O(2) HOPF BIFURCATION OF VISCOUS

SHOCK WAVES IN A CHANNEL

ALIN POGAN, JINGHUA YAO, AND KEVIN ZUMBRUN

Abstract. Extending work of Texier and Zumbrun in the semilinear non-reflection symmetriccase, we study O(2) transverse Hopf bifurcation, or “cellular instability”, of viscous shock waves ina channel, for a class of quasilinear hyperbolic–parabolic systems including the equations of thermo-viscoelasticity. The main difficulties are to (i) obtain Frechet differentiability of the time-T solutionoperator by appropriate hyperbolic–parabolic energy estimates, and (ii) handle O(2) symmetry inthe absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandardquasilinear hyperbolic-parabolic form) the requisite framework for treatment by spatial dynamics onthe space of time-periodic functions, the two standard treatments for this problem. The latter issueis resolved by Lyapunov–Schmidt reduction of the time-T map, yielding a four-dimensional problemwith O(2) plus approximate S1 symmetry, which we treat “by hand” using direct Implicit FunctionTheorem arguments. The former is treated by balancing information obtained in Lagrangian coor-dinates with that from associated constraints. Interestingly, this argument does not apply to gasdynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangiansymmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.

1. Introduction

In this paper, we treat transverse Hopf bifurcation, or time-periodic “cellular instability”, ofplanar viscous shock waves in an infinite channel with periodic boundary conditions, for a classof hyperbolic–parabolic systems including the equations of thermoviscoelasticity. Transverse Hopfbifurcation has been treated in [TZ2] for semilinear equations. The main differences here are partialparabolicity/lack of parabolic smoothing and reflection symmetry of the physical equations. Theformer adds considerable technical difficulty to do with the basic issue of regularity of the time-Tsolution map, as discussed for the 1D case in [TZ3]. The latter implies that the underlying bifurca-tion is not of planar Hopf type, but, rather, a four-dimensional O(2) Hopf bifurcation as discussedfor example in [GSS]: roughly speaking, a “doubled” Hopf bifurcation coupled by nonlinear terms.1

O(2) Hopf bifurcation is typically treated by center manifold reduction followed by transforma-tion to a (doubly) angle invariant normal form, and thereby to a planar stationary bifurcation withD4 symmetry in the (two) radial coordinates. Here, however, the linearized operator about thewave has no spectral gap, hence standard center manifold theorems do not apply; indeed, exis-tence of a center manifold is unclear. Instead, we proceed by the Lyapunov–Schmidt reductionframework of [TZ2], applied to the time-T evolution map of the underlying perturbation equations,resulting in a 4-dimensional stationary bifurcation problem with O(2) symmetry plus an additional“approximate S1 symmetry” induced by the underlying rotational linearized flow. The latter isthen analyzed “by hand”, using direct rescaling/Implicit Function Theorem arguments.2

Research of A.P. was partially supported under NSF grant no. DMS-0300487.Research of J.Y. was partially supported under NSF grant no. DMS-0300487.Research of K.Z. was partially supported under NSF grant no. DMS-0300487.1Corresponding address: 831 East 3rd St, Bloomington, IN 47405, Phone: (812) 855-4591, Fax: (812) 855-0046.2E-mail addresses: pogana@miamioh.edu (A Pogan), jinghua-yao@uiowa.edu(J. Yao), kzumbrun@indiana.edu(K.

Zumbrun

1

We note that, though there exist other methods suitable to treat related problems withoutspectral gap, notably the spatial dynamics approach used by Iooss, Sandstede–Scheel, and others-see in particular [GSS, SS])- the “reverse temporal dynamics” approach of [TZ3] is the only onethat has so far been successfully applied (in the 1-D case) to the viscous shock solutions of physical,partially parabolic systems; see Section 1.5 for further discussion. Indeed, the main advantage ofthis method is that it typically applies whenever there is an existing time-evolutionary stabilitytheory for the background equilibrium solution, which in this case has already been developed. Adisadvantage of the method is, simply, that it is not the standard one, and so, as turns out to bethe case here, one cannot always appeal to existing theory to treat the resulting reduced system.Though elementary, the treatment of the nonstandard finite-dimensional reduced system is thus asignificant part of our analysis.

Regarding regularity, a critical aspect, as in the 1D case [TZ3], is to work in Lagrangian ratherthan Eulerian coordinates, in which hyperbolic transport modes become constant-coefficient linearrather than quasilinear as in the Eulerian case, and certain key variational energy estimates do notlose derivatives. For our argument, we require also other favorable properties related to stabilityof constant solutions that are evident in the Eulerian formulation by existence of a convex entropy,but for multi-D are in the Lagrangian formulation are less clear. Fortunately, this issue may beaddressed by a slight extension of ideas of Dafermos [Da] regarding involution and contingententropy, in particular yielding the necessary properties for the equations of thermoviscoelasticity.

Lest one conclude that Eulerian and Lagrangian formulations share identical properties, wepoint out that for gas dynamics and MHD, this is far from the case. Since the stress tensor inthese cases depends on the strain tensor through density alone, that is, only through the Jacobianof the displacement map, it follows that the Lagrangian equations are invariant under any volume-preserving diffeomorphism, an infinite-dimensional family of symmetries preventing asymptoticstability of constant solutions. Meanwhile, the Eulerian equations, possessing a convex entropy,automatically do have the property of asymptotic stability. This represents a genuine differencebetween Eulerian and Lagrangian formulations for gas dynamics and MHD, and an obstruction tothe methods of this paper. We discuss in Section 5 various ideas how this might be overcomed.

Notation: In what follows u : R × [−π, π] → Rn is a smooth function, periodic on [−π, π].Unless otherwise indicated, indices j, k are in the range {1, 2}, and summations in j, k are from 1to 2. We denote ∂j := ∂

∂xjand ∂t := ∂

∂t , α ∈ N2, and Dα := ∂αx . We denote the symmetric part of

a matrix or linear operator N by ReN := 12(N +N∗) and its spectrum by σ(N).

1.1. Equations and assumptions. Consider a smooth one-parameter family of standing viscousplanar shock solutions

(1.1) u(x, t) = uε(x1), limx1→±∞

uε(x1) = uε± (constant for fixed ε),

of a smoothly-varying family of conservation laws

(1.2)

ut = F(ε; u) :=∑jk

∂j(Bjk(ε; u)∂ku)−

∑j

∂jFj(ε; u)

=∑jk

∂j(Bjk(ε; u)∂ku)−

∑j

∂jFj(ε; u) + G(u, ε)mu u ∈ Rn

on the periodic channel x1 ∈ R, x2 ∈ [−π, π]per subject to (possibly vanishing) constraints

(1.3) mu :=∑j

mj∂ju = 0, mj = constant ∈ Rq×n, for some q ∈ Z+,

2

preserved by the flow of (1.2), with associated linearized operators(1.4)

L(ε)v :=∑jk

∂j

(Bjk(ε; uε)∂kv − ∂uBjk(ε; uε)v∂ku

ε)−∑j

∂j(∂uFj(ε; uε)v) + G(ε; uε)mv.

In the case of a nontrivial constraint m 6= 0, following [Da], we assume further that

(1.5) mL(ε) = Γ(ε)m

for some family of second-order linear partial differential operators Γ(ε) generating a C0 semigroup.This is sufficient but not necessary for preservation of the condition mu = 0 under the linearizedflow ut = L(ε)u, by well-posedness of the induced flow µt = Γ(ε)µ, governing µ := mu.

In most examples the constraints (1.3) are given as divergence free or curl free conditions, asfor many equations of mathematical-physics; see examples 1.4 and 1.6 below. Example 1.7 doesnot involve a constraint. Typically, the bifurcation parameter ε measures shock amplitude orother physical parameters. Here, the linear operator L(ε) is considered as a closed linear operatoron L2(R × [−π, π],Cn) with domain dom(L(ε)) = H2(R × [−π, π],Cn), and the functions Bjk :(−δ, δ) × Rn → Rn×n and F j : (−δ, δ) × Rn → Rn are smooth in u, see also Hypothesis (H0)below. Equations (1.2) are typically shifts Bjk(ε; u) = Bjk(u), F 1(ε; u) := f1(u)−s(ε)u of a singleequation

ut =∑jk

∂j(Bjk(u)∂ku)−

∑j

∂jFj(u) + G(u)m

written in coordinates x = (x1 − s(ε)t, x2) moving with traveling-wave solutions u(x, t) = uε(x1 −s(ε)t) of varying speeds s(ε). Profiles uε satisfy the standing-wave ODE

(1.6) B11(ε; u)u′ = F 1(ε; u)− F 1(ε; uε−).

We assume, further, that there is an invertible coordinate change u→ w yielding the “partiallysymmetric hyperbolic–parabolic form” of [Z2]:

(1.7) A0(ε;w)wt =∑jk

∂j(Bjk(ε;w)∂kw)−∑j

Aj(ε;w)∂jw + Gmw +

(0g

),

with g = O(|∇xw|2), together with conditions (A3) below.We shall use the notation

(1.8) Aj : (−δ, δ)× Rn → Rn×n, Aj(ε; u) = F ju(ε; u);

Aj±, Bjk± : (−δ, δ)→ Rn×n

(1.9) Aj±(ε) = F ju(ε; uε±) = Aj(ε; uε±), Bjk± (ε) = Bjk(ε; uε±).

Aj±,Bjk± : (−δ, δ)→ Rn×n

(1.10) Aj±(ε) = Aj(ε;wε±), Bjk± (ε) = Bjk(ε;wε±).

In what follows, if A is an n × n matrix we will use lower subscripts for the block decomposition

A =

(A11 A12

A21 A22

), where A11 is an (n− r)× (n− r) matrix and A22 is an r × r matrix.

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1.1.1. Structural conditions. We make the following structural assumptions on the equations:

(A1) For every j, k ∈ {1, 2} there exist functions bjk : (−δ, δ) × Rn → Rr×r and function bjk :(−δ, δ)× Rn → Rr×r such that Bjk and Bjk have the representations

(1.11) Bjk(ε; u) =

(0 00 bjk(ε; u),

), Bjk(ε;w) =

(0 0

0 bjk(ε;w),

).

Moreover, the first (n− r) components of F j(ε,u), j ∈ {1, 2} are linear in u.(A2) There exists a matrix-valued function A0 : (−δ, δ)×Rn → Rn×n smooth in u ∈ Rn, positive

definite and having a block-diagonal structure such that

(i) A011(ε; u)Aj11(ε; u) is symmetric for any ε ∈ (−δ, δ), u ∈ Rn;

(ii) A011(ε; u)A1

11(ε; u) is either positive definite or negative definite for any ε ∈ (−δ, δ),u ∈ Rn;

(iii) there exists a constant θ > 0 such that∑j,k

vj ·(A0

22(ε; u)bjk(ε; u)vk

)≥ θ

∑j

|vj |2 for all v1,v2 ∈ Rr.

(A3) A0 is block-diagonal, symmetric positive definite and:

(i) Aj11(ε;w) is symmetric for any ε ∈ (−δ, δ), w ∈ Rn;(ii) A1

11(ε;w) is either positive definite or negative definite for any ε ∈ (−δ, δ), w ∈ Rn;

(iii) Aj±(ε) is symmetric (mod m) for any ε ∈ (−δ, δ), in the sense that Aj±(ε) + vT±mj issymmetric for some v± ∈ Rr×n;

(iv) there exists a constant θ > 0 such that∑j,k

vj ·(A0

22(ε;w)bjk(ε;w)vk

)≥ θ

∑j

|vj |2 for all v1,v2 ∈ Rr;

(v) There exist skew matrices Kj±(ε) ∈ Rn×n such that

Re∑jk

(Kj±(ε)(A0

±(ε))−1Ak±(ε) + Bjk± (ε))ξjξk ≥ θ|ξ|2 > 0 (mod m(ξ))

for all ξ ∈ R2 and ε ∈ (−δ, δ), in the sense that the displayed quadratic form is definitewhen restricted to the kernel of the symbol m(ξ) :=

∑jmjξj of constraint m, or,

equivalently, its sum with C(m∗m)(ξ) is ≥ θ for C > 0 sufficiently large.(B1) Equations (1.2) are invariant under S : x2 → −x2, u→Mu, for M ∈ Rn×n constant, such

that M2 = Id.

Conditions (A1)–(A2) are analogous to the 1D conditions of [TZ3], used to obtain Frechet dif-ferentiability of the nonlinear source term in the time-T solution map for the perturbation equa-tions of (1.2) about a standing shock, themselves strengthened versions of the standard conditionsof [Z2] for systems of conservation laws with physical viscosity. Condition (A1) expresses thefact that coordinate u2 experiences second-order parabolic smoothing– i.e., the principal partof the u2 equation is a parabolic equation u2,t =

∑jk ∂j(b

jk(ε; u)∂ku2)– while coordinate u1

does not– i.e., the principal part of the u1 equation is a first-order hyperbolic conservation law

u1,t +∑d

j=1Aj11(ε; u)∂ju1 = 0. The fact that these equations are of parabolic (hyperbolic) type

follows from the block-symmetrizability assumptions (A2), where (A2)(ii) guarantees uniform ellip-ticity of bjk terms and (A2)(iii) is the familiar condition of Friedrichs symmetrizability guaranteeing

hyperbolicity of A011∂t +

∑dj=1A

j11(ε; ·)∂j .

The key second condition of (A1), together with the assumed block structure of Bjk, imposesfurther that the u1 equation be linear constant-coefficient, yielding easily Frechet differentiability

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in u of the “hyperbolic” (i.e., u1) part of the time-evolution map, which in the quasilinear hyper-bolic case would otherwise not hold (see Appendix A [TZ3] for further discussion/computations).Condition (A2)(ii) is a technical condition that is generally satisfied and has proved to be use-ful in all aspects of viscous shock analysis from existence of profiles to stability [Z2]. Condition(A3) is equivalent (mod m) to the multi-D conditions of [Z1, Z2], used to obtain damping-typetime-evolutionary energy estimates and high-frequency resolvent bounds.

To summarize: from the special linear structure of the hyperbolic part of the u equations,together with block hyperbolic-parabolic structure, we are able to obtain the Frechet differentiabilityof the time-evolution map needed even to begin our analysis. For the high-frequency dampingestimates needed to carry out our ultimate Lyapunov–Schmidt reduction argument, we require theadditional symmetry information (A3)(iii), (v) at the end states of the shock. Condition (A3)(v)may be recognized as a (mod m(ξ)) version of Kawashima’s condition for hyperbolic-parabolicsmoothing [Kaw], implying in particular a (mod m(ξ)) version of his genuine coupling condition:

there is no eigenvector of∑

j Aj±(ε)ξj lying in ker

∑j,k B

jk± (ε)ξjξk ∩ kerm(ξ), for any ξ ∈ R2.

Together, (A3) (iii) and (v) imply, by the restriction to kerm(ξ) of the energy estimates used byKawashima in the unconstrained case (see, e.g., [KaS] or the proof of Lemma 3.18 [Z2]), the spectralbound

(1.12) Reσ(L±(ξ; ε)| kerm(ξ)) ≤ −θ|ξ|2

1 + |ξ|2 , ξ ∈ R2

on the Fourier symbol L±(ξ; ε) of the limiting constant–coefficient operators L±(ε) as x1 → ±∞,when restricted to its invariant subspace kerm(ξ).

Condition (B1) together with translation-invariance in x2, implies O(2) symmetry in the per-turbation equations around the (symmetric, since constant in x2-direction) background shock so-lutions, with R(θ) : u(x1, ·, t)→ u(x1, (·+ θ)mod 2π, t) corresponding to rotation and S reflection,R(θ)S = SR(−θ). That is, “rotation” in this context should be thought of as x2-translation.

Remark 1.1. With slight further effort, we may replace as in [Z1, Z2] the uniform ellipticity condi-

tions (A2)(iii) and (A3)(iv) with the symbolic conditions Re(∑

j,k A022(ε; u)bjk(ε; u)ξjξk

)≥ θ|ξ|2,

for all ξ ∈ R2, using Garding’s inequality instead of direct integration by parts in the energy esti-mates of Section 2; see for example the proof of [Z1, Proposition 5.9]. However, this is not neededfor the physical applications from continuum mechanics that we have in mind.

1.1.2. Technical hypotheses. To (A1)–(A3), (B1), we add the following technical conditions. Hereand elsewhere, σ(M) denotes the spectrum of a matrix or linear operator M .

(H0) For any j, k ∈ {1, 2} the functions F j ,Bjk, Aj , Bj,k are of class Cν , for some ν ≥ 5.(H1) The eigenvalues of A1

11(ε; uε(x1)) have multiplicity independent of x1 ∈ R.(H2) σ(A1

±(ε)) is real, semisimple, and nonzero for A1±(ε) := F 1

u(ε; uε±).(H3) The Fourier symbol Γ±(ξ; ε) of the limiting constant–coefficient operators Γ±(ε) as x→ ±∞

of the operator Γ(ε) defined in (1.5) satisfies Reσ(Γ±(ξ; ε)|Rangem(ξ)) ≤ −θ |ξ|2

1+|ξ|2 , when

restricted to its invariant subspace Rangem(ξ), whence3 L±(ξ; ε) satisfies the (unrestricted)bound

Reσ(L±(ξ; ε)) ≤ −θ |ξ|21 + |ξ|2 , ξ ∈ R2, ε ∈ (−δ, δ), θ > 0.

