ξ2B)±V = λV,

take the real part of the inner product with (A0ξ2 − ηiξK +A0)V , η > 0 sufficiently small,to obtain

<λ(1 + |ξ|2)|V |2 ≤ <〈(A0ξ2 − ηiξK +A0)V, λV 〉= 〈(A0ξ2 − ηiξK +A0)V, (−iξA− ξ2B)V 〉= −|ξ|2〈<((A0B +KA)V, V 〉≤ −θ|ξ|2|v|2,

giving the result.

Corollary 4.6. Under (A1)–(A3), (H0)–(H2), the essential spectrum σess(L) lies to theleft of a curve λ(ξ) = − θ|ξ|2

1+|ξ|2 , θ > 0, there exist essential spectra λ∗(ξ) lying to the right of

λ(ξ) = − θ−2|ξ|21+|ξ|2 , θ2 > θ > 0, and the eigenvalue equations satisfy (h0)–(h1).

Proof. The first two assertions follow by Lemma 4.5 and the standard fact [He] that theboundary of the essential spectrum of an asymptotically constant operator is the rightenvelope of the essential spectra of the limiting constant-coefficient operators as x→ ±∞.Consistent splitting for <λ ≥ 0 \ 0 then follows by the fact that the curves λ(ξ) ∈σ(−iξA − ξ2B)± are the values of λ at which the spatial eigenvalues cross the imaginaryaxis, so that the rightmost envelope is the boundary of consistent splitting [AGJ, GZ],verifying the first part of (h0). The second part, continuous extension of the stable/unstableprojectors to λ = 0, then follow by standard matrix perturbation theory [K]; see, e.g.,[GZ, MaZ3, Z1]. Condition (h1) follows by Remark 4.3.

Corollary 4.7. The Evans function D(λ) exists and is analytic on <λ ≥ 0, with at least` zeros at λ = 0.

25

Proof. Immediate consequence of Proposition 4.6 and Lemma 4.4, noting that decayingsolutions of the eigenvalue equations correspond with zeros of the Evans function.

Definition 4.1. We define the Evans stability condition as:(D) There exist precisely ` zeros of D(·) on <λ ≥ 0, necessarily at λ = 0.

4.3 Basic stability theorem

Theorem 4.8 ([MaZ4, R]). For Lax or overcompressive shocks, under (A1)–(A3), (H0)–(H3), and (D), U is nonlinearly orbitally stable in L1 ∩H3 under perturbations with L1 ∩H3 norm and L1-first moment E0 := ‖U(·, 0) − U‖L1∩H3 and E1 := ‖|x| |U(·, 0) − U |‖L1

sufficiently small, in the sense that, for some α(·) ∈ R`, any ε > 0, all Lp,

(4.10)

|U(x, t)− Uα(t)(x)|Lp ≤ C(1 + t)−12(1− 1

p)(E0 + E1),

|U(x, t)− Uα(t)(x)|H3 ≤ C(1 + t)−14 (E0 + E1),

α(t) ≤ C(1 + t)−1+ε(E0 + E1),

α(t) ≤ C(1 + t)−1/2+ε(E0 + E1).

Proof. The proof, carried out in [MaZ4, Z1, R], is beyond the scope of these lectures.However, see the more detailed result of Theorem 7.2 for Laplacian (identity) viscosity.

Remark 4.9. For stability results in the undercompressive case, see the more complicatedpointwise analyses of [HZ, RZ], again confirming that Evans implies nonlinear stability.

4.4 Meaning of the Evans condition

As λ = 0 is an embedded eigenvalue in the essential spectrum of L (see Corollary 4.6), themeaning of the Evans function at λ = 0 is not immediately clear. The answer is given by thefollowing fundamental lemma describing the low-frequency behavior of the Evans function.

Recall in the Lax case that inviscid shock stability is determined [Er4, M1, M2, M3, Me]by nonvanishing of the Lopatinski determinant

(4.11) ∆(λ) := λ det(r−1 , . . . , r−p−1, r

+p+1, . . . , r

+n , (U+ − U−)),

where r−1 , . . . r−p−1 denote eigenvectors of dF (U−) associated with negative eigenvalues and

r+p+1, . . . r+n denote eigenvectors of dF (U+) associated with positive eigenvalues.

Lemma 4.10 ([GZ, ZS]). Under (A1)–(A3), (H0)–(H3),

(4.12) D(λ) = β∆(λ) + o(|λ|`),

where β is a constant transversality coefficient which in the Lax or overcompressive case isa Wronskian of the linearized traveling-wave ODE measuring transverality of the connect-ing orbit U(·), with β 6= 0 equivalent to transversality, and ∆(·) is a positive homogeneous

26

degree ` function which in the Lax or undercompressive case is exactly the Lopatinski de-terminant determing inviscid stability. In the undercompressive case, β measures “maximaltransversality”, or transversality of connections within the subspace spanned by the tangentspaces of stable and unstable manifolds.

Proof. Straightforward Lyapunov–Schmidt reduction using the variational equations withrespect to λ, Leibnitz rule, and judicious block-determinant reduction; see [PW, GZ, ZS].

Corollary 4.11. Under (A1)–(A3), (H0)–(H3), Evans stability is equivalent to the threeconditions of (i) spectral stability (defined as nonexistence of eigenvalues in <λ ≥ 0\0),(ii) maximal transversality of the traveling-wave connection, and (iii) hyperbolic stability ofthe associated inviscid shock (Lax or undercompressive case) or nonvanishing of an inviscidstability-determinant-like function (overcompressive case).

4.5 Heuristic discussion

Note absence of spectral gap, etc... Compare standard stability result in reaction-diffusioncase, follows by simple ODE-type Proposition E.5, and requires only transverality = simpic-ity of the translational zero-eigenvalue. The conservation-law case requires the additionalcondition of inviscid stability, has to do with constants of integration in integrating up thetraveling-wave ODE. Connection with the inviscid theory through hidden singular pertur-bation, low-frequency limit; see discussion, [ZS, Z1, Z2].

4.6 Verification for small amplitudes

For small amplitude (without loss of generality stationary) shock waves |U+ − U−| suffi-ciently small, and U± approaching a base state U0 for which dF (U0) has a single “genuinelynonlinear” zero eigenvalue, the existence problem may be studied as a bifurcation from theconstant solution, both for inviscid [L, Sm] and viscous [MP, Pe] shock waves, reducingby Implicit Function Theorem (center manifold reduction) to the scalar, Burgers case– inparticular, a classical Lax-type wave. In this case, conditions (ii)–(iii) of Corollary 4.11may be verified via the same reduction. Condition (i) may then be verified by energy esti-mates [HuZ1] or by singular perturbation analysis in spirit of the center manifold reduction[PZ, FS]; see [HuZ1, Z1] for further discussion. Small-amplitude waves bifurcating frommultiple zero eigenvalues are typically of nonclassical over- or undercompressive type, andmay be unstable; see, for example [AMPZ, GZ, Z6]. However, their stability should bedeterminable in principle by similar reduction/singular perturbation techniques to thoseused in the Lax case, an interesting open problem.

4.7 Verification for large amplitudes

Except in isolated special cases (e.g., [Z6]), stability of large-amplitude shock waves for themoment must be verified either numerically or else in certain asymptotic limits. Whether

27

or not there exist simple structural conditions for large-amplitude stability, connected withexistence of a convex entropy perhaps, remains a fundamental open question, as does eventhe profile existence problem for large-amplitude shocks; see [Z1, Z2] for further discussion.In what follows, we describe a workable general approach based on numerical approximationand asymptotic ODE theory.

5 Numerical approximation of the Evans function

We now discuss the general question of numerical stability analysis by approximation of theEvans function. It is more difficult to show stability than instability, since it is necessary toverify stability on an uncountable set of λ where, by contrast, to show instability it sufficesto find a single unstable eigenvalue <λ > 0. Accordingly, it requires a bit of organizationin order to efficiently determine stability numerically.

An efficient method introduced in [EF, Er1, Er2] is the “Nyquist diagram” approachfamiliar from control theory, based on the Principle of the Argument of Complex Analysis.Specifically, taking the winding number around a contour Γ = ∂Λ ⊂ <λ ≥ 0, where Λ is aset outside which eigenvalues may be excluded by other methods (e.g. energy estimates orasymptotic ODE theory), counts the number of unstable eigenvalues in Λ of the linearizedoperator about the wave, with zero winding number corresponding to stability. See, e.g.,[Br1, Br2, BrZ, BDG, HSZ, HuZ2, BHRZ, HLZ, CHNZ, HLyZ1, HLyZ2, BHZ]. Alterna-tively, one may use Mueller’s method or any number of root-finding methods for analyticfunctions to locate individual roots; see, e.g., [OZ1, LS].

In this context, the numerical approximation of the Evans function breaks into twosteps: (i) the computation of analytic bases for stable (resp. unstable) subspaces of A+

(resp. A−) and (ii) the propagation of these bases by ODE (3.2) on a sufficiently largeinterval x ∈ [M, 0] (resp. x ∈ [−M, 0]). In both steps, it is important to preserve thefundamental property of analyticity in λ, which is extremely useful in computing roots bywinding number or other methods. Both problems concern (different aspects of) numericalpropagation of subspaces, the first in λ and the second in x, thus tying into large bodies oftheory in both numerical linear algebra [ACR, DDF, DE1, DE2, DF] and hydrodynamicstability theory [Dr, Da, NR1, NR2, NR3, NR4, B].

5.1 Shooting algorithms

Problem (i) has been examined in [HSZ, Z2, BHZ]; for completeness, we gather the (existingbut dispersed) conclusions in Section 5.2. We now focus on problem (ii). Our basic principlesfor efficient numerical integration of (3.2) are readily motivated by consideration of thesimpler constant-coefficient case

(5.1) Wx = AW, A ≡ constant.

We must avoid two main potential pitfalls:

28

1. Wrong direction of integration. Consider the simplest case that the dimension ofthe stable subspace of A is one, so that we seek to resolve a single decaying eigenmode, withall other modes exponentially growing with increasing x. Evidently, the correct direction ofintegration is the backwards direction, from x = +M back to x = 0, in which the desiredmode is exponentially growing, and errors in other modes exponentially decay. Integratingin the forward direction would be numerically disastrous, with exponential error growtheηM , where η is the spectral gap between decaying and growing modes (recall, M is large):the analog for Evans function computations of integrating a backward heat equation. Ingeneral, we must always integrate from infinity toward zero.

2. Parasitic modes (related to 1). For general systems of equations, the dimensionof the stable subspace of A typically involves two or more eigenmodes, with distinct decayrates µ1 < µ2 < 0. Integrating in backward direction as prescribed in part 1 above, weresolve the fastest decaying µ1 mode without difficulty. However, in trying to resolve theslower decaying µ2 mode, we experience the problem that errors in the direction of the µ1

mode grow exponentially relative to the desired µ2 mode, at relative rate e(µ2−µ1)M . Thatis, parasitic faster-decaying modes will tend to take over slower-decaying modes, preventingtheir resolution. Degradation of results from parasitic modes is of the same rough order asthat resulting from integrating in the wrong spatial direction, differing only in the fact thatthe maximum spectral gap between two decaying modes is typically one-half or less of themaximum gap between decay and growing modes.

5.1.1 Solution one: the centered exterior product method

Problem 1 is easily avoided by integrating in the correct direction. Problem 2 may beovercome by working in the exterior product space W+

1 ∧· · ·∧W+k ∈ C(n

k) (resp. W−k+1∧· · ·∧

W−n ∈ C( n

n−k)), for which the desired subspace appears as a single, maximally stable (resp.unstable) mode, the Evans determinant then being recovered through the isomorphism

(5.2) det(W+

1 · · · W+k W−

k+1 · · · W−n

)∼ (W+

1 ∧ · · · ∧W+k ) ∧ (W−

k+1 ∧ · · · ∧W−n );

see [AS, Br1, Br2, BrZ, BDG, AlB] and ancestors [GB, NR1, NR2, NR3, NR4]. This reducesthe problem to the case k = 1. We may further optimize by factoring out the expectedasymptotic decay rate eµx of the single decaying mode and solving the “centered” equation

(5.3) Z ′ = (A− µI)Z, Z(+∞) = r : A+r = µr

for Z := e−µxW , which is now asymptotically an equilibrium as x → +∞. With thesepreparations, one obtains excellent results [BDG, HuZ2]; however, omitting any one ofthem leads to a loss of efficiency of at least an order of magnitude in our experience [HuZ2].

Exercise 5.1. If W ′j = AWj, show that W := W1 ∧ · · · ∧Wk satisfies the “lifted ODE”

W ′ = A′W, where A is defined by the Leibnitz formula

A(W1 ∧ · · · ∧Wk) := (AW1 ∧ · · · ∧Wk) + · · ·+ (W1 ∧ · · · ∧AWk).

29

5.1.2 Solution two: the polar coordinate method

Unfortunately, the dimension(nk

)of the phase space for the exterior product grows ex-

ponentially with n, since k is ∼ n/2 in typical applications. This limits its usefulness ton ≤ 10 or so, whereas the Evans system arising in compressible MHD is size n = 15, k = 7[BHZ], giving a phase space of size

(nk

)= 6, 435: clearly impractical. A more compact,

but nonlinear, alternative is the polar coordinate method of [HuZ2], in which the exteriorproducts of the columns of W± are represented in “polar coordinates” (Ω, γ)±, where thecolumns of Ω+ ∈ Cn×k and Ω− ∈ C(n−k)×k are orthonormal bases of the subspaces spannedby the columns of W+ :=

(W+

1 · · · W+k

)and W− :=

(W−

k+1 · · · W−n

), W±

j defined asin (3.1), i.e., W+ = Ω+α+, W− = Ω−α−, and γ± := detα±, so that

W+1 ∧ · · · ∧W+

k ∧ = γ+(Ω1+ ∧ · · · ∧ Ωk

+),

where Ωj± denotes the jth column of Ω±, and likewiseW−

k+1∧· · ·∧W−n = γ−(Ω1

−∧· · ·∧Ωn−k− ).

Imposing the choice Ω∗Ω′ to fix a specific realization (similar as the choice PP ′ = 0determining Kato’s ODE), we obtain after a brief calculation (exercise, below; see [HuZ2])the block-triangular system

(5.4)Ω′ = (I − ΩΩ∗)AΩ,

(log γ)′ = trace (Ω∗AΩ)− trace (Ω∗AΩ)(±∞),

γ := γe−trace (Ω∗AΩ)(±∞)x, for which the “angular” Ω-equation is exactly the continuousorthogonalization method of Drury [Dr, Da], and the “radial” γ-equation, given Ω, may besolved by simple quadrature. Ignoring the numerically trivial radial equation, we see thatproblem 2 by fiat does not occur. Moreover, for constant A, it is easily verified that invariantsubspaces Ω of A are equilibria of the flow, so no centering is needed to improve efficiency.Indeed, for A constant, solutions of the γ-equation are constant, so that any first-order orhigher numerical scheme resolves log γ exactly; thus, the γ-equation may for simplicity besolved together with the Ω-equation, with no need for a final quadrature sweep.5 The Evansfunction is recovered, finally, through the relation

(5.5) D(λ) = det(W+

1 · · · W+k W−

k+1 · · · W−n

)|x=0 = γ+γ− det(Ω+,Ω−)|x=0.

This method performs comparably to the exterior product method in simple examplesinvolving small systems [HuZ2], but does not suffer from exponential growth in complexitywith respect to the size of the system.

Exercise 5.2. Multiplying the defining relation (Ωα)′ = AΩα on the left by (I −ΩΩ∗) andusing Ω∗Ω′ = 0 and Ω∗Ω = I, derive (5.4)(i). Multiplying on the left by Ω∗, show thatα′ = Ω∗AΩα, yielding γ′ = trace (Ω∗AΩ)γ by Abel’s Theorem.

Numerical stability. An important detail is to explain the apparent contradictionbetween observed good results for the polar coordinate method [HuZ2] and the well-known

5See Section 5.3.2 for numerical prescriptions (Ω, γ)± of (Ω, γ) at ±∞.

30

instability of continuous orthogonalization [Da, BrRe]. The simple resolution is that, thoughcontinuous orthogonalization is in general unstable, it is in the present context stable!

Heuristically, this is quite simple to see. Intuitively, it is is clear that the stable manifoldof A+ is asymptotically attracting in backward x under the flow of (5.4) for Ω confined tothe Stiefel manifold S = Ω : Ω∗Ω = Ik, and in fact this is well known. Thus, we needonly verify that the Stiefel manifold, likewise, is attracting in backward x. Defining theStiefel error E(Ω) := Ω∗Ω− I, we obtain after a brief computation the error equation

(5.6) E ′ = −E(Ω∗AΩ)− (Ω∗AΩ)∗E

of [HuZ2]. Linearizing about the exact solution Ω → Ω+, E → 0 and replacing coefficients bytheir asymptotic limits, we obtain a linear equation E ′ = A+E , whereAE := −EA+−(A+)∗Eis a Sylvester operator, A+ := (Ω∗+A+Ω+), with eigenvalues and eigenmatrices −(aj + a∗k),rjr

∗k, where aj and rj are eigenvalues and eigenvectors of A+. Noting that the eigenvalues

of A+ are exactly the eigenvalues of A+ restricted to its stable subspace, i.e., the stableeigenvalues of A+, we find that A+ has positive real part eigenvalues, and so E decays inbackward x.

That is, the Stiefel manifold is attracting under the backward flow of continuous orthog-onalization (repelling under the forward flow) if Ω is a stable subspace of A, an observationthat previously seems to have gone unremarked. This confirms that, as suggested by nu-merical results of [HuZ2], continuous orthogonalization is roughly equally well-conditionedas the centered exterior product method. For details and further discussion, see [Z12].

5.1.3 Numerical implementation

Good results are obtained for either the exterior product or polar coordinate method witha standard RK45 integration scheme (as supported for example in MATLAB). The factthat both methods are “centered” in the sense that solutions are constant for constant-coefficient equations means that an adaptive A-stable scheme like RK45 can take largesteps in the approximately constant-coefficient far field while maintaining accuracy; for arigorous discussion of this issue, see [Z12], Section 3.5.1.

Truncation of the computational domain. An important detail is the choice ofapproximate plus and minus spatial infinities L±, which must be sufficiently small thatcomputational costs remain reasonable, but sufficiently large that the error from truncatingthe computational domains from (±∞, 0] to [L±, 0] remains within desired tolerance: moreprecisely, that the solution of the variable-coefficient Evans system remains within a desiredrelative error of its value at ±∞. Recall that this limiting value is a constant solution ofthe limiting constant-coefficient system, so may be expected to be a good approximationfor large |x|; indeed, the same fixed-point argument by which the exact, variable coefficientis defined (the gap and conjugation lemmas of Appendix C.1) shows that a good rule ofthumb for |A−A±| ∼ C±e

−θ±|x|, x ≷ 0, is

(5.7) L± ∼ (logC± + | log TOL|)/θ±

31

where TOL is desired relative error tolerance. That is, it is relatively insensitive to toleranceor coefficient C, depending mainly on the rates θ± of exponential convergence of A→ A±;see [HLyZ1], Section 5.3.3, for further details. This should be supplemented by convergencestudy.

5.2 Initialization of eigenbases at infinity

For completeness, we describe, following [BrZ, HSZ, HuZ2, Z2], a simple and effectivemethod for computing analytically-chosen initializing eigenbases at plus and minus spatialinfinity. Combined with the integration methods described in the rest of the paper, thisgives a basic working Evans solver that performs quite well in practice.6

5.2.1 Numerical implementation of Kato’s ODE

Choose a set of mesh points λj , j = 0, . . . , J along a path Γ ⊂ Λ and denote by Πj := Π(λj)and Rj the approximation of R(λj). Typically, λ0 = λJ , i.e., Γ is a closed contour.

Computing Πj. Given a matrix A, one may efficiently (∼ 32n3 operations; see [GvL,SB]) compute by “ordered” Schur decomposition7, i.e., Schur decomposition A = QUQ−1,Q orthogonal and U upper triangular, for which also the diagonal entries of U are orderedin increasing real part, an orthonormal basis

Ru := (Qk+1, . . . , Qn),

of its unstable subspace, where Qk+1, . . . , Qn are the last n − k columns of Q, n − k thedimension of the unstable subspace. Performing the same procedure for −A, A∗, and−A∗ we obtain orthonormal bases Rs, Lu, Ls also for the stable subspace of A and theunstable and stable subspaces of A∗, from which we may compute the stable and unstableeigenprojections in straightforward and numerically well-conditioned manner via

(5.8) Πs := Rs(L∗sRs)−1L∗s, Πu := Ru(L∗uRu)−1L∗u.

Applying this to matrices A±j := A±(λj), we obtain the projectors Π±j := Π±(λj). Hereafter,

we consider Π±j as known quantities.

Remark 5.3. It is tempting to instead simply call an eigenvalue–eigenvector solver andexpress Πs =

∑ rj l∗jl∗j rj

, where rj, lj are left and right eigenvalues associated with stable eigen-vectors. However, this becomes ill-conditioned near points where stable eigenvalues collide,as frequently happens for cases resulting from other than simple scalar equations.

First-order integration scheme. Approximating Π′(λj) to first order by the finitedifference (Πj+1−Πj)/(λj+1−λj) and substituting this into a first-order Euler scheme gives

Rj+1 = Rj + (λj+1 − λj)Πj+1 −Πj

λj+1 − λjRj ,

6Optimized versions may be found in the STABLAB package developed by J. Humpherys.7Supported, for example, in MATLAB and LAPACK.

32

or Rj+1 = Rj + Πj+1Rj −ΠjRj , yielding by the property ΠjRj = Rj (preserved exactly bythe scheme) the simple greedy algorithm

(5.9) Rj+1 = Πj+1Rj .

It is a remarkable fact [Z2] (a consequence of Lemma 3.3) that, up to numerical error,evolution of (5.9) about a closed loop λ0 = λJ yields the original value RJ = R0.

Second-order scheme. To obtain a second-order discretization of (3.3), we approx-imate Rj+1 − Rj ≈ ∆λjΠ′

j+1/2Rj+1/2, good to second order, where ∆λj := λj+1 − λj .Noting that Rj+1/2 ≈ Πj+1/2Rj to second order, by (5.9), and approximating Πj+1/2 ≈12(Πj+1 + Πj), also good to second order, and Π′

j+1/2 ≈ (Πj+1 − Πj)/∆λj , we obtain,combining and rearranging,

Rj+1 = Rj +12(Πj+1 −Πj)(Πj+1 + Πj)Rj .

Stabilizing by following with a projection Πj+1, we obtain after some rearrangement thereduced second-order explicit scheme

(5.10) Rj+1 = Πj+1[I +12Πj(I −Πj+1)]Rj .

This is the version we recommend for serious computations. For individual numerical ex-periments the simpler greedy algorithm (5.9) will usually suffice [Z2].

5.3 Initialization of Evans function ODE

Finally, we describe the conversion of analytic bases R±(λ) into initial data for the centeredexterior product or polar coordinate method.

5.3.1 Centered exterior product scheme

Denote R+ = (R+1 , . . . , R

+k ) and R− = (R−1 , . . . , R

−n−k). Then, the initializing wedge prod-

ucts RS± of Section 5.1.1 are given simply by

(5.11) RS+ := R+1 ∧ · · · ∧R

+k and RS− := R−1 ∧ · · · ∧R

−n−k.

5.3.2 Polar coordinate scheme

For each λ ∈ Λ, we may efficiently compute matrices Ω±(λ) whose columns form orthonor-mal bases for S±, by the same ordered Schur decomposition used in the computation of Π±.This need not even be continuous with respect to λ. Equating

Ω+α+(λ) = R+(λ), Ω−α−(λ) = R−(λ),

for some α±, we obtain

α+(λ) = Ω∗+R+(λ), α−(λ) = Ω∗−R−(λ),

33

and therefore the exterior product of the columns of R± is equal to the exterior product ofthe columns of Ω± times

(5.12) γ±(λ) := det(Ω∗R)±(λ)).

Thus, we may initialize the polar coordinate ODE (5.4) with

(5.13) Ω = Ω±, γ = det(Ω∗R)±.

Remark 5.4. The wedge products represented by polar coordinates (γ,Ω)±(λ), with γ±defined as in (5.12), are the same products RS± defined in (5.11). In particular, they areanalytic with respect to λ, though coordinates γ and Ω in general are not.

5.4 Error control

5.4.1 Shooting integration

The integration in x of the Evans system is numerically quite well-conditioned. For, inintegrating from (approximate) positive spatial infinity toward zero, we are trying to ap-proximate a neutral mode of the Evans system, with all other modes either neutral orgrowing in the forward x direction,8 hence neutral or exponentially decaying in the back-ward x direction of integration.

In the case of the exterior product method, all other modes are exponentially decayingin backward direction and so we see that errors are exponentially supressed. The case of thepolar coordinate method is less clear, since there are other neutral modes as well; however,as a consequence of a “numerical gap lemma”, they can be shown to be exponentiallydamped as well. For a rigorous treatment, see [Z12]. As a consequence, no error controlseems to be required beyond that built into a standard adaptive ODE solver like RK45.

5.4.2 Kato integration

As the integration of Kato’s ODE is carried out on a bounded closed curve, standard errorestimates apply and convergence is essentially automatic, and we shall not discuss it. Inapplications, we are often interested in determining the winding number of D about sucha curve. For this purpose, following [Br1, Br2, BrZ], we introduce a simple a posteriori“Rouche” check limiting the step-size in λ by the requirement that the relative change inD(λ) be less than a conservative 0.1. (By Rouche’s Theorem, relative error less than one issufficient to obtain the correct winding number.) In contrast to integration in x of the Evanssystem, integration in λ of the Kato system is a one-time cost, so not a rate-determiningfactor in the performance of the overall code. However, the computation time is sensitive(proportional) to the number of mesh points in λ at which the Evans function is computed,so this should be held down as much as possible.

8Here we are using the observation of Section 5.1.2, Numerical stability, in an important way.

34

6 Global stability analysis: a case study

To show the full power of the Evans function approach, we now carry out a global stabilityanalysis for the equations of isentropic gas dynamics (1.1.3) by a combination of energyestimates, numerical Evans function evaluation, and asymptotic ODE (the relevant asymp-totic theory being gathered in Appendix C). The result is that for an isentropic γ-law gas,γ ∈ [1, 3], viscous shock profiles are unconditionally stable.

6.1 The rescaled equations

Taking (x, t, v, u, a0) → (−εs(x − st), εs2t, v/ε,−u/(εs), a0ε−γ−1s−2), with ε so that 0 <

v+ < v− = 1, we consider stationary solutions (v, u)(x) of

vt + vx − ux = 0,

ut + ux + (av−γ)x =(ux

v

)x.

(6.1)

6.1.1 Profile equation

Steady shock profiles of (6.1) satisfy

(6.2) v′ = H(v, v+) := v(v − 1 + a(v−γ − 1)),

where a is found by H(v+, v+) = 0, yielding the Rankine-Hugoniot condition

(6.3) a = − v+ − 1v−γ+ − 1

= vγ+

1− v+1− vγ

+

.

Evidently, a→ γ−1 in the weak shock limit v+ → 1, while a ∼ vγ+ in the strong shock limit

v+ → 0. In this scaling, the large-amplitude limit corresponds to the limit as v+ → 0, ordensity ρ+ := 1/v+ →∞.

6.1.2 Eigenvalue equations

Linearizing (6.1) about the profile (v, u) and integrating with respect to x, we obtain theintegrated eigenvalue problem

λv + v′ − u′ = 0,(6.4a)

λu+ u′ − h(v)vγ+1

v′ =u′′

v,(6.4b)

where h(v) = −vγ+1+a(γ−1)+(a+1)vγ . Spectral stability of v corresponds to nonexistenceof solutions of (6.4) decaying at x = ±∞ for <λ ≥ 0 [HuZ2, BHRZ, HLZ].

35

6.2 Preliminary estimates

Proposition 6.1 ([BHRZ]). For each γ ≥ 1, 0 < v+ ≤ 1, (6.2) has a unique (up to trans-lation) monotone decreasing solution v decaying to its endstates with a uniform exponentialrate. For 0 < v+ ≤ 1

12 and v(0) := v+ + 112 ,

|v(x)− v+| ≤( 1

12

)e−

3x4 x ≥ 0,(6.5a)

|v(x)− v−| ≤(1

4

)e

x+122 x ≤ 0.(6.5b)

Proof. Existence and monotonicity follow trivially by the fact that (6.2) is a scalar first-order ODE with convex righthand side. Exponential convergence as x → +∞ follows by

H(v, v+) = (v − v+)(v −

(1−v+

1−vγ+

)(1−(

v+v

)γ

1−(

v+v

) )), whence v − γ ≤ H(v,v+)v−v+

≤ v − (1 − v+) by

1 ≤ 1−xγ

1−x ≤ γ for 0 ≤ x ≤ 1. See [BHRZ].

Proposition 6.2 ([MN]). Viscous shocks of (1.1.3) are spectrally stable whenever

(6.6)(vγ+1

+

aγ

)2 + 2(γ − 1)(vγ+1

+

aγ

)− (γ − 1) ≥ 0,

in particular, for |v+ − 1| << 1.

Proof. Writing (6.4) as Ut + A(x)Ux = B(x)Uxx, with A =(

1 −1− h(v)

vγ+1 1

), B =

(0 00 1

v

),

we see that S =

(1 00 vγ+1

h(v)

)symmetrizes A, B. Taking the L2 complex inner product of

SU against the equations yields

<λ〈U, SU〉+ 〈U ′, SBU ′〉 = −〈u, g(v)u〉,

where the righthand side, coming from commutator terms, is of favorable sign for v+ satis-fying the condition of the proposition. See [MN, BHRZ] or Appendix F.1.

Proposition 6.3 ([BHRZ]). Nonstable eigenvalues λ of (6.4), i.e., eigenvalues with non-negative real part, are confined for any γ ≥ 1, 0 < v+ ≤ 1 to the region Λ defined by

(6.7) R(λ) + |=(λ)| ≤(√

γ +12

)2.

Proof. Energy estimates related to those of Proposition 6.2; see Appendix F.2.

36

6.3 Evans function formulation

Following [BHRZ], we may express (6.4) as a first-order system W ′ = A(x, λ)W,

(6.8) A(x, λ) =

0 λ 10 0 1λv λv f(v)− λ

, W =

uvv′

, ′ = d

dx,

(6.9) f(v) = 2v − (γ − 1)(1− v+

1− vγ+

)(v+v

)γ−(1− v+

1− vγ+

)vγ+ − 1.

Eigenvalues of (6.4) correspond to nontrivial solutions W for which the boundary con-ditions W (±∞) = 0 are satisfied. Because A(x, λ) as a function of v is asymptoticallyconstant in x, the behavior near x = ±∞ of solutions of (6.8) is governed by the limitingconstant-coefficient systems

(6.10) W ′ = A±(λ)W, A±(λ) := A(±∞, λ),

from which we readily find on the (nonstable) domain <λ ≥ 0, λ 6= 0 of interest that thereis a one-dimensional unstable manifold W−

1 (x) of solutions decaying at x = −∞ and a two-dimensional stable manifold W+

2 (x) ∧W+3 (x) of solutions decaying at x = +∞, analytic in

λ, with asymptotic behavior

(6.11) W±j (x, λ) ∼ eµ±(λ)xV ±j (λ)

as x → ±∞, where µ±(λ) and V ±j (λ) are eigenvalues and associated analytically choseneigenvectors of the limiting coefficient matrices A±(λ).

A standard choice of eigenvectors V ±j [BrZ], uniquely specifying D (up to constant fac-tor) is obtained by Kato’s ODE, a linear, analytic ODE whose solution can be alternativelycharacterized by the property that there exist corresponding left eigenvectors V ±j such that,denoting d/dλ by “ ˙ ”

(6.12) (V · V )± ≡ constant, (V · V )± ≡ 0,

Defining the Evans function D associated with operator L as

(6.13) D(λ) := det(W−1 W

+2 W

+3 )|x=0,

we find that D is analytic on <λ ≥ 0, with eigenvalues of L corresponding in location andmultiplicity to zeroes of D. See [Z2] for further details.

37

6.4 Main results

6.4.1 The strong shock limit

Taking a formal limit as v+ → 0 of the rescaled equations (6.1) and recalling that a ∼ vγ+,

we obtain a limiting evolution equation

vt + vx − ux = 0,

ut + ux =(ux

v

)x

(6.14)

corresponding to a pressure-less gas, or γ = 0.The associated limiting profile equation v′ = v(v − 1) has explicit solution v0(x) =

1−tanh(x/2)2 ; the limiting eigenvalue system is W ′ = A0(x, λ)W,

(6.15) A0(x, λ) =

0 λ 10 0 1λv0 λv0 f0(v0)− λ

,

where f0(v0) = 2v0 − 1 = − tanh(x/2).Observe that the limiting coefficient matrix A0

+(λ) := A0(+∞, λ) is nonhyperbolic (inODE sense) for all λ, having eigenvalues 0, 0,−1 − λ; in particular, the stable manifolddrops to dimension one in the limit v+ → 0, and so the prescription of an associated Evansfunction is underdetermined.

This difficulty is resolved by a careful boundary-layer analysis in [HLZ], determining aspecial “slow stable” mode V +

2 ± (1, 0, 0)T – the common limiting direction of slow stableand unstable modes as v+ → 0, collapsing to a Jordan block– augmenting the “fast stable”mode V3 := (a−1(λ/a+ 1), a−1, 1)T associated with the single stable eigenvalue a = −1− λof A0

+. This determines a limiting Evans function D0(λ) by the prescription (6.13), (6.11)of Section 6.3.

Theorem 6.4 ([HLZ]). For λ in any compact subset of <λ ≥ 0, D(λ) converges uniformlyto D0(λ) as v+ → 0.

Proof. Careful boundary layer analysis/asymptotic ODE estimates The precise estimatesused are special to this case, and of a singular type outside the scope of these notes; see[BHZ]. However, see Appendix C.2 for related convergence results in more generic sit-uations, which suffice for nonisentropic gas and MHD analyses of [HLyZ1, HLyZ2] and[BHZ].

Proposition 6.5 ([HLZ]). The limiting function D0 is nonzero on <λ ≥ 0.

Proof. The same estimate as in the proof of Proposition 6.2, now yielding commutator termg ≡ 0. See Appendix F.3 for further details.

Corollary 6.6. For any γ ≥ 1, isentropic Navier–Stokes shocks are stable in the strongshock limit, i.e., for v+ sufficiently small.

38

6.4.2 The weak shock limit

Stability in the weak shock limit is known [MN]. However, combining the calculation ofSection 3.5 with asymptotic ODE estimates estimates of [PZ], we obtain the new observationthat the Evans function converges in the weak shock limit to a constant function.

6.4.3 Intermediate strength shocks

Having disposed analytically of the weak and strong shock limits, we have reduced theproblem of shock stability to a bounded parameter range on which the Evans functionmay be effiently computed numerically, in uniformly well-conditioned fashion; see [BrZ].Specifically, we may map a semicircle ∂<λ ≥ 0 ∩ |λ| ≤ 10 enclosing Λ for γ ∈ [1, 3] byD0 and compute the winding number of its image about the origin to determine the numberof zeroes of D0 within the semicircle, and thus within Λ. For details of the numericalalgorithm, see [BHRZ, BrZ].

Such a study was carried out systematically in [BHRZ] on the parameter range γ ∈[1, 3], for shocks with Mach number M ∈ [1, 3, 000], which corresponds on γ ∈ [1, 2.5]to v+ ≥ 10−3, with the result of stability for all values tested. In combination with theresults of Sections 6.4.1 and 6.4.2, this gives convincing numerical evidence, as claimed,of unconditional stability of isentropic Navier–Stokes shocks for γ ∈ [1, 2.5] and arbitraryamplitude.

6.4.4 Numerical convergence

Having established analytically convergence of D to D0 as v+ → 0, we turn finally to numer-ics to obtain quantitative information yielding a concrete stability threshold. Specifically,for fixed γ, we numerically compute the “Rouche bound” v+ at which the maximum rela-tive error |D −D0|/|D0| over the semicircular contour ∂Rλ ≥ 0, |λ| ≤ 10 around whichwe perform our winding number calculations becomes 1/2. Recall that Rouche’s Theoremguarantees for relative error < 1 that the winding number of D is equal to the windingnumber of D0, which we have shown to be zero, hence we may conservatively concludestability for v+ less than or equal to this bound. Computations are performed using thealgorithm of [BHRZ]; results are displayed in Table 1. More detailed results are displayedfor the monatomic gas case, γ = 5/3, in Table 2. Results are similar for other γ ∈ [1, 3]; see[HLZ].

6.4.5 Global picture

In Figure 1, we superimpose on the numerically computed image of the semicircle by D0 its(numerically computed) image by the full Evans functionD, for a monatomic gas γ ≈ 1.66 atsuccessively higher Mach numbers v+ =1e-1,1e-2,1e-3,1e-4,1e-5,1e-6, showing bothconvergence of D to D0 in the strong shock limit as v+ approaches zero and convergence ofD to a constant in the weak shock limit v+ → 1.

39

Relative Machγ v+ Error Number

3.0 1.27e-3 .5009 127652.5 1.36e-3 .5006 24232.0 1.49e-3 .5001 4741.5 1.75e-3 .4999 95.51.0 2.8e-3 .4995 18.9

Table 1: Rouche bounds for various γ.

Mach Relative Absolutev+ Number Difference Difference

1.0(-6) 7.71(4) 0.1221 0.06011.0(-5) 1.13(4) 0.1236 0.14451.0(-4) 1.64(3) 0.1487 0.47141.0(-3) 2.44(2) 0.4098 1.34641.0(-2) 36.1 0.9046 2.82531.0(-1) 5.50 1.2386 3.8688

Table 2: Maximum relative and absolute differences between D and D0, for γ = 5/3 and λon the semicircle of radius 10.

40

!2 !1 0 1 2 3 4 5!4

!3

!2

!1

0

1

2

3

4

Re

Im

Figure 1: Convergence to the limiting Evans function as v+ → 0 for a monatomic gas,γ = 1.66... The contours depicted, going from inner to outer, are images of the semicircleunder D for v+ =1e-1,1e-2,1e-3,1e-4,1e-5,1e-6. The outermost contour is the imageunder D0, which is nearly indistinguishable from the image for v+ =1e-6.

More, the displayed contours are, to the scale visible by eye, “monotone” in v+, ornested, one within the other, with lower-Mach number contours essentially “trapped” withinhigher-Mach number contours, and all contours interpolating smoothly between this andthe inner, constant limit. Behavior for other γ ∈ [0, 3] is entirely similar; see [HLZ]. That is,a great deal of topological information is encoded in the analytic family of Evans functionsindexed by v+, from which stability may be deduced almost by inspection. Such topologicalinformation does not seem to be available from other methods of investigating stability suchas direct discretation of the linearized operator about the wave or studies based on linearizedtime-evolution or power methods, and represents in our view a significant advantage of theEvans function formulation.

6.5 Nonisentropic gas dynamics

The case of nonisentropic gas dynamics (not discussed here), may be handled similarly, andin some ways more easily; see [HLyZ1, HLyZ2]. Since specific volume v remains uniformlybounded from zero in the (rescaled) strong-shock limit, we may use the standard convergenceresults of Section C.2 unmodified to deduce convergence to a limiting Evans function; on theother hand, other aspects become slightly more complicated due to higher dimensionalityof the system. The result is the same: for an ideal polytropic gas, viscous shocks areunconditionally stable.

41

6.6 MHD

6.6.1 The case of infinite resistivity/permeability

Theorem 6.1. In the degenerate case µ0 = ∞ or σ = ∞, parallel MHD shocks are transver-sal, Lopatinski stable (resp. low-frequency stable), and spectrally stable with respect to trans-verse modes (u, B), for all physical parameter values, hence are Evans (and thus nonlinearly)stable whenever the associated gas-dynamical shock is Evans stable.

Proof. We verify transvers spectral stability, or nonexistence of decaying solutions of (2.15).For transversality/Lopatinski stability, see [BHZ]. For σ = ∞, we may rewrite (3.1) insymmetric form as

(6.16)µ0vλw + µ0vw

′ −B∗1α′ = µµ0w

′′,

λα+ α′ −B∗1w′ = 0.

Taking the real part of the complex L2-inner product of w against the first equation and αagainst the second equation and summing gives

<λ(∫

(vµ0|w|2 + |α|2) = −∫µµ0|w′|2 +

∫vx|w|2 < 0,

a contradiction for <λ ≥ 0 and w not identically zero. If w ≡ 0 on the other hand, wehave a constant-coefficient equation for α, which is therefore stable. The µ0 = ∞ case goessimilarly; see Appendix, [BHZ].

Notably, this includes all three cases: fast Lax, overcompressive, and slow Lax typeshock. Further, the same proof yields the result for the more general class of equations ofstate p(·) satisfying p(v+)−p(v−)

v+−v−< 0, so that vx < 0 for s < 0. With the analytical results of

[HLZ], we obtain in particular the following asymptotic results.

Corollary 6.2. For σ = ∞ or µ0 = ∞, parallel isentropic MHD shocks with ideal gasequation of state, whether Lax or overcompressive type, are linearly and nonlinearly stablein the small- and large-amplitude limits v+ → 1 and v+ → 0, for all physical parametervalues.

6.6.2 The general case

The general case σ, µ0 6= 0 is considerably more complicated, involving issues beyond thoseconsidered here, associated with branch singularities and Riemann surface computations.However, the basic tools/approach are the same, and the result again is unconditionalstability. For details, see [BHZ].

42

6.7 Stability of detonations in the ZND limit

We remark that a similar limiting analysis has been carried out recently in [Z12] for strongdetonation solutions of reactive Navier–Stokes (rNS) equations in the small viscosity, orZND, limit, with the result that the (rNS) Evans function, appropriately rescaled, convergesto the (ZND) version for bounded frequencies λ as the coefficient of viscosity goes to zero,giving a rigorous relation between nonlinear (rNS) stability (see [TZ4]) and the spectralZND stability much studied by practitioners.

7 Conditional stability of viscous shock waves

We next consider the situation of a viscous shock substantially after the onset of instability,i.e., with one or more strictly unstable but no neutrally unstable eigenvalues, and seek todescribe the nearby phase portrait in terms of invariant manifolds and behavior therein.Specifically, for shock waves of systems of conservation laws with artificial viscosity, weconstruct a center stable manifold and show that the shock is conditionally (nonlinearly)stable with respect to this codimension p set of initial data, where p is the number ofunstable eigenvalues. As discussed for example in [AMPZ, GZ], such conditionally stableshock waves can play an important role in asymptotic behavior as metastable states. Notethat this includes a rigorous nonlinear instability result in the case of unstable eigenvalues,since solutions not on the center stable manifold about an equilibrium leave a neighborhoodof the equilibrium in finite time.

For simplicity of exposition, we here treat the easier case of a Lax shock of a semilinear,identity viscosity system. For treatment of the general case (including in particular the classof equations described in Section 4), see [Z8, Z9]. For a treatment of the present, semilinearcase, see [Z7]. Consider a viscous shock solution u(x, t) = u(x), limz→±∞ u(z) = u±,without loss of generality stationary, of a semilinear parabolic system of conservation laws

(7.1) ut + f(u)x = uxx,

u, f ∈ Rn, x , t ∈ R, under the basic assumptions:

(H0) f ∈ Ck+2, k ≥ 2.(H1) A± := df(u±) have simple, real, nonzero eigenvalues.

Linearizing (7.1) about u yields linearized equations

(7.2) ut = Lu := −(df(u)u)x + uxx,

for which the generator L possesses [He, Sa, ZH] both a translational zero-eigenvalue andessential spectrum tangent at zero to the imaginary axis.

The absence of a spectral gap between neutral (i.e., zero real part) and stable (nega-tive real part) spectra of L prevents the usual ODE-type decomposition of the flow nearu into invariant stable, center, and unstable manifolds. Our first result asserts that wecan still determine a center stable manifold, and that this may be chosen to respect the

43

underlying translation-invariance of (7.1). See [TZ1] for closely related results on existenceof translational-invariant center manifolds.

Theorem 7.1. Under assumptions (H0)–(H1), there exists in an H2 neighborhood of theset of translates of u a codimension-p translation invariant Ck (with respect to H2) centerstable manifold Mcs, tangent at u to the center stable subspace Σcs of L, that is (locally)invariant under the forward time-evolution of (7.1) and contains all solutions that remainbounded and sufficiently close to a translate of u in forward time, where p is the (necessarilyfinite) number of unstable, i.e., positive real part, eigenvalues of L.

Next, specializing a bit further, we add to (H0)–(H1) the additional hypothesis that ube a Lax-type shock:

(H2) The dimensions of the unstable subspace of df(u−) and the stable subspace ofdf(u+) sum to n+ 1.

We assume further the following spectral genericity conditions.

(D1) L has no nonzero imaginary eigenvalues.(D2) The orbit u(·) is a transversal connection of the associated standing wave equation

ux = f(u)− f(u−).(D3) The associated inviscid shock (u−, u+) is hyperbolically stable, i.e.,

(7.3) det(r−1 , . . . , r−p−1, r

+p+1, . . . , r

+n , (u+ − u−)) 6= 0,

where r−1 , . . . r−p−1 denote eigenvectors of df(u−) associated with negative eigenvalues and

r+p+1, . . . r+n denote eigenvectors of df(u+) associated with positive eigenvalues.

As discussed in [ZH, MaZ1], (D2)–(D3) correspond in the absence of a spectral gap to ageneralized notion of simplicity of the embedded eigenvalue λ = 0 of L. Thus, (D1)–(D3)together correspond to the assumption that there are no additional (usual or generalized)eigenvalues on the imaginary axis other than the translational eigenvalue at λ = 0; thatis, the shock is not in transition between different degrees of stability, but has stabilityproperties that are insensitive to small variations in parameters.

With these assumptions, we obtain our second and main result characterizing stabilityproperties of u. In the case p = 0, this reduces to the nonlinear orbital stability resultestablished in [ZH, MaZ1, MaZ2, MaZ3, Z2, Z4].

Theorem 7.2. Under (H0)–(H2) and (D1)–(D3), u is nonlinearly orbitally stable undersufficiently small perturbations in L1∩H2 lying on the codimension p center stable manifoldMcs of u and its translates, where p is the number of unstable eigenvalues of L, in the sensethat, for some α(·), all Lp,

(7.4)

|u(x, t)− u(x− α(t))|Lp ≤ C(1 + t)−12(1− 1

p)|u(x, 0)− u(x)|L1∩H2 ,

|u(x, t)− u(x− α(t))|H2 ≤ C(1 + t)−14 |u(x, 0)− u(x)|L1∩H2 ,

α(t) ≤ C(1 + t)−12 |u(x, 0)− u(x)|L1∩H2 ,

α(t) ≤ C|u(x, 0)− u(x)|L1∩H2 .

44

Moreover, it is orbitally unstable with respect to small H2 perturbations not lying in Mcs,in the sense that the corresponding solution leaves a fixed-radius neighborhood of the set oftranslates of u in finite time.

7.1 Existence of Center Stable Manifold

Proof of Theorem 7.1. By Proposition E.1, it is sufficient to verify that (H0)–(H1) imply(A0)–(A2), for h(u, ux) := f(u)x = df(u)ux. Clearly, (H0) implies (A0) by the form of h.Plugging u = u(x) into (7.1), we obtain the standing-wave ODE f(u)x = uxx, or, integratingfrom −∞ to x, the first-order system

ux = f(u)− f(u−).

Linearizing about the assumed critical points u± yields linearized systems wt = df(u±)w,from which we see that u± are nondegenerate rest points by (H1), as a consequence of which(A2) follows by standard ODE theory [Co].

Finally, linearizing PDE (E.1) about the constant solutions u ≡ u±, we obtain wt =L±w := −df(u±)wx − wxx. By Fourier transform, the limiting operators L± have spectraλ±j (k) = −ika±j − k2, where the Fourier wave-number k runs over all of R; in particular,L± have spectra of nonpositive real part. By a standard result of Henry [He], the essentialspectrum of L lies to the left of the rightmost boundary of the spectra of L±, hence wemay conclude that the essential spectrum of L is entirely nonpositive. As the spectra of Lto the right of the essential spectrum by sectoriality of L, consists of finitely many discreteeigenvalues, this means that the spectra of L with positive real part consists of p unstableeigenvalues, for some p, verifying (A1).

7.2 Conditional stability analysis

Define similarly as in Section E.1 the perturbation variable

(7.5) v(x, t) := u(x+ α(t), t)− u(x)

for u a solution of (7.1), where α is to be specified later in a way appropriate for the taskat hand. Subtracting the equations for u(x + α(t), t) and u(x), we obtain the nonlinearperturbation equation

(7.6) vt − Lv = N(v)x + ∂tα(φ+ ∂xv),

where L := −∂xdf(u) + ∂2x as in (7.2) denotes the linearized operator about u, φ = ux, and

(7.7) N(v) := −(f(u+ v)− f(u)− df(u)v)

where, so long as |v|H1 (hence |v|L∞ and |u|L∞) remains bounded,

(7.8)

N(v) = O(|v|2).∂xN(v) = O(|v||∂xv|),∂2

xN(v) = O(|∂x|2 + |v||∂2xv|).

45

7.2.1 Projector bounds

Let Πu denote the eigenprojection of L onto its unstable subspace Σu, and Πcs = Id − Πu

the eigenprojection onto its center stable subspace Σcs.

Lemma 7.3. Assuming (H0)–(H1), there is Πj defined in (7.11) such that

(7.9) Πj∂x = ∂xΠj

for j = u, cs and, for all 1 ≤ p ≤ ∞, 0 ≤ r ≤ 4,

(7.10)|Πu|Lp→W r,p , |Πu|Lp→W r,p ≤ C,

|Πcs|Wr,p→W r,p , |Πcs|Wr,p→W r,p ≤ C.

Proof. Recalling (see the proof of Theorem 7.1) that L has at most finitely many unstableeigenvalues, we find that Πu may be expressed as

Πuf =p∑

j=1

φj(x)〈φj , f〉,

where φj , j = 1, . . . p are generalized right eigenfunctions of L associated with unstableeigenvalues λj , satisfying the generalized eigenvalue equation (L− λj)rjφj = 0, rj ≥ 1, andφj are generalized left eigenfunctions. Noting that L is divergence form, and that λj 6= 0,we may integrate (L − λj)rjφj = 0 over R to obtain λ

rj

j

∫φjdx = 0 and thus

∫φjdx = 0.

Noting that φj , φj and derivatives decay exponentially by standard theory [He, ZH, MaZ1],we find that

φj = ∂xΦj

with Φj and derivatives exponentially decaying, hence

(7.11) Πuf =∑

j

Φj〈∂xφ, f〉.

Estimating

|∂jxΠuf |Lp = |

∑j

∂jxφj〈φjf〉|Lp ≤

∑j

|∂jxφj |Lp |φj |Lq |f |Lp ≤ C|f |Lp

for 1/p + 1/q = 1 and similarly for ∂rxΠuf , we obtain the claimed bounds on Πu and Πu,

from which the bounds on Πcs = Id−Πu and Πcs = Id− Πu follow immediately.

7.2.2 Linear estimates

Let Gcs(x, t; y) := ΠcseLtδy(x) denote the Green kernel of the linearized solution operator

on the center stable subspace Σcs. Then, we have the following detailed pointwise boundsestablished in [TZ2, MaZ1].

46

Proposition 7.1 ([TZ2, MaZ1]). Assuming (H0)–(H2), (D1)–D(3), the center stableGreen function may be decomposed as Gcs = E + G, where

(7.12) E(x, t; y) = ∂xu(x)e(y, t),

(7.13) e(y, t) =∑

a−k >0

(errfn

(y + a−k t√4(t+ 1)

)− errfn

(y − a−k t√4(t+ 1)

))l−k (y)

for y ≤ 0 and symmetrically for y ≥ 0, l−k ∈ Rn constant, and

(7.14)

|G(x, t; y)| ≤ Ce−η(|x−y|+t)

+( n∑

k=1

t−12 e−(x−y−a−k t)2/Mte−ηx+

+∑

a−k >0, a−j <0

χ|a−k t|≥|y|t−1/2e−(x−a−j (t−|y/a−k |))

2/Mte−ηx+

+∑

a−k >0, a+j >0

χ|a−k t|≥|y|t−1/2e−(x−a+

j (t−|y/a−k |))2/Mte−ηx−

),

(7.15)

|∂yG(x, t; y)| ≤ Ce−η(|x−y|+t)

+ Ct−12

( n∑k=1

t−1/2e−(x−y−a−k t)2/Mte−ηx+

+∑

a−k >0, a−j <0

χ|a−k t|≥|y|t−1/2e−(x−a−j (t−|y/a−k |))

2/Mte−ηx+

+∑

a−k >0, a+j >0

χ|a−k t|≥|y|t−1/2e−(x−a+

j (t−|y/a−k |))2/Mte−ηx−

)

for y ≤ 0 and symmetrically for y ≥ 0, for some η, C, M > 0, where a±j are the eigenvaluesof df(u±), x± denotes the positive/negative part of x, and indicator function χ|a−k t|≥|y| is

1 for |a−k t| ≥ |y| and 0 otherwise.

Proof. As observed in [TZ2], it is equivalent to establish decomposition

(7.16) G = Gu + E + G

for the full Green function G(x, t; y) := eLtδy(x), where

Gu(x, t; y) := ΠueLtδy(x) = eγt

p∑j=1

φj(x)φj(y)t

47

for some constant matrix M ∈ Cp×p denotes the Green kernel of the linearized solutionoperator on Σu, φj and φj right and left generalized eigenfunctions associated with unstableeigenvalues λj , j = 1, . . . , p.

The problem of describing the full Green function has been treated in [ZH, MaZ3],starting with the Inverse Laplace Transform representation

(7.17) G(x, t; y) = eLtδy(x) =∮

Γeλt(λ− L(ε))−1δy(x)dλ ,

whereΓ := ∂λ : <λ ≤ η1 − η2|=λ|

is an appropriate sectorial contour, η1, η2 > 0; estimating the resolvent kernel Gελ(x, y) :=

(λ − L(ε))−1δy(x) using Taylor expansion in λ, asymptotic ODE techniques in x, y, andjudicious decomposition into various scattering, excited, and residual modes; then, finally,estimating the contribution of various modes to (7.17) by Riemann saddlepoint (Station-ary Phase) method, moving contour Γ to optimal, “minimax” positions for each mode,depending on the values of (x, y, t).

In the present case, we may first move Γ to a contour Γ′ enclosing (to the left) all spectraof L except for the p unstable eigenvalues λj , j = 1, . . . , p, to obtain

G(x, t; y) =∮

Γ′eλt(λ− L)−1dλ+

∑j=±

Residueλj(ε)

(eλt(λ− L)−1δy(x)

),

where Residueλj(ε)

(eλt(λ − L)−1δy(x)

)= Gu(x, t; y), then estimate the remaining term∮

Γ′ eλt(λ−L)−1dλ on minimax contours as just described. See the proof of Proposition 7.1,

[MaZ3], for a detailed discussion of minimax estimates E+G and of Proposition 7.7, [MaZ3]for a complementary discussion of residues incurred at eigenvalues in <λ ≥ 0 \ 0. Seealso [TZ2].

Corollary 7.4 ([MaZ1]). Assuming (H0)–(H2), (D1)–(D3),

(7.18) |∫ +∞

−∞∂s

xG(·, t; y)f(y)dy|Lp ≤ C(1 + t−s2 )t−

12( 1

q− 1

p)|f |Lq ,

(7.19) |∫ +∞

−∞∂s

xGy(·, t; y)f(y)dy|Lp ≤ C(1 + t−s2 )t−

12( 1

q− 1

p)− 1

2 |f |Lq ,

for all t ≥ 0, 0 ≤ s ≤ 2, some C > 0, for any 1 ≤ q ≤ p (equivalently, 1 ≤ r ≤ p) andf ∈ Lq, where 1/r + 1/q = 1 + 1/p.

Proof. Standard convolution inequalities together with bounds (7.14)–(7.15); see [MaZ1,MaZ2, MaZ3, Z2] for further details.

48

Corollary 7.5 ([Z4]). The kernel e satisfies

|ey(·, t)|Lp , |et(·, t)|Lp ≤ Ct−12(1−1/p),

|ety(·, t)|Lp ≤ Ct−12(1−1/p)−1/2,

for all t > 0. Moreover, for y ≤ 0 we have the pointwise bounds

|ey(y, t)|, |et(y, t)| ≤ C(t+ 1)−12

∑a−k >0

(e−

(y+a−k

t)2

M(t+1) + e−

(y−a−k

t)2

M(t+1)

),

|ety(y, t)| ≤ C(t+ 1)−1∑

a−k >0

(e−

(y+a−k

t)2

M(t+1) + e−

(y−a−k

t)2

(t+1)t

),

for M > 0 sufficiently large, and symmetrically for y ≥ 0.

Proof. Direct computation using definition (7.13); see Appendix G.

7.2.3 Reduced equations II

Recalling that ∂xu is a stationary solution of the linearized equations ut = Lu, so thatL∂xu = 0, or ∫ ∞

−∞G(x, t; y)ux(y)dy = eLtux(x) = ∂xu(x),

we have, applying Duhamel’s principle to (7.6),

v(x, t) =∫ ∞

−∞G(x, t; y)v0(y) dy

−∫ t

0

∫ ∞

−∞Gy(x, t− s; y)(N(v) + αv)(y, s) dy ds+ α(t)∂xu(x).