(H4) Considered as connecting orbits of (1.6), uε lie in an `-dimensional manifold, ` ≥ 1, ofsolutions of 1.1), obtained as a transversal intersection of the unstable manifold at uε− andthe stable manifold at uε+. (In the most typical case of a Lax-type shock [L], ` = 1 and themanifold of solutions consists simply of the set of x1-translates of a single wave.)

3Using (1.12) for u ∈ kerm(ξ) and for all else the fact that (L±(ξ; ε))− λ)u = 0 implies (Γ±(ξ; ε)− λ)m(ξ)u = 0.

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(H5) The full (unconstrained) 1D system ut = ∂1(B11(ε; u)∂1u) − ∂1F1(ε; u) has the structure

(1.11) of (A1) with possibly larger r ≥ r, along with the corresponding analog of (A2).(H6) The eigenvalue equation (λ − Lk(ε)u = 0 of the operator-valued symbol Lk(ε) obtained

from L(ε) by Fourier transform with respect to the x2 coordinate (see Section 1.3 forfurther details) is expressible as an equivalent first-order ODE ∂1W = R(x1, k, λ; ε)W withrespect to x1.

Hypotheses (H0)–(H5) are the standard ones imposed in the stability theory for planar viscousshock waves [Z2], and are used here to allow us to bring to bear tools from the stability literature,following the philosophy of [TZ2, TZ3]; hypothesis (H6), automatic in the case without constraints,is also needed to apply those tools. The most substantial of these conditions is (H3), which togetherwith symmetry assumptions (A3) (iii) and (v), guarantees stability of constant solutions u ≡ uε± atdissipative rates: more precisely, that the spectra λ of the linearized operators about these constantsolutions obey bounds

(1.13) Reλ ≤ −c(|Imλ|2 + k2)/(d+ (|Imλ|2 + k2)), c, d > 0,

where k denotes the Fourier wave number in x2 direction. (H6) is a subtle but important hypothesisallowing us to construct the resolvent (λ−Lk(ε))−1 using standard ordinary-differential techniques[He]. In the case m ≡ 0 without a constraint, this follows automatically from (H1)–(H5).

1.2. O(2) Hopf bifurcation. Before stating our results, we recall the standard O(2) Hopf bifur-cation scenario in finite dimensions, following Crawford and Knobloch [CK]. After Center mani-fold/Normal form reduction, this takes the form, to cubic order, of

(1.14)z1 = (εκ(ε) + χ(ε)i)z1 + (Λ|z1|2 + Γ|z2|2)z1,

z2 = (εκ(ε) + χ(ε)i)z2 + (Λ|z2|2 + Γ|z1|2)z2,

where ε ∈ R is a bifurcation parameter, κ, χ ∈ R are nonzero bifurcation coefficients, and zj , Λ, Γ ∈C. Model (1.14) has O(2) symmetry group consisting of rotation R(θ) : (z1, z2) → (z1e

iθ, z2e−iθ)

and reflection S : (z1, z2) → (z2, z1), with SR(θ) = R(−θ)S, and also an additional S1 symmetryT (β) : (z1, z2) → (z1e

iβ, z2eiβ) associated with normal form. The linearization about the trivial

equilibrium solution (z1, z2) = (0, 0) features a pair of double eigenvalues

λ±(ε) = κ(ε)ε± iχ(ε)

crossing the imaginary axis at ε = 0: an equivariant Hopf bifurcation with double multiplicityforced by reflection symmetry. Noting that radial equations decouple from angular equations as

(1.15)r1 = εκ(ε)r1 + (ReΛ|r1|2 + ReΓ|r2|2)r1,

r2 = εκ(ε)r2 + (ReΛ|r2|2 + ReΓ|r1|2)r2,

rj := |zj |, we find that periodic solutions are exactly equilibria for the planar radial system (1.15).Under the nondegeneracy conditions

(1.16) ReΛ|ε=0 6= 0, Re(Λ + Γ)|ε=0 6= 0, Re(Λ− Γ)|ε=0 6= 0,

it is readily seen that the periodic solutions consist, besides the trivial solution (r1, r2) = (0, 0)exactly of “traveling” (or “rotating”) wave solutions (r1, r2) ≡ (r∗, 0) or (0, r∗) and “standing” (or“symmetric”) wave solutions (r1, r2) ≡ (r∗, r∗) consisting of a nonlinear superposition of counter-rotating traveling waves, r∗ 6= 0, with associated radial bifurcations of pitchfork type |r| ∼ √ε.

Restricting attention to periodic solutions with period T (ε) near the linear period T∗(ε) :=

2π/χ(ε), and noting that spurious radial equilibria introduced by Re(Λ− Γ) = 0 will have different

periods in z1, z2 unless Im(Λ− Γ) = 0 as well, we find that (1.16) may be weakened to

(1.17) ReΛ|ε=0 6= 0, Re(Λ + Γ)|ε=0 6= 0, (Λ− Γ)|ε=0 6= 0.6

1.2.1. Alternative treatment via the displacement map. Alternatively, we show in §4 that O(2) Hopfbifurcation after Lyapunov–Schmidt reduction of the time-T displacement map Fj := zj(T )− zj(0)for zj(0) := aj takes the form, to cubic order, of a two-parameter stationary bifurcation:

(1.18)0 = F1(a1, a2, ε, µ) = (εκ(ε, µ) + χ(ε)µi)a1 + (Λ|a1|2 + Γ|a2|2)a1,

0 = F2(a1, a2, ε, µ) = (εκ(ε, µ) + χ(ε)µi)a2 + (Λ|a2|2 + Γ|a1|2)a2

in four dimensions, where aj ,Λ,Γ ∈ C, κ, χ ∈ R are nonzero bifurcation coefficients, and ε, µ ∈ Rare bifurcation parameters, with µ measuring the difference between T and the linear period T∗(ε).It is readily checked that, again, zeros of (1.18), corresponding to periodic solutions for the originalproblem, consist, besides the trivial solution (a1, a2) = (0, 0), exactly of traveling waves (a1, a2) =(a∗, 0) or (0, a∗) and standing waves (a1, a2) = (a∗, e

iθa∗), a∗ 6= 0, θ ∈ R, each of pitchfork type|a| ∼ √ε, under the nondegeneracy conditions

(1.19) ReΛ|ε=0 6= 0, Re(Λ + Γ)|ε=0 6= 0, (Λ− Γ)|ε=0 6= 0.

This gives a different, more direct (though higher-dimensional), route to O(2) Hopf bifurcationavoiding Center Manifold or Normal form reductions, the simple form of the truncated system(1.18) being forced rather by symmetry/time-averaging. The extension to the full system thenproceeds by rescaling/Implicit Function Theorem arguments, as described in §4 and Appendix B.

1.3. Statement of the main result. We are now ready to describe our main results. Notefirst that, by independence of coefficients of L(ε) on the x2-coordinate, the spectra of L(ε) may bedecomposed into spectra associated with invariant subspaces of functions eikx2f(x1) given by Fourierdecomposition, on which L(ε) acts as an ordinary differential operator Lk(ε) in x1. The operatorL0(ε) is exactly the linearized operator for the associated one-dimensional problem, while theoperators Lk(ε), k ∈ Z\{0}, govern the evolution of “transverse modes” with Fourier wave numberk. See Section 3 for further discussion/computations. We perform a corresponding decompositionon the operator Γ defined in (1.5), obtaining a family of ordinary differential operators Γk(ε).

By the standard connection [He] between essential spectrum of asymptotically constant-coefficientordinary differential operators and the spectra of their constant-coefficient limits as x1 → ±∞, theessential spectra of Lk(ε) are governed by the sharp bounds (1.13) implied by (A3), (H3), lyingasymptotic to Reλ = −c/d for |λ| → ∞ and bounded to the right by Reλ ≤ −ck2/(d + k2)), asdepicted in Figure (Here, we are using also assumption (H6) that the eigenvalue equation, hencealso the resolvent equation, of Lk(ε) can be expressed as a first-order ODE, in order to apply thegeneral result of [He], for which such reduction is an implicit first step- the sole point at whichwe use (H6).) In particular, Lk(ε) are not sectorial operators as in the semilinear case treated in[TZ2, SS], but merely generators of C0 semigroups, making the technical aspects of the analysis-particularly those concerned with regularity- significantly more difficult in this case.

We are interested in the case that there exists a conjugate pair of eigenvalues

(1.20) λ±(ε) = γ(ε)± iω(ε), γ′(0), ω(0) 6= 0

crossing the imaginary axis as ε crosses the bifurcation value ε = 0, associated with transverseFourier modes k = ±k∗ 6= 0, and that these are each of the minimal (in the presence of reflectivesymmetry) multiplicity two. By the assumed reflective symmetry in x2, λ+(ε) and λ−(ε) are thusassociated to subspaces with eigenbases

(1.21) e±ik∗x2wε(x1) and e∓ik∗x2wε(x1),

respectively, where “¯” denotes complex conjugate. Together, they comprise a spectral transverseO(2) Hopf bifurcation.

Noting that the one-dimensional operator L0(ε) has essential spectra passing through the originλ = 0, we see as mentioned earlier that the linearized operator L(ε) has no spectral gap, preventing

7

the application of standard center manifold techniques. Moreover, there is the additional technicaldifficulty that L0(ε) has at least ` embedded eigenvalues at λ = 0 due to the presence of nearbyshock profiles, as recorded in Hypothesis(H4), which, as zero-eigenvalues, are also resonant withbifurcation scenario (1.20). In the stability literature [Z1, Z2], such embedded eigenvalues areusefully expressed as zeros of an Evans function, or Wronskian of the linearized eigenvalue ODEsassociated with the wave, a notion that extends point spectrum to allow in some cases eigenfunctionsthat are merely bounded and not decaying, but play a role in time-asymptotic behavior of thelinearized evolution equations. As in [TZ3], we follow that convention here as well.4

Collecting the above discussion, we identify the followingHypothesis (Dε). We assume that the family of operators L(ε), ε ∈ (−δ, δ), satisfies:

(i) For each ε there exists an open set Ξ(ε) such that5

{λ ∈ C : Reλ ≥ 0} \ {0} ⊂ Ξ(ε) ⊆ ρess(L(ε));

(ii) λ = 0 is a zero of algebraic multiplicity ` (defined in (H4)) of the Evans function D(·, 0, ε);(iii) There exists a pair of eigenvalues λ±(ε) = γ(ε)± iω(ε) of L(ε) of multiplicity 2, for which

γ(0) = 0 and γ′(0) > 0, associated with operators L±k∗(ε), k∗ 6= 0;(iv) The operators Lk(ε), k ∈ Z, have no imaginary eigenvalues other than the eigenvalues 0

and λ±(ε) associated respectively with L0(ε) and L±k∗(ε);(v) The operators Γk(ε)| kerm, k ∈ Z, have no eigenvalues Reλ ≥ 0, except possibly for (λ, k) =

(0, 0).

Condition (Dε)(i) follows in fact from (H3) and (A3) as discussed above, hence technicallyspeaking is redundant; however, we include it for clarity of the spectral description. Condition(Dε)(ii) expresses in a generalized sense that the multiplicity-` zero-eigenvalues guaranteed by (H4),which are also embedded in the essential spectrum of L0(ε), are the only spectra of L0(ε) containedin the nonstable complex half-plane C+ = {λ ∈ C : Reλ ≥ 0}. As noted in [Z1, Z2], this is equivalentto transversality of the background shock profile as a solution of the one-dimensional traveling-waveODE together with inviscid stability in the sense of Majda [Ma1, Ma2]. Note that this condition isexpressed with respect to the unconstrained system (1.2), ignoring the condition Mu = 0, so mustbe checked in that context, in particular taking into account the form of G. Condition (Dε)(iii)expresses O(2) Hopf bifurcation while Condition (Dε)(iv) expresses the standard accompanyingnonresonance assumption apart from the zero eigenvalues of L0(ε) described in (Dε)(ii). See Figurefor a pictorial illustration of this scenarios for the different values k = 0, k = ±k∗, and k 6= 0,±k∗.Finally, condition (v), plus the observation that (Lk(ε)−λ)u = 0 implies (Γk(ε)−λ)mu = 0, ensuresthat mu = 0 for any nonstable eigenvalue λ of Lk(ε): in particular, that bifurcating eigenvaluesλ±(ε) are associated with “physical” eigenmodes satisfying constraint (1.3).

Condition (Dε) describes, not the standard O(2) Hopf bifurcation of Section 1.2, but an “`-fold” version involving ` additional resonant eigenvalues at λ = 0. Thus, we expect not isolatedsolutions bifurcating from a unique trivial solution, but `-dimensional families bifurcating from the`-dimensional manifold of trivial (i.e., planar) solutions given in (H4). Likewise, we expect not asingle reduced equation (1.18), but a family of such systems indexed by a parameter b ∈ R`.

4Existence/analyticity at λ = 0 of the 1D Evans function follows from (H0)–(H5) as described, e.g., in [Z1, Z2].5ρess denoting the essential resolvent set, consisting of resolvent points and isolated finite-multiplicity eigenvalues.

8

Define now the exponentially-weighted space X ′1 := H4(R×T,Cn; eη(1+|x1|2)1/2dx1dx2), η << 1,

with its natural Hilbert space norm and scalar product

‖f‖2X′1 =∑

α∈N2,|α|≤4

∫R×T

eη(1+|x1|2)1/2 |Dαf)(x1, x2)|2dx1dx2 and

〈f, g〉X′1 =∑

α∈N2,|α|≤4

∫R×T

eη(1+|x1|2)1/2(Dαf)(x1, x2) · (Dαg)(x1, x2)dx1dx2.

Then, the main result of this paper is:

Theorem 1.2. Let uε, ε ∈ (−δ, δ), be a one-parameter family of standing viscous planar shocksolutions of (1.2)–(1.3) satisfying Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). Then:

(i) Existence of a time-periodic solutions uε close in X ′1 norm to uε with fundamental period Tclose to the linearized value T∗(ε) = 2π/ω(ε) is equivalent to satisfaction of one of an `-parameterfamily (4.11) of equations of form (1.18) plus higher-order perturbations, indexed by b ∈ R` suf-ficiently small, relating ε, a parameter µ measuring the difference between T and T∗(ε), and theprojection Π(ε)(uε − uε)|t=0, appropriately coordinatized as (a1, a2) ∈ C2, where Π(ε) is the totaleigenprojection of L(ε) onto the eigenspace (1.21) associated with eigenvalues λ±(ε).6

(ii) Under the genericity assumption (4.9) (Hypothesis 1, §4) analogous to (1.19), equations(1.2), (1.3) exhibit an `-fold O(2)-Hopf bifurcation from uε. Namely, the set of X ′1-close (nontrivial)time-periodic solutions with periods nearby T∗(ε) consists precisely of 3 smooth families indexed byb sufficiently small, of bifurcating solutions uε,b satisfying

√ε/C ≤ ‖uε,b − uε‖X′1 ≤ C

√ε.

The values (a1, a2) for the families described in (ii) are ε-close to the “traveling-” and “standing-wave” solutions (a∗(b), 0), (0, a∗(b)) and (a\(b), a\(b)) of the associated cubic truncated system,as described in §1.2. These are, variously, of supercritical (ε > 0) or subcritical (ε < 0) type,depending on model parameters Λ(b), Γ(b) at b = 0. In the Lax case ` = 1, the “traveling-wave”type solutions are actual traveling waves uε,b(x, t) = hε,b(x1, x2 − dt) with respect to the transversex2 direction.

(a)

-

6Imλ

Reλz

z

z

z

z

z

zu

@@

@@I

∂σess(L0(ε))

(b)

-

6Imλ

Reλ

z

z

z

z

z

z

u

@@

@@I

∂σess(Lk(ε))

(c)

-

6Imλ

Reλ

→

→

λ+(ε)′s

λ−(ε)′s

z

z

z

z

z

z

z

zu

−η

@@

@@I

∂σess(Lk(ε))

Figure 1. Schematic plot of spectrum of Lk(ε) for (a) k = 0, (b) k 6= 0,±k∗ and(c) k = ±k∗. Dots denote eigenvalues of Lk(ε), curves boundaries of σessLk(ε).

6Here, the choice of b ∈ R` is effectively a phase condition, fixing a nearby representative from the `-fold familyof profiles described in (H4).

9

Remark 1.3. It is interesting to note that the case ω(0) = 0 of a transverse stationary bifurcationcan be converted to the case ω(0) 6= 0 of a transverse Hopf bifurcation, and vice versa, by theintroduction of a moving coordinate frame x2 → x2−dt, inducing a shift λ→ λ− ikd in eigenvaluesassociated with Fourier number k. Setting d = ω(0)/k∗, this converts the scenario (1.20) to that ofan ordinary (non-reflective symmetric) bifurcation involving a pair of roots crossing at γ(0)±2iω(0)plus a pair of roots crossing at λ = 0, i.e., a translationally-invariant stationary bifurcation in themoving coordinate frame. This type of bifurcation has been treated recently in [M] for the strictlyparabolic semilinear case. Likewise, a stationary O(2) bifurcation involving a double eigenvalueλ(ε) = γ(ε) with wave numbers k = ±k∗ can be converted by the change of coordinates x2 → x2−dtto an ordinary (non-O(2)-symmetric) Hopf bifurcation λ±(ε) = γ(ε)± idk∗, and treated as in [TZ2]to yield a time-periodic solution in the moving coordinate frame. In the Lax case ` = 1, theuniqueness/shift invariance argument of §4 yields the further information that this is a travelingwave in x2, as shown by direct (stationary) argument in [M]. Thus, there is some overlap in theresults obtainable by the methods here and those of [M]; the difference in the O(2) Hopf case isthat we obtain full information on all time-periodic solutions and not only traveling waves.

1.4. Applications to example systems. We now give some examples of systems coming fromcompressible continuum mechanics to which our results do and do not apply, in the sense thatstructural conditions (A), (B), (H) are or are not satisfied. We leave as a separate issue thequestion whether or when bifurcation conditions (Dε) are satisfied, to be examined numerically oranalytically in specific circumstances. Accordingly, we mainly suppress the bifurcation parameterε in the discussion below.

Example 1.4. The equations of isothermal viscoelasticity in Lagrangian coordinates are

(1.22) Ψtt −∇X ·(DW (∇XΨ) + Z(∇XΨ,∇XΨt)

)= 0, X ∈ Rd, d = 2, 3

where Ψ ∈ Rd denotes deformation of an initial reference configuration of constant temperature anddensity, and ∇X · denotes divergence, taken row-wise across a matrix field. The elastic potential Wis a function of the deformation gradient F := ∇XΨ and the viscous stress tensor Z a function ofF and Ft obeying the Claussius-Duhem inequality Z(F,Q) : Q ≥ 0, where “:” denotes Frobeniusmatrix inner product; see [A, Da, BLeZ]. Here and below, gradient and curl, like divergence aretaken row-wise. We restrict attention to solutions depending on (X1, X2) ∈ R× [−π, π)periodic.

Expressing these equations as a second-order system in F and u := Ψt, we obtain the system

(1.23) Ft −∇Xu = 0, ut −∇X ·DW (F ) = −∇X · Z(F,∇Xu)),

consisting of a first-order linear hyperbolic equation in F coupled with a second-order parabolicequation in u, together with the constraints

(1.24) m(F, u) := ∇X × F = 0

ensuring that F is a gradient, The d2 constraints (1.24), induced by the reduction from second- tofirst-order system, are not full rank, but rather have Fourier symbol m(ξ) of rank d2 − d, with d-

dimensional kernel consisting of rank-one matrices {vT ξ}, v ∈ Rd. Adding a term Gm :=

(m∗m

0

)to

the righthand side, or, equivalently, substituting for (1.23)(i) the partially parabolic regularisationFt −∇Xu = ∇X ×∇X × F , we arrive at a system (1.2) of form:

(1.25) Ft −∇Xu = ∇X ×∇X × F, ut −∇X ·DW (F ) = −∇X · Z(F,∇Xu)).

With this choice, we have Γ = curl2, hence, recalling that ∆ = ∇Xdiv + curl2,

(1.26) Γ| kerm = ∆, (Γk)| kerm = ∂21 − k2,

from which we readily verify the conditions (H3) and (Dε)(v) on Γ.10

In the commonly occurring case that the Claussius-Duhem inequality holds strictly [A], or

(1.27) Q : Z(F,Q) & θ|Q|2 > 0,

we find, examining the reduced system (1.23), that the second equation satisfies the uniform ellip-ticity condition (A2)(iii) and the first equation is linear, hyperbolic, hence for shock speed c 6= 0 weobtain (A1)–(A2) for A0 = Id, with, evidently, F playing the role of the hyperbolic variable and uthat of the parabolic variable. A more delicate point is to verify (A3), in particular symmetrizabil-ity condition (A3)(iii) for an appropriate augmented system (1.7), which amounts to symmetrizablehyperbolicity of the first-order, or inviscid, part of the equations:

(1.28) Ft −∇Xu = 0, ut −∇X ·DW (F ) = 0,

in the vicinity of the endstates (F, u)± of the shock profile, and the Kawashima condition (A3)(v).For, evidently the 2nd-order part of the original system (1.22) is hyperbolic if and only if W

is strictly rank-one convex. Likewise, hyperbolicity of (1.28) is equivalent to rank-one convexity;however, symmetrizability by a block diagonal symmetrizer is equivalent to strict convexity of W ,which since incompatible with the principle of rotational frame-indifference cannot occur [A, BLeZ].A related issue is existence of a convex entropy, a condition implying and in many physical situationsequivalent to symmetrizable hyperbolicity [L]. It is readily verified [BLeZ, p.5] that η(F, u) :=W (u) + |u|2/2 is an entropy for both (1.23) and (1.28) but convex if and only if W is convex.

Thus, the equations are not symmetrizable in standard sense for physically relevant elastic po-tentials W satisfying the principles of frame indifference. However, as shown by Dafermos usingthe method of contingent entropies [Da, Chapter 2, pp. 30-31 and Chapter 5, pp. 111-116], theequations are block-diagonally symmetrizable (mod m), provided that W is strictly polyconvex,i.e., expressible as a strictly convex function of (F, u, F ],detF ), where F ] denotes the adjugate, ortransposed matrix of minors of F . That is, imposing the physically reasonable condition of poly-convexity of W , a common assumption in elasticity intermediate between between convexity andrank-one convexity [B], at endstates (F, u)±, we obtain satisfaction of (A3)(iii) for symmetrizersA0±. From these values we may then construct a global interpolant A0 by a construction similar

to that of [Z2, Eg. 1.25] for the case of gas dynamics. Likewise, the Kawashima condition (A3)(v)

may be verified by exhibition of explicit (mod m) skew-symmetrizers Kj± for W rank-one convex

at (F, u)±, completing the verification of (A3). For further details/computations, see Appendix A.Finally, (H5)-(H6) follow by inspection for shock speed c 6= 0, while (H4) and (D)(ii) are equiv-

alent to the corresponding conditions for the reduced 1D problem

(1.29) at − uX = 0, ut − (∂ω/∂a)TX = (β(a)uX)X , a, u ∈ Rd, d = 2, 3

treated in [BLeZ], where ω(a) := W (F ), F1j = aj , and Fj2, Fj3 ≡ F ∗j2, F∗j3 = constant; see again

Appendix A. Summarizing, we have for W polyconvex at F± := ∇, Z satisfying the strict Clausius-Duhem inequality, and a family of planar viscoelastic shocks uε with nonzero speed c(ε), satisfying(H4) and (Dε)(ii) for (1.29), that (Dε)(iii)-(iv) imply a nonlinear O(2) transverse Hopf bifurcationas described in Theorem 1.2.

Remark 1.5. For rank-one convex W (F±), equations (1.28) admit (mod m) block-diagonal sym-metrizers A0

11,±(ξ) = d2W (F±), A022,± = Id at endstates (F, u)± that are positive definite (mod m).

Constructing a global A0 as a sufficiently slowly varying convex combination of A0± as we are free

to do (noting that any such convex combination satisfies condition (A2)), we find by pseudodiffer-ential techniques (specifically, Garding’s estimate) that 〈v,A0v〉 is definite as a quadratic form onL2 when restricted to kerm, by convexity of the symbol on kerm(ξ). Noting that this property isthe only one needed for the high-frequency energy estimates for which assumptions (A3) are used,we thus find that the requirement of polyconvexity may be reduced to rank-one convexity at theendstates, the same (essentially minimal) condition required in the 1D analysis of [BLeZ].

11

Example 1.6. Similar considerations as in Example 1.4 yield that the equations of general (non-isothermal) thermoviscoelasticity [Da] satisfy our assumptions, so long as the thermoelastic poten-tial e = e(F, S) is a strictly convex function of (F, F ], detF ) and entropy S, at endstates (F, u, e)±of the shock. However, for ordinary gas dynamics, e is a convex function of τ := detF and S alone,where τ denotes specific volume. Thus, considered as a function of (F, F ], detF, S), it is nonstrictlyconvex, and indeed it is readily verified that (H4) along with (A3)(iii) fails; see §5. The constraintsin both cases are m(F, u, e) := ∇X × F = 0 as in Example 1.4, expressing that F is a gradient.

Example 1.7. Eulerian gas dynamics or MHD with artificial viscosity, i.e., strictly parabolicsecond-order terms, satisfy conditions (A1)–(A3), (H0)–(H6) automatically with z(u) = u. Thisadmits a much simpler treatment of regularity, as, e.g., in [TZ2, §6]. For these examples, m ≡ 0.

1.5. Discussion and open problems. The Lyapunov-reduction/time-T displacement map argu-ment used here serves as a substitute for the Center Manifold/Normal form reduction or Lyapunov-reduction/spatial dynamics methods that have been used in other contexts. It is interesting to con-trast the important use of additional S1 symmetry in these arguments, corresponding roughly toinvariance with respect to time-evolution. This is imposed by force in normal form reduction, andappears naturally in the spatial dynamics approach framed in the space of time-periodic solutions.In our argument, we use the fact that time-evolution is an approximate S1 symmetry in a similarway, to restrict the possible forms arising at the level of cubic approximation; see Remark 4.6.

An interesting open problem would be to carry out a similar analysis for physical, partiallyparabolic, systems either in one- or multi-dimensions using spatial dynamics techniques as donein [SS] in the one-dimensional semilinear strictly parabolic case. This appears to require bothadditional care in the choice of spaces/functional analytical framework, and additional analysis toestablish Frechet differentiability in the absence of parabolic smoothing of the nonlinear part ofthe resulting ill-posed evolution system.7 However, a possible advantage might be to remove thedependence on Lagrangian coordinates that prevents for the moment the treatment of gas dynamicsand MHD. Some other ideas using the present framework are mentioned in Section 5.

We note that the same issues obstructing multi-dimensional bifurcation analysis obstruct also theproof of a multi-dimensional conditional stability result similar to that obtained in [Z5] in the one-dimensional case- specifically, incompatibility between Lagrangian form needed to obtain regularityneeded for the center-stable manifold reduction step and high-frequency resolvent estimates neededfor time-asymptotic decay estimates- making this issue one of independent interest.

Though we do not carry it out here, spectral stability information on bifurcating solutions shouldbe in principle available via the same reduced displacement map; indeed, it should be “reverse-engineerable” from the standard normal-form analysis via the relation (through time-integration)

of Λ, Γ to Λ, Γ. A very interesting open problem is to prove a full nonlinear stability result under theassumption of spectral stability for a class of time-periodic multi-dimensional solutions includingthe bifurcating time-periodic waves established here, similarly as was done in one dimension in[BeSZ] in the strictly parabolic case. Another interesting direction is the treatment of spinningshocks and detonations in a cylindrical duct [KS], which should be treatable by similar arguments.

Finally, we note that, apart from the verification of structural conditions (A1)–(A3) and technicalHypotheses (H0)–(H6), there is the question whether and when for any particular model the spectralbifurcation scenario (Dε) in fact occurs. This is in general a difficult question, as is any descriptionof the spectral properties of strongly variable-coefficient differential operators, that is typicallystudied either numerically (as in the related numerical Evans function study of longitudinal Hopfbifurcations in [BFZ]) or in special asymptotic limits.

7Though we think it is fairly straightforward there, we note that the latter step has so far not been carried outeven for the quasilinear strictly parabolic case, and does require additional computation beyond those of [SS]

12

As pointed out in [M], steady transverse O(2) spectral bifurcation arises in MHD with artificialor physical viscosity, as can be seen by combining asymptotic inviscid stability computations of [FT]in the large-magnetic field limit with the observations of [ZS, Z1, Z6] relating viscous and inviscidstability in the low-frequency limit. However, so far as we know,the phenomenon of cellular insta-bility/transverse O(2) bifurcation of shock waves has so far been demonstrated either numericallyor mathematically only for this single example from MHD [FT, M] and only in the steady and notthe Hopf bifurcation case. The systematic cataloguing of this phenomenon for other waves andmodels by either numerical or analytic means we regard as an extremely interesting open problem.

Acknowledgement. Thanks to Arnd Scheel for helpful conversation improving the exposition.Thanks also to the anonymous referees for their careful reading and several helpful suggestions.

2. Variational energy estimates

In this section we introduce a useful energy functional associated with the perturbation equationsfor (1.2), and prove the key energy estimate it satisfies; see Proposition 2.2 below. We start bylinearizing equation(1.2) about uε. The linearized equation reads as follows:

(2.1) ∂tv = L(ε)v,

where the linear operators L(ε) are defined in (1.4). Next, we note that if u is a solution of (1.2)and if we denote by v(x, t) := u(x, t)− uε(x1), then v satisfies the perturbed equation:

(2.2)

∂tv =L(ε)v +∑jk

∂j

(Bjk(v + uε)∂k(v + uε)−Bjk(uε)∂ku

ε −Bjk(uε)∂kv

− ∂uBjk(uε)v∂kuε)−∑j

∂j

(F j(v + uε)− F j(uε)− ∂uF j(uε)v

).

Using assumptions (A1), (A2) and (H0), we infer that

(2.3)(F j(v + uε)− F j(uε)− ∂uF j(uε)v

)=

(0

qj,ε(v)

);

(2.4) Bjk(v + uε)∂k(v + uε)−Bjk(uε)∂kuε −Bjk(uε)∂kv − ∂uBjk(uε)v∂ku

ε =

(0

pj,ε(v)

).

The functions pjk,ε, qj,ε, j, k = 1, 2, are defined as follows: for v ∈ Hs(R× [−π, π],Cn), s ≥ 2,

(2.5) qj,ε(v) =

∫ 1

0(1− t)∂2

uFj2 (uε + tv)vv dt;

(2.6) pjk,ε(v) =

∫ 1

0∂ub

jk(uε + tv)v∂kv2 dt+

∫ 1

0(1− t)∂2

ubjk(uε + tv)vv∂ku

ε2 dt.

Using the functions introduced in (2.5) and (2.6), we obtain from (2.1), (2.3) and (2.4) that theperturbed equation (2.2) can be written as

(2.7) vt − L(ε)v =∑j,k

∂j

(0

pjk,ε(v)

)−∑j

∂j

(0

qj,ε(v)

).

We point out that in equation (2.7) the first block-component does not have nonlinear terms, whichis a direct consequence of assumption (A1), in particular the lack of parabolic structure of the firstblock-component. This is an extremely useful property, actually the main reason why the smoothnessissue does not cause problem later in our analysis.

13

We assume throughout that the initial perturbation lies in kerm, hence, by invariance of kermunder time-evolution, the perturbation lies always in kerm and we may apply our (mod m) struc-tural assumptions without care. By standard energy estimates, given any T > 0 and s ≥ 2, thereexists δ∗(T ) > 0, such that, if ‖v0‖Hs ≤ δ∗(T ), the perturbation system has a unique solutionv ∈ C0([0, T ], Hs(R× [−π, π],Cn)), with v(·, 0) = v0, and the bound

‖v(·, t)‖Hs ≤ C‖v0‖Hs ,

holds for any t ∈ [0, T ] and some C > 0 that depends on T , but neither on ε nor on t. Likewise,we have a formally quadratic linearized truncation error |qj,ε(v)|, |pjk,ε(v)| = O(|v|(|v| + |∇xv|))for v ∈ Hs(R × [−π, π],Cn) with ‖v‖Hs ≤ C. Our goal in this section is to establish a quadraticbound on the linearization error:

(2.8) ‖v(·, T )− eL(ε)Tv0‖Hs ≤ C‖v0‖2Hs .

Here {eL(ε)t}t≥0 denotes the C0-semigroup generated by L(ε), see, e.g., [Lun, Z1, Z2]. This inequal-ity is far from evident in the absence of parabolic smoothing as shown in [TZ3]. The correspondingbound does not hold for quasilinear hyperbolic equations, nor as discussed in [TZ3, Appendix A],for systems of general hyperbolic–parabolic type, due to loss of derivatives. However, it followseasily for systems satisfying assumptions (A1)–(A2).

Applying the differential operator Dα to the perturbation system and multiplying the result

system by A0,ε :=

(A0

11(uε) 00 A0

22(uε)

):=

(A0

11 00 A0

22

), we obtain that

(2.9)

A0,εDα∂tv =A0,εDα∑jk

∂j

(Bjk(uε)∂kv

)+A0,εDα

∑jk

∂j

(∂uB

jk(uε)v∂kuε)

−A0,εDα∑j

∂j(Ajv)−A0,εDα

∑j

∂j

(0

qj,ε(v)

)

+A0,εDα∑j,k

∂j

(0

pjk,ε(v)

).