Defining

(7.20)α(t) = −

∫ ∞

−∞e(y, t)v0(y) dy

+∫ t

0

∫ +∞

−∞ey(y, t− s)(N(v) + α v)(y, s)dyds,

following [ZH, Z4, MaZ2, MaZ3], where e is defined as in (7.13), and recalling the decom-position G = E +Gu + G of (7.16), we obtain the reduced equations

(7.21)v(x, t) =

∫ ∞

−∞(Gu + G)(x, t; y)v0(y) dy

−∫ t

0

∫ ∞

−∞(Gu + G)y(x, t− s; y)(N(v) + αv)(y, s)dy ds,

49

and, differentiating (7.20) with respect to t, and observing that ey(y, s) → 0 as s → 0, asthe difference of approaching heat kernels,

(7.22)α(t) = −

∫ ∞

−∞et(y, t)v0(y) dy

+∫ t

0

∫ +∞

−∞eyt(y, t− s)(N(v) + αv)(y, s) dy ds.

We emphasize that this (nonlocal in time) choice of α and the resulting reduced equationsare completely different from those of Section E.1, according to their respective purposes.A third possible choice has been introduced in [GMWZ1, GMWZ2] for the study of theinviscid limit problem. As discussed further in [Go, Z4, MaZ2, MaZ3, GMWZ1, BeSZ], αmay be considered in the present context as defining a notion of approximate shock location.

7.2.4 Nonlinear damping estimate

Proposition 7.2 ([MaZ3]). Assuming (H0)-(H3), let v0 ∈ H2, and suppose that for0 ≤ t ≤ T , the H2 norm of v remains bounded by a sufficiently small constant, for v as in(7.5) and u a solution of (7.1). Then, for some constants θ1,2 > 0, for all 0 ≤ t ≤ T ,

(7.23) ‖v(t)‖2H2 ≤ Ce−θ1t‖v(0)‖2

H2 + C

∫ t

0e−θ2(t−s)(|v|2L2 + |α|2)(s) ds.

Proof. Subtracting the equations for u(x+α(t), t) and u(x), we may write the perturbationequation for v alternatively as

(7.24) vt +(∫ 1

0df(u(x) + τv(x, t)

)dτ v

)x

− vxx = α(t)∂xu(x) + α(t)∂xv.

Observing that ∂jx(∂xu)(x) = O(e−η|x|) is bounded in L1 norm for j ≤ 2, we take the

L2 inner product in x of∑2

j=0 ∂2jx v against (7.24), integrate by parts and rearrange the

resulting terms to arrive at the inequality

∂t‖v‖2H2(t) ≤ −θ‖∂3

xv‖2L2 + C

(‖v‖2

H2 + |α(t)|2),

θ > 0, for C > 0 sufficiently large, so long as ‖v‖H2 remains bounded. Using the Sobolevinterpolation

‖v‖2H2 ≤ C−1‖∂3

xv‖2L2 + C‖v‖2

L2

for C > 0 sufficiently large, we obtain

∂t‖v‖2H2(t) ≤ −θ‖v‖2

H2 + C(‖v‖2

L2 + |α(t)|2),

from which (7.23) follows by Gronwall’s inequality.

50

7.2.5 Proof of nonlinear stability

Decompose now the nonlinear perturbation v as

(7.25) v(x, t) = w(x, t) + z(x, t),

where

(7.26) w := Πcsv, z := Πuv.

Applying Πcs to (7.21) and recalling commutator relation (7.9), we obtain an equation

(7.27)w(x, t) =

∫ ∞

−∞G(x, t; y)w0(y) dy

−∫ t

0

∫ ∞

−∞Gy(x, t− s; y)Πcs(N(v) + αv)(y, s)dy ds

for the flow along the center stable manifold, parametrized by w ∈ Σcs.

Lemma 7.6. Assuming (H0)–(H1), for v lying initially on the center stable manifold Mcs,

(7.28) |z|W r,p ≤ C|w|2H2

for some C > 0, for all 1 ≤ p ≤ ∞ and 0 ≤ r ≤ 4, so long as |w|H2 remains sufficientlysmall.

Proof. By tangency of the center stable manifold to Σcs, we have immediately |z|H2 ≤C|w|2H2 , whence (7.28) follows by equivalence of norms for finite-dimensional vector spaces,applied to the p-dimensional subspace Σu. (Alternatively, we may see this by direct com-putation using the explicit description of Πuv afforded by Lemma 7.3.)

Proof of Theorem 7.2. Recalling by Theorem 7.1 that solutions remaining for all time in asufficiently small radius neighborhood N of the set of translates of u lie in the center stablemanifold Mcs, we obtain trivially that solutions not originating in Mcs must exit N infinite time, verifying the final assertion of orbital instability with respect to perturbationsnot in Mcs.

Consider now a solution v ∈ Mcs, or, equivalently, a solution w ∈ Σcs of (7.27) withz = Φcs(w) ∈ Σu. Define

(7.29) ζ(t) := sup0≤s≤t

(|w|H2(1 + s)

14 + (|w|L∞ + |α(s)|)(1 + s)

12

).

We shall establish:Claim. For all t ≥ 0 for which a solution exists with ζ uniformly bounded by some fixed,

sufficiently small constant, there holds

(7.30) ζ(t) ≤ C2(E0 + ζ(t)2) for E0 := |v0|L1∩H2 .

51

From this result, provided E0 < 1/4C22 , we have that ζ(t) ≤ 2C2E0 implies ζ(t) <

2C2E0, and so we may conclude by continuous induction that

(7.31) ζ(t) < 2C2E0

for all t ≥ 0, from which we readily obtain the stated bounds. (By standard short-time Hs

existence theory, v ∈ H2 exists and ζ remains continuous so long as ζ remains bounded bysome uniform constant, hence (7.31) is an open condition.)

Proof of Claim. By (7.10), |w0|L1∩H2 = |Πcsv0|L1∩H2 ≤ CE0. Likewise, by Lemma 7.6,(7.29), (7.8), and Lemma 7.3, for 0 ≤ s ≤ t,

(7.32) |Πcs(N(v) + αv)(y, s)|L2 ≤ Cζ(t)2(1 + s)−34 .

Combining the latter bounds with representations (7.27) and (7.22) and applying Corol-lary 7.4, we obtain

(7.33)

|w(x, t)|Lp ≤∣∣∣ ∫ ∞

−∞G(x, t; y)w0(y) dy

∣∣∣Lp

+∣∣∣ ∫ t

0

∫ ∞

−∞Gy(x, t− s; y)Πcs(N(v) + αv)(y, s)dy ds

∣∣∣Lp

≤ E0(1 + t)−12(1− 1

p)

+ Cζ(t)2∫ t

0(t− s)−

34+ 1

2p (1 + s)−34dy ds

≤ C(E0 + ζ(t)2)(1 + t)−12(1− 1

p)

and, similarly, using Holder’s inequality and applying Corollary 7.5,

(7.34)

|α(t)| ≤∫ ∞

−∞|et(y, t)||v0(y)| dy

+∫ t

0

∫ +∞

−∞|eyt(y, t− s)||(N(v) + αv)(y, s)| dy ds

≤ |et|L∞ |v0|L1 + Cζ(t)2∫ t

0|eyt|L2(t− s)|(N(v) + αv)|L2(s)ds

≤ E0(1 + t)−12 + Cζ(t)2

∫ t

0(t− s)−

34 (1 + s)−

34ds

≤ C(E0 + ζ(t)2)(1 + t)−12 .

By Lemma 7.6,

(7.35) |z|H2(t) ≤ C|w|2H2(t) ≤ Cζ(t)2.

52

In particular, |z|L2(t) ≤ Cζ(t)2(1 + t)−12 . Applying Proposition 7.2 and using (7.33) and

(7.34), we thus obtain

(7.36) |w|H2(t) ≤ C(E0 + ζ(t)2)(1 + t)−14 .

Combining (7.33), (7.34), and (7.36), we obtain (7.30) as claimed.

As discussed earlier, from (7.30), we obtain by continuous induction (7.31), or ζ ≤2C2|v0|L1∩H2 , whereupon the claimed bounds on |v|Lp and |v|H2 follow by (7.33) and (7.36),and on |α| by (7.34). Finally, a computation parallel to (7.34) (see, e.g., [MaZ3, Z2]) yields|α(t)| ≤ C(E0 + ζ(t)2), from which we obtain the last remaining bound on |α(t)|.

Remark 7.7. We point out that the finite-dimensional part z of v is in fact controlledpointwise by its L2 norm and thus by |w|H2 , satisfying |z(x, t)| ≤ Ce−θ|x||z(·, t)|L2(x) ≤Ce−θ|x||w(·, t)|2H2(x), making possible a pointwise version of the argument above. This isa key point in treating the nonclassical over- or undercompressive cases, which appear torequire pointwise bounds [HZ, RZ, Z8].

8 Longitudinal bifurcation of viscous shock waves

A well-known phenomenon in combustion is the appearance of “galloping”, “spinning”,and “cellular” instabilities of traveling detonation fronts, apparently corresponding withHopf bifurcation of the background solution; see, e.g., [KS, LyZ1, LyZ2, TZ1, TZ2] andreferences therein. At very high energies (shock strengths), similar phenomena appear tooccur for viscous shock waves in the form of cellular instabilities and Mach stem formation[BE, MR1, MR2].

Mathematically, there are two distinct issues associated with this phenomenon. The firstis to verify the spectral scenario associated with Hopf bifurcation, consisting of a conjugatepair λ±(ε) = γ(ε) + iτ(ε) of complex eigenvalues of the linearized operator about the wavecrossing the imaginary axis from stable (negative real part) to unstable side as bifurcationparameter ε varies from negative to positive values. Here, ε measures variation in physicalparameters such as heat release, rate of reaction, or strength of the detonation. For studiesfrom various points of view, see [KS, Er5, FW, BMR, LyZ1, LyZ2, Z11, AM, MR1, MR2].We will return to this topic in Section 9.

The second issue is to show that this spectral scenario indeed corresponds at the non-linear level to Hopf bifurcation: i.e., apparition of nearby time-periodic solutions Ur ofapproximate period 2π/τ(ε), ε = ε(r), branching from the steady solution at ε(0) = 0.What makes this nontrivial is the absence of a spectral gap between zero and the essen-tial spectrum of the linearized operator about the wave, which circumstance prevents theapplication of standard bifurcation theorems for partial differential equations, as in, e.g.,[C, He, MM, VI]. Here, we discuss this second issue in the context of viscous shock waves,following the “reverse temporal dynamics” approach of [TZ2, TZ3]. See also the relatedanalysis of [SS] by “spatial dynamics” techniques. For a treatment of the detonation case,see [TZ4]. For a simple center manifold treatment in the case of a spectral gap, see [TZ1].

53

8.0.6 Equations and assumptions

Consider a one-parameter family of standing viscous shock solutions

(8.1) U(x, t) = U ε(x), limz→±∞

U ε(z) = U ε± (constant for fixed ε),

of a smoothly-varying family of conservation laws

(8.2) Ut = F(ε, U) := (B(ε, U)Ux)x − F (ε, U)x, U ∈ Rn,

with associated linearized operators

(8.3) L(ε) := ∂F/∂U |U=Uε = ∂xBε(x)∂x − ∂xA

ε(x),

whereBε(x) := B(ε, U ε(x)), Aε(x) := FU (ε, U ε(x))− (BU (ε, U ε)·)∂xU

ε.

Let Dε(λ) denote the Evans functions associated with L(ε).Equations (8.2) are typically shifts B(ε, U) = B(U), F (ε, U) := f(U)−s(ε)U of a single

equationUt = (B(U)Ux)x − f(u)x

written in coordinates x = x− s(ε)t moving with traveling-wave solutions U(x, t) = U ε(x−s(ε)t) of varying speeds s(ε). Profiles U ε satisfy the standing-wave ODE

(8.4) B(U)U ′ = F (ε, U)− F (ε, U ε−).

Assumption 1. We assume that, for each value of ε, system (8.2) satisfies conditions(A1)–(A3) and (H0)–(H3) of Section 4.

Assumption 2. We assume that U ε are of classical Lax or overcompressive type.

Assumption 3. We assume the Hopf bifurcation criterion

(Dε) On a neighborhood of <λ ≥ 0, the only zeroes of Dε are (i) a zero ofmultiplicity ` at λ = 0, and (ii) a crossing conjugate pair of zeroes λ±(ε) = γ(ε) + iτ(ε)with γ(0) = 0, ∂εγ(0) > 0, and τ(0) 6= 0.

By the analog for real viscosity systems of Theorem 7.2 (see [Z9]), condition Dε togetherwith (A1)–(A3), (H0)–(H3) implies that U ε is linearly and nonlinearly stable for ε < 0 andlinearly and nonlinearly unstable for ε > 0; that is, there is a transition from stability toinstability at ε = 0 of the background shock.

8.0.7 Results

Definition 8.1. Let B2 ⊂ B1 and X2 ⊂ X1 denote the Banach spaces determined bynorms ‖U‖B1 := ‖U‖H1 , ‖∂xU‖B2 := ‖∂xU‖B1 + ‖U‖L1 and

(8.5)‖U‖X1 := ‖eη〈x〉U‖H2 ,

‖∂xU‖X2 := ‖∂xU‖X1 + ‖e2η〈x〉U‖H1 ,

where η > 0 and 〈x〉 := (1 + |x|2)1/2.

54

Our main result is the following theorem establishing Hopf bifurcation from the steadysolution U ε at ε = 0 under bifurcation criterion (Dε), corresponding to “longitudinal” or“galloping” (“pulsating”) instabilities.

Theorem 8.2. Let U ε, (8.2) be a family of traveling-waves and systems satisfying assump-tions (A1)–(A3), (H0)–(H3), and (Dε). Then, for η > 0 sufficiently small, there existC > 0, r0 > 0, a C1 function r → ε(r), defined for |r| < r0 and such that ε(0) = 0,and a C1 family of time-periodic solutions Ur(x, t) of (8.2) with ε = ε(r), of period T (r),T (·) ∈ C1, T (0) = 2π/τ(0), with

(8.6) C−1r ≤ ‖Ur − U ε(r)‖X1 ≤ Cr

for all t. For Lax shocks, up to fixed translations in x, t, these are the only time-periodicsolutions nearby in X1 with period T ∈ [T0, T1] for any fixed 0 < T0 < T1 < +∞; indeed,they are the only nearby solutions of the more general form Ur(x − σrt, t) with Ur(x, ·)periodic. For overcompressive shocks, they are part of a C1 (` − 1)-parameter family ofsolutions that are likewise unique up to translation in x, t.

Remark 8.3. Bound (8.6), by Sobolev embedding, includes also the result of exponentiallocalization, |Ur − U ε| ≤ Cre−η|x|.

Remark 8.1. Under suitable spectral assumptions, a stability analysis of general time-periodic shock waves is carried out in [BeSZ] in the case of Laplacian viscosity B ≡ I.These assumptions are verified in [SS] for the bifurcating waves described in Theorem 8.2in the supercritical case, as expected from standard finite-dimensional ODE (Hopf’s secondbifurcation theorem, concerning stability; see the interesting historical discussion in [Gol]).

It is an interesting question when and whether bifurcation condition (Dε) is satisfiedfor viscous shock waves in physically interesting situations. The results of [HLyZ1] indicatethat longitudinal instability does not occur at all for ideal gas dynamics, nor any other kindof instability. However, it seems quite possible that (Dε) can occur in the richer setting ofMHD, where Lax-type shock waves are known to be sometimes unstable even for an idealgas equation of state [T], or for phase-transitional gas dynamics with a van der Waals-typeequation of state. In the detonation context, such longitudinal bifurcations are well knownto occur. Numerical investigation of (Dε) across a range of shock and detonation waves isan important direction for future investigation.

8.1 Linearized estimates

Assuming (A1)–(A3), (H0)–(H3), let L(ε) be the linearized operators (8.3), and λ±(ε) thecrossing eigenvalues of (Dε). Let L(X,Y ) denote the space of bounded linear operatorsfrom Banach space X to Y , equipped with the usual operator norm | · |L(X,Y ).

55

Lemma 8.4 ([TZ2]). Under assumptions (A1)–(A3), (H0)–(H2), (Dε), associated witheigenvalues λ±(ε) of L(ε) are right and left eigenfunctions φε

± and φε± ∈ Ck(x, ε), k ≥ 2,

exponentially decaying in up to k derivatives as x→ ±∞, and L(ε)-invariant projection

(8.7) Πf :=∑j=±

φεj(x)〈φε

j , f〉

onto the total (oscillatory) eigenspace Σε := Spanφε±, bounded from Lq or B2 to W k,p∩X2

for any 1 ≤ q, p ≤ ∞. Moreover,

(8.8) φε± = ∂xΦε

±,

with Φε ∈ Ck+1 exponentially decaying in up to k + 1 derivatives as x→ ±∞.

Proof. From simplicity of λ±, and the fact [MaZ3] that they are bounded away from theessential spectrum of L(ε), we obtain either by standard spectral perturbation theory [K] orby direct Evans-function calculations [GJ1, GJ2, MaZ3, Z2] that there exist λ±(·), φε

±(·) ∈L2 with the same smoothness Ck(ε), k ≥ 5, assumed on F . The exponential decay propertiesin x then follow by standard asymptotic ODE theory; see, e.g., [GZ, Z2, Z3]. Finally, recallthe observation of [ZH] that, by divergence form of L(ε), we may integrate L(ε)φ = λφfrom x = −∞ to x = +∞ to obtain λ

∫ +∞−∞ φ(x)dx = 0, and thereby (since λ± 6= 0 by

assumption)

(8.9)∫ +∞

−∞φ±(x)dx = 0,

from which we obtain by integration (8.8) with the stated properties of Φ±. From (8.8)and representation (8.7), we obtain by Holder’s inequality the stated bounds on projectionΠ.

Defining Πε := Id−Πε, Σε := Range Πε, and L(ε) := L(ε)Πε, denote by

(8.10) G(x, t; y) := eL(ε)tδy(x)

the Green kernel associated with the linearized solution operator eLt of the linearized evo-lution equations ut = L(ε)u, and

(8.11) G(x, t; y) := eL(ε)tΠδy(x)

the Green kernel associated with the transverse linearized solution operator eL(ε)tΠ. Bydirect computation, G = O + G, where

(8.12) O(x, t; y) := e(γ(ε)+iτ(ε))tφ+(x)φt+(y) + e(γ(ε)−iτ(ε))tφ−(x)φt

−(y).

56

8.1.1 Short time estimates

Lemma 8.5. Under assumptions (A1)–(A3), (H0)–(H3), (Dε), for 0 ≤ t ≤ T , 1 ≤ p ≤ ∞,η > 0,

(8.13) ‖eL(ε)tf‖Lp , ‖eL(ε)tΠf‖Lp ≤ C‖f‖Lp .

(8.14) ‖eη〈x〉∫eL(ε)t∂xf‖Lp , ‖eη〈x〉

∫eL(ε)tΠ∂xf‖Lp ≤ C‖eη〈x〉f‖Lp ,

(8.15) ‖eη〈x〉∫∂ε,te

L(ε)t∂xf‖Lp , ‖eη〈x〉∫∂ε,te

L(ε)tΠ∂xf‖Lp ≤ C‖eη〈x〉f‖W 2,p ,

where 〈x〉 := (1 + |x|2)1/2.

Proof. From the standard C0 semigroup bound |eL(ε)t|L(Lp,Lp) ≤ C and properties eL(ε)t =eL(ε)tΠ + eL(ε)tΠ and ‖Πf‖Lp ≤ ‖f‖Lp , we obtain (8.13). Likewise, we may obtain inte-grated bounds (8.14)(η = 0) using the divergence form of L(ε) by integrating the linearizedequations with respect to x to obtain linearized equations Ut = L(ε)U for the integratedvariable

U(x, t, ε) :=∫ x

−∞u(z, t, ε)dz, u(·, t) := eL(ε)t∂xf,

with the linearized operator L(ε) := −Aε(x)∂x + Bε(x)∂2x of the same parabolic form as

L(ε), then applying standard C0 semigroup estimates (alternatively, more detailed pointwisebounds as in [MaZ3]) to the bound

‖eη〈x〉U(·, t, ε)‖Lp = ‖eη〈x〉eL(ε)tf‖Lp ≤ C‖eη〈x〉f‖Lp .

The eLtΠ bound then follows by relation eL(ε)t = eL(ε)tΠ+eL(ε)tΠ together with ‖eη〈x〉∫

Π∂xf‖Lp ≤‖f‖Lp . We obtain (8.14) (η > 0) by the change of variables V = Ueη〈x〉, which, sinceα := eη〈x〉 satisfies |(d/dx)kα| ≤ C|α| for k ≥ 0, converts the linearized equations to anequation with the same principal part plus lower-order terms with bounded coeffients.

Finally, (8.15) follows from ∂teL(ε)t = eL(ε)tL(ε) and the variational equation (∂t −

L)∂εU = (∂εL)U , U(t) := eL(ε)tU0, together with ‖L(ε)U0‖Lp , ‖L(ε)U‖Lp ≤ C‖U0‖W 2,p .

8.1.2 Pointwise Green function bounds

We now develop a key cancellation estimates adapting the pointwise semigroup methods of[ZH, MaZ3, Z2] to the present case. Our starting point is the inverse Laplace transformrepresentation

(8.16) G(x, t; y) =1

2πiP.V.

∫ η+i∞

η−i∞eλtGλ(x, y)dλ,

57

η > 0 sufficiently large, established in [MaZ3].Deforming the contour using analyticity of Gλ [MaZ3] across oscillatory eigenvalues

λ±(ε) we obtain G = G + O, where O, defined in (8.12), is the sum of the residues of theintegrand at λ±, and, for ν, r > 0 sufficiently small,

(8.17)

G(x, t; y) =1

2πi

∮γeλtGλ(x, y)dλ

+1

2πiP.V.

(∫ −ν−ri

−ν−i∞+∫ −ν+i∞

−ν+ri

)eλtGλ(x, y)dλ

=: GI + GII ,

where γ is the counterclockwise arc of circle ∂B(0, r) connecting −ν − ri and −ν + ri, andGI as in [Z3] is the low-frequency and GII the high-frequency component of G. Defineassociated solution operators SI(t, ε) and SII(t, ε) by

(8.18) SI(t)f(x) :=∫ +∞

−∞GI(x, t; y)f(y) dy, SII(t)f(x) :=

∫ +∞

−∞GII(x, t; y)f(y) dy

andS := SI + SII .

Suppressing the parameter ε, denote by a±j , r±j , l±j the eigenvalues and associated rightand left eigenvectors of Aε

± = Fu(uε±, ε). Following [MaZ3], let a∗j (x), j = 1, . . . , J ≤ (n− r)

denote the eigenvalues of A119.

Proposition 8.6. Under assumptions (A1)–(A3), (H0)–(H3), (Dε), for 0 ≤ q, r, s, 0 ≤2q + s ≤ 4, and ν > 0,

(8.19)∂q

ε∂rt ∂

sxG

II(x, t; y) =∑

p≤(2q+r)+s

( J∑j=1

O(e−νt)δx−a∗j t(−y)

+O(e−ν(|x−y|+t)))∂p

y .

Proof. The case (q = r = 0) is immediate from the bounds of [MaZ3]. Derivatives withrespect to ε may be converted using the variational equation

(8.20) (λ− L)(∂εGλ) = (∂εL)Gλ

into two spatial derivatives, thus extending to case (r = 0). Likewise, time-derivatives,appearing as powers of λ at the level of the resolvent kernel, have the effect of single spatialderivatives on “hyperbolic” blocks, and double spatial derivatives on “parabolic” blocks, inthe notation of [MaZ3], yielding by the same estimates as in case (q = r = 0) the assertedbounds. This completes the proof of the general case.

9In the notation of [MaZ3], a±j are eigenvalues of A∗ := A11 −B21B−122 A12 = A11.

58

Corollary 8.7. Under assumptions (A1)–(A3), (H0)–(H3), (Dε),∑∞

j=0 SII(jT ) converges

uniformly and absolutely in operator norm | · |L(X1,X1), for ε sufficiently small and T in anycompact set, for any 0 < η < ν, to a limit that is C1 in (ε, T ) with respect to the operatornorm L(B1, B1).

Proof. Straightforward computation using (8.19).

Proposition 8.8. Under assumptions (A1)–(A3), (H0)–(H3), (Dε),∑∞

j=0 SI(jT ) con-

verges uniformly (but conditionally) in the operator norm | · |L(X2,B1), for ε sufficiently smalland T in any compact set bounded away from the origin, with limiting kernel satisfying for0 ≤ q, r, s, 0 ≤ 2q + s ≤ 4, ν > 0,

(8.21) ∂qε∂

rT∂

rx

∞∑j=0

GIy(x, jT ; y) = O(e−ν|x−y| + e−ν|x|).

Proof. (Convergence). Since (by Sobolev embedding) X2 ⊂ ∂x‖e2η〈x〉 · ‖L∞ , it is sufficientto show convergence in L(∂x‖e2η〈x〉 · ‖L∞ , L

2), or, equivalently, convergence in L(‖e2η〈x〉 ·‖L∞ , L

2) of the operator∑∞

j=0 SI∂x with kernel

J∑j=0

GIy(x, jT ; y),

where, by (8.17),

(8.22) GIy(x, jT ; y) =

12πi

∮γeλjT∂yGλ(x, y)dλ,

hence (summing under the integral as described in the introduction)

(8.23)

J∑j=0

GIy(x, jT ; y) =

12πi

∮γ

( 11− eλT

)∂yGλ(x, y)dλ

− 12πi

∮γe(J+1)Tλ

( 11− eλT

)∂yGλ(x, y)dλ

=: I + II.

Taking y ≤ 0 for definiteness, recall from [MaZ3, Z2] that, for λ sufficiently small,Gλ(x, y) can in the Lax or overcompressive case be expanded analytically at λ = 0 as a sumof “excited” terms

(8.24) λ−1φm(x)l−trk eλµ−k (λ)y, µ−k = −(a−k )−1 +O(λ),

a−k 6= 0, where φm(x) = O(e−ν|x|) are stationary modes of the linearized equations; “scat-tering terms”

(8.25) r±j l−trk eλµ−k (λ)y+λµ±j (λ)x, µ±β = −(a±β )−1 +O(λ);

59

and error terms

(8.26) O(e−ν(|x|+|y|))

and

(8.27) O(λeλµ−k (λ)y+λµ±j (λ)x),

with symmetric expansions for y ≥ 0. These bounds may be converted by the RiemannSaddlepoint (∼ Stationary Phase) estimates to the pointwise Green function bounds of[ZH, MaZ3, Z2].

Taking y-derivatives, we find that( 11− eλT

)∂yGλ = −T−1(λ−1 +O(1))∂yGλ

expands as (Ta−k )−1 times the same excited and scattering terms (8.24) and (8.25), pluserror terms of the same order (8.26), (8.27), plus new pole terms of order

(8.28) λ−1O(e−ν(|x|+|y|)).

Thus, by the same Riemann saddlepoint estimates used to bound GI(x, t; y) in [ZH, MaZ3,Z2], we find that the contribution to term II of all except the new terms (8.28) satisfiesexactly the same bounds as GI(x, (J + 1)T ; y), expanding for t = (J + 1)T and y ≤ 0 asthe sum of excited terms

(8.29) (Ta−k )−1φj(x)l−trk

(errfn

y + a−k t√4β−k t

− errfn

y − a−k t√4β−k t

),β−k > 0, scattering terms

(8.30)

n∑k=1

(1 + t)−1/2O(e−(x−y−a−k t)2/Mte−ηx+)

+∑

a−k >0, a−j <0

χ|a−k t|≥|y|(1 + t)−1/2O(e−(x−a−j (t−|y/a−k |))2/Mte−ηx+

),

+∑

a−k >0, a+j >0

χ|a−k t|≥|y|(1 + t)−1/2O(e−(x−a+j (t−|y/a−k |))

2/Mte−ηx−)

bounded by convected heat kernels, where x+ (resp. x−) denotes the positive (resp. nega-tive) part of x, and a negligible error

(8.31) O(e−ν(|x−y|+t)).

60

Terms (8.28) by the same argument give a time-independent contribution of

(8.32) O(e−ν(|x|+|y|))

up to negligible error (8.31).Noting that all terms except (8.29) and (8.32) decay in L2(x) at least at Gaussian

rate (1 + t)−1/4, independent of y, while kernels (8.29) integrated against an exponentiallylocalized function converge exponentially to

φj(x)l−trk = O(e−ν|x|),

we find that II converges as J →∞ to a kernel with the claimed bound O(e−ν|x|+ eν|x−y|).Term I (independent of J) by the elementary bound |∂yGλ(x, y)| ≤ Ce−ν|x−y| for λ boundedaway from essential spectrum of L, and the fact that γ is bounded away from both λ = 0and the essential spectrum of L, satisfies trivially the bound I = O(e−η|x−y|), completingthe proof of convergence while at the same time establishing (8.21) for q, r, s = 0.