To prove our energy estimate we need the following identities:

(2.10)

A0,εDα∂j

(Bjk(uε)∂kv

)= ∂j

(A0,ε

22 bjk(uε)Dα∂kv2

)− (∂jA

0,ε22 )bjk(uε)Dα∂kv2

+A0,ε22 ∂j

( ∑β≤α;|β|=1

(αβ

)Dβbjk(uε)Dα−β∂kv2

)+A0,ε

22 ∂j

( ∑β≤α;|β|>1

(αβ

)Dβbjk(uε)Dα−β∂kv2

);

(2.11)

A0,εDα∂j(Ajv) =A0,εAjDα∂jv +A0,εDα(∂jA

jv)

+A0,ε∑

β≤α;|β|≥1

(αβ

)DβAjDα−β∂jv.

Next, we briefly mention the weak Moser inequality in a channel, another tool needed in our analysis.

If κ ≥ 1, α1, . . . , αm are multi-indexes, s =m∑i=1|αi| and h1, . . . , hm ∈ Hmax{s,κ}(R × [−π, π],Cn),

then

(2.12) ‖(∂α1h1) · · · · · (∂αmhm)‖L2 ≤( m∑i=1

‖hi‖Hs

)(∏j 6=i‖hj‖L1

)≤ C

( m∑i=1

‖hi‖Hs

)(∏j 6=i‖hj‖Hκ

).

14

The proof of (2.12) is based on the Hausdorff-Young inequality and the strong Sobolev embeddingprinciple, see, e.g., [Z2, Lemma 1.5] for the whole-space case. As an application of the weak Moserinequality, we can prove the following lemma.

Lemma 2.1. Assume Hypotheses (A1)–(A2) and (H0). Then, for any multi-index α ∈ N2 with

|α| ≥ 2 and any v ∈ H |α|(R×[−π, π],Cn) with v2 ∈ H |α|+1(R×[−π, π],Cr), the functions Dαqj,ε(v)and Dαpjk,ε(v) belong to L2(R× [−π, π],Cn) and the following estimate of their norms holds:

max{‖Dαqj,ε(v)‖L2 , ‖Dαpjk,ε(v)‖L2} .(‖v‖H|α| + ‖v2‖H|α|+1

)(‖v‖H|α| + ‖v‖|α|

H|α|

).

The proof of the lemma follows directly from the week Moser inequality, (2.12), and the propertiesof the functions F j and bjk, j, k ∈ {1, 2}, stated in Section 1.1, see, e.g., [Ta, TZ3, Z2]. The mainresult of this section reads as follows:

Proposition 2.2. Assume Hypotheses (A1)–(A2) and (H0). Then for any 2 ≤ s ≤ ν − 1, T0 > 0,there exists some C = C(T0) > 0 such that for any v satisfying (2.7) with initial data v(·, 0) = v0

sufficiently small in Hs(R× [−π, π],Rn) ∩ kerm, the following inequalities hold true:

(2.13) ‖v(·, T )‖2Hs +

∫ T

0‖v2(·, t)‖2Hs+1 dt ≤ C‖v0‖2Hs for all T ∈ [0, T0],

(2.14) ‖v(·, T )− eL(ε)Tv0‖Hs ≤ C‖v0‖2Hs for all T ∈ [0, T0].

Proof. As noted above, from the assumption that v lies initially in kerm, we obtain that v liesalways in kerm, and so the (mod m) assumptions of symmetrizability and Kawashima skew-symmetrizability may be applied; we use this freely in the argument below. Since A0,ε is symmetricand positive definite, we obtain that the energy functional

(2.15) E(v) :=1

2

∑|α|≤s

〈Dαv, A0,εDαv〉L2

defines a norm equivalent to ‖ · ‖Hs , i.e., E(·)1/2 ∼ ‖ · ‖Hs . Using (2.10), it follows that

(2.16) ∂tE(v) =∑|α|≤s

〈Dαv, A0,εDα∂tv〉L2 .

Next, we estimate the right-hand side of (2.16), using (2.2), (2.10) and (2.11) . We break this longestimate into three separate parts. Using the hypotheses from Section 1.1 we have that for anyv ∈ Hs(R× [−π, π],Rn) with ‖v‖Hs ≤ δ0 ≤ δ∗(T0) the following holds:(2.17)∑jk

∑|α|≤s

⟨Dαv, A0,εDα∂j

(Bjk(uε)∂kv

)⟩L2

=∑jk

∑|α|≤s

⟨Dαv2, ∂j

(A0,ε

22 bjk(uε)Dα∂kv2

)⟩L2

+ δ0‖v2‖2Hs+1 + C(δ0)‖v2‖2Hs +O(1)‖v‖2Hs

= −∑jk

∑|α|≤s

⟨Dα∂jv2,

(A0,ε

22 bjk(uε)Dα∂kv2

)⟩L2

+ δ0‖v2‖2Hs+1 + C(δ0)‖v2‖2Hs +O(1)‖v‖2Hs

≤ −θ‖v2‖2Hs+1 + δ0‖v2‖2Hs+1 + C(δ0)‖v2‖2Hs +O(1)‖v‖2Hs

≤ (−θ + δ0)‖v2‖2Hs+1 + (C(δ0) +O(1))‖v‖2Hs .15

In addition, one readily checks that(2.18)∑jk

∑|α|≤s

⟨Dαv, A0,εDα∂j

(∂uB

jk(uε)v∂kuε)⟩

L2=∑jk

∑|α|≤s

〈Dαv2, A0,ε22 D

α∂j

(∂ub

jk(uε)v∂kuε2

)⟩L2

=∑jk

∑|α|≤s

⟨∂j

((A0,ε

22 )∗Dαv2

), Dα

(∂ub

jk(uε)v∂kuε2

)⟩L2

≤ O(1)‖v‖2Hs + δ0‖v2‖2Hs+1 + C(δ0)‖v‖2Hs .

Moreover, we infer that(2.19)∑j

∑|α|≤s

〈Dαv, A0,εDα∂j(Ajv)〉L2 =

∑j

∑|α|≤s

〈Dαv, A0,εAjDα∂jv〉L2 +O(1)‖v‖2H|α|

=∑j

∑|α|≤s

〈Dαv1, A0,ε11 A

j11D

α∂jv1〉L2 +∑j

∑|α|≤s

〈Dαv1, A0,ε11 A

j12D

α∂jv2〉L2 +O(1)‖v‖2H|α|

+∑j

∑|α|≤s

〈Dαv2, A0,ε11 A

j21D

α∂jv1〉L2 +∑j

∑|α|≤s

〈Dαv2, A0,ε11 A

j22D

α∂jv2〉L2

=∑j

∑|α|≤s

−1

2〈Dαv1, ∂j(A

0,ε11 A

j11)Dαv1〉L2 +

∑j

∑|α|≤s

〈Dαv1, A0,ε11 A

j12D

α∂jv2〉L2 +O(1)‖v‖2Hs

+∑j

∑|α|≤s

〈∂j(

(A0,ε11 A

j21)∗Dαv2

), Dαv1〉L2 +

∑j

∑|α|≤s

〈Dαv2, A0,ε11 A

j22D

α∂jv2〉L2

≤ O(1)‖v1‖2Hs + δ0‖v2‖2Hs+1 + C(δ0)‖v1‖2Hs + δ0‖v2‖2Hs+1 + C(δ0)‖v‖2Hs

+ δ0‖v2‖2Hs+1 + C(δ0)‖v2‖2Hs +O(1)‖v‖2Hs ≤ 3δ0‖v2‖2Hs+1 + C(δ0)‖v‖2Hs .

Next, we introduce the function Qε : Hs(R× [−π, π],Rn)→ L2(R× [−π, π],Rn) defined by

(2.20) Qε(v) := −A0,εDα∑j

∂j

(0

qj,ε(v)

)+A0,εDα

∑j,k

∂j

(0

pjk,ε(v)

).

From Lemma 2.1, we conclude that for any v ∈ Hs(R× [−π, π],Rn) the following estimate holds:

(2.21)

∑|α|≤s

〈Dαv, Qε(v)〉L2 =∑|α|≤s

〈Dαv2, Qε(v)〉L2

≤ O(1)‖v2‖Hs+1

(‖v‖Hs + ‖v2‖Hs+1

)(‖v‖Hs + ‖v‖sHs

).

Finally, from (2.17)–(2.21) we obtain that

(2.22)∂tE(v) ≤ O(1)‖v2‖Hs+1

(‖v‖Hs + ‖v2‖Hs+1

)(‖v‖Hs + ‖v‖sHs

)− θ‖v2‖2Hs+1 + 5δ0‖v2‖2Hs+1 +O(1)‖v‖2Hs .

We choose δ0 > 0 such that 5δ0 < θ/2. So long as ‖v‖Hs remains sufficiently small, we infer that

(2.23)∂tE(v) ≤ −(θ/2)‖v2‖2Hs+1 +O(‖v‖2Hs)

≤ −(θ/2)‖v2‖2Hs+1 + CE(v),

from which (2.13) follows by Gronwall’s inequality since

E(v(T )) + (θ/2)

∫ T

0‖v2(·, t)‖2Hs+1dt ≤ C2E(v0).

16

To prove (2.14), we first note that the error function E(x, t) := v(x, t) − eLεtv0(x) satisfies theequation

(2.24) ∂tE = L(ε)E −∑j

∂j

(0

qj,ε(v)

)+∑j,k

∂j

(0

pjk,ε(v)

),

with initial condition E(·, 0) = 0. We note that equation (2.24) has a structure similar to that of(2.2). Using the same argument as in (2.10) and (2.11) one can show that

(2.25)

A0,εDα∂tE = A0,εDα

[∑jk

∂j

(Bjk(uε)∂kE

)+∑jk

∂j

(∂uB

jk(uε)E∂kuε)−∑j

∂j(AjE)

−∑j

∂j

(0

qj,ε(v)))+

∑j,k

∂j

(0

pjk,ε(v)

)]

=∑jk

A0,εDα∂j

(Bjk(uε)∂kE

)+∑jk

A0,εDα∂j

(∂uB

jk(uε)E∂kuε)

−∑j

A0,εDα∂j(AjE)−

∑j

A0,εDα∂j

(0

qj,ε(v)

)+∑j,k

A0,εDα∂j

(0

pjk,ε(v)

)=∑jk

A0,εDα∂j

(Bjk(uε)∂kE

)+∑jk

A0,εDα∂j

(∂uB

jk(uε)E∂kuε)

−∑j

A0,εDα∂j(AjE)−Qε(v).

From the definition of the energy functional in (2.15) we note that E(E) ∼ ‖E‖2Hs . In addition, itstime evolution ∂tE(E) satisfies the identity(2.26)

∂tE(E) =∑|α|≤s

〈DαE,A0,εDα∂tE〉L2 =∑|α|≤s

⟨DαE,

∑jk

A0,εDα∂j

(Bjk(uε)∂kE

)⟩L2

+∑|α|≤s

⟨DαE,

∑jk

A0,εDα∂j

(∂uB

jk(uε)E∂kuε)⟩

L2−∑|α|≤s

〈DαE,∑j

A0,εDα∂j(AjE)〉L2

−∑|α|≤s

〈DαE,Qε(v)〉L2 .

To estimate the first three terms in the identity above we argue in the same way as in (2.17)–(2.19).The fourth term can be controlled by

(2.27)∑|α|≤s

〈DαE,−Qε(v)〉L2 ≤ O(1)‖E2‖Hs+1

(‖v‖Hs + ‖v2‖Hs+1

)(‖v‖Hs + ‖v‖sHs

).

Combining all of these estimates together and using the weighted Young’s inequality, we have, aslong as ‖v‖Hs remains sufficiently small, that

(2.28) ∂tE(E) ≤ −θ2‖E2‖2Hs+1 +O(1)‖E‖2Hs +O(1)‖v‖2Hs

(‖v‖2Hs + ‖v2‖2Hs+1

).

Also, since ‖v(·, t)‖Hs ≤ C‖v0‖Hs , we obtain that

(2.29) ∂tE(E) ≤ −θ2‖E2‖2Hs+1 +O(1)‖E‖2Hs +O(1)‖v0‖2Hs

(‖v‖2Hs + ‖v2‖2Hs+1

).

17

Using (2.13), Gronwall’s inequality and the fact that the error function satisfies the initial conditionE(·, 0) = 0, we conclude that

E(E(T )) ≤ O(1)‖v0‖2Hs

∫ T

0

(‖v(·, t)‖2Hs + ‖v2‖2Hs+1

)dt ≤ O(1)‖v0‖4Hs ,

which implies that ‖E(·, T )‖Hs ≤ O(1)‖v0‖2Hs , proving the lemma. �

3. O(2) bifurcation for the general case

In this section we prove various preliminary results needed to prove our main result of this paper,the existence of O(2)-Hopf bifurcation under a spectral criterion (Hypothesis Dε) described in detailin the introduction. More precisely, we are looking to prove the existence of periodic solution ofequation (2.2) (subject to the constraint mv=0), of period T > 0, by solving for T as a function ofthe initial data v(0) and the bifurcation parameter ε in the fixed point equation associated to thereturn map of (2.2)(mod m). Furthermore, we reduce this infinite dimensional nonlinear systemto a finite dimensional system by using the a special variant of the Lyapunov-Schmidt reductionmethod, introduced in [TZ2] and refined in [TZ3].

In what follows we are going to consider the operator L(ε) defined in (1.4) as a second orderdifferential operator from H2(R× [−π, π],Cn) to L2(R× [−π, π],Cn). By taking Fourier Transformin x2 ∈ [−π, π], we can identify L2(R× [−π, π],Cn) with `2(Z, L2(R,Cn)) and Hp(R× [−π, π],Cn)

with⊕

k∈Z Hpk(R,Cn), p = 1, 2, where the Hilbert space Hp

k(R,Cn) is the Sobolev space Hp(R,Cn),p = 1, 2, with the scalar products

(3.1) 〈f, g〉H1k(R,Cn) = (1 + k2)〈f, g〉L2 + 〈∂x1f, ∂x1g〉L2 ;

(3.2) 〈f, g〉H2k(R,Cn) = (1 + k2 + k4)〈f, g〉L2 + (1 + 2k2)〈∂x1f, ∂x1g〉L2 + 〈∂2

x1f, ∂2

x1g〉L2 .

The operator L(ε) can be identified with (Lk(ε))k∈Z, where Lk(ε) : H2k(R,Cn) → L2(R,Cn) are

defined by Lk(ε) = L(ε)(k). A simple computation shows that

(3.3) Lk(ε) = L0(ε) + ikJ(ε)− k2B22

(ε, x1),

where

(3.4) L0(ε) = ∂x1

[B

11(ε, x1)∂x1 −A

1(ε, x1)

]and J(ε) is the first order operator defined by

(3.5) J(ε) =[B

12(ε, x1) +B

21(ε, x1)

]∂x1 +B(ε, x1)−A2

(ε, x1).

Here the functions Aj, B

jk: (−δ, δ) × R → Rn×n are defined by composing the functions Aj and

Bjk, respectively, with (ε,uε(x1)).

Remark 3.1. Since the operator Lk(ε), k ∈ Z, are one-dimensional differential operators ((3.3),(3.4)), we note that its eigenvalues can be obtained, away from the essential spectrum, as zeros ofthe classical Evans function, denoted D(·, k, ε), see, e.g [Z1, Z2]. At the special eigenvalues λ = 0of L0(ε), which are embedded in essential spectra, the zeros of the Evans function carry additionalinformation determining asymptotic stability [Z1].

The next step in constructing the O(2) bifurcation is to construct the time-T evolution mapof the equation (2.2)(mod m). An crucial role in this construction is played by the rotationalinvariance and by the x2 → −x2 invariance of this equation. More precisely, we note that there

18

exists a non-degenerate rotation group of linear operators {R(θ)}θ∈R on L2(R× [−π, π],Cn), withR(θ)∗ = R(−θ), for all θ ∈ R8, and a bounded symmetry S on L2(R× [−π, π],Rn) satisfying

(3.6) F(ε;R(θ)u) = R(θ)F(ε; u), F(ε;Su) = SF(ε; u),

for all u ∈ H2(R × [−π, π],Rn), θ ∈ R and ε ∈ (−δ, δ). The group {R(θ}θ∈R and S satisfy thefollowing condition

(3.7) R(θ)S = SR(−θ) for all θ ∈ R.

Since L(ε) = ∂F∂u (ε; uε), from (3.6) we obtain that H2(R × [−π, π],Cn) = dom(L(ε)) is invariant

under S and R(θ), for all θ ∈ R and

(3.8) L(ε)R(θ) = R(θ)L(ε), L(ε)S = SL(ε) for all θ ∈ R, ε ∈ (−δ, δ).We introduce G the generator of the rotation group {R(θ)}θ∈R and we note that dom(L(ε)) ⊂dom(G) for all ε ∈ (−δ, δ). Moreover, from (3.7) and (3.8) and since {R(θ)}θ∈R is a rotation group,we infer that

(3.9) G∗ = −G, GL(ε) = L(ε)G, GS = −SG for all ε ∈ (−δ, δ).Very important to our reduction are the eigenspaces Σ±(ε) associated to the eigenvalues λ±(ε) ofL(ε). In the next lemma we summarize a few basic properties of these eigenspaces.

Lemma 3.2. Assume Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). For any ε ∈ (−δ, δ) thefollowing assertions hold true:

(i) The subspaces Σ±(ε) are invariant under G, S and R(θ) for any θ ∈ R;(ii) There exits α(ε) ∈ R \ {0} such that σ(G|Σ+(ε)) = {±iα(ε)};

(iii) Let wε ∈ Σ+(ε) be the eigenfunction (unique up to a scalar multiple) of G|Σ+(ε) correspond-ing to the eigenvalue iα(ε). Then, the eigenspaces Σ±(ε) can be represented as follows:

(3.10) Σ+(ε) = Sp{wε, Swε}, Σ−(ε) = Sp{wε, Swε}.Proof. Assertion (i) follows from the fact that the operator L(ε) commutes with R(θ), S and G by(3.8) and (3.9).

(ii) From (3.9) we have that G∗ = −G and since by (i) Σ+(ε) is invariant under G, we infer that(G|Σ+(ε))

∗ = −G|Σ+(ε). It follows that σ(G|Σ+(ε)) ⊂ iR. Since dim(Σ+(ε)) = 2, we conclude thatthere exists α(ε) ∈ R \ {0} such that σ(G|Σ+(ε)) = {±iα(ε)}. Taking into account that the group ofrotations {R(θ)} is non-degenerate, we infer that kerG = {0}, which implies that α(ε) 6= 0, proving(ii).

(iii) We note that since L(ε)wε = λ+(ε)wε, from (3.8) it follows that

L(ε)Swε = SL(ε)wε = λ+(ε)Swε,

and thus Swε ∈ Σ+(ε). Next, we will show that wε and Swε are linearly independent. From(3.9) we obtain that G(Swε) = −SGwε = −iα(ε)Swε. Thus, wε and Swε are eigenfunctionsof the same operator corresponding to different eigenvalues, which proves that wε and Swε arelinearly independent. Using again that dim(Σ+(ε)) = 2 we have that Σ+(ε) = Sp{wε, Swε}. Since

L(ε)u = L(ε)u for any u ∈ H2(R× [−π, π],Cn) we readily infer that Σ−(ε) = Sp{wε, Swε}. �

Since λ±(ε) is an eigenvalue of L(ε) and λ+(ε) = λ−(ε) for any ε ∈ (−δ, δ), we know that λ±(ε)

are also eigenvalues of L(ε)∗ for any ε ∈ (−δ, δ). Moreover, if we denote by Σ±(ε) the eigenspaces

associated to eigenvalues λ±(ε) of L(ε)∗, we have that dim(Σ±(ε)) = 2 for any ε ∈ (−δ, δ). The

properties satisfied by the eigenspaces Σ±(ε) are similar to the ones described in Lemma 3.2 asshown in the lemma below.

8We can naturally extend the operator S to the complexification of its domain such that S∗ = S

19

Lemma 3.3. Assume Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). For any ε ∈ (−δ, δ) thefollowing assertions hold true:

(i) The subspaces Σ±(ε) are invariant under G, S and R(θ) for any θ ∈ R;(ii) σ(G|Σ−(ε)) = {±iα(ε)}. The function α(ε) is the one introduced in Lemma 3.2(ii);

(iii) Let wε ∈ Σ−(ε) be the eigenfunction (unique up to a scalar multiple) of G|Σ+(ε) corre-

sponding to the eigenvalue iα(ε). Without loss of generality we can choose wε such that

〈wε, wε〉L2 = 1. Then, the eigenspaces Σ±(ε) can be represented as follows:

Σ−(ε) = Sp{wε, Swε}, Σ+(ε) = Sp{wε, Swε}.Proof. First, we note that by taking adjoint in (3.8) and (3.9) we obtain that the operator L(ε)∗

commutes with R(θ), S and G for any θ ∈ R and ε ∈ (−δ, δ). Since, in addition dim(Σ±(ε)) = 2for any ε ∈ (−δ, δ), we can obtain all properties above by using the same arguments we usedin Lemma 3.2. The only thing left to prove is that the operators G|Σ+(ε) and G|Σ+(ε) have the

same eigenvalues. Since G∗ = −G and the eigenspace Σ+(ε) is invariant under G we have that(G|Σ+(ε))

∗ = −G|Σ+(ε), and thus σ(G|Σ+(ε)) = {±iβ(ε)} for some β(ε) ∈ R. To finish the proof all

we need to do is to show that |α(ε)| = |β(ε)|. Indeed, one can readily check that

iα(ε)〈wε, wε〉L2 = 〈Gwε, wε〉L2 = 〈wε, G∗wε〉L2 = 〈wε,−Gwε〉L2 = −iβ(ε)〈wε, wε〉L2 ,

finishing the proof. �

Lemma 3.4. Assume Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). For any ε ∈ (−δ, δ) thefollowing assertion holds true:

(3.11) 〈Swε, wε〉L2 = 0.

(3.12) R(θ)wε = eiα(ε)θwε;R(θ)Swε = e−iα(ε)θSwε;R(θ)wε = eiα(ε)θwε;R(θ)Swε = e−iα(ε)θSwε.

Proof. To prove (3.11) we use (3.9), Lemma 3.2 and Lemma 3.3 as follows: first we compute

〈SGwε, wε〉L2 = 〈S(iα(ε)wε), wε〉L2 = iα(ε)〈Swε, wε〉L2 .

In addition,

〈SGwε, wε〉L2 = −〈GSwε, wε〉L2 = −〈Swε, G∗wε〉L2 = −〈Swε,−Gwε〉L2

= 〈Swε, Gwε〉L2 = 〈Swε, iα(ε)wε〉L2 = −iα(ε)〈Swε, wε〉L2 .

Using the group property of {R(θ)}θ∈R, one readily infers (3.12) from Lemma 3.2 and Lemma 3.3.�

In the next lemma we describe the relation between the eigenspaces Σ±(ε) and the kernel of m.

Lemma 3.5. Assume Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). Then, the followinginclusion holds true:

(3.13) Σ±(ε) ⊂ kerm for all ε ∈ (−δ, δ).Proof. Let v be a vector contained in Σ+(ε) or Σ−(ε) and λ = λ+(ε) or λ = λ−(ε)}, respectively.Since Σ±(ε) are eigenspaces of L(ε) we obtain that L(ε)v = λv. Using (1.5) we obtain that

(Γ(ε)− λ)mv = mL(ε)v − λmv = 0.

From Hypothesis (Dε)(ii) and (v) we infer that λ is not an eigenvalue of Γ(ε), which implies thatmv = 0, proving the lemma. �

20

Throughout this paper we denote by Π±(ε) the orthogonal projection onto Σ±(ε) parallel to

Σ±(ε)⊥ and by Π(ε) = I−Π+(ε)−Π−(ε). Also, we introduce Σ(ε) := Range(Π(ε)) the complementof Σ+(ε) ⊕ Σ−(ε). From (3.10) and (3.11) we know that the projectors Π±(ε) have the followingrepresentation:

(3.14) Π+(ε)u = 〈u, wε〉L2wε + 〈u, Swε〉L2Swε, Π−(ε)u = 〈u, wε〉L2wε + 〈u, Swε〉L2Swε

for any u ∈ L2(R × [−π, π],Cn). Using the fact that wε and Swε are eigenfunctions of L(ε)associated to the eigenvalue λ+(ε) and wε and Swε are eigenvalues of L(ε)∗ associated to the

eigenvalue λ−(ε) = λ+(ε) for any ε ∈ (−δ, δ), one can readily check that

(3.15) Π±(ε)L(ε) = L(ε)Π±(ε), Π(ε)L(ε) = L(ε)Π(ε) for any ε ∈ (−δ, δ).Next, we note that equation (2.2) is of the form

(3.16) v′(t) = L(ε)v(t) +N (ε; v(t)), t ≥ 0,

where the non-linear function N : (−δ, δ) ×H2(R × [−π, π],Cn) → L2(R × [−π, π],Cn) is definedby

(3.17) N (ε; u) = F(ε; u)− L(ε)u.

From (3.6) and (3.8) we conclude that

(3.18) N (ε;R(θ)u) = R(θ)N (ε; u), N (ε;Su) = SN (ε; u),

for all u ∈ H2(R × [−π, π],Rn), θ ∈ R and ε ∈ (−δ, δ). To construct the return map of (2.2)(mod m), we start by coordinatizing equation (3.16) as follows:

(3.19) v±(t) = Π±(ε)v(t), v(t) = Π(ε)v(t), t ≥ 0, ε ∈ (−δ, δ).We introduce the functions φ, ψ : R+ → C by φ(t) = 〈v(t), wε〉L2 and ψ(t) = 〈v(t), Swε〉L2 . Sincewe are looking for real-valued solutions of our PDE-system we are interested in finding solution vof equation (3.16) satisfying v(t) = v(t) for each t ≥ 0. It follows that

(3.20) φ(t) = 〈v(t), wε〉L2 , ψ(t) = 〈v(t), Swε〉L2 for all t ≥ 0,

which implies that

(3.21) v−(t) = v+(t) for all t ≥ 0.

Next, we rewrite the system (3.16) in the new variables (φ, ψ, v). From Lemma 3.3(iii) andLemma 3.4 we conclude that (3.16), subject to the constraint mv = 0, is equivalent to the system

(3.22)

φ′ = λ+(ε)φ+ 〈N (ε;φwε + ψSwε + v), wε〉L2

ψ′ = λ+(ε)ψ + 〈N (ε;φwε + ψSwε + v), Swε〉L2

v′ = LΠ(ε)v + Π(ε)N (ε;φwε + ψSwε + v),

where LΠ(ε) = L(ε)|Range(Π(ε)).Since the linear operator L(ε) generates a C0-semigroup, (see, e.g., [Lun] or [Z2]), from (3.15)

we infer that LΠ(ε) generates a C0-semigroup. Next, we integrate (3.22) in t ∈ [0, T ] using thevariation of constants formula to obtain the system

(3.23)

φ(T ) = eTλ+(ε)φ(0) +

∫ T0 e(T−t)λ+(ε)Φ(φ(s), ψ(s), v(s), ε) ds

ψ(T ) = eTλ+(ε)ψ(0) +∫ T

0 e(T−t)λ+(ε)Ψ(φ(s), ψ(s), v(s), ε) ds

v(T ) = eTLΠ(ε)Π(ε)v(0) +

∫ T0 e(T−t)LΠ(ε)V(φ(s), ψ(s), v(s), ε) ds

.

Here we denoted by {etLΠ(ε)}t≥0 the C0-semigroup generated by the operator LΠ(ε). The nonlin-earities Φ,Ψ : C2 ×H2(R× [−π, π],Cn)× (−δ, δ)→ C, are defined by

Φ(z1, z2, v, ε) = 〈N (ε; z1wε+z2Swε+v), wε〉L2 , Ψ(z1, z2, v, ε) = 〈N (ε; z1w

ε+z2Swε+v), Swε〉L2 .21

In addition, the nonlinear map V : C2 × H2(R × [−π, π],Cn) × (−δ, δ) → L2(R × [−π, π],Cn) isdefined by

V(z1, z2, v, ε) = Π(ε)N (ε; z1wε + z2Swε + v).

To prove existence of periodic solutions of period T of (2.2)(mod m), it is enough to show thatwe can solve for T in the T -return map of (3.23) in terms of the initial conditions. This is equiv-alent with finding fixed points of the period map defined by (3.23) or with finding zeros of thedisplacement map Disp = (Disp1,Disp2,Disp3) : C2 ×H2(R× [−π, π],Cn)× (−δ, δ)× (0,∞)→C2 × L2(R× [−π, π],Cn) defined by

(3.24) Disp1(a1, a2, v0, ε, T ) = (eTλ+(ε) − 1)a1 +

∫ T

0e(T−t)λ+(ε)Φ(φ(s), ψ(s), v(s), ε) ds;

(3.25) Disp2(a1, a2, v0, ε, T ) = (eTλ+(ε) − 1)a2 +

∫ T

0e(T−t)λ+(ε)Ψ(φ(s), ψ(s), v(s), ε) ds;

(3.26) Disp3(a1, a2, v0, ε, T ) = (eTLΠ(ε)Π(ε)− I)v0 +

∫ T

0e(T−t)LΠ(ε)V(φ(s), ψ(s), v(s), ε) ds,

satisfying the condition v0 ∈ Range(Π(ε)), where (φ, ψ, v) is a solution of (3.22) with initialcondition (φ, ψ, v)(0) = (a1, a2, v0). At this moment it is crucial to eliminate v0 from the system

(3.27) Disp(a1, a2, v0, ε, T ) = 0

using a special form of the Lyapunov–Schmidt reduction in order to obtain a finite dimensionalsystem. To apply the Lyapunov-Schmidt reduction method, following [TZ1]–[TZ4], we need toinvestigate some of the properties of LΠ(ε) = L(ε)|Range(Π(ε)). More precisely, we need to investigate

the (right) invertibility of eTLΠ(ε) − I for T > 0 to be chosen later. From Hypothesis (Dε) we infer

that LΠ(ε) has only finitely many eigenvalues with positive real part of finite multiplicity. Wedefine

θ(ε) = min{Reλ > 0 : λ ∈ σpoint(L(ε))},and we introduce the spectral projectors(3.28)

Πpos(ε) = spectral projection of σ(LΠ(ε)) ∩ {λ ∈ C : Reλ > θ(ε)}, Πneg(ε) = Π(ε)−Πpos(ε).

Moreover, if we define Lpos(ε) = L(ε)|Range(Πpos(ε)) and Lneg(ε) = L(ε)|Range(Πneg(ε)), from the invari-ance of the spectral projectors we obtain the following diagonal decomposition on Range(Π(ε)) =Range(Πpos(ε))⊕ Range(Πneg(ε)):

(3.29) LΠ(ε) =

(Lpos(ε) 0

0 Lneg(ε)

).

Since Range(Πpos(ε)) is finite dimensional and σ(Lpos(ε)) ⊂ {λ ∈ C : Reλ > 0}, we infer

(3.30) eTLpos(ε) − I is invertible on Range(Πpos(ε)) for all T > 0, ε ∈ (−δ, δ).

Taking again Fourier Transform in x2 ∈ [−π, π], we can identify Lneg(ε) with (Lnegk (ε))k∈Z on⊕

k∈Z H2k(R,Cn) ∩ Fk(Range(Πneg(ε))). Here Lneg

k (ε) = Lneg(ε)(k) and Fku = u(k). In addition,using that Range(Πpos(ε)) and Σ±(ε) are finite dimensional spaces we have that there exists Zneg(ε)a finite subset of Z such that for any ε ∈ (−δ, δ), the following assertions hold true:

(3.31) dom(Lnegk (ε)) = H2

k(R,Cn) ∩Fk(Range(Πneg(ε))) for all k ∈ Z \ Zneg(ε);

(3.32)

dom(Lnegk (ε)) is a finite codimension subspace H2

k(R,Cn)∩Fk(Range(Πneg(ε))) for all k ∈ Zneg(ε).22

From the definition of spectral projections and Hypothesis (Dε) we have that σ(Lneg(ε)) ⊂ {λ ∈C : Reλ < 0} ∪ {0}, which implies that

(3.33) {λ ∈ C : Reλ > 0} ⊂ ρ(Lnegk (ε)) for all k ∈ Z, ε ∈ (−δ, δ).

In the next step we are going to prove that the semigroups generated by Lnegk (ε), k ∈ Z \ {0}, are

uniformly exponentially stable. In order to formulate the result, we introduce the spaces

(3.34) Tpk(ε) = Hp

k(R,Cn) ∩Fk(Range(Πneg(ε))), p = 1, 2.

Lemma 3.6. Assume Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε). Then, the followingassertions hold true:

(i) There exists C > 0 sufficiently large such that for any k ∈ Z \ {0}, ε ∈ (−δ, δ) the followingestimate holds

(3.35) ‖(λ− Lneg

k (ε))−1‖

T1k(ε)→T1

k(ε)≤ C for any λ ∈ C with Reλ > 0;

(ii) There exists a constant ν > 0, small enough such that for any k ∈ Z \ {0}, ε ∈ (−δ, δ) thefollowing estimate holds

(3.36) ‖etLnegk (ε)‖

T1k(ε)→T1

k(ε)≤ Ce−ζt for any t ≥ 0.

Proof. First, we note that we can apply the results from [Z2, Prop. 4.7] to conclude that there aretwo positive constants R and C sufficiently large and θ00 > 0 sufficiently small such that for anyε ∈ (−δ, δ)

(3.37) ‖(λ− Lk(ε)

)−1‖T

1k(ε)→T1

k(ε)≤ C whenever |(k, λ)| ≥ R, Reλ > −θ00.