(Derivative bounds). The remaining bounds (8.21)(q, r, s 6≡ 0) follow easily, by theobservation that the limiting summands in term II are independent of T and satisfy thesame bounds after x- or ε- differentiation as before, while I, by (either using the variationalequations (8.20) to convert ε-derivatives to two spatial derivatives as above, or estimatingdirectly as in [TZ2])

∂qε∂

sx∂yGλ(x, y) = O(e−ν|x−y|)

and|∂T

( 11− eλT

)| ≤ C

for λ ∈ γ likewise satisfies the same bounds after as before.

Corollary 8.9. Under assumptions (A1)–(A3), (H0)–(H3), (Dε),∑∞

j=0 S(jT ) converges inoperator norm | · |L(X2,B1), uniformly in ε, T, for ε sufficiently small and T in any compactset bounded away from the origin, to a limit that is bounded in | · |L(X2,X1)∩L(B2,B1) andC1(ε, T ) in | · |L(B2,B1) on the subspace X2.

Proof. By Corollary 8.7 and Proposition 8.8, it is sufficient to establish the second statementfor SI using bounds (8.21). Further, since ∂xL

1 ⊂ B2 and (by Sobolev embedding) ∂x‖e2η〈x〉·‖L∞ ⊂ X2, it is sufficient to show boundedness in L(∂xL

1, L2) and L(∂x‖e2η〈x〉 · ‖L∞ , X1),or, equivalently, boundedness in L(L1, L2) of the operator ∂q

(ε,T )

∑∞j=0 S

I∂x, with kernel

∞∑j=0

∂qε,TG

Iy(x, jT ; y),

q = 0, 1, and boundedness in L(‖e2η〈x〉 · ‖L∞ , ‖eη〈x〉 · ‖L∞) of the operators ∂rx

∑∞j=0 S

I∂x,0 ≤ r ≤ 2 with kernels

∂rx

∞∑j=0

GIy(x, jT ; y),

both routine consequences of bound (8.21).

61

From Corollary 8.9 we obtain the following important conclusion.

Proposition 8.10. Under assumptions (A1)–(A3), (H0)–(H3), (Dε), for ε sufficientlysmall and T in any compact set bounded away from the origin, Id − S(ε, T ) has a rightinverse

(Id− S(ε, T ))−1 ∈ L(X2, X1) ∩ L(B2, B1)

that, restricted to its domain X2, is C1 in ε, T in the L(B2, B1) norm.

Proof. Given f ∈ X2, let

(8.33) (I − S(ε, T ))−1f := limN→∞

N∑j=0

S(ε, jT )f,

where the limit is taken in B1. For all N,

(8.34) (Id− S(ε, T ))N∑

j=0

S(ε, jT )f = f − S(ε, (N + 1)T )f.

Because S(ε, T ) ∈ L(B1) (as a consequence of Lemma 8.5),

(Id− S(ε, T )) limN→∞

N∑j=0

S(ε, jT )f = limN→∞

(Id− S(ε, T ))N∑

j=0

S(ε, jT )f.

From the above equality and (8.34), the operator defined in (8.33) is a right inverse ofId− S(ε, T ) on X2. The other assertions follow from Corollary 8.9.

8.2 Nonlinear energy estimates

We next carry out Hs-energy estimates on the perturbation equations

(8.35) Ut − L(ε)U = Q(ε, U, Ux)x

of (8.2) about U ε, to be used to control higher derivatives in the fixed point iteration usedto carry out our Lyapunov–Schmidt reduction and bifurcation analyses.

By standard energy estimates, given any T > 0 and s ≥ 2, there exists δ(T ) > 0, suchthat, if ‖U0‖Hs ≤ δ, the system (8.35) has a unique solution U ∈ C0([0, T ],Hs), and thebound

‖U(·, t)‖Hs ≤ C‖U0‖Hs ,

holds for t ∈ [0, T ], for some C > 0 which depends on T, but neither on ε nor on t.Likewise, we have a formally quadratic linearized truncation error |Q| = O(|U |(|U |+ |Ux|))for |U | ≤ C. Our goal in this section, and what is far from evident in the absence ofparabolic smoothing, is to establish a quadratic bound on the linearization error:

(8.36) ‖U(·, T )− eL(ε)TU0‖Hs ≤ C‖U0‖2Hs .

The corresponding bound does not hold for quasilinear hyperbolic equations, nor as dis-cussed in A, [TZ3], for systems of general hyperbolic–parabolic type, due to loss of deriva-tives. However, it follows easily for systems satisfying assumptions (A1)–(A2).

62

8.2.1 Hs linearization error

By the strong block structure assumption (A1), we may write (8.35) more precisely as

(8.37) Ut + (AεU)x − (BεUx)x = ∂x

(0

qε1(U)(U,U) + qε

2(U)(U, ∂xU2)

),

where Bε =(

0 00 bε

)and qε

j (·) are Ck−2 bilinear forms. Appealing to (A2), we may

differentiate r times and multiply by

A0,ε := A0(U ε) =(A0,ε

11 00 A0,ε

22

)to obtain

(8.38) A0,ε∂rxUt + Aε∂r

xUx − (Bε∂r+1x U)x = Θr +A0,ε∂x

(0qr

),

where A0,ε is symmetric positive definite, Aε11 is symmetric, Bε =

(0 00 bε

)with bε sym-

metric positive definite, and

(8.39)‖Θr‖L2 = O(‖U‖Hr),

qr = ∂rx

(qε1(U)(U,U) + qε

2(U)(U, ∂xU2)).

Lemma 8.11. For r ≥ 1, ‖qr‖L2 ≤ C(‖U‖Hr + ‖U2‖Hr+1

)(‖U‖Hr + ‖U‖r

Hr

).

Proof. Standard application of Moser’s inequality; see, e.g., [Ta, Z3].

Proposition 8.12. Assuming (A1), (A2), (H0), for 1 ≤ s ≤ k − 1, 0 ≤ T ≤ T0 uniformlybounded, some C = C(T0) > 0, and U satisfying (8.35) with initial data U(·, 0) = U0

sufficiently small in Hs,

(8.40) ‖U(·, T )‖2Hs +

∫ T

0‖U2(·, t)‖2

Hs+1 dt ≤ C‖U0‖2Hs ,

(8.41) ‖U(·, T )− eLTU0‖Hs ≤ C‖U0‖2Hs .

Proof. By symmetric positive definiteness of A0,ε,

E(U) := (1/2)s∑

r=0

〈∂rxU,A

0,ε∂rxU〉

63

defines a norm equivalent to ‖ · ‖Hs , i.e., E(·)1/2 ∼ ‖ · ‖Hs . Applying (8.38), we find that

(8.42)

∂tE(U) = −〈∂sxU, A

ε∂s+1x U〉+ 〈∂s

xU, ∂x(Bε∂sxU)〉

+ 〈∂sxU2, A

0,ε22 ∂xqs〉+O(‖U‖2

Hs)

= 〈∂sxU, (1/2)Aε

x∂sxU〉+ 〈∂s+1

x U2, bε∂s+1

x U2〉− 〈∂s+1

x U2, A0,ε22 qs〉 − 〈∂

sxU2, ∂xA

0,ε22 qs〉+O(‖U‖2

Hs)

≤ −θ‖U2‖2Hs+1 +O(‖U‖2

Hs)+O(‖U2‖Hs+1)

(‖U‖Hs + ‖U2‖Hs+1

)(‖U‖Hs + ‖U‖s

Hs

)for some θ > 0. So long as ‖U‖Hs remains sufficiently small, this gives

(8.43)∂tE(U) ≤ −(θ/2)‖U2‖2

Hs+1 +O(‖U‖2Hs)

≤ −(θ/2)‖U2‖2Hs+1 + CE ,

from which (8.40) follows by Gronwall’s inequality, in the form

E(U(T )) + (θ/2)∫ T

0‖U2‖2

Hs+1(t)dt ≤ C2E(U0).

To obtain (8.41), observe that E(·, t) := U(·, t)− eL(ε)tU0 satisfies

(8.44) Et + (AεE)x − (BεEx)x = ∂x

(0

qε1(U)(U,U) + qε

2(U)(U, ∂xU2)

),

with E(·, 0) = 0, and thus

(8.45) A0,ε∂rxEt + Aε∂r

xEx +−(Bε∂r+1x E)x = ΘE

r +A0,ε∂x

(0qεr

),

similarly as in (8.38), with ‖ΘEr ‖L2 = O(‖E‖Hr). Thus, calculating as in (8.42), we obtain

(8.46)

∂tE(E) = −〈∂sxE, A

ε∂s+1x E〉+ 〈∂s

xE, ∂x(Bε∂sxE)〉

+ 〈∂sxE2, A

0,ε22 ∂xqs〉+O(‖E‖Hs(‖E‖Hs + ‖U‖2

Hs)

= 〈∂sxE, (1/2)Aε

x∂sxE〉+ 〈∂s+1

x E2, bε∂s+1

x E2〉− 〈∂s+1

x E2, A0,ε22 qs〉 − 〈∂

sxU2, ∂xA

0,ε22 qs〉+O(‖E‖Hs(‖E‖Hs + ‖U‖2

Hs)

≤ −θ‖E2‖2Hs+1 + +O(‖E‖Hs(‖E‖Hs + ‖U‖2

Hs)+O(‖E2‖Hs+1)

(‖U‖Hs + ‖U2‖Hs+1

)(‖U‖Hs + ‖U‖s

Hs

),

which, so long as ‖U‖Hs remains sufficiently small, gives

∂tE(E) ≤ −θ‖E2‖2Hs+1 + C‖E‖2

Hs + C‖U‖2Hs

(‖U‖2

Hs + ‖U2‖2Hs+1

)64

and thus, using ‖U‖Hs(t) ≤ C‖U0‖Hs by (8.40),

∂tE(E) ≤ +CE(E) + C2‖U0‖2Hs

(‖U‖2

Hs + ‖U2‖2Hs+1

).

Applying Gronwall’s inequality, and using E(0) = 0, we thus obtain

E(E(T )) ≤ C3‖U0‖2Hs

∫ T

0

(‖U‖2

Hs + ‖U2‖2Hs+1

)(t)dt,

yielding the result by (8.40) and E(E) ∼ ‖E‖2Hs .

Remark 8.2. Note that the above, finite-time estimate is considerably simpler than the“Kawashima-type” global-in-time estimates used in the nonlinear stability analysis [MaZ4,Z2, Z3], which require also additional assumptions of, among other things, symmetrizabilityand “genuine noncoupling” of Aε

± and Bε±.

8.2.2 Weighted variational bounds

Define now

(8.47) N (U0, ε, T ) := U(·, T )− eL(ε)TU0.

where U satisfies (8.35) with initial data U0.

Proposition 8.13. Assuming (A1), (A2), (H0), for 1 ≤ s ≤ k − 1, 0 ≤ T ≤ T0 uniformlybounded, some C = C(T0) > 0, N is uniformly bounded in X2 and C1 in B2 with respectto U0, T , ε for ‖U0‖X1, |ε| sufficiently small, with

(8.48)

‖N (U0, ε, T )‖X2 ≤ C‖U0‖2X1,

‖∂U0N (U0, ε, T )‖X2 ≤ C‖U0‖X1 ,

‖∂ε,TN (U0, ε, T )‖B2 ≤ C‖U0‖2X1.

Proof. Similarly as in the proof of (8.14), we may make the change of variables V = Uα(x),α(x) := eη〈x〉, 〈x〉 := (1 + |x|2)1/2, to convert both (8.35) and the corresponding lin-earized equations to equations for which the same principal part plus lower-order termswith bounded coeffients and the nonlinear part is α−1 times the same principal part plusbounded factors times lower-order terms, thus recovering ‖U(·, T )− eLTU0‖X1 ≤ C‖U0‖2

X1,

or

(8.49) ‖N (U0, ε, T )‖X1 ≤ C‖U0‖2X1,

by the same argument used to obtain (8.40). Here, we are using the fact that α satisfiesboth |α−1| ≤ C and |(d/dx)kα| ≤ C|α| for k ≥ 0. Bound (8.48)(i) then follows by (8.49)and the Duhamel formulation∫

N =∫ T

0

( ∫eL(ε)(T−t)∂x

)Q(U,Ux)(t) dt,

65

bounds (8.14) with p = ∞, and Sobolev embedding

(8.50)‖eη〈x〉U(·, t)‖L∞ , ‖eη〈x〉∂xU(·, t)‖L∞ ≤ ‖eη〈x〉U(·, t)‖W 1,∞

≤ C‖eη〈x〉U(·, t)‖H2 .

Bound (8.48)(ii) follows, similarly, by repeating in weighted norm X1 the arguments for(8.40) and (8.41) with U replaced by W := U1 − U2, where U j are solutions of (8.35) withdifferent initial data U1

0 and U20 . Bound (8.48)(iii) follows by (8.48)(i) together with the

equation for F := W − eL(ε)TW0.

8.3 Bifurcation analysis

We now carry out the bifurcation analysis following the framework of [TZ2], with a fewslight modifications to simplify the analysis.

8.3.1 Construction of the period map

Given a solution U of (8.2), define the perturbation variable

(8.51) U(x, t, ε) := U ε(x, t)− U ε(x),

satisfying nonlinear perturbation equations

(8.52) Ut − L(ε)U = Qε(U,Ux)x, U(x, 0, ε) = U0(x, ε).

By (8.35), (8.37),

(8.53) Qε(U, ε) = O(|U |2 + |U ||Ux|)

so long as U satisfies a uniform L∞ bound: in particular, for ‖U‖X1 ≤ C.Decomposing

(8.54) U = w1<φε+ + w2=φε

+ + v

where w1<φε+ +w2=φε

+ ∈ Σε (so that, in particular, wj ∈ R), and coordinatizing as (w, v),we obtain after a brief calculation

(8.55)w =

(γ(ε) τ(ε)−τ(ε) γ(ε)

)w +Nw(w, v, ε),

v = L(ε)v +Nv(w, v, ε),

where L = LΠ and

(8.56)Nw,1φ+ +Nw,2φ− = ΠQε(U,Ux)x,

Nv = ΠQε(U,Ux)x.

66

Lemma 8.14. Nw is Ck, k ≥ 2, from (w, v) ∈ R2 × Lq to R2, for any 0 ≤ q ≤ ∞, with

(8.57) |Nv| ≤ C(|w|2 + ‖v‖2Lq).

Nv is Ck, k ≥ 2, from (w, v) ∈ R2 × X1 to X ′2 := ∂xf : ‖e2η〈x〉f‖H1 < +∞ and, for

(|w|+ ‖v‖X1) ≤ C, C1 from (w, v) ∈ R2 ×B1 to ∂x(L1), with

(8.58) ‖Nv‖X′2≤ C(|w|2 + ‖v‖2

X1) and ‖DNv‖∂x(L1) ≤ C(|w|+ ‖v‖X1).

Proof. Direct calculation, using (8.53) and the Π-bounds of Lemma 8.4.

Proposition 8.15. For 0 ≤ t ≤ T , any fixed C1, T > 0, some C > 0, and |a|, ‖b‖X1,|ε| sufficiently small, system (8.55) with initial data (w0, v0) = (a, b) sufficiently small inR2 ×X1 possesses a solution

(8.59) (w, v)(a, b, ε, t) ∈ R2 ×X1

that is Ck+1 in t and Ck in (a, b, ε), k ≥ 2, with respect to the weaker norm B1, with

(8.60)C−1|a| − C‖b‖2

X1≤ |w(t)| ≤ C(|a|+ ‖b‖2

X1),

‖v(t)‖X1 ≤ C(‖b‖X1 + |a|2),

and

(8.61) |D(a,b)(w, v)(t)|L(R2×B1,B1) ≤ C.

In particular, for ‖b‖X1 ≤ C1|a|, all 0 ≤ t ≤ T ,

(8.62) ‖v(t)‖X1 ≤ C|w(t)|.

Likewise, for ‖b‖X1 ≤ C1|a|2, all 0 ≤ t ≤ T ,

(8.63) ‖v(t)‖X1 ≤ C|w(t)|2.

Proof. Existence and uniquess follow by a standard Contraction–mapping argument, usinga priori bounds (8.48), which also imply (8.60) (by decoupling of linear parts) and (8.61).Combining (8.60)(i)–(ii), we obtain evidently (8.62) and (8.63) for |a| sufficiently small.

Setting t = T in (8.59) and applying Duhamel’s principle/variation of constants, wemay express the period map

(8.64) (a, b, ε) → (a, b) := (w, v)(a, b, ε, T )

as a discrete dynamical system

(8.65)a = R(ε, T )a+N1(a, b, ε, T ),

b = S(ε, T )b+N2(a, b, ε, T ),

67

with ε, T ∈ R1, a, N1 ∈ R2 and b ∈ B1, N2 ∈ B2, where

(8.66)R(ε, T ) := e(γ(ε)Id+τ(ε)J)T ,

S(ε, T ) := eL(ε)T ,

J :=(

0 1−1 0

), are the linearized solution operators in w, v and

(8.67)N1(a, b, ε, T ) :=

∫ T

0R(ε, T − s)Nw(w, v, ε)(s)ds,

N2(a, b, ε, T ) :=∫ T

0S(ε, T − s)Nv(w, v, ε)(s)ds

the differences between nonlinear and linear solution operators: equivalently,

(8.68)N1(a, b, ε, T ) = ΠN (a1<φε

+ + a2=φε+ + b, ε, T ),

N2(a, b, ε, T ) = (Id− Π)N (a1<φε+ + a2=φε

+ + b, ε, T ),

where N is the linearization error N defined in (8.47).Evidently, periodic solutions of (8.55) with period T correspond to fixed points of the

period map (equilibria of (8.65)) or, equivalently, zeroes of the displacement map

(8.69)∆1(a, b, ε, T ) := (R(ε, T )− Id)a+N1(a, b, ε, T ),∆2(a, b, ε, T ) := (S(ε, T )− Id)b+N2(a, b, ε, T ).

8.3.2 Lyapunov–Schmidt reduction

We now carry out a nonstandard Lyapunov–Schmidt reduction following the “inverse tem-poral dynamics” framework of [TZ2], tailored for the situation that (S(ε, T ) − Id) is notuniformly invertible, or, equivalently, σ(L) is not bounded away from jπ/T, j ∈ Z. Inthe present situation, L has both an `-dimensional kernel (Lemma 8.18 below) and es-sential spectra accumulating at λ = 0, and no other purely imaginary spectra, so that(S(ε, T )− Id) = (eLT Π− Id) inherits the same properties; see [TZ2] for further discusssion.

Our goal, and the central point of the analysis, is to solve ∆2(a, b, ε, T ) = 0 for b as afunction of (a, ε, T ), eliminating the transverse variable and reducing to a standard planarbifurcation problem in the oscillatory variable a. A “forward” temporal dynamics techniquewould be to rewrite ∆2 = 0 as a fixed point map

(8.70) b = S(ε, T )b+N2(a, b, ε, T ),

then to substitute for T an arbitrarily large integer multiple jT . In the strictly stablecase <σ(L) ≤ −η < 0, |S(ε, jT )|L(X1,X1) < 1/2 for j sufficiently large. Noting that N2 isquadratic in its dependency, we would have therefore contractivity of (8.70) with respect tob, yielding the desired reduction. However, in the absence of a spectral gap between σ(L)

68

and the imaginary axis, |S(ε, jT )|L(X1,X1) does not decay, and may be always greater thanunity; thus, this naive approach does not succeed.

The key idea in [TZ2] is to rewrite ∆2 = 0 instead in “backward” form

(8.71) b = (Id− S(ε, T ))−1N2(a, b, ε, T ),

then show that (Id−S)−1 is well-defined and bounded on Range N2, thus obtaining contrac-tivity by quadratic dependence of N2. Since (Id− S)−1N2 is formally given by

∑∞j=0 S

jN2

this amounts to establishing convergence: a stability/cancellation estimate. Quite similarestimates appear in the nonlinear stability theory, where the interaction of linearized evo-lution S and nonlinear source N2 are likewise crucial for decay. The formulation (8.71) canbe viewed also as a “by-hand” version of the usual proof of the standard implicit functiontheorem [TZ2].

Lemma 8.16. Assuming (A1)–(A3), (H0)–(H3), (Dε),

N1 : (a, b, ε, T ) ∈ R2 × Lq × R2 → N1(a, b, ε, T ) ∈ R1,

is quadratic order and C1, for any 1 ≤ q ≤ ∞, and

N2 : (a, b, ε, T ) ∈ R2 ×X1 × R2 → N1(a, b, ε, T ) ∈ X2,

is quadratic order, and C1 as a map from R2×B1×R2 to B2 for ‖b‖X1 uniformly bounded,with

(8.72)|N1|, |∂ε,TN1|L(R2,R2) ≤ C(|a|+ ‖b‖B1)

2,

|∂aN1|L(R2,R2) + |∂bN1|L(B1,R2) ≤ C(|a|+ ‖b‖B1),

(8.73)‖N2‖X2 , |∂ε,TN2|L(R2,B2) ≤ C(|a|+ ‖b‖X1)

2,

|∂aN2|L(R2,B2) + |∂bN2|L(B1,B2) ≤ C(|a|+ ‖b‖X1).

Proof. Bounds (8.72) follow from representation (8.68), variational bounds (8.48) of Propo-sition 8.13, and the Π-bounds of Lemma 8.4, from which we likewise obtain

‖N2‖X1 , |∂ε,TN2|L(R2,B1) ≤ C(|a|+ ‖b‖X1)2,

|∂aN2|L(R2,B1) + |∂bN2|L(B1,B1) ≤ C(|a|+ ‖b‖X1).

The remaining bounds

‖N2‖∂xe−2η〈x〉H1 ≤ C(|a|+ ‖b‖X1)2,

|∂ε,TN2|L(R2,∂xe−2η〈x〉H1) ≤ C(|a|+ ‖b‖X1)2,

|∂aN2|L(R2,∂xL1) + |∂bN2|L(B1,∂xL1) ≤ C(|a|+ ‖b‖X1)

follow easily from Duhamel representation (8.67)(ii), bounds (8.60)(ii), (8.61), and (8.58),and bounds (8.14), (8.15) on the linearized solution operator eLtΠ.

69

Corollary 8.17 ([TZ2]). Assuming (A1)–(A3), (H0)–(H3), (Dε),

(8.74) ∆2(a, b, ε, T ) = 0 (a, b, ε, T ) ∈ R2 ×X1 × R2,

is equivalent to

(8.75) b = (Id− S(ε, T ))−1N2(a, b, ε, T ) + ω

for

(8.76) ω ∈ Ker(Id− S(ε, T )) ∩X1.

Proof. Applying to the left of (8.74) the right inverse (Id−S(ε, a, b))−1 given by Proposition8.10, we obtain by Lemma 8.16

b := (Id− S(ε, T ))−1(Id− S(ε, T ))b

= (Id− S(ε, T ))−1N2(ε, a, b) ∈ X1.

Observing that b− b belongs to Ker(Id− S(ε, a, b)) ∩X1 by the right inverse property, weobtain (8.75). Conversely, (8.75) implies (8.74) by application on the left of Id−S(a, b, ε, T ).

Lemma 8.18. The kernel of (Id−S(ε, T )) is of fixed dimension ` as in (H4), is independentof T , and has a smooth basis ω =

(ω1 . . . ω`

)(ε). In the Lax case, it is generated entirely

by translation invariance.

Proof. This follows by the corresponding properties of KerL assumed in (H4).

Corollary 8.19. Assuming (A1)–(A3), (H0)–(H3), (Dε), the map

(8.77) T (a, b, ε, T, α) := (Id− S(ε, T ))−1N2(a, b, ε, T ) + ω(ε)α,

(Id − S)−1 : X2 → X1 as defined in Proposition 8.10 is bounded from R2 × X1 × R2+` toX1, and C1 from R2 × B1 × R2+` to X1, for |α| bounded and |a| + ‖b‖X1 + |ε| sufficientlysmall, with

(8.78)

‖T ‖X1 ≤ C(|a|+ ‖b‖2X1

),|∂a,bT |L(B1,B1) ≤ C(|a|+ ‖b‖X1),

|∂TT |L(B1,B1) ≤ C(|a|2 + ‖b‖2X1

),

|∂εT |L(B1,B1) ≤ C(|a|2 + ‖b‖2X1

+ |α|),|∂αT |L(B1,B1) ≤ C(|a|2 + ‖b‖2

X1+ 1).

Proof. Immediate, from Proposition 8.10 and Lemmas 8.16 and 8.18.

70

Proposition 8.20 ([TZ2]). Under assumptions (A1)–(A3), (H0)–(H3), (Dε), there existsa function β(a, ε, T, α), bounded from R4+` to X1 and C1 from R4+` to B1, with

(8.79) ∆2(a, β(a, ε, T, α), ε, T ) ≡ 0,

(8.80)

‖β‖X1 , ‖∂ε,Tβ‖L(R,B1) ≤ C(|a|2 + |α|),‖∂aβ‖L(R2,B1) ≤ C|a|,‖∂αβ‖L(R2,B1) ≤ C,

for |(a, ε, α)| sufficiently small. Moreover, for |(a, ε)|, ‖b‖X1 sufficiently small, all solutionsof (8.74) lie on the `-parameter manifold b = β(a, ε, T, α).

Proof. By Corollary 8.17, (8.74) is equivalent to the fixed-point problem

b = T (a, b, ε, T, α)

for some α ∈ R`. By (8.78)(i)–(ii), for |(a, ε, α)|, sufficiently small, T preserves a small ballin X1 on which it is contractive in b with respect to the weaker norm ‖ · ‖B1 . Observingthat closed balls in X1 are closed also in B1, we may conclude by the contraction-mappingprinciple the existence of a unique solution β, which, moreover, inherits the regularity of Tin its dependence on parameters (a, ε, T, α).

Remark 8.3. At the expense of further bookkeeping, we may replace (8.5) by ‖U‖X1 :=‖eη〈x〉U‖H4 , ‖∂xU‖X2 := ‖∂xU‖X1 +‖e2η〈x〉U‖H1 and carry one further derivative in (a, ε, T )throughout the analysis, to obtain C2 dependence of β(a, ε, T, α). Indeed, strengthening(H0) to k = 2r + 1 in (H0), we may obtain arbitrary smoothness Cr of reduction function(nullcline) β(·).

8.3.3 Proof of the main theorem

The bifurcation analysis is straightforward now that we have reduced to a finite-dimensionalproblem, the only tricky point being to deal with the `-fold multiplicity of solutions (parametrizedby α). Define to this end

(8.81) β(a, ε, T, α) := β(a, ε, T, |a|α),

with α restricted to a ball in R`, noting, by (8.80), that

(8.82) ‖β‖X1 , |∂a,ε,T,αβ|L(R,B1) ≤ C|a|,

with β Lipshitz in (a, ε, T, α) and C1 away from a = 0. Solutions (w, v) of (8.55) originatingat (a, b) = (a, β), by (8.62), remain for 0 ≤ t ≤ T in a cone

(8.83) C := (w, v) : |v| ≤ C1|w|,

C1 > 0. Likewise, any periodic solution of (8.55) originating in C, since it necessarilysatisfyies ∆2 = 0, must originate from data (a, b) of this form.

71

Proof of Theorem 8.2. Defining b ≡ β(a, ε, T, α), and recalling invariance of C under flow(8.55), we may view v(t) as a multiple

(8.84) v(x, t) = c(a, ε, T, α, x, t)w(t)

of w(t), where c is bounded, Lipschitz in all arguments, and C1 away from a = 0. Substi-tuting into (8.55)(i), we obtain a planar ODE

(8.85) w =(γ(ε) τ(ε)−τ(ε) γ(ε)

)w + N(w, ε, T, t, α, a)

in approximate Hopf normal form, with nonlinearity N := Nw(w, v, ε) now nonautonomousand depending on the additional parameters (T, α, a), but still satisfying the key bounds

(8.86) |N |, |∂ε,T,α,aN | ≤ C|w|2; |∂wN | ≤ C|w|

along with planar bifurcation criterion (Dε)(ii). From (8.86), we find that N is C1 in allarguments, also at a = 0. By standard arguments (see, e.g., [HK, TZ1]), we thus obtaina classical Hopf bifurcation in the variable w with regularity C1, yielding existence anduniqueness up to time-translates of an `-parameter family of solutions originating in C,indexed by r and δ with r := a1 and (without loss of generality) a2 ≡ 0. It remains only toestablish uniqueness up to spatial translates.

In the Lax case, we observe, first, that, by dimensional considerations, the one-parameterfamily constructed must agree with the one-parameter family of spatial translates. Second,we argue as in [TZ2] that any periodic solution has a spatial translate originating in C,yielding uniqueness up to translation among all solutions and not only those originating inC; see Proposition 2.17 and Corollary 2.18 [TZ2] for further details.