It follows that

(3.38) ‖(λ− Lneg

k (ε))−1‖

T1k(ε)→T1

k(ε)≤ C whenever |λ| ≥ R, Reλ ≥ 0

for any ε ∈ (−δ, δ) and any k ∈ Z \ {0}. From Hypothesis (Dε) we conclude that σ(Lneg(ε)) =σess(L(ε)), which implies that the operator Lneg(ε) has only essential spectrum, that is σ(Lneg

k (ε)) =σess(L

negk (ε)) for all k ∈ Z \ {0}. Moreover, from (A1)–(A3) and (H3) we infer that

sup Reσess(Lnegk (ε)) ≤ sup

ξ∈R

[− θ0(ξ2 + k2)

1 + ξ2 + k2

]≤ −θ0

2< 0

for any ε ∈ (−δ, δ) and any k ∈ Z \ {0}. We conclude that {λ ∈ C : Reλ ≥ 0, |λ| ≤ R} is containedin ρ(Lneg

k (ε)) for any ε ∈ (−δ, δ) and any k ∈ Z \ {0}, which implies that

(3.39) ‖(λ− Lneg

k (ε))−1‖

T1k(ε)→T1

k(ε)≤ C whenever |λ| ≤ R, Reλ ≥ 0

for any ε ∈ (−δ, δ) and any k ∈ Z \ {0}. Assertion (i) follows shortly from (3.38) and (3.39).Assertion (ii) follows from the Gearhart-Pruss Spectral Mapping theorem for C0-semigroups onHilbert spaces and the estimate (3.35). �

The (right) invertibility problem for eTLneg0 (ε)Π(ε)−I was settled in [TZ3, Prop.4] (see also [TZ2,

Lemma 5.10]). To formulate this result we need to introduce the function spaces X1, B1, X2 andB2 as follows:

(3.40) X1 = H2η (R,Cn) = H2(R,Cn; eη(1+|x1|2)1/2

dx1), B1 = H1(R,Cn),

with their natural Hilbert space scalar product. Furthermore, we define

(3.41) X2 = ∂x1H12η(R,Cn) ∩X1, B2 = ∂x1L

1(R,Cn) ∩B1.23

and note that X2 is a Hilbert space while B2 is a Banach space. The scalar product on X2 and thenorm on B2 are defined by

(3.42) 〈∂x1f, ∂x1g〉X2 = 〈∂x1f, ∂x1g〉H2η

+ 〈f, g〉H12η, ‖∂x1f‖B2 = ‖f‖L1 + ‖∂x1f‖H11.

Remark 3.7. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), we can choose a δ > 0 small

enough such that the operator eTLneg0 (ε)Π(ε) − I has a right inverse, bounded from X2 to X1 and

from B2 to B1, uniformly in (ε, T ) for T ∈ [T0, T1], 0 < T0 < T1 < ∞, and ε ∈ (−δ, δ). Moreover,the function

(3.43) (ε, T )→ (eTLneg0 (ε)Π(ε)− I)†|X2

: (−δ, δ)× [T0, T1]→ L(X2, X1) ∩ L(B2, B1)

is C1 in the L(B2, B1) norm9.

In the following lemma we collect the results from the previous lemmas on the invertibility ofeTL

neg(ε)Π(ε)− I. To formulate the result we define the spaces

(3.44) X1 =⊕

k∈Z\{0}

H1k(R,Cn)⊕X1, B1 =

⊕k∈Z\{0}

H1k(R,Cn)⊕B1,

(3.45) X2 =⊕

k∈Z\{0}

H1k(R,Cn)⊕X2, B2 =

⊕k∈Z\{0}

H2k(R,Cn)⊕B2.

We recall the definition of Hpk(R,Cn), k ∈ Z, p = 1, 2, given in (3.1) and (3.2) and the definition

of X1, B1, X2 and B2 in (3.40) and(3.41). We point out that a standard inverse of the operator

eTLΠ(ε)Π(ε)− I does not exist, since 0 is an eigenvalue of LΠ(ε), and thus, by the spectral mapping

theorem for point spectrum, of eTLΠ(ε)Π(ε)− I. One can hope to obtain a right inverse, since the

lack of invertibility is due to group invariance. However, due to the lack of spectral gap, this issueis nontrivial; see [TZ2, TZ3] for further discussion.

Lemma 3.8. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), we can choose δ > 0 small

enough and T0 > 0 large enough, such that the operator eTLΠ(ε)Π(ε)−I has a right inverse, bounded

from X2 to X1 and from B2 to B1, uniformly in (ε, T ) for T ∈ [T0, T1], T1 < ∞, and ε ∈ (−δ, δ).Moreover, the function

(3.46) (ε, T )→ (eTLΠ(ε)Π(ε)− I)†|X2

: (−δ, δ)× [T0, T1]→ L(X2,X1) ∩ L(B2,B1)

is C1 in the L(B2,B1) norm.

Proof. Without loss of generality we can choose T0 > 0 large enough so that Ce−ζT0 ≤ 12 , where C

and ν are defined in (3.35) and (3.36). The invertibility result follows from Lemma 3.6, Remark 3.7,(3.29) and (3.30). Next, we note that using the regularity properties of L(ε) we have that(3.47)

(ε, T )→ (eTLnegk (ε)Π(ε))k∈Z\{0} : (−δ, δ)× [T0, T1]→ L

( ⊕k∈Z\{0}

H2k(R,Cn),

⊕k∈Z\{0}

H1k(R,Cn)

)is C1. The lemma follows shortly from Remark 3.7, (3.29), (3.30) and (3.47). �

Now we are ready to apply the Lyapunov-Schmidt reduction on (3.27). We follow the proceduredescribed in detail in [TZ2] and further developed in [TZ3]. Using the result from Lemma 3.8, wenote that the infinite-dimensional part of equation (3.27), Disp3(a1, a2, v0, ε, T ) = 0, is equivalentto

(3.48) v0 = (I − eTLΠ(ε)Π(ε))†|X2N3(a1, a2, v0, ε, T ) + h,

9Throughout this paper we use A† to denote the right inverse of linear operator A

24

for h ∈ ker(I − eTLΠ(ε)Π(ε)) ∩ X1, where Nj , j = 1, 2, 3, denote the nonlinear integral terms fromthe definition of Disp in (3.24)-(3.26). Another key element of the analysis in [TZ2] is to use the

(right) invertibility result from Lemma 3.8 to show that (I − eTLΠ(ε)Π(ε))†|X2is well-defined and

bounded on RangeN3 and then prove contractivity by the quadratic bounds of N3. In the nextlemma we collect some estimates satisfied by the nonlinearities Nj , j = 1, 2, 3.

Lemma 3.9. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), the function Nj : C2 ×H2(R × [−π, π],Cn) × (−δ, δ) × (0,∞) → C, j = 1, 2, is quadratic order and C1. Moreover, forany M > 0 the map N3 : C2 ×BX1(0,M)× (−δ, δ)× (0,∞)→ X2 is quadratic order and C1 fromC2 × B1 × (−δ, δ)× (0,∞) to B2. More precisely the following estimates hold true:

|Nj(a1, a2, v0, ε, T )|+ |∂ε,TNj(a1, a2, v0, ε, T )| ≤ c(|a1|+ |a2|+ ‖v0‖B1)2

|∂akNj(a1, a2, v0, ε, T )|+ ‖∂v0Nj(a1, a2, v0, ε, T )‖B1 ≤ c(|a1|+ |a2|+ ‖v0‖B1)

|N3(a1, a2, v0, ε, T )|+ ‖∂ε,TN3(a1, a2, v0, ε, T )‖B2 ≤ c(|a1|+ |a2|+ ‖v0‖X1)2

‖∂akN3(a1, a2, v0, ε, T )‖B2 + ‖∂v0N3(a1, a2, v0, ε, T )‖L(B1,B2) ≤ c(|a1|+ |a2|+ ‖v0‖X1),

for any j = 1, 2 and k = 1, 2, whenever ‖v0‖X1 ≤M .

Proof. These estimates follow shortly from the estimates from Proposition 2.2 and the variation ofconstants formulas of Nj , given in (3.24)-(3.26). �

We conclude this section with this a result describing ker(I − eTLΠ(ε)).

Lemma 3.10. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), there exist smooth func-tions h1, . . . , h` : (−δ, δ) → H2(R × [−π, π],Cn) such that {hj(ε) : j = 1, . . . , `} is a basis of

ker(I − eTLΠ(ε)Π(ε)) of dimension ` (defined in (H4)) for all T > 0.

Proof. From hypothesis (Dε) (ii) one can readily infer the existence of an `-dimensional, ε-smoothbasis of kerLΠ(ε). The lemma follows shortly by using the intricate connections between kerLΠ(ε)

and ker(I − eTLΠ(ε)Π(ε)) that one can readily check by using elementary semigroup theory. �

4. Proof of the main result

In this section we collect the result from the previous sections to prove the existence of an O(2)-Hopf bifurcation from our one-parameter family of standing viscous planar shocks. First, we solvefor v0 in the infinite-dimensional equation Disp3 = 0.

Lemma 4.1. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), there exists δ > 0 smallenough and a map z : C2×(−δ, δ)× [T0, T1]×R` → X1 that is C1 from C2×(−δ, δ)× [T0, T1]×R` toB1 such that for any a1, a2 ∈ BC(0, δ), ε ∈ (−δ, δ) and T ∈ [T0, T1], the local solutions of equationDisp3(a1, a2, v0, ε, T ) = 0 are given by v0 = z(a1, a2, ε, T, b1, . . . , b`) for some (b1, . . . , b`) ∈ R`.

Proof. We define the map F : C2 ×X1 × (−δ, δ)× [T0, T1]× R` → X1 by

(4.1) F(a1, a2, v0, ε, T, b1, . . . , b`) = (I − eTLΠ(ε))†|X2N3(a1, a2, v0, ε, T ) +

∑j=1

bjhj(ε).

From (3.48) we have that the equation Disp3(a1, a2, v0, ε, T ) = 0 is equivalent to the fixed pointequation

(4.2) v0 = F(a1, a2, v0, ε, T, b1, . . . , b`),

for some b = (b1, . . . , b`) ∈ R`. Using the results from [TZ2, Section 2] and [TZ3, Section 4] wecan show that the map F is bounded from C2 × X1 × (−δ, δ) × [T0, T1] × R` to X1 and C1 from

25

C2×B1×(−δ, δ)×[T0, T1]×R` to X1. Moreover, using the results from Lemma 3.8 and Lemma 3.9 wereadily obtain appropriate estimates on F and its partial derivatives. Using again the results from[TZ2, Section 2] we infer by a Contraction Mapping/Implicit Function Theorem type argument thatthere exists a map z : C2×(−δ, δ)× [T0, T1]×R` → X1 that is C1 from C2×(−δ, δ)× [T0, T1]×R` toB1 such that v0 = z(a1, a2, ε, T, b1, . . . , b`) uniquely solves (4.2) locally. This proves the lemma. �

Remark 4.2. Since the function z is C1 from C2 × (−δ, δ) × [T0, T1] × R` to B1, we can use theresults from Lemma 3.8 and Lemma 3.9 to infer that

(4.3) z(0, 0, 0, T, 0, . . . , 0) = 0.

Moreover, by differentiating with respect to T in (4.2), one can easily check that

(4.4) ∂T z(0, 0, 0, T, 0, . . . , 0) = 0.

At this point we note that to solve the equation Disp = 0 it is enough to solve a system of twoscalar (complex) equations, with variables a1, a2, ε, T and parameters b1, . . . , b`. Next, we choosek∗ ∈ Z, with |k∗| large enough such that 2k∗π

ω(0) ∈ (T0, T1). Let T∗ : (−δ, δ) → R be the function

defined by T∗(ε) = 2k∗πω(ε) . We plug in

(4.5) v0 = z(a1, a2, ε, T, b1, . . . , b`) and T = T∗(ε)(1 + µ), µ ∈ (−δ, δ)in (3.24) and (3.25) to obtain the system

(4.6)

{N1(a1, a2, ε, µ, b1, . . . , b`) = 0

N2(a1, a2, ε, µ, b1, . . . , b`) = 0.

To solve the remaining C2 system in variables a1, a2, with bifurcation parameters ε and µ, andinvolving parameters (b1, . . . , b`) ∈ (−δ, δ)`, we need to identify the symmetries that are satisfiedby this system, which are inherited from the original system (2.2) or its reformulation (3.22).

Lemma 4.3. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), the finite-dimensionalsystem (4.6) is invariant in the variables (a1, a2) under the symmetry (z1, z2) → (z2, z1) and therotation (z1, z2)→ (z1e

iθ, z2e−iθ) for any θ ∈ R.

Proof. We recall that the system (2.2) is invariant under the symmetry S and the non-degeneraterotation group {R(θ)}θ∈R. Therefore, the group actions of S and the C0-group {R(θ)}θ∈R areinherited on the eigenspaces Σ±(ε) associated to the crossing eigenvalues λ±(ε). From Lemma 3.2and since S is a symmetry, we infer that the group action of S on the eigenspaces Σ±(ε) is isomorphic

to the transformation S : C2 → C2 defined by S(z1, z2) = (z2, z1). Using Lemma 3.2 and (3.12),we conclude that for any θ ∈ R the group action of R(θ) on the eigenspaces Σ±(ε) is isomorphic to

the transformation R(θ) : C2 → C2 defined by R(θ)(z1, z2) = (z1eiα(ε)θ, z2e

−iα(ε)θ). It follows that

the system (4.6) is invariant in the variables (a1, a2) ∈ C2 under the transformations S and R(θ)for all θ ∈ R. Making the change of variables θ → θ

α(ε) , the lemma follows shortly. �

It was shown in [TZ3, Rmk 13] that one can easily improve the C1-regularity of the map z bychoosing the spaces Xj , j=1,2 (defined in (3.40) and (3.41)), appropriately.

Remark 4.4. By choosing the space X1 = H4η (R,C) = H4(R,Cn; eη(1+|x1|2)1/2

dx1) and X2 =

∂x1H12η(R,Cn) ∩ X1, one can use the same analysis from [TZ2, TZ3] to prove that z is of class

C2. More generally, we can strengthen the result by proving that the map z is of class Cm whereν = 2m+1 in (H0). Since ν ≥ 5, we have that z is (at least) of class C3 from C2×(−δ, δ)×[T0, T1]×R`to B1. Therefore, we conclude that the functions Nj , j = 1, 2, are of class (at least) C3 from

BC(0, δ)2 × (−δ, δ)2+` to C.26

In the next lemma we exploit the fact that Nj , j = 1, 2, are of class C3 by expanding thesefunctions to cubic order.

Lemma 4.5. Under Hypotheses (A1)–(A3), (B1), (H0)–(H6) and (Dε), there exists δ > 0 smallenough, two non-zero real constants κ, χ 6= 0 and smooth functions Λ,Γ : (−δ, δ)` → C such that

N1(a1, a2, ε, µ, b1, . . . , b`) = a1

(κε+ iχµ+ Λ(b1, . . . , b`)|a1|2 + Γ(b1, . . . , b`)|a2|2

)+O(4),

N2(a1, a2, ε, µ, b1, . . . , b`) = a2

(κε+ iχµ+ Γ(b1, . . . , b`)|a1|2 + Λ(b1, . . . , b`)|a2|2

)+O(4).(4.7)

Proof. To find an expansion for the functions Nj , j = 1, 2, we use the definition of the functionsDispj , j = 1, 2 given in (3.24) and (3.25) and the invariance properties of the system (4.6) provedin Lemma 4.3. From Lemma 3.9, (4.3), (4.4) and the substitution (4.5) it follows that the leadingorder terms in ε and µ are obtained from the leading order terms of Dispj , j = 1, 2, namely

(4.8)(eT∗(ε)(1+µ)λ+(ε) − 1

)aj =

(eT∗(ε)(1+µ)γ(ε)+2k∗πiµ − 1

)aj , j = 1, 2.

Taking κ = γ′(0)T∗(0) = 2k∗πγ′(0)ω(0) 6= 0 (by Hypothesis (Dε)) and χ = 2k∗π 6= 0, we infer that there

exist smooth functions Λj ,Γj ,Υj : (−δ, δ)` → R such that

N1(a1, a2, ε, µ, b1, . . . , b`) = a1

(κε+ iχµ+ Λ1(b1, . . . , b`)|a1|2 + Υ1(b1, . . . , b`)a1a2

+ Γ1(b1, . . . , b`)|a2|2)

+ a2

(+Λ2(b1, . . . , b`)|a1|2 + Υ2(b1, . . . , b`)a1a2 + Γ2(b1, . . . , b`)|a2|2

)+O(4),

N2(a1, a2, ε, µ, b1, . . . , b`) = a2

(κε+ iχµ+ Γ1(b1, . . . , b`)|a1|2 + Υ1(b1, . . . , b`)a1a2

+ Λ1(b1, . . . , b`)|a2|2)

+ a1

(Γ2(b1, . . . , b`)|a1|2 + Υ2(b1, . . . , b`)a1a2 + Λ2(b1, . . . , b`)|a2|2

)+O(4).