In the overcompressive case, we observe, likewise, that the `-parameter family con-structed consists of spatial translates of a smooth (`−1)-parameter family of distinct orbitsof traveling-wave ODE (8.4), then argue as in [TZ2] that any periodic solution has a spatialtranslate lying within a corresponding cone C′ about some member of this (`−1)-parameterfamily. Constructing solutions about each such member by the same technique, we thusobtain from their union an (` − 1) parameter family containing translates of all periodicsolutions on an entire neighborhood of U , and not only in cone C. See Proposition 2.22 andCorollary 2.23 [TZ2] for further details.

Remarks 8.4. The apparent restriction to C1 regularity caused by factor |a| in the defini-tion of β is illusory, since we restrict eventually to the ray a2 ≡ 0 and a1 ≥ 0, on which β isas smooth as β. By Remark 8.3, the latter may be made as smooth as desired by assumingsufficient regularity in (H0), and so we can carry out a bifurcation analysis to arbitrarydesired regularity.

Restricting to the “central” solutions α ≡ 0 of the constructed cone of solutions yieldsβ = O(|a|2), and thus the exact Hopf normal form

(8.87) w =(γ(ε) τ(ε)−τ(ε) γ(ε)

)w +Nw(w, 0, ε) + M(w, ε, T, t, α, a),

72

where M = O(|w|3) and ∂w,aM = O(|w|2). For these central solutions, we thus recoverall of the standard description of Hopf bifurcation: in particular, that ε(r) is a pitchforkbifurcation with (dε/dr)(0) = 0 (hence sgn(d2ε/dr2)(0) is expected to determine stability;see Remark 8.1).

9 Cellular instability for flow in a duct

We conclude by briefly discussing multidimensional stability and bifurcation of flow in afinite cross-sectional duct, or “shock tube”, extending infinitely in the axial direction.

It is well known both experimentally and numerically [BE, MT, BMR, FW, MT, ALT,AT, F1, F2, KS] that shock and detonation waves propagating in a finite cross-sectionduct can exhibit time-oscillatory or “cellular” instabilities, in which the initially nearlyplanar shock takes on nontrivial transverse geometry. Majda et al [MR1, MR2, AM] havestudied the onset of such instabilities by weakly nonlinear optics expansion of the associatedplanar inviscid shock in the whole space. More recently, Kasimov–Stewart [KS] and Texier–Zumbrun [TZ2, TZ3, TZ3] have studied these instabilities as Hopf bifurcations of flow ina finite-cross-section duct, associated with passage across the imaginary axis of eigenvaluesof the linearized operator about the wave.

In this section, following [Z11], we combine the analyses of [BSZ, TZ4, Z4] to makean explicit connection between stability of planar shocks on the whole space, and Hopfbifurcation in a finite cross-section duct, by a variant of the mechanism proposed by Majdaet al. Specifically, we point out that violation of the refined stability condition of [ZS, Z1,BSZ], a viscous correction of the inviscid planar stability condition of Majda [M1]–[M4], isgenerically associated with Hopf bifurcation in a finite cross-section duct corresponding tothe observed cellular instability, for cross-sectionM sufficiently large. Indeed, we show more,that this is associated with a cascade of bifurcations to higher and higher wave numbers andmore and more complicated solutions, with features on finer and finer length/time scales.

9.1 Equations and assumptions

Consider a planar viscous shock solution

(9.1) u(x, t) = u(x1 − st)

of a two-dimensional system of viscous conservation laws

(9.2) ut +∑

f j(u)xj = ∆xu, u ∈ Rn, x ∈ R2, t ∈ R+

on the whole space. This may be viewed alternatively as a planar traveling-wave solutionof (9.2) on an infinite channel

C := x : (x1, x2) ∈ R1 × [−M,M ]

73

under periodic boundary conditions

(9.3) u(x1,M) = u(x1,−M).

We take this as a simplified mathematical model for compressible flow in a duct, inwhich we have neglected boundary-layer phenomena along the wall ∂Ω in order to isolatethe oscillatory phenomena of our main interest.

Following [TZ2], consider a one-parameter family of standing planar viscous shock so-lutions uε(x1) of a smoothly-varying family of conservation laws

(9.4) ut = F(ε, u) := ∆xu−2∑

j=1

F j(ε, u)xj , u ∈ Rn

in a fixed channel C, with periodic boundary conditions (typically, shifts∑F j(ε, u)xj :=∑

f j(u)xj − s(ε)ux1 of a single equation (9.2) written in coordinates x1 → x1 − s(ε)tmoving with traveling-wave solutions of varying speeds s(ε)), with linearized operatorsL(ε) := ∂F/∂u|u=uε . Profiles uε satisfy the standing-wave ODE

(9.5) u′ = F 1(ε, u)− F 1(ε, u−).

Let

(9.6) A1±(ε) := lim

z→±∞F 1

u (ε, uε).

Following [Z1, TZ2, Z4], we make the assumptions:(H0) F j ∈ Ck, k ≥ 2.

(H1) σ(A1±(ε)) real, distinct, and nonzero, and σ(

∑ξjA

j±(ε)) real and semisimple

for ξ ∈ Rd.

For most of our results, we require also:(H2) Considered as connecting orbits of (9.5), uε are transverse and unique up to

translation, with dimensions of the stable subpace S(A1+) and the unstable subspace U(A1

−)summing for each ε to n+ 1.

(H3) det(r−1 , . . . , r−c , r

+c+1, . . . , r

+n , u+−u−) 6= 0, where r−1 , . . . , r

−c are eigenvectors of

A1− associated with negative eigenvalues and r−c+1, . . . , r

−n are eigenvectors of A1

+ associatedwith positive eigenvalues.

Hypothesis (H2) asserts in particular that uε is of standard Lax type, meaning that theaxial hyperbolic convection matrices A1

+(ε) and A1−(ε) at plus and minus spatial infinity

have, respectively, n−c positive and c−1 negative real eigenvalues for 1 ≤ c ≤ n, where c isthe characteristic family associated with the shock: in other words, there are precisely n−1outgoing hyperbolic characteristics in the far field. Hypothesis (H3) may be recognizedas the Liu–Majda condition corresponding to one-dimensional stability of the associatedinviscid shock. In the present, viscous, context, this, together with transverality, (H2),plays the role of a spectral nondegeneracy condition corresponding in a generalized sense[ZH, Z1] to simplicity of the embedded zero eigenvalue associated with eigenfunction ∂x1 uand translational invariance.

74

9.2 Stability conditions

Our first set of results, generalizing the one-dimensional analysis of [Z4], characterize stabil-ity/instability of waves uε in terms of the spectrum of the linearized operator L(ε). Fixingε, we suppress the parameter ε. We start with the routine observation that the semilinearparabolic equation (9.2) has a center-stable manifold about the equilibrium solution u.

Proposition 9.1. Under assumptions (H0)–(H1), there exists in an H2 neighborhood of theset of translates of u a codimension-p translation invariant Ck (with respect to H2) centerstable manifold Mcs, tangent at u to the center stable subspace Σcs of L, that is (locally)invariant under the forward time-evolution of (9.2)–(9.3) and contains all solutions thatremain bounded and sufficiently close to a translate of u in forward time, where p is the(necessarily finite) number of unstable, i.e., positive real part, eigenvalues of L.

Proof. By standard considerations [He, TZ2], L(ε) possesses no essential spectrum and atmost a finite set of positive real part eigenvalues on <λ > 0. With this observation, theproof goes word for word as in the proof of Theorem 7.1 in the one-dimensional case, whichdepends only on the properties of L as a sectorial second-order elliptic operator, and onsemilinearity and translation-invariance of the underlying equations (9.2).

Introduce now the nonbifurcation condition:(D1) L has no nonzero imaginary eigenvalues.

As discussed above, (H2)–(H3) correspond to a generalized notion of simplicity of theembedded eigenvalue λ = 0 of L. Thus, (D1) together with (H2)–(H3) correspond to theassumption that there are no additional (usual or generalized) eigenvalues on the imaginaryaxis other than the translational eigenvalue at λ = 0; that is, the shock is not in transitionbetween different degrees of stability, but has stability properties that are insensitive tosmall variations in parameters.

Theorem 9.2. Under (H0)–(H3) and (D1), u is nonlinearly orbitally stable as a solutionof (9.2)–(9.3) under sufficiently small perturbations in L1 ∩ H2 lying on the codimensionp center stable manifold Mcs of u and its translates, where p is the number of unstableeigenvalues of L, in the sense that, for some α(·), all Lp,

(9.7)

|u(x, t)− u(x− α(t))|Lp ≤ C(1 + t)−12(1− 1

p)|u(x, 0)− u(x)|L1∩H2 ,

|u(x, t)− u(x− α(t))|H2 ≤ C(1 + t)−14 |u(x, 0)− u(x)|L1∩H2 ,

α(t) ≤ C(1 + t)−12 |u(x, 0)− u(x)|L1∩H2 ,

α(t) ≤ C|u(x, 0)− u(x)|L1∩H2 .

Moreover, it is orbitally unstable with respect to small H2 perturbations not lying in Mcs,in the sense that the corresponding solution leaves a fixed-radius neighborhood of the set oftranslates of u in finite time.

Proof. See Section 9.7.

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Remark 9.3. Theorem 9.2 includes in passing the result that existence of unstable eigen-values implies nonlinear instability, hence completey characterizes stability/instability ofwaves under the nondegeneracy condition (D1). The rates of decay (9.7) are exactly thoseof the one-dimensional case [Z4].

9.3 Bifurcation conditions

Define the Hopf bifurcation condition:(D2) Outside the essential spectrum of L(ε), for ε and δ > 0 sufficiently small, the

only eigenvalues of L(ε) with real part of absolute value less than δ are a crossing conjugatepair λ±(ε) := γ(ε)± iτ(ε) of L(ε), with γ(0) = 0, ∂εγ(0) > 0, and τ(0) 6= 0.

Proposition 9.4 ([TZ2, TZ3]). Let uε, (9.4) be a family of traveling-waves and systemssatisfying assumptions (H0)–(H3) and (D2), and η > 0 sufficiently small. Then, for a ≥ 0sufficiently small and C > 0 sufficiently large, there are C1 functions ε(a), ε(0) = 0, andT ∗(a), T ∗(0) = 2π/τ(0), and a C1 family of solutions ua(x1, t) of (9.4) with ε = ε(a),time-periodic of period T ∗(a), such that

(9.8) C−1a ≤ supx1∈R

eη|x1|∣∣ua(x, t)− uε(a)(x1)

∣∣ ≤ Ca for all t ≥ 0.

Up to fixed translations in x, t, for ε sufficiently small, these are the only nearby solutionsas measured in norm ‖f‖X1 := ‖(1 + |x1|)f(x)‖L∞(x) that are time-periodic with periodT ∈ [T0, T1], for any fixed 0 < T0 < T1 < +∞. Indeed, they are the only nearby solutionsof form ua(x, t) = ua(x− σat, t) with ua periodic in its second argument.

Proof. See Section 9.7.

Remark 9.5. Together with Theorem 9.2, Proposition 9.4 implies that, under the Hopfbifurcation assumption (D2) together with the further assumption that L(ε) have no strictlypositive real part eigenvalues other than possibly λ±, waves uε are linearly and nonlinearlystable for ε < 0 and unstable for ε > 0, with bifurcation/exchange of stability at ε = 0.

9.4 Longitudinal vs. transverse bifurcation

The analysis of [TZ2] in fact gives slightly more information. Denote by

(9.9) Πεf :=∑j=±

φεj(x)〈φε

j , f〉

the L(ε)-invariant projection onto the oscillatory eigenspace Σε := Spanφε±, where φε

± arethe eigenfunctions associated with λ±(ε). Then, we have the following result, proved butnot explicitly stated in [TZ2].

Proposition 9.6. Under the assumptions of Proposition 9.4, also

(9.10) supx1

eη|x1||ua − u−Πε(ua − u)| ≤ Ca2 for all t ≥ 0.

76

Proof. The weaker bound

(9.11) supx1

(1 + |x1|)|ua − u−Πε(ua − u)| ≤ Ca2 for all t ≥ 0,

is established in the course of the Lyapunov reduction of [TZ2]; see (2.17), Proposition 2.9,case ω ≡ 0. The stronger version (9.10) follows by the same argument together with thestrengthened cancellation estimates of Proposition 2.5 [TZ3].

Bounds (9.8) and (9.10) together yield the standard finite-dimensional property thatbifurcating solutions lie to quadratic order in the direction of the oscillatory eigenspace ofL(ε). From this, we may draw the following additional conclusions about the structure ofbifurcating waves. By separation of variables, and x2-independence of the coefficients ofL(ε), we have that the eigenfunctions ψ of L(ε) decompose into families

(9.12) eiξx2ψ(x1), ξ =πk

M,

associated with different integers k, where M is cross-sectional width. Thus, there are twovery different cases: (i) (longitudinal instability) the bifurcating eigenvalues λ±(ε) are asso-ciated with wave-number k = 0, or (ii) (transverse instability) the bifurcating eigenvaluesλ±(ε) are associated with wave-numbers ±k 6= 0.

Corollary 9.7. Under the assumptions of Proposition 9.4, ua depend nontrivially on x2

if and only if the bifurcating eigenvalues λ± are associated with transverse wave-numbers±k 6= 0.

Proof. For k 6= 0, the result follows by the fact that, by (9.8) and (9.10), Π(ua − u) is thedominant part of ua − u, and the fact that Πf by inspection depends nontrivially on x2

whenever Πf 6= 0. For k = 0, the result follows by uniqueness, and the fact that, restrictedto the one-dimensional case, the same argument yields a bifurcating solution dependingonly on x1.

Bifurcation through longitudinal instability corresponds to “galloping” or “pulsating” in-stabilities described in detonation literature, while symmetry-breaking bifurcation throughtransverse instability corresponds to “cellular” instabilities introducing nontrivial transversegeometry to the structure of the propagating wave.

9.5 The refined stability condition and bifurcation

Longitudinal or “galloping” bifurcation, though almost certainly occurring for detonations(see [TZ2, TZ4] and references therein), has up to now not been observed for shock waves asfar as we know (though we see no reason why they should not in general be possible), nor hasthere been proposed any specific mechanism by which this might occur. The main purposeof the present paper, as we now describe, is to point out that for transverse or “cellular”bifurcations, to the contrary, there is a simple and natural mathematical mechanism, closelyrelated to the inviscid stability theory for shocks in the whole space, by which they can andlikely do occur.

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9.5.1 The inviscid stability condition

Inviscid stability analysis for shocks in the whole space centers about the Lopatinski deter-minant

(9.13) ∆(ξ, λ) :=(R−

1 · · · R−p−1 R+

p+1 · · · R+n λ[u] + iξ[f2]

),

ξ ∈ R1, λ = γ + iτ ∈ C, τ > 0, a spectral determinant whose zeroes correspond tonormal modes eλteiξx2w(x1) of the constant-coefficient linearized equations about the dis-continuous shock solution. Here, R+

p+1, . . . ,R+n and R−

1 , . . . ,R−p−1 denote bases for the

unstable/resp. stable subspaces of

(9.14) A+(ξ, λ) := (λI + iξdf2(u±))(df1(u±))−1.

Weak stability |∆| > 0 for τ > 0 is clearly necessary for linearized stability, while strong,or uniform stability, |∆|/|(ξ, λ)| ≥ c0 > 0, is sufficient for nonlinear stability. Betweenstrong instability, or failure of weak stability, and strong stability, there lies a region ofneutral stability corresponding to the appearance of surface waves propagating along theshock front, for which ∆ is nonvanishing for <λ > 0 but has one or more roots (ξ0, λ0)with λ0 = τ0 pure imaginary. This region of neutral inviscid stability typically occupies anopen set in physical parameter space [M1, M2, M3, BRSZ, Z1, Z2]. For details, see, e.g.,[Er1, M1, M2, M3, Me, Se1, Se2, Se3, ZS, Z1, Z2, Z3, BRSZ], and references therein.

It has been suggested [MR1, MR2, AM] that nonlinear hyperbolic evolution of surfacewaves in the region of neutral linear stability might explain the onset of complex behaviorsuch as Mach stem formation/kinking of the shock. We pursue here a variant of this ideabased instead on interaction between neglected viscous effects and transverse spatial scales.

9.5.2 The refined stability condition

Viscous stability analysis for shocks in the whole space centers about the Evans function

D(ξ, λ),

ξ ∈ R1, λ = γ + iτ ∈ C, τ > 0, a spectral determinant analogous to the Lopatinskideterminant of the inviscid theory, whose zeroes correspond to normal modes eλteiξx2w(x1),of the linearized equations about u (now variable-coefficient), or spectra of the linearizedoperator about the wave. The main result of [ZS], establishing a rigorous relation betweenviscous and inviscid stability, was the asymptotic expansion

(9.15) D(ξ, λ) = γ∆(ξ, λ) + o(|(ξ, λ)|)

of D about the origin (ξ, λ) = (0, 0), where γ is a constant measuring tranversality of u as aconnecting orbit of the traveling-wave ODE. Equivalently, consideringD(ξ, λ) = D(ρξ0, ρλ0)as a function of polar coordinates (ρ, ξ0, λ0), we have

(9.16) D|ρ=0 = 0 and (∂/∂ρ)|ρ=0D = γ∆(ξ0, λ0).

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An important consequence of (9.15) is that weak inviscid stability, |∆| > 0, is necessaryfor weak viscous stability, |D| > 0 (an evident necessary condition for linearized viscousstability). For, (9.15) implies that the zero set of D is tangent at the origin to the cone∆ = 0 (recall, (9.13), that ∆ is homogeneous, degree one), hence enters τ > 0 if ∆ = 0does. Moreover, in case of neutral inviscid stability ∆(ξ0, iτ0) = 0, (ξ0, iτ0) 6= (0, 0), onemay extract a further, refined stability condition

(9.17) β := −Dρρ/Dρλ|ρ=0 ≥ 0

necessary for weak viscous stability. For, (9.16) then implies Dρ|ρ=0 = γ∆(ξ0, iτ0) = 0,whence Taylor expansion of D yields that the zero level set of D is concave or convextoward τ > 0 according as the sign of β; see [ZS] for details. As discussed in [ZS, Z1],the constant β has a heuristic interpretation as an effective diffusion coefficient for surfacewaves moving along the front.

As shown in [ZS, BSZ], the formula (9.17) is well-defined whenever ∆ is analyticat (ξ0, iτ), in which case D considered as a function of polar coordinates is analytic at(0, ξ0, iτ0), and iτ0 is a simple root of ∆(ξ0, ·). The determinant ∆ in turn is analyticat (ξ0, iτ0), for all except a finite set of branch singularities τ0 = ξ0ηj . As discussed in[BSZ, Z2, Z3], the apparently nongeneric behavior that the family of holomorphic functions∆ε associated with shocks (uε

+, uε−) have roots (ξε

0, iτ0(ε)) with iτ0 pure imaginary on anopen set of ε is explained by the fact that, on certain components of the complement on theimaginary axis of this finite set of branch singularities, ∆ε(ξε

0, ·) takes the imaginary axisto itself. Thus, zeros of odd multiplicity persist on the imaginary axis, by consideration ofthe topological degree of ∆ε as a map from the imaginary axis to itself.

Moreover, the same topological considerations show that a simple imaginary root ofthis type can only enter or leave the imaginary axis at a branch singularity of ∆ε(ξε, ·) orat infinity, which greatly aids in the computation of transition points for inviscid stability[BSZ, Z1, Z2, Z3]. As described in [Z2, Z3, Se1], escape to infinity is always associated withtransition to strong instability. Indeed, using real homogeneity of ∆, we may rescale by |λ|to find in the limit as |λ| → ∞ that 0 = |λ0|−1∆(ξ0, λ0) = ∆(ξ0/|λ0|, λ0/|λ0|) → ∆(0, i),which, by the complex homogeneity ∆(0, λ) ≡ λ∆(0, 1) of the one-dimensional Lopatinskideterminant ∆(0, ·), yields one-dimensional instability ∆(0, 1) = 0. As described in [Z1],Section 6.2, this is associated not with surface waves, but the more dramatic phenomenonof wave-splitting, in which the axial structure of the front bifurcates from a single shock toa more complicated multi-wave Riemann pattern.

Example 9.8. For gas dynamnics, complex symmetry, ∆(ξ, λ) = ∆(−ξ, λ), and rotationalinvariance, ∆(ξ, λ) = ∆(−ξ, λ), imply that

(9.18) ∆(ξ, iτ) = ∆(|ξ|2, |τ |2).

Explicit computation [Er1, M1, Z1] yields that ∆(ξ0, ·) has a pair of branch points of square-root type, located at |τ0|2 = |ξ0|2(M2− 1), where M is the downstream Mach number c2/u2,where c is sound speed and u the axial particle velocity of the shock on the downstream side,

79

defined as the side in the direction of particle velocity. Transition from strong stability toneutral stability occurs through a pair of simple imaginary zeros entering the imaginary axisat the branch points, and transition from neutral stability to strong instability occurs throughescape of these zeros to infinity, with associated one-dimensional instability/wave-splitting.

Remark 9.9. We note in passing that one-dimensional inviscid stability ∆ε(0, 1) 6= 0 isequivalent to (H3) through the relation

(9.19) ∆ε(0, λ) = λ det(r−1 , . . . , r−c , r

+c+1, . . . , r

+n , u+ − u−).

9.5.3 Transverse bifurcation of flow in a duct

We now make an elementary observation connecting cellular bifurcation of flow in a ductto stability of shocks in the whole space: specifically, to violation of the refined stabilitycondition. Assume for the family uε the stability conditions:

(B1) For ε sufficiently small, the inviscid shock (uε+, u

ε−) is weakly stable; more

precisely, ∆ε(1, λ) has no roots <λ ≥ 0 but a single simple pure imaginary root λ(ε) =iτ∗(ε) 6= 0 lying away from the singularities of ∆ε.

(B2) The refined stability coefficient β(ε) defined in (9.17) satisfies <β(0) = 0,∂ε<β(0) < 0.

Lemma 9.10 ([ZS, Z1]). Assuming (H0)–(H2) and (B1), for ε, ξ sufficiently small, thereexist a smooth family of roots (ξ, λε

∗(ξ)) of D(ξ, λ) with

(9.20) λε∗(ξ) = iξτ∗(ε)− ξ2β(ε) + δ(ε)ξ3 + r(ε, ξ)ξ4, r ∈ C1(ε, ξ).

Moreover, these are the unique roots of D satisfying <λ ≥ −|ξ|/C for some C > 0 andρ = |(ξ, λ)| sufficiently small.

Proof. This follows by the Implicit Function Theorem applied to the function D(ρ, λ0) :=ρ−1D(ρξ0, ρλ0), about the values (ρ, λ0) = (0, iξ0τ∗(ε), where ξ0 is without loss of generalityheld fixed, using the facts thatD expressed in polar coordinates (ρ, ξ0, λ0) satisfiesD|ρ=0 ≡ 0and ∂ρD|ρ=0 ≡ ∆, so that Dρ, Dλλ, and Dλ all vanish at (0, ξ0, ξ0iτ∗(ε)). For details, seethe proof of Theorem 3.7, [Z1].

Corollary 9.11. Assuming (H0)–(H2) and (B1)-(B2), for ε, ξ sufficiently small, there is aunique C1 function E(ξ) 6= 0, E(0) = O, such that <λε

∗(ξ) = 0 for ε = E(ξ). In the genericcase <δ(0) 6= 0, moreover,

(9.21) E(ξ) ∼ (<δ(0)/∂εβ(0))ξ.

Proof. As a consequence of (9.20), we have for some smooth G

(9.22) <(λε

∗(ξ)ξ2

)= −<β(ε) + ξG(ε, ξ),

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G := (δ(ε) + r(ε, ξ)), whence the equation 0 = <(

λε∗(ξ)

ξ2

)= −<β(ε) + ξG(ξ, ε) has a

unique root ε = E(ξ) by assumption (B2) and standard scalar bifurcation theory. FromG = δ(ε)ξ+O(|ξ|2), we find, in the generic case <δ(0) 6= 0, that ∂ξ(ξG)|ξ,ε=0,0 = <δ(0) 6= 0,yielding (9.21).

Remark 9.12. As described further in [BSZ], the above results on the refined stabilitycondition apply also in the case of “real” or partial viscosity, in particular to the physicalNavier–Stokes equations of compressible gas dynamics and MHD.

To (B1) and (B2), adjoin now the additional assumptions:(B3) δ(0) 6= 0.

(B4) At ε = 0, the Evans function D(ξ, λ) has no roots ξ ∈ R, <λ ≥ 0 outside asufficiently small ball about the origin.

Then, we have the following main result.

Theorem 9.13. Assuming (H0)–(H2) and (B1)-(B4), for εmax > 0 sufficiently small andeach cross-sectional width M sufficiently large, there is a finite sequence 0 < ε1(M) < · · · <εk(M) < · · · ≤ εmax, with εk(M) ∼ (<δ(0)/∂εβ(0))πk

M , such that, as ε crosses successiveεk from the left, there occur a series of transverse (i.e., “cellular”) Hopf bifurcations of uε

associated with wave-numbers ±k, with successively smaller periods Tk(ε) ∼ τ∗(0)2Mk .

Proof. By (B4), for |ε| ≤ εmax sufficiently small, we have by continuity that there exist noroots of D(ξ, λ) for <λ ≥ −1/C, C > 0, outside a small ball about the origin. By Lemma9.10, within this small ball, there are no roots other than possibly (ξ, λε(ξ)) with <λ ≥ 0:in particular, no nonzero purely imaginary spectra are possible other than at values λε(ξ)for operator L(ε) acting on functions on the whole space.

Considering L instead as an operator acting on functions on the channel C := x :(x1, x2) ∈ R1 × [−M,M ], we find by discrete Fourier transform/separation of variablesthat its spectra are exactly the zeros of D(ξk, λ), as ξk = πk

L runs through all integer wave-numbers k; see (9.12). Applying Corollary 9.11, and using (B3), we find, therefore, thatpure imaginary eigenvalues of L(ε) with |ε| ≤ εmax sufficiently small occur precisely atvalues ε = εk, and consist of crossing conjugate pairs λk

±(ε) associated with wave-numbers±k, satisfying Hopf bifurcation condition (D2) with

=λk±(ε) ∼ τ∗(0)

)πkL.

Applying Proposition 9.4, we obtain the result.

Remark 9.14. As evidenced by decreasing periods Tk, this phenomenon of increasing-complexity solutions is completely different from the more familiar one of period-doubling.

Remark 9.15. Lemma 9.10 and Corollary 9.11 are readily generalized to the case with(B1) replaced by (B1’) For ε sufficiently small, the inviscid shock (uε

+, uε−) is weakly stable,

81

with all pure imaginary roots simple and lying away from the singularities of ∆ε. In thiscase we obtain a famiy of roots/crossings of the imaginary axis, one for each imaginaryroot of ∆ε. In particular, for gas dynamics, due to rotational invariance (see example 9.8),we obtain families of four crossing eigenvalues λ±(εk), with each of λ+ and λ− occurringat both wave-numbers k and −k. This is not a standard Hopf bifurcation, but a morecomplicated version with O(2) symmetry, and so we cannot apply directly Theorem 9.13.

9.6 Discussion and open problems

We have presented in a simple setting a rigorous mathematical demonstration of a mech-anism by which destabilization of hyperbolic surface waves arising in the inviscid shockstability problem in the whole space can, at appropriate transverse length scales, lead toHopf bifurcation of a viscous shock in a finite-cross-section duct: specifically, destabilizationof the effective transverse viscosity β investigated in [ZS, BSZ], or violation of the refinedstability condition. Though in a similar spirit, this appears to be a fundamentally differentmechansim than the hyperbolic ones proposed by Majda et al [MR1, MR2, AM] via weaklynoninear geometric optics.

We point out that as shock parameters cross the inviscid strong instability boundary, gas-dynamical shocks undergo one-dimensional instability, or wave-splitting, a more dramaticchange in front topology than the cellular instabilities we seek to investigate. Thus, cellularinstability must occur before the strong instability boundary is reached. Experimentalobservations, though not conclusive, indicate that nonetheless it occurs near the stronginstability boundary [BE], suggesting that it lies in the region of neutral inviscid stabilityas we have conjectured.

More, if the transition to cellular instability occurs at low frequencies, it must occur bythe scenario described, or else the viscous shock would remain stable up to the point ofwave-splitting. If, on the other hand, it occurs at high frequencies, then as pointed out in[BSZ], then it necessarily involves Hopf bifurcation, by one-dimensional inviscid stability,(H2) (satisfied for typical equations of state). Thus, it would appear quite promising tosearch for Hopf bifurcations in the region of neutral inviscid stability, whether of the “low-frequency” type studied here or a “high-frequency” type involving unknown mechanisms.This would be a very interesting direction for numerical investigations, for example by thenumerical Evans function techniques of [Br1, Br2, BrZ, BDG, HuZ2, BHRZ].

Another interesting direction would be investigation of the stability coefficient β, bothnumerically and analytically. As pointed out in [BSZ], this is numerically quite well-conditioned. One might also consider attempting to carry out an asymptotic analysis nearthe endpoints of the region of neutral stability, at which the imaginary root τ∗ approacheseither a branch point of ∆ or else infinity.

At a technical level, an interesting open problem is to carry out a bifurcation analysisin the rotationally symmetric case, for example, for gas dynamics, in which the bifurcationassociated with crossing λ± no longer a standard Hopf bifurcation but a more complicatedtype involving O(2) symmetry. For a description of Hopf bifurcation with O(2) symmetry,see, for example, [W]. For a circular cross-section, there is besides axial translation an

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additional continuous group invariance of rotation in the transverse direction, leading tothe possibility of “spinning” instabilities. These degenerate cases require further analysis atthe level of the finite-dimensional reduced equations; however, the initial reduction to finitedimensions, as carried out in [TZ2], is essentially the same. Other open problems are toextend to detonations, as done for the one-dimensional case in [TZ4] and to treat also real,or partial viscosities. As discussed in [TZ2, TZ3], the latter problem involves interestingissues involving Lagrangian vs. Eulerian formulations.

9.7 Analysis

Nonlinear stability and bifurcation follow quite similarly as in the one-dimensional case, de-composing behavior into a one-dimensional (averaged in x2) flow driving time-exponentiallydamped transverse modes.10 Define the perturbation variable

(9.23) v(x, t) := u(x+ α(t), t)− u(x)

for u a solution of (9.2)–(9.3), where α is to be specified later. Subtracting the equationsfor u(x+ α(t), t) and u(x), we obtain the nonlinear perturbation equation

(9.24) vt − Lv =2∑

j=1

Nj(v)xj + ∂tα(ux1 + ∂x1v),

where

(9.25) L := ∆x −2∑

j=1

∂xjAj(x), Aj := dfj(u)

denotes the linearized operator about u and Nj(v) := −(f j(u+v)−f j(u)−df j(u)v), where,so long as |v|H1 (hence |v|L∞ and |u|L∞) remains bounded,

(9.26) N j(v) = O(|v|2), ∂xNj(v) = O(|v||∂xv|), ∂2

xNj(v) = O(|∂x|2 + |v||∂2

xv|).

9.7.1 Projector bounds

Let Πu denote the eigenprojection of L onto its unstable subspace Σu, and Πcs = Id − Πu

the eigenprojection onto its center stable subspace Σcs.

Lemma 9.16. Assuming (H0)–(H1), there is Πj defined in (9.29) such that

(9.27) Πj∂x = ∂xΠj

for j = u, cs and, for all 1 ≤ p ≤ ∞, 0 ≤ r ≤ 4,

(9.28)|Πu|Lp→W r,p , |Πu|Lp→W r,p ≤ C,

|Πcs|Wr,p→W r,p , |Πcs|Wr,p→W r,p ≤ C.

10Indeed, though we do not do it here, this prescription could be followed quite literally at the nonlinearlevel.

83

Proof. Recalling that L has at most finitely many unstable eigenvalues, we find that Πu

may be expressed as

Πuf =p∑

j=1

φj(x)〈φj , f〉,

where φj , j = 1, . . . p are generalized right eigenfunctions of L associated with unstableeigenvalues λj , satisfying the generalized eigenvalue equation (L− λj)rjφj = 0, rj ≥ 1, andφj are generalized left eigenfunctions. Noting that L is divergence form, and that λj 6= 0,we may integrate (L − λj)rjφj = 0 over R to obtain λ

rj

j

∫φjdx = 0 and thus

∫φjdx = 0.

Noting that φj , φj and derivatives decay exponentially in x1 by separation of variables andstandard one-dimensional theory [He, ZH, MaZ1], we find that φj = ∂xΦj with Φj andderivatives exponentially decaying in x1, hence

(9.29) Πuf =∑

j

Φj〈∂xφ, f〉.

Estimating |∂jxΠuf |Lp = |

∑j ∂

jxφj〈φjf〉|Lp ≤

∑j |∂

jxφj |Lp |φj |Lq |f |Lp ≤ C|f |Lp for 1/p +

1/q = 1 and similarly for ∂rxΠuf , we obtain the claimed bounds on Πu and Πu, from which

the bounds on Πcs = Id−Πu and Πcs = Id− Πu follow immediately.

9.7.2 Linear estimates

Let Gcs(x, t; y) := ΠcseLtδy(x) denote the Green kernel of the linearized solution operator

on the center stable subspace Σcs. Then, we have the following detailed pointwise boundsestablished in [TZ2, MaZ1].

Proposition 9.1 ([TZ2, MaZ1]). Assuming (H0)–(H2), (D1)–D(3), the center stableGreen function may be decomposed as Gcs = E + G, where

(9.30) E(x, t; y) = ∂x1 u(x1)e(y1, t),

(9.31) e(y1, t) =∑

a−k >0

(errfn

(y1 + a−k t√4(t+ 1)

)− errfn

(y1 − a−k t√4(t+ 1)

))l−k (y1)

for y1 ≤ 0 and symmetrically for y1 ≥ 0, l−k ∈ Rn constant, and a±j are the eigenvalues ofdf(u±), and

(9.32) |∫C∂s

xG(·, t; y)f(y)dy|Lp ≤ C(1 + t−s2 )t−

12( 1

q− 1

p)|f |Lq ,

(9.33) |∫C∂s

xGy(·, t; y)f(y)dy|Lp ≤ C(1 + t−s2 )t−

12( 1

q− 1

p)− 1

2 |f |Lq ,

for all t ≥ 0, 0 ≤ s ≤ 2, some C > 0, for any 1 ≤ q ≤ p and f ∈ Lq ∩ Lp.

84

Proof. As observed in Section 7, it is equivalent to establish decomposition

(9.34) G = Gu + E + G

for the full Green function G(x, t; y) := eLtδy(x), where

Gu(x, t; y) := ΠueLtδy(x) = eγt

p∑j=1

φj(x)φj(y)t

for some constant matrix M ∈ Cp×p denotes the Green kernel of the linearized solutionoperator on Σu, φj and φj right and left generalized eigenfunctions associated with unstableeigenvalues λj , j = 1, . . . , p.

Using separation of variables, moreover, we may decompose G =∑

k Gk, where Gk is the

Green function acting on Fourier modes of wave number k, i.e., Gk = F−1Gδ(ξ−πk/M)F ,where F denotes Fourier transform in x2, and ξ Fourier frequency. This reduces the problemto that of deriving the asserted bounds separately on the one-dimensional Green functionG0 and on the complement

∑k 6=0G

k, where the difficulty, due to lack of spectral gap, isconcentrated in the estimation of the one-dimensional part G0. Here, we are using also thefact that the Fourier projection Π0f(x) ≡

∫ L−L f(y2)dy2 of a function in x2 onto the subspace

of zero wave number functions satisfies |Π0f |Lp(x2) ≤ |f |L1(x2) ≤ |f |Lp(x2) for any p ≥ 1, byHolder’s inequality, so that Π0 and its complementary projection I − Π0 are bounded asoperators on Lp(x2).

The bounds on the one-dimensional Green function G0 have already been stated inProposition 7.1; see [TZ2, Z4] for further discussion. The bounds on the complement∑

k 6=0Gk follow by the straightforward semigroup estimate |eLtf |Lp ≤ Ce−ηt|f |Lp , η > 0,

where L denotes the projection of L onto the intersection of its center stable subspace andthe subspace of functions with transverse Fourier wave numbers 6= 0, which evidently has anonzero spectral gap σ(L) ≤ −2η < 0 for some η > 0; see [TZ2] for related computations.

Corollary 9.17 ([Z4]). The kernel e satisfies for all t > 0

|ey(·, t)|Lp , |et(·, t)|Lp ≤ Ct−12(1−1/p),

|ety(·, t)|Lp ≤ Ct−12(1−1/p)−1/2.

Proof. Direct computation using definition (9.31).

9.7.3 Nonlinear damping estimate

Proposition 9.2 ([MaZ3]). Assuming (H0)-(H3), let v0 ∈ H2, and suppose that for0 ≤ t ≤ T , the H2 norm of v remains bounded by a sufficiently small constant, for v as in(9.23) and u a solution of (9.2)–(9.3). Then, for some constants θ1,2 > 0, for all 0 ≤ t ≤ T ,

(9.35) ‖v(t)‖2H2 ≤ Ce−θ1t‖v(0)‖2

H2 + C

∫ t

0e−θ2(t−s)(|v|2L2 + |α|2)(s) ds.

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Proof. Energy estimates identical with those of the one-dimensional proof in Section 7.2.4,using the fact that boundary terms in x2 are identically zero due to periodic boundaryconditions.

9.7.4 Proof of nonlinear stability

Proof of Theorem 9.2. With these preparations, the proof goes exactly as in the proof ofTheorem 7.2 in the one-dimensional case. Specifically, decomposing the nonlinear pertur-bation v as v(x, t) = w(x, t) + z(x, t), where w := Πcsv, z := Πuv, applying Πcs to (7.21),and recalling commutator relation (9.27), we obtain again the equation

(9.36)w(x, t) =

∫CG(x, t; y)w0(y) dy

−∫ t

0

∫CGy(x, t− s; y)Πcs(N(v) + αv)(y, s)dy ds

for the flow along the center stable manifold, parametrized by w ∈ Σcs. Following word forword the proof of Theorem 7.2, we obtain the result.

9.7.5 Bifurcation analysis

Proof of Proposition 9.4. With the linearized estimates, projector bounds, etc., already es-tablished for the proof of conditional stability, the proof of bifurcation goes, likewise, wordfor word as in the proof of Theorem 8.2 in the one-dimensional case.

Acknowledgement. Thanks to Olivier Lafitte and Benjamin Texier for their interestand encouragement in this work, in an earlier, much abbreviated, version of the coursepresented at Paris XIII in Summer, 2008, focused on the topic of numerical Evans functionevaluation.

A Traveling-wave equations for isentropic MHD

Viscous shock profiles of (1.9) with v− = 1, u1,− = 0, s = −1, must satisfy the system ofordinary differential equations

(A.1)

v′ − u′1 = 0,u′1 + (p+ (1/2µ0)|B|2)′ = (((1/v)u′1)

′,

u′ − ((1/µ0)B∗1B)′ = ((µ/v)u′)′,(vB)′ − (B∗1 u)

′ = ((1/σµ0v)B′)′,

together with the boundary conditions (v, u1, u, B)(±∞) = (v, u1, u, B)±, where B =(B1, B), u = (u1, u).

Evidently, we can integrate each of the differential equations from −∞ to x, and usingthe boundary conditions (in particular v− = 1 and u1,− = 0), we find, after some elementary

86

manipulations, the profile equations (after having introduced shorthand notation u = u1,w := u, B = B):

v′ = v(v − 1) + v(p− p−) +v

2µ0(B2 −B2

−),(A.2)

µw′ = vw − vB∗1µ0

(B −B−),(A.3)

1σµ0

B′ = v2B − vB− −B∗1vw,(A.4)

with u ≡ v − 1. By a preliminary change of independent variables, we can also arrangewithout loss of generality that µ = 1.

A.1 The case σ = ∞

When σ = ∞, we obtain in place of the final equation of (A.1), (vB)′− (B∗1 u)′ = 0, yielding

after integration the relation

(A.5) B =B− +B∗1w

v.

Substituting in (A.2)–(A.4), we obtain the reduced, planar, ODE (1.23) in (v, w) alone.

B Further example systems

Some additional example systems satisfying (A1)–(A3), (H0)–(H3):

1. The general Navier–Stokes equations of compressible gas dynamics, written in La-grangian coordinates, appear as

(B.1)

vt − ux = 0,ut + px = ((ν/v)ux)x,(e+ u2/2)t + (pu)x = ((κ/v)Tx + (µ/v)uux)x,

where v > 0 denotes specific volume, u velocity, e > 0 internal energy, T = T (v, e) > 0temperature, p = p(v, e) pressure, and µ > 0 and κ > 0 are coefficients of viscosity and heatconduction, respectively. We assume an ideal temperature dependence

(B.2) T = T (e),

independent of v.11 Defining U1 := (v), U2 := (u, e + |u|2/2), we find that the singlehyperbolic mode is v, and the associated equation is linear by inspection; likewise, the

viscosity matrix b =(ν/v 0∗ κ/νTe

)is lower triangular with positive real diagonal entries,

11This assumption can be removed with further effort; see Remark 5.1, [TZ3].

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hence its spectrum is positive real and (by Lyapunov’s Lemma) there exists A022 symmetric

positive definite such that A022b > 0. Conditions (A1)–(A3) and (H1)–(H3) are thus satisfied

under the mild assumptions of monotone temperature-dependence,

(B.3) Te > 0,

and thermodynamic stability of the endstates,

(B.4) (pv)± < 0, (Te)± > 0;

see, e.g., [MaZ4, Z2, Z3] for further discussion. Notably, this allows the interesting case of avan der Waals-type equation of state, with pv > 0 for some values of v along the connectingprofile.

2. The full (nonisentropic) equations of MHD are

(B.5)

vt − u1x = 0,u1t + (p+ (1/2µ0)(B2

2 +B23))x = ((ν/v)u1x)x,

u2t − ((1/µ0)B∗1B2)x = ((ν/v)u2x)x,u3t − ((1/µ0)B∗1B3)x = ((ν/v)u3x)x,(vB2)t − (B∗1u2)x = ((1/σµ0v)B2x)x,(vB3)t − (B∗1u3)x = ((1/σµ0v)B3x)x,(e+ (1/2)(u2

1 + u22 + u2

3) + (1/2µ0)v(B22 +B2

3))t

+[(p+ (1/2µ0)(B22 +B2

3))u1 − (1/µ0)B∗1(B2u2 +B3u3)]x= [(ν/v)u1u1x + (µ/v)(u2u2x + u3u3x)

+(κ/v)Tx + (1/σµ20v)(B2B2x +B3B3x)]x,

where v denotes specific volume, u = (u1, u2, u3) velocity, p = P (v, e) pressure, B =(B∗1 , B2, B3) magnetic induction, B∗1 constant, e internal energy, T = T (v, e) > 0 tem-perature, and µ > 0 and ν > 0 the two coefficients of viscosity, κ > 0 the coefficient of heatconduction, µ0 > 0 the magnetic permeability, and σ > 0 the electrical resistivity. Underassumptions (B.2), (B.3), and (B.4), conditions (A1)–(A3) are again satisfied, and condi-tions (H1)–(H3) are satisfied (see [MaZ4]) under the generically satisfied assumptions thatthe shock be of Lax or overcompressive type, the endstates U ε

± be strictly hyperbolic, andthe speed s be nonzero, i.e., the shock move with nonzero speed relative to the backgroundfluid velocity, with U1 := (v), U2 := (u,B, e+ |u|2/2 + v|B|2/2µ0). (For gas dynamics, onlyLax-type shocks and nonzero speeds can occur, and all points U are strictly hyperbolic.)

3. (MHD with infinite resistivity/permeability) An interesting variation of (B.5) that isof interest in certain astrophysical parameter regimes is the limit in which either electricalresistivity σ, magnetic permeability µ0, or both, go to infinity, in which case the righthandsides of the fifth and sixth equations of (B.5) go to zero and there is a three-dimensional setof hyperbolic modes (v, vB2, vB3) instead of the usual one. By inspection, the associatedequations are still linear in the conservative variables. Likewise, (A1)–(A3), (H1)–(H3) holdunder assumptions (B.2), (B.3), and (B.4) for nonzero speed Lax- or overcompressive-typeshocks with strictly hyperbolic endstates.

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4. (multi-species gas dynamics or MHD) Another simple example for which the hy-perbolic modes are vectorial is the case of miscible, multi-species flow, neglecting speciesdiffusion, in either gas dynamics or magnetohydrodynamics. In this case, the hyperbolicmodes consist of k copies of the hyperbolic modes for a single species, where k is the numberof total species. Again, the associated equations are linear, and (A1)–(A3), (H1)–(H3) holdfor nonzero speed Lax- or overcompressive-type shocks with strictly hyperbolic endstates,under assumptions (B.2), (B.3), and (B.4).

C Asymptotic linear ODE theory

In this appendix, we gather the linear ODE theory needed for our analyses.

C.1 The gap and conjugation lemmas

Our first set of results concerns systems with bounded, exponentially convergent coefficientmatrices. Consider a general first-order system

(C.1) W ′ = Ap(x, λ)W

with asymptotic limits Ap± as x→ ±∞, where p ∈ Rm denote model parameters and λ ∈ C.

Lemma C.1 (The gap lemma [GZ, ZH]). Suppose for fixed θ > 0 and C > 0 that

(C.2) |(∂/∂x)k(Ap −Ap±)|(x, λ) ≤ Ce−θ|x| 0 ≤ k ≤ K,

for x ≷ 0 uniformly for (λ, p) in a neighborhood of (λ0, p0) and that A varies analyticallyin λ and smoothly (resp. continuously) in p as a function into L∞(x). If V −(λ) is aneigenvector of A− with eigenvalue µ(λ), both analytic in λ, then there exists a solution of(C.1) of form

(C.3) W (λ, x) = V (x, λ)eµ(λ)x,

where V is C1 in x and locally analytic in λ and, for any fixed θ < θ, satisfies

(C.4) V (x, λ) = V −(λ) +O(e−θ|x||V −(λ)|), x < 0.

Proof. Setting W (x) = eµxV (x), we may rewrite W ′ = AW as

(C.5) V ′ = (A− − µI)V + ΘV, Θ := (A−A−) = O(e−θ|x|),

and seek a solution V (x, λ) → V −(x) as x → ∞. Choose θ < θ1 < θ such that there isa spectral gap |R

(σA− − (µ + θ1)

)| > 0 between σA− and µ + θ1. Then, fixing a base

point λ0, we can define on some neighborhood of λ0 to the complementary A−-invariantprojections P (λ) and Q(λ) where P projects onto the direct sum of all eigenspaces of A−with eigenvalues µ satisfying R(µ) < R(µ) + θ1, and Q projects onto the direct sum of

89

the remaining eigenspaces, with eigenvalues satisfying R(µ) > R(µ) + θ1. By basic matrixperturbation theory (eg. [Kat]) it follows that P and Q are analytic in a neighborhood ofλ0, with

(C.6)∣∣∣e(A−−µI)xP

∣∣∣ ≤ C(eθ1x), x > 0,∣∣∣e(A−−µI)xQ

∣∣∣ ≤ C(eθ1x), x < 0.

It follows that, for M > 0 sufficiently large, the map T defined by

(C.7)T V (x) = V − +

∫ x

−∞e(A−−µI)(x−y)PΘ(y)V (y)dy

−∫ −M

xe(A−−µI)(x−y)QΘ(y)V (y)dy

is a contraction on L∞(−∞,−M ]. For, applying (C.6), we have

(C.8)|T V1 − T V2|(x) ≤ C|V1 − V2|∞

(∫ x

−∞eθ1(x−y)eθydy +

∫ −M

xeθ1(x−y)eθydy

)= C|V1 − V2|∞

eθ1xe−(θ−θ1)M

θ − θ1<

12|V1 − V2|∞.

By iteration, we thus obtain a solution V ∈ L∞(−∞,−M ] of V = T V with V ≤ C3|V −|;since T clearly preserves analyticity V (λ, x) is analytic in λ as the uniform limit of analyticiterates (starting with V0 = 0). Differentiation shows that V is a bounded solution ofV = T V if and only if it is a bounded solution of (C.5). Further, taking V1 = V , V2 = 0 in(C.8), we obtain from the second to last inequality that

(C.9) |V − V −| = |T (V )− T (0)| ≤ C2eθx|V | ≤ C4e

θx|V −|,

giving (C.4). Analyticity, and the bounds (C.4), extend to x < 0 by standard analyticdependence for the initial value problem at x = −M .

Remark C.2. The title “gap lemma” alludes to the fact that we do not make the usualassumption of a spectral gap between µ(λ) and the remaining eigenvalues of A−, as in stan-dard results on asymptotic behavior of ODE [Co]; that is, the lemma asserts that exponentialdecay of A can substitute for a spectral gap.

Remark C.3. In the case <σ(A+) > −θ, we may take Q = ∅ and θ = θ, improving (C.4).

Lemma C.4 (The conjugation lemma [MeZ1, PZ]). Suppose for fixed θ > 0 andC > 0 that

(C.10) |(∂/∂x)k(Ap −Ap±)|(x, λ) ≤ Ce−θ|x| 0 ≤ k ≤ K,

for x ≷ 0 uniformly for (λ, p) in a neighborhood of (λ0, p0) and that A varies analyticallyin λ and smoothly (resp. continuously) in p as a function into WK,∞(x). Then, there exist

90

in a neighborhood of (λ0, p0) invertible linear transformations P p+(x, λ) = I + Θp

+(x, λ) andP p−(x, λ) = I + Θp

−(x, λ) defined on x ≥ 0 and x ≤ 0, respectively, analytic in λ and smooth(resp. continuous) in p as functions into L∞[0,±∞), such that

(C.11) |(∂/∂λ)j(∂/∂x)kΘp±| ≤ C(j, k)e−θ|x| for x ≷ 0, 0 ≤ j, 0 ≤ k ≤ K,

for any 0 < θ < θ, some Cj,k = Cj,k(θ, θ) > 0, and the change of coordinates W =: P p±Z

reduces (C.1) to the constant-coefficient limiting systems

(C.12) Z ′ = Ap±Z for x ≷ 0.

Proof. Substituting W = P−Z into (C.1), equating to (C.12), and rearranging, we obtainthe defining equation

(C.13) P ′− = A−P− − P−A, P− → I as x→ −∞.

Viewed as a vector equation, this has the form P ′− = AP−, where A approaches exponen-tially as x→ −∞ to its limit A−, defined by

(C.14) A−P := A−P − PA−.

The limiting operator A− evidently has analytic eigenvalue, eigenvector pair µ ≡ 0, P− ≡ I,whence the result follows by Lemma C.1 for j = k = 0. The x-derivative bounds 0 < k ≤K + 1 then follow from the ODE and its first K derivatives, and the λ-derivative boundsfrom standard interior estimates for analytic functions. Finally, invertibility of P− followsfor x large and negative from (C.11) and for x ≤ 0 by global existence of a solution to

(P−1)′ = −P−1P ′P−1 = A+P−1 − P−1A.

A symmetric argument gives the result for P+.

Remark C.5. In the case that A is block diagonal or triangular, the conjugators P± mayevidently be taken block diagonal or triangular as well, by carrying out the same fixed-pointargument on the invariant subspace of (C.13) consisting of matrices with this special form.This can be of use in problems with multiple scales; see, for example, Thm 1.16, [BHZ].

Definition C.1 (Abstract Evans function). Suppose that on the interior of a set Ω inλ, p, the dimensions of the stable and unstable subspaces of Ap

±(λ) remain constant, andagree at ±∞ (“consistent splitting” [AGJ]), and that these subspaces have analytic basesR±j extending continuously to boundary points of Ω. Then, the Evans function is definedon Ω as

(C.15) Dp(λ) := det(P+R+1 , · · · , P

+R+k , P

−R−k+1, P−R−n )|x=0,

where P p± are as in Lemma C.4.

91

C.2 The convergence lemma

Our next set of results explores in the same context of systems bounded, exponentiallyconvergent coefficient matrices, the question of convergence with respect to parameters ofthe associated flow (resp. Evans function). Specifically, we develop the notion of a regularlyperturbed system of parametric ODE. Consider a family of first-order equations

(C.16) W ′ = Ap(x, λ)W

indexed by a parameter p, and satisfying exponential convergence condition (C.2) uniformlyin p. Suppose further that

(C.17) |(Ap −Ap±)− (A0 −A0

±)| ≤ C|p|e−θ|x|, θ > 0

and

(C.18) |(Ap −A0)±)| ≤ C|p|.

Lemma C.2 ([PZ, BHZ]). Assuming (C.2) and (C.17)–(C.18), for |p| sufficiently small,there exist invertible linear transformations P p

+(x, λ) = I + Θp+(x, λ) and P 0

−(x, λ) = I +Θp−(x, λ) defined on x ≥ 0 and x ≤ 0, respectively, analytic in λ as functions into L∞[0,±∞),

such that

(C.19) |(P p − P 0)±| ≤ C1|p|e−θ|x| for x ≷ 0,

for any 0 < θ < θ, some C1 = C1(θ, θ) > 0, and the change of coordinates W =: P p±Z

reduces (C.16) to the constant-coefficient limiting systems

(C.20) Z ′ = Ap±(λ)Z for x ≷ 0.

Proof. Applying the conjugating transformation W → (P 0+)−1W for the p = 0 equations,

we may reduce to the case that A0 is constant, and P 0+ ≡ I, noting that the estimate (C.17)

persists under well-conditioned coordinate changes W = QZ, Q(±∞) = I, transforming to

(C.21)|(Q−1ApQ−Q−1Q′ −Ap

±)−(Q−1A0Q−Q−1Q′ −A0

±)|

≤ |Q((Ap −Ap

±)− (A0 −A0±))Q−1|+ |Q−1(Ap −A0)±Q− (Ap −A0)±|,

where

(C.22) |Q−1(Ap −A0)±Q− (Ap −A0)±| = O(|Q− I|)|(Ap −A0)±| = O(e−θ|x|)|p|.

In this case, (C.17) becomes just

|Ap −Ap±| ≤ C1|p|e−θ|x,

and we obtain directly from the conjugation lemma, Lemma C.4, the estimate

|P p+ − P 0

+| = |P p+ − I| ≤ CC1|p|e−θ|x|

for x > 0, and similarly for x < 0, verifying the result.

92

Remark C.6. In the case Ap± ≡ constant, or, equivalently, for which (C.17) is replaced by

|Ap−A0| ≤ C1|p|e−θ|x, we find that the change of coordinates W = P p±Z, P p

± := (P 0)−1± P p

±,converts (C.16) to Z ′ = A0Z, where P p

± = I + Θp± with

(C.23) |Θp±| ≤ CC1|p|e−θ|x| for x ≷ 0.

That is, we may conjugate not only to constant-coefficient equations, but also to exponen-tially convergent variable-coefficient equations, with sharp rate (C.23).

In the general case Ap± 6= A0±, we may still conjugate (C.16) to Z ′ = A0Z by thechange of coordinates W = P p

±Z, P p± := (P 0)−1

± Q±Pp±, where Q± defined by

(C.24) Q′ = Ap±Q−QA0

±

conjugates the constant-coefficient equation Y ′ = Ap ± Y to X ′ = A0±X, obtaining bounds

(C.25) P p± = I + Θp

±, |Θp±| ≤ CC1|p| for |x| ≤ C

valid for finite values of x. However, in general, Q± grow without bound as x→ ±∞.

Corollary C.7 (PZ). Provided that the bases R+j (R−j ) appearing in (C.15) of the stable

(unstable) subspaces of Ap+ (Ap

−) converge at rate O(p) to those of A0+ (A0

−) as p→ 0,12 theEvans functions Dp converge uniformly to D0 at rate O(p) as p→ 0 on compact sets of Ω.

Proof. Immediate by (C.19) and Definition C.1.

Exercise C.8. Show that (C.10) and continuity of A with respect to p imply continuityof (A − A±) in an eθ|x| weighted norm, connecting the result of Corollary C.7 with theearlier-observed continuity of Dp in Lemma C.4.

C.2.1 Regular perturbation of traveling waves

(C.2), (C.17), and (C.18) together give a notion of regular perturbation of systems of form(C.1), from the point of view of Evans function computations. Noting that the coefficientmatrix Ap(x, λ) in applications often depends on x through a family of traveling-waveconnections up(x) satisfying a family of nonlinear ODE

(C.26) u′ = f(u, p), f(up±, p) = 0,

let us examine briefly the meaning of our regularity criteria in this context.

Proposition C.9. If up± are nondegenerate rest points, in the sense that df(up

±, p) haveno center subspace, and up are transversal connections, then, for any smooth matrix-valuedfunction A(u, p), Ap(x, λ) := A(up, p) satisfies the regular perturbation conditions (C.2),(C.17), and (C.18).

12This as typically holds given (C.18); in particular, it holds by standard matrix perturbation theory [K]if the stable and unstable eigenvalues of A0

± are spectrally separated.

93

Proof. This follows readily by the fixed-point construction of the stable (unstable) manifoldgiven in Theorem D.9, which, by smooth dependence of fixed-point solutions expresses thestable (unstable) manifold at up

+ (up−), written in perturbation variables u := u − u±, as

a smooth map from Rk (Rn+1−k) cross p into the set of functions decaying exponentiallyin x as x → +∞ (−∞), with full rank differential in Rk × Rn+1−k. Adding an arbitraryphase condition to eliminate translation-invariance, one obtains the tangent directions asx → +∞ (−∞) as smooth functions of p, and the profile solutions as smooth functions ofthe tangent directions into the space of exponentially decaying function, from which theasserted conditions readily follow.

That is, regular perturbation in the profile problem leads as one would hope to regularperturbation in Evans sense of the associated linearized eigenvalue ODE.