Noting that there always exist reflectionally symmetric solutions a1 ≡ a2, for which the situationreduces to that of a standard Hopf bifurcation, recalling that the system originates from a rotatingevolutionary system in which a1, a2 rotate in a common direction with common (to linear order)speed ω(ε), and noting that periodic solutions are preserved under influence of the flow, we inferthat Υ1 = Λ2 = Υ2 = Γ2 = 0 must hold. For, otherwise it is easy to see that solutions a1 ≡ a2

are not preserved under an approximate common rotation. Dropping the index of the functions Λ1

and Γ1, the lemma follows immediately. �

To finish the proof of our main result we introduce the following genericity assumption:

Hypothesis 1. If Λ,Γ : (−δ, δ)` → C are the functions from expansion (4.7), we assume that thereexists δ > 0 small enough such that

(4.9) Λ 6= Γ, Re(Λ + Γ) 6= 0, ReΛ 6= 0.

Proof of Theorem 1.2. From Lemma 4.1 it follows that to prove the theorem it is enough to solvethe finite-dimensional system (4.6). Let b = (b1, . . . , b`) ∈ (−δ, δ)` with |b| ≤ C|(a1, a2)| for someconstant C > 0. Making the substitution

(4.10) a1 = a, a2 = ρa, a ∈ C and ρ ∈ C ∪ {∞}.in (4.6) and using the results from Lemma 4.5, we obtain the equivalent system:

(4.11)

a(κε+ iχµ+ Λ(b)|a|2 + Γ(b)|a|2|ρ|2

)+O(4) = 0

aρ(κε+ iχµ+ Γ(b)|a|2 + Λ(b)|a|2|ρ|2

)+O(4) = 0

.

There are three cases of interest: when a 6= 0 and ρ is bounded and bounded away from 0 andwhen |ρ| << 1 or |ρ| >> 1. We are going to treat all of these cases separately.

27

Case 1. a 6= 0 and there exists C > 0 such that 1C ≤ |ρ| ≤ C.

Multiplying the first equation of the system (4.11) by ρ and subtracting it from the second equation,we obtain the equation:

aρ(

(Λ(b)− Γ(b))|a|2(1− |ρ|2) +O(4) = 0.

Since in this case ρ is bounded and bounded away from 0 we can divide this equation by aρ|a|2 toobtain the equation:

(4.12)((Λ(b)− Γ(b)

)(1− |ρ|2) +O(a) = 0.

Since by Hypothesis 1 we have that Λ(b) 6= Γ(b) from (4.12) we infer that |ρ| = 1 + O(a).Substituting back in the first equation of (4.11) and using that a 6= 0 we obtain that

(4.13) κε+ iχµ+(Λ(b) + Γ(b)

)|a|2 +O(a3) = 0.

Depending on model parameters (specifically, the relative signs of κ and Re(Λ+Γ)), this will occurfor ε positive (supercritical case) or negative (subcritical case). Taking the real part in the aboveequation and since κ 6= 0 by Lemma 4.5 and Re(Λ(b) + Γ(b)) 6= 0 by Hypothesis 1, we infer that

|a| = O(|ε|1/2). From Lemma 4.5 we have that χ 6= 0, therefore by taking imaginary part in (4.13)we conclude that µ = O(ε).Case 2. a 6= 0, |ρ| << 1.From the first equation of (4.11), by using the fact that ρ is small, we obtain the equation

(4.14) κε+ iχµ+(Λ(b) + |ρ|2Γ(b)

)|a|2 +O(a3) = 0.

Since ReΛ(b) 6= 0, by Hypothesis 1, we have that Re(Λ(b) + |ρ|2Γ(b)

)6= 0, by continuity. Taking

real and imaginary part in (4.14) and using that κ, χ 6= 0, we infer that |a| = O(|ε|1/2) andµ = O(ε).Case 3. a 6= 0, |ρ| >> 1.From the second equation of (4.11) and since 1/ρ is small, we obtain the equation

(4.15) κε+ iχµ+(

Λ(b) +1

|ρ|2 Γ(b))|a|2 +O(a3) = 0.

Using Hypothesis 1 again, we have that Re(

Λ(b) + 1|ρ|2 Γ(b)

)6= 0, by continuity. Arguing just

as in the previous cases, by taking real and imaginary part in (4.15), we conclude again that

|a| = O(|ε|1/2) and µ = O(ε). Summarizing, we conclude that any solution of (4.6) satisfies one ofthe conditions

(i) |a1| = O(|ε|1/2) and a2 = eiθa1 +O(a21),

(ii) |a1| = O(|ε|1/2) and |a2| << |a1| (iii) |a2| = O(|ε|1/2) and |a1| << |a2|.(4.16)

Moreover, whenever we have a solution of (4.6) we have that µ = O(ε). Making the change ofvariables

(4.17) a1 =a1

|ε|1/2 , a2 =a2

|ε|1/2 , µ =µ

ε

in (4.6) we obtain the system:

(4.18)

a1

(κsgn(ε) + iχµ+ Λ(b)|a1|2 + Γ(b)|a2|2

)+O(|ε|1/2) = 0

a2

(κsgn(ε) + iχµ+ Γ(b)|a1|2 + Λ(b)|a2|2

)+O(|ε|1/2) = 0

.

Since the functions Nj , j = 1, 2 are of class C3, by Remark 4.4, it follows that O(|ε|1/2) terms in(4.18) are of C3 class in a1, a2 and µ. Since system (4.18) is equivalent to the system (4.6) which

28

is rotationally invariant by Lemma 4.3, it inherits the property of rotational invariance, and inaddition the property that (a1, a2) = (0, 0) is always a solution.

Setting ε = 0 in (4.18), we reduce to (a rescaled version of) the truncated cubic system discussedin §1.2.1, which has under nondegeneracy assumptions (4.9) precisely four families of solutions:

(a1, a2) = (0, 0), (a∗, 0), (0, a∗), (a\, eiθa\),

θ ∈ R, of which the last three are nontrivial equilibria bifurcating from the first, zero equilibrium.Depending on model parameters, these will occur variously for ε > 0 (supercritical case) or ε < 0(subcritical case), according to the rule sgn(ε) = −(ReΛ(b)|a1|2 + ReΓ(b)|a2|2)/κ, where therighthand side is, variously, −(ReΛ(b) + ReΓ(b))a2

∗/κ, −(ReΛ(b))/κ, or −(ReΓ(b)/κ. Fixingb = 0, we find by a combination of Implicit Function arguments and symmetry considerations thateach of these families continues uniquely under small perturbations in |ε|1/2, to give the claimedexact solutions of the full system obtained by Lyapunov–Schmidt reduction using the choice ofright inverse corresponding to b = 0. These computations are carried out in Appendix B.

It remains to deal with the `-fold indeterminacy associated with parameter b, induced by the`-fold kernel of the one-dimensional linearized operator L0(ε). This may be accounted for asfollows, using the method introduced for that purpose in [TZ2]. First, we make the change of

variables bj =bj

|(a1,a2)| , j = 1, 2, ensuring for fixed b that |b| ≤ C|(a1, a2)| as required for our

arguments above; by the same arguments, we obtain thereby a unique family of solutions perturbingfrom the ε = 0 solutions for each choice of b, thus obtaining an `-dimensional cone of distinctsolutions above each solution for b = 0, which are the unique solutions lying within the cone{(a1, a2, v0) : ‖v0‖X1 ≤ C|(a1, a2)|}, for some C > 0; moreover, these cones may be extended tosmooth `-dimensional manifolds of solutions by perturbing about different choices of backgroundwave in the `-dimensional manifold of stationary shock solutions with the same endpoints, thusobtaining a family of nearby problems to which the same arguments uniformly apply. Finally,applying [TZ2, Prop. 2.20] we find that any periodic solution of (3.22) can be shifted by such achange of coordinates so as to originate in the cone {(a1, a2, v0) : ‖v0‖X1 ≤ C|(a1, a2)|}, for somesuch nearby problem. Thus, we can infer uniqueness of the full `-parameter families of solutions asin [TZ2, Cor. 2.21].

Characterization as traveling waves. Finally, we verify in the Lax case, ` = 1, that“traveling-wave” type solutions close to (a1, a2) = (a∗, 0) or (0, a∗) are indeed traveling waveswith respect to x2. This may be seen by the computation carried out in Appendix B showing thatthese types of solutions are unique up to rotational invariance, i.e., up to translation with respect tox2, for each fixed b. On the other hand, time-translates of any such solution give a nearby periodicsolution (specifically, nearby in rescaled coordinates), which must therefore be a “traveling-type”solution for some nearby choice of b. Recalling, for the Lax case ` = 1, that change in b correspondssimply to translation in x1, we find therefore that time-translates of “traveling-type” solutions cor-respond to translates in x = (x1, x2), from which it is readily seen (by substitution in the originalPDE) that they are traveling waves hε(x1−ct, x2−dt) in x1 and x2. But, time-periodicity, plus thefact that the background standing shocks by our assumptions are not periodic in x1, implies thatc must vanish, leaving the conclusion that the solution must be a traveling wave in x2 alone. �

Remark 4.6. The above displacement map argument substitutes in our setting for the standard ap-proaches used to treat O(2) Hopf bifurcation in other settings, namely, the center-manifold/normalforms approach, not available to us because in absence of spectral gap we have no readily availablecenter manifold (possibly even nonexisting as far as we know); and the spatial dynamics approach,or Lyapunov-Schmidt reduction of the evolution equations recast in the class of time-periodic func-tions, (the original approach for ODE given in [GSS]), again not applicable to our framework basedon the time-T map and the “reverse temporal dynamics” approach of [TZ3]. Both of these standard

29

approaches rely on O(2)×S1 symmetry, with the additional S1 symmetry imposed, respectively, bynormal form reduction and time-periodicity. It is interesting that we do not need full S1 symmetryin our argument, substituting similar but less detailed information coming from the origins of thetime-T map as an evolution problem to obtain the key property Υ1 = Λ2 = Υ2 = Γ2 = 0. Thoughwe do not treat it in our analysis, spectral stability information for the bifurcating waves should inprinciple be readily available from the reduced time-T solution map (4.6) with ε, µ held fixed.

5. Gas dynamics and MHD

Finally, we comment briefly on the cases of gas dynamics or MHD, to which our analysis does notapply. In Eulerian coordinates, the 2-D compressible Navier–Stokes equations are [Ba, Da, Sm]:

(5.1)

ρt + (ρu)x + (ρv)y = 0,

(ρu)t + (ρu2 + p)x + (ρuv)y = (2µ+ η)uxx + µuyy + (µ+ η)vxy,

(ρv)t + (ρuv)x + (ρv2 + p)y = µvxx + (2µ+ η)vyy + (µ+ η)uyx,

(ρE)t + (ρuE + up)x + (ρvE + vp)y =(κTx + (2µ+ η)uux + µv(vx + uy) + ηuvy

)x

+(κTy + (2µ+ η)vvy + µu(vx + uy) + ηvux

)y,

where ρ is density, u and v are the fluid velocities in x and y directions, p is pressure, T is

temperature, E = e + u2

2 + v2

2 is specific energy, e is specific internal energy, u2

2 + v2

2 is kineticenergy, and the constants µ > 0 and |η| ≤ µ and κ > 0 are coefficients of first (“dynamic”) andsecond viscosity and heat conductivity. The equations are closed by equations of state

(5.2) p = p(ρ, T ), e = e(ρ, T ).

Here, x and y, are coordinates in a rest frame, and t ≥ 0 is time. For common fluids and gasesin normal conditions, the polytropic gas laws p = Γρe, e = CvT give a good fit to experimentalobservations, where Γ > 0 and Cv > 0 are constants depending on the gas [Ba]. The MHD equationsfeature an additional coupling to a magnetic field vector; our discussion applies also in that case.

5.1. Lagrangian formulation. Equations (5.1) may be converted to Lagrangian coordinates asfollows [A, Da, DM, HT]. Let X,Y denote a reference configuration, and T = t, and define particlepaths (x, y)(X,Y, t) by ∂t(x, y) = (u, v), (x, y)|t=0 = (x0, y0)(X,Y ) for some choice of (x0, y0)(·)Denote by Fjk the entries of

F :=∂(x, y)

∂(X,Y ):=

(∂x∂X

∂x∂Y

∂y∂X

∂y∂Y

).

A great simplification is obtained if one can choose (x0, y0) so that detF |t=0 ≡ (1/ρ)|t=0, as canalways be done for L1

loc data in 1-D or an L1 multi-D perturbation thereof- in particular for theperturbed planar shocks considered here. In this case, the values (x, y)(X,Y, t) may be consideredas deformations of an initially uniform-density rest configuration, and equations (5.1) reduce to aspecial case of thermoviscoelasticity, with e depending only on detF = 1/ρ and entropy S:

(5.3)

(F11)t − uX = 0, (F12)t − uY = 0, (F21)t − vX = 0, (F22)t − vY = 0,

ut + (F22p)X − (F21p)Y = 2nd-order derivative terms,

vt − (F12p)X + (F11p)Y = 2nd-order derivative terms,

Et − (F12vp)X + (F11vp)Y + (F22up)X − (F21up)Y = 2nd-order derivative terms,

together with the compatibility conditions (preserved by time evolution):

(5.4) ∂Y F11 − ∂XF12 = 0; ∂Y F21 − ∂XF22 = 0.30

Here, we have omitted the description of the complicated (but divergence-form) righthand sidesinvolving second-order derivative terms involving transport effects, not needed for our discussion.

5.2. Coordinate-ambiguity. The augmented system approach followed in this paper works forthe difficult case of thermoviscoelasticity, but it does not work for the apparently simpler case ofgas dynamics. The reason: for gas dynamics, the “contingent” entropy of the enlarged system isonly nonstrictly convex and the equations fail to be symmetric, etc. At an operational level, thisis because there is no penalty on shear strains, giving neutral directions in the associated entropyfunction. In fact, the problem is deeper than that- it reflects the fact that Lagrangian gas dynamicequations have an infinite-dimensional family of invariances consisting of all volume preserving mapsof the spatial coordinate. (Since pressure depends only on density= reciprocal of the Jacobian ofthe deformation map, there can be no change under Jacobian-preserving transformations.)

This massive ambiguity in Lagrangian coordinatization means that there is a correspondinginfinite-dimensional family of neutral perturbations in the stress tensor F that do not decay to zerotime-asymptotically, but remain constant without affecting the evolution of other, gas-dynamical,variables, and, as a consequence, there can be no coordinate system, extended or otherwise, inwhich perturbations of F decay. Thus, the strategy followed here will fail; a side-consequence isthat gas-dynamical shocks are never asymptotically orbitally stable in Lagrangian coordinates, afundamental distinction between Eulerian and Lagrangian formulations.

5.3. Possible remedies. The treatment of gas dynamics/MHD is an important direction for fu-ture investigation, perhaps by “factoring out” invariances in linearized estimates, then trying toshow periodicity modulo these invariant transformations. Another approach might be to treat theproblem instead by spatial dynamics techniques plus the standard O(2) reduction argument onthe space of time-periodic functions. Here, one must confront similar issues of regularity as facedhere, but without having time-evolutionary stability machinery as a guide. A further idea is tocarry out a Nash-Moser type iteration in Eulerian coordinates, dealing with loss of regularity inthe quasilinear hyperbolic modes by the iteration scheme instead of re-coordinatization.

Finally, an alternative approach modifying our present method would be to establish a “Korn-type” inequality showing that there exists a volume-preserving transformation under which |F | iscontrolled by | detF | in Hs norms. For, we could then carry out our bifurcation analysis in theusual (Eulerian) gas dynamics variables, including τ = detF , choosing an optimal Lagrangiancoordinatization in order to control nonlinear variational estimates on the time-T evolution map.However, we have been unable to establish such an inequality, and suspect that one does not hold.The resolution of this question seems an interesting mathematical problem in its own right.

Appendix A. Explicit computations for 2D isentropic viscoelasticity

In this appendix, we illustrate further the structural assumptions of Section 1.1, suppressingbifurcation parameter ε, by complete computations in a particular case. Expanding coordinate-wise the vectorial representation of the equations of isentropic viscoelasticity given in (1.23), in thesimple case d = 2, we obtain

(A.1)

(F11)t − uX = 0, (F12)t − uY = 0, (F21)t − vX = 0, (F22)t − vY = 0,

ut − (∂W/∂F11)X − (∂W/∂F12)Y = (Z11(F,∇X,Y (u, v)T ))X + (Z12(F,∇X,Y (u, v)T ))Y ,

vt − (∂W/∂F21)X − (∂W/∂F22)Y = (Z21(F,∇X,Y (u, v)T ))X + (Z22(F,∇X,Y (u, v)T ))Y ,

together with compatibility conditions ∂Y F11 − ∂XF12 = 0, ∂Y F21 − ∂XF22 = 0.31

A.1. Connection with gas dynamics. In the case W = W (V ) of gas dynamics, where V :=detF = F11F22−F12F21 denotes specific volume, the Carnot relation pdV = dW between pressure p,specific volume V , and elastic potential W (equivalently, work of deformation) yields p = dW

dV (V ),from which we recover for appropriate choice of Z the isentropic version of (5.3), through thecomputation ∂W/∂Fij = (dW/dV )(∂V/∂Fij) = P (V )(∂ detF/∂Fij) together with ∂ detF/∂F11 =F22, ∂ detF/∂F22 = F11, ∂ detF/∂F12 = −F21, and ∂ detF/∂F21 = −F12. Specifically, the choice

(A.2) Z(F,Q) = (detF )(µSym(QF−1) + ηTrace(QF−1)

)F−1,T

corresponds in Eulerian coordinates to Cauchy stress tensor T := (detF )−1ZF T = µSym(QF−1)+ηTrace(QF−1)Id, or, substituting Q = ∇X,Y (u, v)T ,

(A.3) T = µSym(∇x,y(u, v)T ) + ηTrace(∇x,y(u, v)T )Id,

where SymM := 12(M +MT ) symmetric part of matrix M . This yields in Eulerian coordinates the

familiar Navier–Stokes viscosity divx,yT = µ∆x,y(u, v)T + (µ+ η)divx,y(u, v)T )Id corresponding tothe righhand side of (5.3). See [Da, pp. 46-47] for further discussion, along with derivation of (A.3)as the simplest choice satisfying the principles of frame-indifference associated with the problem.