C.3 The tracking lemma

Our last set of results concern possibly unbounded but slowly varying coefficient matrices.For bounded coefficients and uniform spectral gap, such calculations are classical; see forexample [Sa]. Consider an approximately block-diagonal system

(C.27) W ′ =(M1 00 M2

)(x, p) + δ(x, p)Θ(x, p)W,

where Θ is a uniformly bounded matrix, δ(x) scalar, and p a vector of parameters, satisfyinga pointwise spectral gap condition

(C.28) minσ(<M ε1 )−maxσ(<M ε

2 ) ≥ η(x) for all x.

(Here as usual <N := (1/2)(N +N∗) denotes the “real”, or symmetric part of N .) Then,we have the following tracking/reduction lemma of [MaZ3, PZ].

Lemma C.3 ([MaZ3, PZ]). Consider a system (C.27) under the gap assumption (C.28),with Θε uniformly bounded and η ∈ L1

loc. If sup(δ/η)(x) is sufficiently small, then thereexist (unique) linear transformations Φ1(x, p) and Φ2(x, p), possessing the same regularitywith respect to p as do coefficients Mj and δΘ, for which the graphs (Z1,Φ2Z1) and(Φ1Z2, Z2) are invariant under the flow of (C.27), and satisfy

(C.29) sup |Φ1|, sup |Φ2| ≤ C sup(δ/η)

and

(C.30) |Φε1(x)| ≤ C

∫ +∞

xe∫ x

y η(z)dzδ(y)dy, |Φε1(x)| ≤ C

∫ x

−∞e∫ x

y −η(z)dzδ(y)dy.

Proof. By the change of coordinates x→ x, δ → δ := δ/η with dx/dx = η(x), we may reduceto the case η ≡ constant = 1 treated in [MaZ3]. Dropping tildes and setting Φ2 := ψ2ψ

−11 ,

where (ψt1, ψ

t2)

t satisfies (C.27), we find after a brief calculation that Φ2 satisfies

(C.31) Φ′2 = (M2Φ2 − Φ2M1) + δQ(Φ2),

94

where Q is the quadratic matrix polynomial Q(Φ) := Θ21 +Θ22Φ−ΦΘ11 +ΦΘ12Φ. Viewedas a vector equation, this has the form

(C.32) Φ′2 = MΦ2 + δQ(Φ2),

with linear operator MΦ := M2Φ − ΦM1. Note that a basis of solutions of the decoupledequation Φ′ = MΦ may be obtained as the tensor product Φ = φφ∗ of bases of solutions ofφ′ = M2φ and φ′ = −M∗

1 φ, whence we obtain from (C.28)

(C.33) eMz ≤ Ce−ηz, for z > 0,

or uniform exponentially decay in the forward direction.Thus, assuming only that Φ2 is bounded at −∞, we obtain by Duhamel’s principle the

integral fixed-point equation

(C.34) Φ2(x) = T Φ2(x) :=∫ x

−∞eM(x−y)δ(y)Q(Φ2)(y) dy.

Using (C.33), we find that T is a contraction of order O(δ/η), hence (C.34) determines aunique solution for δ/η sufficiently small, which, moreover, is order δ/η as claimed. Finally,substituting Q(Φ) = O(1 + |Φ|2) = O(1) in (C.34), we obtain

|Φ2(x)| ≤ C

∫ x

−∞eη(x−y)δ(y) dy

in x coordinates, or, in the original x-coordinates, (C.30). A symmetric argument establishesexistence of Φ1 with the asserted bounds. Regularity with respect to parameters is inheritedas usual through the fixed-point construction via the Implicit Function Theorem.

Remark C.10. For η constant and δ decaying at exponential rate strictly slower that e−ηx

as x → +∞, we find from (C.30) that Φ2(x) decays like δ/η as x → +∞, while if δ(x)merely decays monotonically as x→ −∞, we find that Φ2(x) decays like (δ/η) as x→ −∞,and symmetrically for Φ1.

Remark C.11. 1. A closer look at the proof of Lemma C.3 shows that, in the approximatelyblock lower-triangular case, δΘ21 not necessarily small, there exists a block-triangularizingtransformation Φ1 = O(sup |δ/η|) << 1, under the much less restrictive conditions

sup(|δ/η|(|Θ11|+ |Θ22|)

)< 1 and sup(|δ/η||Θ21|) <<

1sup |δ/η|

.

2. Similarly, in the standard, approximately block-diagonal case, an examination of theproof shows that bounds (C.29) may be sharpened to(C.35)

sup |Φ1| ≤ C sup(δ/η)(

sup |Θ12|+ sup(δ/η) sup(|Θ11|+ |Θ22|) + sup(δ/η)2 sup |Θ21|),

sup |Φ2| ≤ C sup(δ/η)(

sup |Θ21|+ sup(δ/η) sup(|Θ11|+ |Θ22|) + sup(δ/η)2 sup |Θ12|).

95

Remark C.12. An important observation of [MaZ3, PZ] is that hypothesis (C.28) ofLemma C.3 may be weakened to

(C.36) minσ(<M ε1 )−maxσ(<M ε

2 ) ≥ η(x) + α(x, p)

with no change in the conclusions, for any α satisfying a uniform L1 bound |α(·, p)|L1 ≤ C1.(Substitute eMx ≤ CeC1e−ηz for (C.33), with no other change in the proof.) More generally,(C.37) may be replaced in Lemma C.3 by

(C.37) W ′ =(M1 + α1 0

0 M2 + α2

)(x, p) + (δΘ + β)(x, p)W,

where |αj |L1 are bounded and |β|L1 is sufficiently small, with the conclusion that

(C.38) sup |Φ1|, sup |Φ2| ≤ C(sup(δ/η) + |β|L1).

(The additional term∫ x−∞ eM(x−y)Qβ(Φ2)(y) dy, Qβ(Φ2) := β21 + β22Φ − Φβ11 + Φβ12Φ,

now appearing in the righthand side of (C.34) is contractive for |β|L1 small.) These allowus to neglect commutator terms in some of the more delicate applications of tracking: forexample, the high-frequency analysis of [MaZ3], or the analysis of case IIb in [Z11].

D Center Stable Manifold for ODE

The center stable manifold construction in the PDE case follows closely the construction forfinite-dimensional ODE, which we therefore recall here for completeness; see also [B, VI].Consider an ODE

(D.1) u′ = f(u), f ∈ C1,

and an equilibrium f(u∗) = 0, with associated linearized equation

(D.2) v′ = Av, A := df(u∗).

Associated with A, define the center stable subspace Σcs as the direct sum of all eigenspacesof A associated to neutral or stable eigenvalues, i.e., eigenvalues with zero or negativereal part. Likewise, define the unstable subspace Σu as the direct sum of eigenspacesassociated to unstable eigenvalues, i.e., eigenvalues with strictly positive real part, so thatRn = Σcs ⊕ Σu.

Defining the associated (total) eigenprojections Πcs and Πu as the sum of all eigen-projections associated with neutral–stable and unstable eigenvalues, respectively, we have,either by reduction to Jordan form or direct estimation using the inverse Laplace trans-form/resolvent estimates, bounds

(D.3)|eAtΠcs| ≤ C(η)eηt t ≥ 0,

|eAtΠu| ≤ C(θ)e−θ|t|, t ≤ 0,

for any η > 0 strictly smaller than the minimum of the real parts of unstable eigenvaluesand θ > 0 arbitrarily small.

96

Proposition D.1 (Center Stable Manifold Theorem for ODE). For f in Ck+1,1 ≤ k < ∞, there exists local to u∗ a Ck center stable manifold Mcs, tangent at u∗ toΣcs, expressible in coordinates w := u − u∗ as a Ck graph Φcs : Σcs → Σcs ⊕ Σu, that is(locally) invariant under the flow of (D.1) and contains all solutions that remain boundedand sufficiently close to u∗ in forward time. In general it is not unique.

D.1 Frechet differentiability of substitution operators

Standard invariant (e.g., stable, center, center stable) manifold constructions proceed byfixed point iteration on various weighted L∞ spaces

(D.4) ‖f‖η := supt≥t0

eη(t−t0)|f(t)|, Bη := f : ‖f‖η <∞,

with η positive for stable manifolds and negative for center or center stable manifolds. Asdescribed in [B, VI], Frechet differentiability of the associated fixed-point mapping (henceeventual smoothness of the resulting manifold) hinges on Frechet differentiability with re-spect to spaces Bη of the special class of substitution operators, defined for g : Rn → Rn

as

(D.5) G(f)(t) := g(f(t)).

Let Ckb denote the Banach space of Ck functions g : Rn → Rn with |djg| uniformly

bounded for 0 ≤ j ≤ k, with associated norm

‖g‖Ckb

:=∑

0≤j≤k

supRn

|djg|.

Lemma D.2. For η ≥ 0, k ≥ 1, if g ∈ Ckb (Rn → Rn), then G is Ck from Bη → Bη, with

dkGf(t) = (dkg)(f(t)).

Proof. By Taylor’s Theorem,

g(f2(t))− g(f1(t)) = dg(f1(t))(f2(t)− f1(t)) + o(|f2(t)− f1(t)|),

as |f2(t)− f1(t)| → 0. By the assumed uniform boundedness of |dg|, we readily obtain thatf → dgf is a bounded linear operator from Bη → Bη. On the other hand |f2 − f1|(t) ≤e−η(t−t0)‖f2 − f1‖η for t ≥ t0 implies |f2 − f1| → 0 uniformly on t ≥ t0 as ‖f2 − f1‖η → 0,and so we have

(D.6)‖g(f2(t))− g(f1(t))−dg(f1(t))(f2(t)− f1(t))‖η =

supt≥t0

eη(t−t0)o(|f2(t)− f1(t)|) = o(‖f2 − f1‖η,

yielding the result for k = 1. The general result then follows by induction on k, applyingthe result for k = 1 to successively higher derivatives of g.

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Lemma D.3. Let g ∈ Ck+1b and 0 < −η′ < −η/k. Then, the substitution operator G is Ck

from Bη′ → Bη, with dkGf(t) = (dkg)(f(t)).

Proof. More generally, the result of Lemma D.3 holds for any g ∈ Ck+αb , 0 < α ≤ 1,

in the sense that |dkg(x + h) − dkg(x)| ≤ C|h|α for some uniform C > 0, as may beseen by rewriting the (k − 1)th-order Taylor remainder formula g(x + h) − Tk−1g(x, h) =( ∫ 1

0 dkg(x+ θh) (1−θ)k−1

(k−1)! dθ)hk as

g(x+ h)− Tkg(x, h) =(∫ 1

0

(dkg(x+ θh)− dkg(x)

)(1− θ)k−1

(k − 1)!dθ)hk,

where Tkg(x, h) denotes the kth-order Taylor expansion of g about x evaluated at x+h, thenusing the assumed uniform bound on |dkg(x+h)−dkg(x)| to obtain |g(x+h)−Tkg(x, h)| ≤C|h|k+α, and thus

|g(x+ h)− Tkg(x, h)|−(k+α)η′ ≤ C|h|k+αη′ .

Without loss of generality taking α sufficiently small, this yields the result. A similarestimate yields continuity of the kth Frechet derivative.

D.2 Smooth dependence of fixed-point solutions

We next present two general results on smooth dependence of fixed point solutions. LetT (x, y) be continuous in x, y and contractive in y, T : B1 × B2 → B2 for Banach spaces B1

and B2, defining a fixed point map y(x), y : B1 → B2, continuous in the parameter x, suchthat y(x) = T (x, y(x)). Then, we have the following standard result.

Lemma D.4. If T is Lipschitz in (x, y), then y is Lipschitz from B1 → B2. If T is Ck

(Frechet sense), k ≥ 1, in (x, y), then y is Ck from B1 → B2, with

(D.7) (dy/dx)(x0) = (Id− Ty)−1Tx(x0, y(x0))

and higher derivatives (djy/dxj)(x0), 1 ≤ j ≤ k obtained by formal differentiation of (D.7),substituting for lower derivatives wherever they appear.

Proof. (i) Triangulating, we have

(D.8)|y(x2)− y(x1)| = |T (x2, y(x2))− T (x1, y(x1)|

≤ |T (x2, y(x2))− T (x2, y(x1)|+ |T (x2, y(x1))− T (x1, y(x1)|≤ θ|y(x2)− y(x1)|+ L|x2 − x1|,

where 0 < θ < 1 and 0 < L are contraction and Lipschitz coefficients, yielding afterrearrangement |y(x2)− y(x1)| ≤ L

1−θ |x2 − x1|.(ii) Applying Taylor’s Theorem, and using the result of (i), we have

(D.9)y(x2)− y(x1) = Tx(x2 − x1) + Ty(y2 − y1) + o(|x2 − x1|+ |y(x2)− y(x1)|)

= Tx(x2 − x1) + Ty(y2 − y1) + o(|x2 − x1|),

98

where all derivatives are evaluated at (x1, y(x1)). Noting that the operator norm |Ty| isbounded by contraction coefficient 0 < θ < 1, we have by Neumann series expansion that(Id − Ty) is invertible with uniformly bounded inverse |(Id − Ty)−1| ≤ (1 − θ)−1. Thus,rearranging, we have

y(x2)− y(x1) = (Id− Ty)−1Tx(x2 − x1) + o(|x2 − x1|),

yielding the result for k = 1 by definition of (Frechet) derivative. The results for k ≥ 1 thenfollow by induction upon differentiation of (D.7).

The next Lemma shows how we can recover C1 dependence of fixed point solutions inthe case, as in Lemma D.3, that T is differentiable only from a stronger to a weaker space.We discuss higher derivatives later where they appear, since they involve specific chainsof successively weaker spaces that are not convenient for statement of a general theorem.Let B′2 ⊂ B2, with ‖ · ‖B′2 ≥ ‖ · ‖B2 , and T (x, y) be a map B1 × B′2 → B′2 that is Lipschitzcontinuous in (x, y) and contractive (with respect to ‖ · ‖B′2) in y. Denote by y(x) : B1 → B′2the unique Lipschitz fixed-point solution defined by y(x) = T (x, y(x)).

Lemma D.5. If (i) T is continuously differentiable from B1 × B′2 → B2, and (ii) Ty ex-tends to a bounded linear operator from B2 → B2, continuous in operator norm with re-spect to (x, y), with |Ty|B2 < 1, then y is continously differentiable from B1 → B2, with(dy/dx)(x0) = (Id− Ty)−1Tx(x0, y(x0)).

Proof. By Taylor’s Theorem and ‖y(x2)− y(x1)‖B′2 ≤ L‖x2 − x1‖B1 , we have

(D.10)

y(x2)− y(x1) = Tx(x2 − x1) + Ty(y2 − y1)+ o(‖x2 − x1‖B1 + ‖y(x2)− y(x1)‖B′2)

= Tx(x2 − x1) + Ty(y2 − y1) + o(‖x2 − x1‖B1),

where the o(‖x2 − x1‖B1) term is measured in the weaker ‖ · ‖B2 norm. Observing by|Ty|B2→B2 < 1 and Neumann series inversion that (Id− Ty) considered as an operator fromB2 → B2 is invertible with uniformly bounded inverse

|(Id− Ty)−1|B2→B2 ≤ (1− |Ty|B2→B2)−1,

we may solve (D.10) to obtain

y(x2)− y(x1) = (Id− Ty)−1Tx(x2 − x1) + o(‖x2 − x1‖B1),

yielding differentiability as claimed, with (dy/dx) = (Id − Ty)−1Tx continuous by the as-sumed continuity of Tx and Ty as operators from B1 → B1 and B2 → B2.

99

D.3 Global Center Stable Manifold construction

We now establish a global version of Proposition D.1 for small Lipschitz nonlinearity. Asin Section D.2, denote by Ck

b the space of Ck functions that are uniformly bounded in upto k derivatives and consider for a fixed, constant matrix A and an arbitrary nonlinearityN such that N(t, 0) ≡ 0, Nw(t, 0) ≡ 0 the ODE

(D.11) w′ = Aw +N(t, w).

Proposition D.6. For N ∈ Ck+1b with Lipschitz constant ε > 0 sufficiently small, (D.11)

has a unique Ck invariant manifold Mcs tangent at w = 0 to the center stable subspaceΣcs of A, consisting of the union of all orbits whose solutions grow at sufficiently slowexponential rate |w(t)| ≤ Ceθ|t| in positive time, for any fixed θ < θ < η.

Proof. Applying projections Πj , j = cs, u to (D.11), we obtain using the variation of con-stants formula equations

Πjw(t) = eA(t−t0,j)Πjw(t0,j) +∫ t

t0,j

eA(t−s)ΠjN(s, w(s)) ds,

j = cs, u, so long as the solution w exists, with t0,j arbitrary. Assuming growth of at most|w(t)| ≤ Ceθt in positive time, we find for j = u using (D.3) and the bound |N(w)| ≤ ε|w|coming from N(0) = 0 and the assumed Lipschitz bound on N , that as t0,u → +∞, thefirst term eA(t−t0,u)Πuw(t0,u) converges to zero while the second, integral term converges to∫ +∞t eA(t−s)ΠuN(s, w(s)) ds, so that, choosing t0,cs = 0, we have

Πcsw(t) = eAtΠcsw(0) +∫ t

0eA(t−s)ΠcsN(s, w(s)) ds,

Πuw(t) = −∫ +∞

teA(t−s)ΠuN(s, w(s)) ds.

Summing, we obtain for wcs := Πcsw(0) the fixed-point representation

(D.12)w(t) = T (wcs, w) := eAtwcs +

∫ t

0eA(t−s)ΠcsN(s, w(s)) ds

−∫ +∞

teA(t−s)ΠuN(s, w(s)) ds,

valid for solutions growing at rate at most |w(t)| ≤ Ceθt for t ≥ 0.Define now the negatively-weighted sup norm

‖f‖−θ := supt≥0

e−θt|f(t)|,

noting that |f(t)| ≤ eθt‖f‖−θ for all t ≥ 0. We first show that, for |wcs| and δ, ε > 0sufficiently small, the integral operator T is Lipschitz in wcs and contractive in w from the‖ · ‖−θ-ball B(0, δ) to itself.

100

Using (D.3), |N(t, w(t))| ≤ ε|w(t)|, and |w(t)| ≤ eθ|t|‖w‖−θ, we obtain

|T (w)(t)| ≤ Ceθ|t||wcs|+ Cε‖w‖−θ

(∫ t

0eθ|t−s|eθ|s| ds+

∫ +∞

teη(t−s)eθ|s| ds

),

hence, using η ± θ > 0 and taking C1ε < 1/2 and C|w0| ≤ δ/2, that

‖T (w)(t)‖−θ ≤ C|wcs|+ Cε‖w‖−θ

(∫ t

0eθ|t−s|eθ(|s|−|t|) ds

+∫ +∞

teη(t−s)eθ(|s|−|t|) ds

)≤ C|wcs|+ C1ε‖w‖−θ < δ.

Similarly, we find that

‖T (w1)− T (w2)‖−θ ≤ Cε‖w1 − w2‖−θ

(∫ t

0eθ|t−s|eθ|s| ds+

∫ +∞

teη(t−s)eθ|s| ds

)≤ C1ε‖w1 − w2‖−θ < (1/2)‖w1 − w2‖−θ,

yielding contraction on B(0, δ) and thus existence of a unique fixed point w = w(wcs). Asimilar estimate shows that T is Lipschitz in wcs, so that w(·) is Lipshitz from Σcs to B−θ.

We next investigate smoothness of w(·). Note that T decomposes into the sum of abounded, hence C∞, linear map wcs → eAtwcs from Σcs → B−θ and the composition K · Nof a bounded (hence C∞) linear map

K(f) :=∫ t

0eA(t−s)Πcsf(s) ds−

∫ +∞

teA(t−s)Πuf(s) ds

from B−θ → B−θ and a substitution operator N (w)(s) := N(w(s)) with N ∈ Ck+1b that

by Lemma D.3 is Ck from B−θ/(k+1) → B−θ. Moreover, the first derivative dN (w)(s) =dN(w(s)), by |dN | ≤ ε, extends as a bounded linear operator from B−θ′ → B−θ′ , any θ′ > 0,with |dN |B−θ′ ≤ ε, whence Tw extends as a bounded linear operator from B−θ′ → B−θ′ ,that for any given θ′ > 0, in particular θ, is contractive, |Tw|B−θ′ < 1, for ε > 0 sufficientlysmall, independent of w and wcs. Applying Lemma D.5, we find that w(wcs) is C1 fromΣcs → B−θ′ for any θ′ > 0 and ε > 0 sufficiently small, with

(D.13) dw = (Id− Tw)−1Twcs .

Differentiating (D.13) using the chain, product, and inverse linear operator derivativeformulae, validated by the fact that (Id− Tw)−1 is uniformly bounded (since |Tw| ≤ γ < 1)from B−θ′ → B−θ′ for all 0 ≤ θ0 < θ′ < η0 < η, and the observation that higher (mixed)partial derivatives of T exist and are continuous from B−θ′/(k+1) → B−θ′ on the same range,we find that w(wcs) is Ck from B−θ/C → B−θ for C > 0 sufficiently large, so long as θ0 > 0

is chosen ≤ θ/C and ε > 0 is taken sufficiently small.

101

Finally, defining

(D.14) Φ(wcs) := Πuw(ws)|t=0 = −∫ +∞

0e−AsΠuN(s, w(s)) ds,

we obtain as claimed a Ck function from Σcs → Σu, whose graph is the invariant manifoldof orbits growing at exponential rate |w(t)| ≤ Ceθt in forward time. From the latter charac-terization, we obtain evidently invariance in forward and backward time. By uniqueness offixed point solutions, we have w(0) = 0 and thus Φcs(0) = 0. Finally, differentiating (D.14)with respect to ws, we obtain by Lemma D.3

dΦ(0) = −∫ +∞

0e−AsΠuNw(s, 0)(dw/dws)(s) ds = 0,

since Nw(0) ≡ 0, yielding tangency as claimed.

D.4 Local construction: proof of Proposition D.1

We can reduce the general situation, locally, to the case described in the global result bythe following truncation procedure. Consider a general nonlinearity N(t, w). Introducing aC∞ cutoff function

ρ(x) =

1 |x| ≤ 1,0 |x| ≥ 2,

define N ε(t, w) := ρ(|w|/ε)N(t, w).

Lemma D.7. Let N ∈ Ck+1, k ≥ 1, and N(t, 0) ≡ 0, ∂wN(t, 0) ≡ 0. Then, N ε ∈ Ck+1b ,

N ε ≡ N for |u| ≤ ε, and the Lipschitz constant for N ε with respect to w is uniformlybounded by Cε for all 0 < ε ≤ ε0, where

(D.15) C = 2(1 + max |r|ρ′(|r|)

)max|w|≤2ε0

|∂2wN(t, w)|.

Proof. The Lipshitz constant is bounded by max |∂wNε|, where

|∂wNε| = |(ρε)′N + ρε∂wN |

= |(|w|/ε)ρ′(|w|/ε)(N(t, w)/|w|) + ρ(|w|/ε)∂wN(t, w)|

≤ 2ε(

max |r|ρ′(|r|) max|w|≤2ε0

|N(t, w)||w|2

+ max|w|≤2ε0

|∂wN(t, w)||w|

)≤ 2ε

(max |r|ρ′(|r|) max

|w|≤2ε0

+1)

max|w|≤2ε0

|∂2wN(t, w)|,

the final inequality following by N(t, 0) ≡ 0, ∂wN(t, 0) ≡ 0 and the Integral Mean ValueTheorem, or first-order Taylor remainder formula.

102

Proof of Proposition D.1. Defining w := u−u∗, we obtain the nonlinear perturbation equa-tion

(D.16) w′ = Aw +N(w),

with A constant and N ∈ Ck+1 satisfying N(0) = 0, dN(0) = 0. Applying the truncationprocedure, we obtain a modified equation

(D.17) w′ = Aw +N ε(w)

for which N ε ∈ Ck+1b with arbitrarily small Lipschitz norm ε > 0 and N ≡ N ε within a

neighborhood B(0, ε) of w = 0, i.e., with identical local flow. Applying Proposition D.6, weobtain a global center stable manifold for (D.17), which is therefore a local center stablemanifold for (D.16). Noting that solutions that stay uniformly bounded and close to theequilibrium for positive time are also bounded solutions of the truncated equations, we findthat all such belong to the constructed center stable manifold.

Remark D.8. The inclusion of t-dependence of N in (D.11) is not needed for the presentapplication (D.16), but allows also the treatment of time-periodic solutions after Floquettransformation to constant-coefficient linear part.

D.5 Addendum: The Stable (Unstable) Manifold Theorem

Theorem D.9 (Stable/Unstable Manifold Theorem). For f ∈ Ck, 1 ≤ k ≤ ∞, thereexist local to u∗ Ck stable and unstable manifolds Ms and Mu tangent at u∗ to Σs and Σu,expressible in coordinates w := u − u∗ as Ck graphs Φs : Σs → Σc ⊕ Σu and Φu : Σu →Σc ⊕ Σs, that are (locally) invariant under the flow of (D.1) and uniquely determined bythe property that solutions in Ms decay exponentially to u∗ in forward time and solutionsin Mu decay exponentially to u∗ in backward time, at rate |w(t)| ≤ C(η)e−η|t||w(0)| for any0 < η < η, η as in (D.3).

Proof. Without loss of generality, restrict to the stable case. The unstable case may beconverted to the stable case by reversing time. Defining w := u−u∗, we obtain the nonlinearperturbation equation w′ = Aw + N(t, w) with A constant and N satisfying N(t, 0) ≡ 0,Nw(t, 0) ≡ 0. Applying projections Πj , j = s, u, c, and using the fact that these commutewith A, we may coordinatize as three coupled equations (Πjw)′ = A(Πjw) + ΠjN(t, w) invariables ws := Πsw, wc := Πcw, and wu := Πuw. Considering ΠjN as inhomogeneoussource terms and applying the variation of constants formula, we obtain

(D.18) Πjw(t) = eA(t−t0,j)Πjw(t0,j) +∫ t

t0,j

eA(t−s)ΠjN(s, w(s)) ds,

j = s, c, u, so long as the solution w exists. Under the assumption |w(t)| ≤ Ce−ηt, wefind by (D.3) that |eA(t−τ)Πjw(τ)| decays exponentially as τ → +∞ for j = c, u, while∫ τt e

A(t−s)ΠjN(s, w(s)) ds (since |N(w, s)| ≤ C|w(s)| for |w| bounded) converges to a limit.

103

Thus, taking t0,j → +∞ for j = c, u, in (D.18), we find that the first (linear) term onthe righthand side disappears, leaving Πjw(t) = −

∫ +∞t eA(t−s)ΠjN(s, w(s)) ds. Choosing

t0,s = 0 and summing the three equations (D.18), we obtain for ws := Πsw(0) the integralfixed-point representation

w(t) = T (ws, w) := eAtws +∫ t

0eA(t−s)ΠsN(s, w(s)) ds−

∫ +∞

teA(t−s)ΠcuN(s, w(s)) ds,

valid for all solutions decaying exponentially in forward time at rate e−ηt, where Πcu := Πc+Πu denotes the total eigenprojection of A onto the center–unstable subspace Σcu := Σc⊕Σu.

Define the (positively) time-weighted sup norm ‖f‖η := supt≥0 eηt|f(t)|, noting that

|f(t)| ≤ e−ηt‖f‖η for t ≥ t0. We have for any ε > 0 using (D.3), |f(t)| ≤ e−ηt‖f‖η and|N(t, w(t))| ≤ ε|w(t)| for |w(t)| sufficiently small that

(D.19) |T (w)(t)| ≤ Ce−ηt|ws|+ Cε‖w‖η

(∫ t

0e−η(t−s)e−ηs ds−

∫ +∞

te−θ(t−s)e−ηs ds

)for supt≥t0 |w(t)| ≤ ‖w‖η ≤ δ sufficiently small, whence

(D.20)‖T (w)‖η = sup

t≥0eηt)|T (w)(t)| ≤ sup

t≥0

(Ce(η−η)t|ws|+ Cε‖w‖η

∫ t

0e(η−η)(t−s) ds

+ Cε‖w‖η

∫ +∞

te(η−θ)(t−s) ds

)≤ C|ws|+ C1ε‖w‖η.

Taking ε so small that C1ε < 1/2 and C|w0| ≤ δ/2, we thus have ‖T (w)‖η ≤ δ for ‖w‖η ≤ δ,and T takes B(0, δ) to itself.

Similarly, we have by Nw(t, 0) = 0 and the (integral, vector form of the) Mean ValueTheorem that |N(t, w1(t))−N(t, w2(t))| ≤ ε|w1(t)−w2(t)| for ‖wj‖η ≤ δ sufficiently small,whence

(D.21)

‖T (w1)− T (w2)‖η = supt≥t0

eη(t−t0)|(T (w1)− T (w2))(t)|

≤ supt≥t0

Cε‖w1 − w2‖η

(∫ t

0e(η−η)(t−s) ds+

∫ +∞

te(η−θ)(t−s) ds

)

< (1/2)‖w1 − w2‖η,

yielding contraction on B(0, δ) and thus existence of a unique fixed point. From the usualfixed-point estimate ‖w‖η ≤ 2‖T (0)‖η ≤ 2C|w0|, coming from

(D.22) ‖w − T (0)‖η = ‖T (w)− T (0)‖η ≤ (1/2)‖w − 0‖η

and direct estimation of ‖T (0)‖η = ‖eA(t−t0)w0‖η, we obtain the claimed decay rate w(t) ≤Ce−ηtw(0) for t ≥ 0.