A.2. Parabolicity of viscous terms. Note that (A.1)–(A.2) determine the omitted second-orderterms on the righthand side of (5.3). Strict parabolicity of these terms may be deduced eitherdirectly, by the evident parabolicity of their Eulerian counterparts together with the Chain Rule,or, appealing to general principles, by verifying the strict Clausius–Duhamel propertyQ : Z(F,Q) ≤−θ|Q|2 < 0 using the strain rate monotonicity criterion of Antman [A]; see [BLeZ, Appendix A.2].

A.3. Hyperbolicity/symmetrizability at endstates. Linearizing about endstates (F, u)± first-order equations (1.28), for d = 2, 3, and writing in vectorial form in terms of unknown

(f, u) = (F11, . . . , F1d, . . . Fd1, . . . Fdd, u1, . . . , ud),

we obtain

(A.4) Vt +∑

Aj∂jV = 0, Aj :=

(0 Ej

(Ej)T d2Wdf2 0

),

where E1 =

(1 0 0 00 0 1 0

)T, E2 =

(0 1 0 00 0 0 1

)Tfor d = 2, and similarly for d = 3. Here, strict

rank-one convexity is equivalent to E(ξ)T d2Wdf2 E(ξ) > 0 for all ξ ∈ Rd, where E(ξ) :=

∑j E

jξj .

Denoting A(ξ) :=∑

j Ajξj , using column operations to expand

det(A(ξ)− α) = det

(−α E(ξ)

E(ξ)T d2Wdf2 −α

)= det

(−α 0

E(ξ)T d2Wdf2 α−1E(ξ)T d

2Wdf2 E(ξ)− α

)= αd

2−d det(E(ξ)T d

2Wdf2 E(ξ)− α2

)for α 6= 0, and extending by analyticity to all α ∈ C, we find that the eigenvalues of A(ξ) are

α = 0,√µk(ξ), where µk(ξ), real, are the eigenvalues of E(ξ)T d

2Wdf2 E(ξ). Thus, (1.28) is hyperbolic,

i.e., σ(A(ξ)) is real, semisimple, if W is strictly rank-one convex and only if W is rank-one convex.On the other hand, existence of a block-diagonal symmetrizer A0 = blockdiag{M,N} is equiva-

lent to E(ξ)M ≡ NE(ξ)d2Wdf2 , or N−1E(ξ) ≡ E(ξ)d

2Wdf2 M

−1 for all ξ ∈ Rd, or, by linearity:

N−1H ≡ Hd2W

df2M−1 for all H ∈ Cd×d2

,

32

from which we may deduce that N = cId, M = cd2Wdf2 for some c ∈ C. Thus, existence of a block-

diagonal symmetrizer is equivalent to strict convexity of W . As noted in Example 1.4, the weakercondition of polyconvexity is sufficient for existence of a (mod m) block-diagonal symmetrizer [Da].As noted in Remark 1.5, rank-one convexity is sufficient for existence of a symmetrizer greater than0 on kerm.

A.4. Explicit (mod m) Kawashima skew-symmetrizer. By inspection, the choice

Kj± := θ

(0 d2W

df2 Ej

−(Ej)T d2Wdf2 0

).

yields for K±(ξ) := −∑j Kj±ξj and W rank one convex at F± that, for θ > 0 sufficiently small,

(K(ξ)A(ξ) + B(ξ))± = θ

(d2Wdf2 ±

E(ξ)E(ξ)T d2Wdf2 ±

∗∗ ∗

)+

(0 0

0 b(ξ)

)is greater than 0 as a quadratic form when restricted to kerm, verifying (A3)(v), by ker d

2Wdf2 E ∩

kerm = {0}.

A.5. The 1D system. (Verification of (H4)–(H5)) Next, we consider the 1D version of (1.25):

Ft − (uX , 0, 0) = (Id− e1eT1 )FXX , ut − eT1 (dW (F ))X = (β(F )uX)X ,

where e1 = (1, 0, 0) is the first standard basis element. Denoting (a1, a2, a3) := (F11, F21, F31) and

F :=

F12 F13

F22 F23

F32 F33

, we find that this decouples into a triangular system consisting of

(A.5) at − uX = 0, ut − (∂ω/∂a)(a, F )TX = (β(a, F )uX)X , ω(a) := W (F ),

together with heat equation Ft = FXX . For shocks with speed c 6= 0, it is readily verified that theseequations satisfy the structural assumptions of (A1)–(A3), with a playing the role of hyperbolic

variable, and (F , u) of parabolic variable.

Profile equation. Noting that the decoupled F equation does not support nontrivial traveling-wave solutions, we find that F ≡ constant for any profile, reducing existence to the correspondingproblem for the 1D reduced equation (1.29) of the introduction, or (A.5) with F held constant.

Eigenvalue equation and verification of (Dε)(ii). Likewise, the associated 1D eigenvalueequations for the linearized equations about a shock profile decouple into

(λ− c∂x)a− uX = 0, (λ− c∂x)u− (∂ω/∂a)TX = (βuX)X ,

and (λ− c∂x)F = ∂2X F , hence the associated Evans function factors into the product of the Evans

functions for these two systems. But, the F system, being constant-coefficient, is readily seen tohave a nonvanishing Evans function in the vicinity of λ = 0,10 hence condition (Dε)(ii) reduces tothe corresponding condition on the reduced 1D system (1.29) as claimed in the introduction.

10Specifically, the limits as λ → 0 from the positive real part complex half-plane of its stable manifold as X →+∞/unstable manifold as X → −∞ (the limits involved in the definition of the Evans function [GZ, ZH, MaZ])consist either of solutions blowing up exponentially at the other end X → −∞/X → +∞ or of constant solutionsthat are limits as λ → 0 of solutions blowing up at the other end, either of which prevents their appearance ascomponents of a generalized eigenfunction corresponding to a zero of the Evans function.

33

A.6. First-order formulation of the eigenvalue equation for Lk(ε). (Verification of (H6))The eigenvalue equations for the full system (1.25), ignoring constraints, for d = 2 takes form

λF11 − cF ′11 − u′1 = ikF ′12 − k2F11,

λF12 − cF ′12 + iku1 = F ′′12 − ikF ′11,

λF21 − cF ′21 − u′2 = ikF ′22 − k2F21,

λF22 − cF ′22 + iku2 = F ′′22 − ikF ′21,

together with second-order ODE in (u1, u2), and similarly for d = 3. where c is the speed of theshock and “′” denotes ∂1, Evidently, for c 6= 0, these can be rewritten as a first-order system

cF ′11 = −ikF ′12 + k2F11 +−λF11 + u′1,

F ′12 = F ′12,

F ′′12 = λF22 − cF ′22 + iku2,

etc. in phase variable (F11, F12, F21, F22, u1, u2, F′12, F

′22, u

′1, u′2), and similarly for d = 3.

Appendix B. Jacobian computations for the truncated cubic system

In this appendix we carry out the details needed to solve system (4.18), crucial to the proof ofour main result. Relabeling slightly for notational convenience, we may write the rescaled system(4.18) as

(B.1)

{(κsgn(ε) + iχµ+ Λ|a|2 + Γ|h|2)a = |ε|1/2Φ1(ε, µ, a, h),

(κsgn(ε) + iχµ+ Λ|h|2 + Γ|a|2)h = |ε|1/2Φ2(ε, µ, a, h),

where κ, χ is constant, µ is real, a and h complex, and Φj ∈ C are C1 functions, taking withoutloss of generality (using rotational invariance) a = (a1, 0), h = (h1, h2) and assuming the genericityconditions

(B.2) Λ 6= Γ, Re(Λ + Γ) 6= 0, ReΛ 6= 0.

Expressed in real coordinates, these are:(B.3)

(κsgn(ε) + ReΛ|a|2 + ReΓ|h|2)a1 = |ε|1/2ReΦ1(ε, µ, a, h),

(χµ+ ImΛ|a|2 + ImΓ|h|2)a1 = |ε|1/2ImΦ1(ε, µ, a, h),

(κsgn(ε) + ReΛ|h|2 + ReΓ|a|2)h1 − (χµ+ ImΛ|h|2 + ImΓ|a|2)h2 = |ε|1/2ReΦ2(ε, µ, a, h),

(κsgn(ε) + ReΛ|h|2 + ReΓ|a|2)h2 + (χµ+ ImΛ|h|2 + ImΓ|a|2)h1 = |ε|1/2ImΦ2(ε, µ, a, h),

B.1. Case a1 6= 0, h = 0. Denoting the left-hand side of (B.3) by H1, we find easily that at a root(a1, µ, h1, h2) = (a∗, µ∗, 0, 0) of (B.3):(B.4)

J1 := det∂H1

∂(a1, µ, b1, b2)|(a∗,µ∗,0,0,0) =

∣∣∣∣∣∣∣∣2(ReΛ)a2

∗ 0 0 02(ImΛ)a2

∗ χa∗ 0 00 0 ReC∗ −ImC∗0 0 ImC∗ ReC∗

∣∣∣∣∣∣∣∣ = 2χ(ReΛ)(Λ− Γ)a7∗,

where C∗ := κ + iχµ∗ + Γa2∗ = (Λ − Γ)a2

∗ since κ + iχµ∗ + Λa2∗ = 0. Thus, under genericity

assumptions ReΛ 6= 0, Λ 6= Γ, and since χ 6= 0, by Lemma 4.5, we have J1 6= 0 and we canconclude by the Implicit Function Theorem the desired existence and uniqueness of solutions of the|ε|1/2-perturbed system nearby those of the ε = 0 one. The case h1 6= 0, a = 0 goes symmetrically.

34

B.2. Case h = a 6= 0. This case is trickier due to the obvious nonuniqueness induced by rotationalinvariance of the truncated cubic order system (specifically, the additional rotational invariance incommon direction, or S1-symmetry). This will require a little bit different handling. Specifically,note that we may by rotational symmetry plus invariance under forward evolution/approximaterotation take both a and h to be real, and work with three unknowns (a1, h1, µ) and only three ofthe equation (B.3). This is enough to give uniqueness of solutions by Implicit Function Theorem,but not existence; existence on the other hand follows by reflective symmetry guaranteeing thata = h is always a solution.

Denoting the first three lines on the left-hand side of (B.3) by H2, we have at a root (a1, µ, h1) =(a\, µ\, a\) of (B.3):(B.5)

J2 := det∂H2

∂(a1, µ, h1)|(a\,µ\,a\) =

∣∣∣∣∣∣2(ReΛ)a2

\ 0 2(ReΓ)a2\

2(ImΛ)a2\ χa\ 2(ImΓ)a2

\

2(ReΓ)a2\ 0 2(ReΛ)a2

\

∣∣∣∣∣∣ = 4χ(ReΛ + ReΓ)(ReΛ− ReΓ)a5\ .

Alternatively, denoting the first, second, and fourth lines of the left-hand side of (B.3) as H3, wehave(B.6)

J3 := det∂H3

∂(a1, µ, h1)|(a\,µ\,a\) =

∣∣∣∣∣∣2(ReΛ)a2

\ 0 2(ReΓ)a2\

2(ImΛ)a2\ χa\ 2(ImΓ)a2

\

2(ImΓ)a2\ χa\ 2(ImΛ)a2

\

∣∣∣∣∣∣ = 4χ(ReΛ + ReΓ)(ImΛ− ImΓ)a5\ .

Under genericity conditions (B.2), one of J2 or J3 does not vanish, or else Re(Λ + Γ) and (Λ−Γ)would both vanish, a contradiction. But, either of J2 6= 0 or J3 6= 0 is sufficient to give uniquenessby an application of the Implicit Function Theorem.

Remark B.1. The reason the h = 0 case works in standard fashion is that the solution is rotationallyinvariant, so rotation does not induce nonuniqueness. Note that in the h = 0 case we get bothexistence- which we need, since it does not follow from invariance considerations- and uniqueness,while for THE h = a case we get only uniqueness?which is all we need, since existence IS guaranteedby reflectional/rotational symmetry (both required for that).

B.3. Case a = h = 0. Finally, we must treat the trivial solution (a, h) = (0, 0). Passing to realand imaginary parts in (B.1) and denoting the left-hand side of this system by H4, we find that:

(B.7) J4 := det∂H4

∂(a1, a2, h1, h2)|(0,0,0,0) =

∣∣∣∣∣∣∣∣κ −χµ 0 0χµ κ 0 00 0 κ −χµ0 0 χµ κ

∣∣∣∣∣∣∣∣ = (κ2 + χ2µ2)2 6= 0

for arbitrary µ, hence by the Implicit Function Theorem there is a unique solution (a, b)(|ε|1/2, µ)for each (ε, µ) near (0, µ∗). But, this is already accounted for by the trivial solution (a, b) = (0, 0).Thus, we can conclude again in this case the desired existence and uniqueness of solutions of theε-perturbed system nearby those of the ε = 0 one. This completes the argument and the paper.

Appendix C. Alternative treatment of the constrained resolvent problem: themethod of Freistıhler–Plaza

A subtle point in the analysis was to show that {λ : Reλ ≥ 0}\{0} lies in the essential resolventset of L|kerm, done in roundabout fashion by choosing carefully a representative of L (mod m) sothat {λ : Reλ ≥ 0}\{0} lies in the essential resolvent set of the full (unrestricted) operator L, thenarguing that (λ− L)u = f and f ∈ kerm implies u ∈ kerm. It is worth mentioning another, more

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direct approach that suffices for systems like isentropic viscoelasticity for which the constraint marises through reduction from a higher-order equation.

Recall that the linearized equations for isentropic viscoelasticity take the form

(C.1)(λ− c∂1)F −∇V = 0

(λ− c∂1)V −∇ · (d2WF ) = ∇ · (B1∇V, . . . Bd∇V ),

where F ∈ Rd×d and V ∈ Rd, with divergence, gradient, etc., applied row-wise, or, taking Fouriertransform in the x2, . . . , xd directions,

(C.2)

(λ− c∂1)F − (∂1V, iξV ) = 0

(λ− c∂1)V − (∂1(d2WF ), iξ(∂1(d2WF )) = ∂1(B1(∂1V, iξV ) +∑j 6=1

(iξjBj(∂1V, iξV ).

Noting by the assumption of parabolicity that B11 is invertible, this can be written in standardfashion as a first-order ODE in the phase variable (F, V, p), with p := ∂1V : the Evans system for(C.1). Modifying a clever observation of [FP] in the inviscid case, we observe that

(C.3) F = (Z, iξY ), V = (λY − cZ), p = (λZ − cq)Z, V, q ∈ Cd, is an invariant subspace of the resulting first-order system, satisfying the reducedequations

(C.4) Y ′ = Z, Z ′ = q, (B11q)′ = h(λ, ξ, Y, Z),

where h is linear in Y , Z. From (C.3), (C.4), we find that F = (Y ′, iξY ) is the Fourier transformof a gradient, hence obeys the (Fourier transform version of the) curl free constraint. Moreover, itis readily seen for solutions of (C.2) that at any point x1 at which F is a Fourier transform of a

gradient, i.e., F = (Y ′, iξY ), then (F, V, p) lies in this invariant subspace. Hence, kerm is preservedby the flow of the Fourier-transformed eigenvalue equation, and we may construct the resolvent ofL|kerm via study of the reduced flow (C.4), for which the properties of hyperbolicity at infinity, etc.,follow under the assumption of rank-one convexity of W at endstates (F, V )± of the shock. (Weshall not carry out these details here, having already constructed the resolvent otherwise.) Theabove derivation perhaps illuminates a bit the construction of [FP], at the same time showing itsapplication to general systems obtained by reduction from higher- to lower-order derivative form.

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miami University, Oxford, OH 45056E-mail address: pogana@miamioh.edu

University of Iowa, Iowa City, IA 52242E-mail address: jinghua-yao@uiowa.edu

Indiana University, Bloomington, IN 47405E-mail address: kzumbrun@indiana.edu

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