104

Noting that T decomposes into the sum of a bounded, hence C∞, linear map ws → eAtws

from Σs → Bη and the composition K · N of a bounded (hence C∞) linear map

K(f) :=∫ t

0eA(t−s)Πsf(s) ds−

∫ +∞

teA(t−s)Πcuf(s) ds

from Bη → Bη and a substitution operator N (w)(s) := N(w(s)) that is Ck by Lemma D.2,we find that T is Ck from Σs × Bη → Bη, whence w(wk) is Ck from Σs → Bη as claimed.Defining

(D.23) Φs(ws) := Πcuw(ws)|t=0 = −∫ +∞

0eA(t−s)ΠcuN(s, w(s)) ds,

we obtain as claimed a Ck function from Σs → Σcu, whose graph is the invariant manifold oforbits decaying exponentially to u∗ in forward time. By uniqueness of fixed point solutions,we have w(0) = 0 and thus Φs(0) = 0. Finally, differentiating (D.23) with respect to ws,we obtain by Lemma D.2 dΦs(0) = −

∫ +∞0 eA(t−s)ΠcuNw(s, 0)(dw/dws)(s) ds = 0, since

Nw(s, 0) ≡ 0, yielding tangency as claimed.

E Center Stable Manifold for PDE

We now turn to the case of a general semilinear parabolic equation

(E.1) ut = F(u) := h(u, ux) + uxx, u, h ∈ Rn,

with steady state u(x, t) ≡ u(x) and associated group invariance

(E.2) Ψα : Ψα(u)(x, t) := u(x+ α, t),

and construct a local translation-invariant center stable manifold in the vicinity of u.Our approach follows closely that used to construct translation-invariant center mani-

folds in [TZ1], by introduction of a reduced flow on the quotient space induced by groupequivalence. However, we coordinatize differently, using orthogonal projection rather thaneigenprojections, to avoid the difficulty (as in the ultimate application to shock waves)that the zero eigenvalue associated with translation-invariance may be embedded in essen-tial spectrum of the linearized operator about the wave, as a consequence of which theremay not exist a well-defined zero eigenprojection with respect to Hs (see [ZH] for furtherdiscussion).

We make the following assumptions, in practice typically satisfied [TZ1].

(A0) h ∈ Ck+1, k ≥ 2.(A1) The linearized operator L = ∂F

∂u (u) about u has p unstable (positive real part)eigenvalues, with the rest of its spectrum of nonpositive real part.

(A2) |∂jxu(x)| ≤ Ce−θ|x|, θ > 0, for 1 ≤ j ≤ k + 2.

105

Proposition E.1 (Center Stable Manifold Thm. for PDE). Under assumptions(A0)–(A2), there exists in an H2 neighborhood of the set of translates of u a translationinvariant Ck (with respect to H2) center stable manifold Mcs, tangent at u to the centerstable subspace Σcs of L, that is (locally) invariant under the forward time-evolution of(E.1) and contains all solutions that remain bounded and sufficiently close to a translate ofu in forward time. In general it is not unique.

E.1 Reduced equations

Differentiating with respect to α the relation 0 ≡ ∂t(Ψα(u)) = F(Ψα(u)), we recover thestandard fact that

φ :=dΨα(u)dα |α=0

=∂u

∂x

is an L2 zero eigenfunction of L, by the assumed decay of ux.Define orthogonal projections

(E.3) Π2 :=φ 〈φ, ·〉|φ|2

L2

, Π1 := Id−Π2,

onto the range of right zero-eigenfunction φ := (∂/∂x)u of L and its orthogonal complementφ⊥ in L2, where 〈·, ·〉 denotes standard L2 inner product.

Lemma E.1. Under the assumed regularity h ∈ Ck+1, k ≥ 2, Πj, j = 1, 2 are bounded asoperators from Hs to itself for 0 ≤ s ≤ k + 2.

Proof. Immediate, by the assumed decay of φ = ux and derivatives.

Introducing the shifted perturbation variable

(E.4) v(x, t) := u(x+ α(t), t)− u(x)

similarly as in [Z4, MaZ2, TZ1], we obtain the nonlinear perturbation equation

(E.5) ∂tv = Lv + G(v) + ∂tα(φ+ ∂xv),

where L := ∂F∂u (u) and

(E.6) G(v) = g(v, vx, x) := h(u+ v, ux + vx)− h(u, ux)− dh(u, ux)(v, vx)

is a quadratic-order Taylor remainder.Choosing ∂tα so as to cancel Π2 of the righthand side of (E.5), we obtain finally the

reduced equations

(E.7) ∂tv = Π1(Lv + G(v)) + (∂tα)Π1∂xv

106

and

(E.8) ∂tα = −Π2(Lv + G(v))1 + Π2(∂xv)

for v ∈ φ⊥, of the same regularity as the original equations.Here, we have implicitly chosen α(0) so that v(0) ∈ φ⊥, or

〈φ, u0(x+ α)− u(x)〉 = 0.

Assuming that u0 lies in a sufficiently small tube about the set of translates of u, or u0(x) =(u + w)(x − β) with |w|H2 sufficiently small, this can be done in a unique way such thatα := α− β is small, as determined implicitly by

0 = H(w, α) := 〈φ, u(·+ α)− u(·)〉+ 〈φ,w(·+ α)〉,

an application of the Implicit Function Theorem noting that

∂αH(0, 0) = 〈φ, ∂xu(·)〉 = |φ|2L2 6= 0.

With this choice, translation invariance under our construction is clear, with translationcorresponding to a constant shift in α that is preserved for all time.

Clearly, (E.8) is well-defined so long as |∂xv|L∞ ≤ C|v|H2 remains small, hence we maysolve the v equation independently of α, determining α-behavior afterward to determinethe full solution

u(x, t) = u(x− α(t)) + v(x− α(t), t).

Moreover, it is easily seen (by the block-triangular structure of L with respect to this de-composition) that the linear part Π1L = Π1LΠ1 of the v-equation possesses all spectrum ofL apart from the zero eigenvalue associated with eigenfunction φ. Thus, we have effectivelyprojected out this zero-eigenfunction, and with it the group symmetry of translation.

We may therefore construct the center stable manifold for the reduced equation (E.7),automatically obtaining translation-invariance when we extend to the full evolution using(E.8). See [TZ1] for further discussion.

E.2 Preliminary estimates

For ease of notation, introduce L0 := Π1L, G0 := Π1G.

Lemma E.2. Under the assumed regularity h ∈ Ck+1, both G and G0 are Frechet differen-tiable of order (k + 1) considered respectively as functions from H2 to H1 and φ⊥ ⊂ H2 toH1: G on the whole space and G0 for |v|H2 sufficiently small.

Proof. Differentiability of G follows by direct calculation; see [Sa, TZ1]. Differentiabilityof Π1G follows similarly, using also the fact, already discussed, that 1 + π2(∂xv) remainsbounded from zero for |v|H2 small, and the fact (see Lemma E.1) that Πj as bounded linearoperators from each Hs to itself are infinitely differentiable in the Frechet sense.

107

Lemma E.3. L0 generates an analytic semigroup eL0t = Π1eLtΠ1 on φ⊥ ⊂ H2. Moreover,

the unstable (positive real part) spectra of L and L0 agree in both location and multiplicity,with associated total unstable eigenprojections Π0

u and Πu related by Π0u = Π1ΠuΠ1 and

total center stable eigenprojections Π0cs and Πcs related by Π0

cs = Π1ΠcsΠ1. Likewise, exceptpossibly at λ = 0, the resolvent sets of L and L0 agree, with (λ− L0)−1 = Π1(λ− L)−1Π1.

Proof. Direct computation using relations LΠ1 = L and LΠ2 = 0 yields the resolventrelation, whence we obtain the remaining relations by their characterizations in terms ofthe resolvent (for example, the characterization of eigenprojection as residue of the resolventoperator [K]). As L is a sectorial operator, it follows that L0 is as well, and both generateanalytic semigroups given by the inverse Laplace transform of the resolvent.

Corollary E.4 ([TZ1]). Under assumptions (A0)–(A2),

(E.9)‖etL0Πcs‖H1→H2 ≤ Cω(1 + t−1/2)eωt,

‖e−tL0Πu‖H1→H3 ≤ Cωe−βt,

for some β > 0, and for all ω > 0, for all t ≥ 0.

Proof. By Lemma E.3, it is sufficient to prove the corresponding bounds for the purelydifferential operator L. By sectoriality of L, we have the inverse Laplace transform repre-sentations

(E.10)etLΠu :=

∫Γu

eλt(λ− L)−1 dλ,

etLΠcs :=∫

Γcs

eλt(λ− L)−1 dλ,

where Γcs denotes a sectorial contour bounding the center and stable spectrum to the right[Pa], which by (A1) may be taken so that <Γs ≤ ω, and Γu denotes a closed curve enclosingthe unstable spectrum of L, with <Γs ≥ β > 0.

Applying the resolvent formula L(λ−L)−1 = λ(λ−L)−1−Id, we obtain in the standardway

LetLΠj :=∫

Γj

λeλt(λ− L)−1 dλ,

from which we obtain immediately the second stated bound, and, by a scaling argument[Pa], the bound

(E.11) ‖etLΠcs‖H1→H3 ≤ ‖LetLΠs‖H1→H1 ≤ C(1 + t−1)eωt.

Recalling the standard bound ‖etLΠcs‖H1→H1 ≤ Ceωt, and interpolating between | · |H1 and| · |H3 , we obtain the first stated bound.

Let ρ be a smooth truncation function as in §D.4 and Gδ0(v) := ρ

(|v|H2

δ

)G0(v).

108

Lemma E.5 ([TZ1]). The map Gδ0 : H2 → H1 is Ck+1 and its Lipschitz norm with respect

to v is O(δ) as δ → 0.

Proof. The norm in H2 is a quadratic form, hence the map

v ∈ H2 7→ ρ( |v|H2

δ

)∈ R+,

is smooth, and Gδ0 is as regular as G0. Now

|Gδ0(v1)− Gδ

0(v2)|H1 ≤ |ρ( |v1|H2

δ

)− ρ( |v2|H2

δ

)|L∞ |G0(v1)|H1

+ |ρ( |v2|H2

δ

)|L∞ |G0(v1)− G0(v2)|H1

≤ 3|v1 − v2|H2

(sup

|v|H2<δ

|G0(v)|H1

δ+ sup|v|H2<δ

|dG0(v)|H1

),

and sup|v|H2<δ |G0(v)|H1 = O(δ2), sup|v|H2<δ |dG0(v)|H1 = O(δ).

Corollary E.6. Under assumptions (A0)–(A2),

(E.12)‖etL0ΠcsGδ

0‖H2→H2 ≤ Cω(1 + t−1/2)eωt,

‖e−tL0ΠuGδ0‖H2→H2 ≤ Cωe

−βt,

for some β > 0, and for all ω > 0, for all t ≥ 0, with Lipshitz bounds

(E.13)‖etL0ΠcsdGδ

0‖H2→H2 ≤ Cωδ(1 + t−1/2)eωt,

‖e−tL0ΠudGδ0‖H2→H2 ≤ Cωδe

−βt.

E.3 Translation-invariant center stable manifold

Proof of Proposition E.1. Using bounds (E.9), (E.12), and (E.13), and observing that, sincefinite-dimensional, the unstable flow eL0tΠu is well-defined in both forward and backwardtime, we find, applying the finite-dimensional argument word-for-word, that the centerstable manifold of the truncated flow ut = L0u + Gδ

0(u) is the graph of the map Φ definedon Σcs as Φ(ucs) = (vcs(0)), where vcs is the unique solution in φ⊥ ⊂ H2 of

v(x, t) = etL0ucs +∫ t

0e(t−s)L0ΠcsGδ

0(v(x, s))ds−∫ ∞

te(t−s)L0ΠuGδ

0(v(x, s))ds

as guaranteed by the Contraction Mapping Theorem.Indeed, the only deviation from the finite-dimensional case is the appearance of the

new factor (1 + (t − s)−1/2) in the integrand of estimates having to do with integral term∫ t0 e

L0(t−s)ΠcsGδ0 , which, since integrable, does not alter the final estimates. The proof of

smoothness relies on these same bounds, so likewise carries over word-for-word as in the

109

finite-dimensional case. Finally, invariance in forward time follows by the characterizationof the center stable manifold as the set of solutions of the truncated equations growing nofaster than |v(t)| ≤ Ceβt in forward time. (Since the flow of (E.1) is only a semigroup,we cannot conclude invariance in backward time as in the finite-dimensional case.) Sincesufficiently small bounded solutions of the original equations are bounded solutions also ofthe truncated equations, they are contained in the center stable manifold, independent ofthe choice of truncation function.

This gives a center stable manifold with the stated properties for the reduced equation(E.7) for v ∈ φ⊥. Solving for the shift α in terms of v using (E.8), and substituting v, α into(E.4) to obtain u, we obtain a translation-invariant center stable manifold for the originalequations (E.1). Observing that small bounded solutions v of the reduced equations, by(E.4), correspond to solutions u of the (E.1) remaining close to a translate of u, we aredone.

E.4 Application to viscous shock waves

Proof of Theorem 7.1. By Proposition E.1, it is sufficient to verify that (H0)–(H1) imply(A0)–(A2), for h(u, ux) := f(u)x = df(u)ux. Clearly, (H0) implies (A0) by the form of h.Plugging u = u(x) into (7.1), we obtain the standing-wave ODE f(u)x = uxx, or, integratingfrom −∞ to x, the first-order system

ux = f(u)− f(u−).

Linearizing about the assumed critical points u± yields linearized systems wt = df(u±)w,from which we see that u± are nondegenerate rest points by (H1), as a consequence of which(A2) follows by standard ODE theory [Co].

Finally, linearizing PDE (E.1) about the constant solutions u ≡ u±, we obtain wt =L±w := −df(u±)wx − wxx. By Fourier transform, the limiting operators L± have spectraλ±j (k) = −ika±j − k2, where the Fourier wave-number k runs over all of R; in particular,L± have spectra of nonpositive real part. By a standard result of Henry [He], the essentialspectrum of L lies to the left of the rightmost boundary of the spectra of L±, hence wemay conclude that the essential spectrum of L is entirely nonpositive. As the spectra of Lto the right of the essential spectrum by sectoriality of L, consists of finitely many discreteeigenvalues, this means that the spectra of L with positive real part consists of p unstableeigenvalues, for some p, verifying (A1).

E.5 A simple orbital stability theorem

Similar ideas yield an extremely simple stability theorem in the case of a spectral gap.Substitute for (A1):

(A1’) The linearized operator L = ∂F∂u (u) about u has a single isolated eigenvalue at

zero, with the rest of its spectrum of strictly negative real part ≤ −η < 0.

Proposition E.2 (Stable Manifold Thm. for PDE). Under assumptions (A0), (A1’),and (A2), the set of translates of u is exponentially asymptotically orbitally stable in H2.

110

Proof. Performing the same reduction to (E.5), (E.8), we find that the reduced operatorL has spectrum of strictly negative real part. Carrying out the PDE version of the StableManifold Theorem (similarly as, but much more simply than in the proof of PropositionE.1), with Σs equal to the whole space, we obtain the result.

F Miscellaneous energy estimates

F.1 Small-amplitude stability for isentropic gas dynamics

Proof of Proposition 6.2. We note that h(v) > 0. By multiplying (6.4b) by both the con-jugate u and vγ+1/h(v) and integrating along x from ∞ to −∞, we have∫

R

λuuvγ+1

h(v)+∫

R

u′uvγ+1

h(v)−∫

Rv′u =

∫R

u′′uvγ

h(v).

Integrating the last three terms by parts and appropriately using (6.4a) to substitute for u′

in the third term gives us∫R

λ|u|2vγ+1

h(v)+∫

R

u′uvγ+1

h(v)+∫

Rv(λv + v′) +

∫R

vγ |u′|2

h(v)= −

∫R

(vγ

h(v)

)′u′u.

We take the real part and appropriately integrate by parts to get

R(λ)∫

R

[vγ+1

h(v)|u|2 + |v|2

]+∫

Rg(v)|u|2 +

∫R

vγ

h(v)|u′|2 = 0,

where

g(v) = −12

[(vγ+1

h(v)

)′+(vγ

h(v)

)′′].

Thus, to prove stability it suffices to show that g(v) ≥ 0 on [v+, 1].By straightforward computation, we obtain identities:

γh(v)− vh′(v) = aγ(γ − 1) + vγ+1 and(F.1)

vγ−1vx = aγ − h(v).(F.2)

111

Using (F.1) and (F.2), we abbreviate a few intermediate steps below:

g(v) = − vx

2

[(γ + 1)vγh(v)− vγ+1h′(v)

h(v)2+

d

dv

[γvγ−1h(v)− vγh′(v)

h(v)2vx

]]= − vx

2

[vγ ((γ + 1)h(v)− vh′(v))

h(v)2+

d

dv

[γh(v)− vh′(v)

h(v)2(aγ − h(v))

]]= −avxv

γ−1

2h(v)3×[

γ2(γ + 1)vγ+2 − 2(a+ 1)γ(γ2 − 1)vγ+1 + (a+ 1)2γ2(γ − 1)vγ

+ aγ(γ + 2)(γ2 − 1)v − a(a+ 1)γ2(γ2 − 1)]

= −avxvγ−1

2h(v)3[(γ + 1)vγ+2 + vγ(γ − 1) ((γ + 1)v − (a+ 1)γ)2(F.3)

+ aγ(γ2 − 1)(γ + 2)v − a(a+ 1)γ2(γ2 − 1)]

≥ −avxvγ−1

2h(v)3[(γ + 1)vγ+2 + aγ(γ2 − 1)(γ + 2)v − a(a+ 1)γ2(γ2 − 1)]

≥ −γ2a3vx(γ + 1)2h(v)3v+

(vγ+1+

aγ

)2

+ 2(γ − 1)

(vγ+1+

aγ

)− (γ − 1)

.(F.4)

Thus from (6.6), we have spectral stability.

F.2 High-frequency bounds for isentropic gas dynamics

Lemma F.1. For each γ ≥ 1, the following identity holds for Rλ ≥ 0:

(R(λ) + |=(λ)|)∫

Rv|u|2 − 1

2

∫Rvx|u|2 +

∫R|u′|2

≤√

2∫

R

h(v)vγ

|v′||u|+∫

Rv|u′||u|.(F.5)

Proof. We multiply (6.4b) by vu and integrate along x. This yields

λ

∫Rv|u|2 +

∫Rvu′u+

∫R|u′|2 =

∫R

h(v)vγ

v′u.

We get (F.5) by taking the real and imaginary parts and adding them together, and notingthat |R(z)|+ |=(z)| ≤

√2|z|.

Lemma F.2. For each γ ≥ 1, the following identity holds for Rλ ≥ 0:

(F.6)∫

R|u′|2 = 2R(λ)2

∫R|v|2 + R(λ)

∫R

|v′|2

v+

12

∫R

[h(v)vγ+1

+aγ

vγ+1

]|v′|2

112

Proof. We multiply (6.4b) by v′ and integrate along x. This yields

λ

∫Ruv′ +

∫Ru′v′ −

∫R

h(v)vγ+1

|v′|2 =∫

R

1vu′′v′ =

∫R

1v(λv′ + v′′)v′.

Using (6.4a) on the right-hand side, integrating by parts, and taking the real part gives

R[λ

∫Ruv′ +

∫Ru′v′

]=∫

R

[h(v)vγ+1

+vx

2v2

]|v′|2 + R(λ)

∫R

|v′|2

v.

The right hand side can be rewritten as

(F.7) R[λ

∫Ruv′ +

∫Ru′v′

]=

12

∫R

[h(v)vγ+1

+aγ

vγ+1

]|v′|2 + R(λ)

∫R

|v′|2

v.

Now we manipulate the left-hand side. Note that

λ

∫Ruv′ +

∫Ru′v′ = (λ+ λ)

∫Ruv′ −

∫Ru(λv′ + v′′)

= −2R(λ)∫

Ru′v −

∫Ruu′′

= −2R(λ)∫

R(λv + v′)v +

∫R|u′|2.

Hence, by taking the real part we get

R[λ

∫Ruv′ +

∫Ru′v′

]=∫

R|u′|2 − 2R(λ)2

∫R|v|2.

This combines with (F.7) to give (F.6).

Lemma F.3. For each γ ≥ 1, for h(v) as in (1.6), we have

(F.8) supv

∣∣∣∣h(v)vγ

∣∣∣∣ = γ1− v+1− vγ

+

≤ γ.

Proof. Defining

(F.9) g(v) := h(v)v−γ = −v + a(γ − 1)v−γ + (a+ 1),

we have g′(v) = −1− aγ(γ − 1)v−γ−1 < 0 for 0 < v+ ≤ v ≤ v− = 1, hence the maximum ofg on v ∈ [v+, v−] is achieved at v = v+. Substituting (6.3) into (F.9) and simplifying yields(F.8).

113

Proof of Proposition 6.3. Using Young’s inequality twice on the right-hand side of (F.5)together with (F.8), we get

(R(λ) + |=(λ)|)∫

Rv|u|2 − 1

2

∫Rvx|u|2 +

∫R|u′|2

≤√

2∫

R

h(v)vγ

|v′||u|+∫

Rv|u′||u|

≤ θ

∫R

h(v)vγ+1

|v′|2 +(√

2)2

4θ

∫R

h(v)vγ

v|u|2 + ε

∫Rv|u′|2 +

14ε

∫Rv|u|2

< θ

∫R

h(v)vγ+1

|v′|2 + ε

∫R|u′|2 +

[γ

2θ+

14ε

] ∫Rv|u|2.

Assuming that 0 < ε < 1 and θ = (1− ε)/2, this simplifies to

(R(λ) + |=(λ)|)∫

Rv|u|2 + (1− ε)

∫R|u′|2

<1− ε

2

∫R

h(v)vγ+1

|v′|2 +[γ

2θ+

14ε

] ∫Rv|u|2.

Applying (F.6) yields

(R(λ) + |=(λ)|)∫

Rv|u|2 <

[γ

1− ε+

14ε

] ∫Rv|u|2,

or equivalently,

(R(λ) + |=(λ)|) < (4γ − 1)ε− 14ε(1− ε)

.

Setting ε = 1/(2√γ + 1) gives (6.7).

F.3 Nonvanishing of D0 for isentropic gas dynamics

The formal limiting eigenvalue system associated with D0 together with the limiting bound-ary conditions may be expressed equivalently as the integrated eigenvalue problem

λv + v′ − u′ = 0,(F.10a)

λu+ u′ − 1− v

vv′ =

u′′

v.(F.10b)

corresponding to a pressureless gas, γ = 0, with special upstream boundary conditions

(F.11) (u, u′, v, v′)(−∞) = (0, 0, 0, 0), (u, u′, v, v′)(+∞) = (c, 0, 0, 0).

Motivated by this observation, we establish stability of the limiting system by a Matsumura–Nishihara-type spectral energy estimate exactly analogous to that used to prove stabilityfor γ = 1 in [MN, BHRZ].

114

Proof of Proposition 6.5. Multiplying (F.10b) by vu/(1− v) and integrating on some subin-terval [a.b] ⊂ R, we obtain

λ

∫ b

a

v

1− v|u|2dx+

∫ b

a

v

1− vu′udx−

∫ b

av′udx =

∫ b

a

u′′u

1− vdx.

Integrating the third and fourth terms by parts yields

λ

∫ b

a

v

1− v|u|2dx+

∫ b

a

[v

1− v+(

11− v

)′]u′udx

+∫ b

a

|u′|2

1− vdx+

∫ b

av(λv + v′)dx

=[vu+

u′u

1− v

] ∣∣∣ba.

Taking the real part, we have

R(λ)∫ b

a

(v

1− v|u|2 + |v|2

)dx+

∫ b

ag(v)|u|2dx+

∫ b

a

|u′|2

1− vdx

= R[vu+

u′u

1− v− 1

2

[v

1− v+(

11− v

)′]|u|2 − |v|2

2

] ∣∣∣ba,(F.12)

where

g(v) = −12

[(v

1− v

)′+(

11− v

)′′].

Note thatd

dx

(1

1− v

)= − (1− v)′

(1− v)2=

vx

(1− v)2=v(v − 1)(1− v)2

= − v

1− v.

Thus, g(v) ≡ 0 and the third term on the right-hand side vanishes, leaving

R(λ)∫ b

a

(v

1− v|u|2 + |v|2

)dx+

∫ b

a

|u′|2

1− vdx

=[R(vu) +

R(u′u)1− v

− |v|2

2

] ∣∣∣ba.

We show next that the right-hand side goes to zero in the limit as a→ −∞ and b→∞.By Lemma C.4, the behavior of u, v near±∞ is governed by the limiting constant–coefficientsystems W ′ = A0

±(λ)W , where W = (u, v, v′)T and A0± = A0(±∞, λ). In particular,

solutions W asymptotic to (1, 0, 0) at x = +∞ decay exponentially in (u′, v, v′) and arebounded in coordinate u as x → +∞. Observing that 1− v → 1 as x → +∞, we thus seeimmediately that the boundary contribution at b vanishes as b→ +∞.

115

The situation at −∞ is more delicate, since the denominator 1 − v of the second termgoes to zero at rate ex as x → −∞, the rate of convergence of the limiting profile v. Byinspection, the limiting coefficient matrix

(F.13) A0− =

0 λ 10 0 1λ λ 1− λ

,

has eigenvalues

µ = −λ, 1±√

1 + 4λ2

,

hence for Rλ ≥ 0 the unique decaying mode at x = +∞ is the unstable eigenvector corre-sponding to µ = 1+

√1+4λ2 , with growth rate

R(

1 +√

1 + 4λ2

)=

12

+12

R√

1 + 4λ > 1/2.

Thus, |u|, |u′|, |v′|, |v| ≤ Ce(1+ε)x/2 as x→ −∞, ε > 0, and in particular∣∣∣R(u′u)1− v

∣∣∣ ≤ Ce(1+ε)x/ex ≤ Ceεx → 0

as x→ −∞. It follows that the boundary contribution at x = a vanishes also as a→ −∞,hence, in the limit as a→ −∞, b→ +∞,

(F.14) R(λ)∫ +∞

−∞

(v

1− v|u|2 + |v|2

)dx+

∫ +∞

−∞

|u′|2

1− vdx = 0.

But, for Rλ ≥ 0, this implies u′ ≡ 0, or u ≡ constant, which, by u(−∞) = 0, implies u ≡ 0.This reduces (F.10a) to v′ = λv, yielding the explicit solution v = Ceλx. By v(±∞) = 0,therefore, v ≡ 0 for Rλ ≥ 0. It follows that there are no nontrivial solutions of (F.10),(F.11) for Rλ ≥ 0.

Remark F.4. The above energy estimate is equivalent to multiplying the system by the

special symmetrizer(

1 00 v/(1− v)

), then taking the L2 inner product with (v, u)T .

G Miscellaneous proofs

Proof of Corollary 7.5. For definiteness, take y ≤ 0. Then, (7.13) gives

ey(y, t) :=∑

a−k >0

[c0k,−]l−tk

(K(y + a−k t, t)−K(y − a−k t, t)

)et(y, t) :=

∑a−k >0

[c0k,−]l−tk

((K +Ky)(y + a−k t, t)− (K +Ky)(y − a−k

t, t)),

116

ety(y, t) :=∑

a−k >0

[c0k,−]l−tk

((Ky +Kyy)(y + a−k t, t)− (Ky +Kyy)(y − a−k t, t)

),

where

K(y, t) :=e−y2/4t

√4πt

denotes the standard heat kernel. The pointwise bounds (7.5)–(7.5) follow immediately fort ≥ 1 by properties of the heat kernel, in turn yielding (7.5)–(7.5) in this case. The boundsfor small time t ≤ 1 follow from estimates

|Ky(y + at, t, β)−Ky(y − at, t, β)| =∣∣∣∣∫ y−at

y+atKyy(z, t, β) dz

∣∣∣∣≤ Ct−3/2

∫ y−at

y+ate−z2

Mt dz ≤ Ct−1/2e−y2

Mt ,

and, similarly,

|Kyy(y + at, t, β)−Kyy(y − a, t, β)| =∣∣∣∣∫ at

−atKyyy(z, t, β) dz

∣∣∣∣≤ Ct−2

∫ y−at

y+ate−z2

Mt dz ≤ Ct−1e−y2

Mt .

The bounds for |ey| are again immediate.

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