Mathematik in den Naturwissenschaften Leipzig An Evans function approach to spectral stability of small-amplitude shock profiles

We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers equation. In multidimensions, the canonical behavior is described, rather by a 2x2 viscous conservation law; this case will be treated in a companion paper by similar techniques.


Section 1. Introduction
In this paper, we study one-dimensional spectral stability in the small amplitude limit of smooth shock profiles, i.e., traveling wave solutions (1.1) u =ū(x − st), lim z→±∞ū (z) = u ± , arising under various regularizations of a hyperbolic system of conservation laws (1.2) u t + f (u) x = 0 : specifically, viscous conservation laws of form where Re σ(B) ≥ 0, and relaxation systems of form where (1.5) Re σ q v (u, v * (u)) < 0 Research of the second author was supported in part by the National Science Foundation under Grant No. DMS-0070765. along a smooth equilibrium manifold defined by (1.6) q(u, v * (u)) ≡ 0, and f (u) =f (u, v * (u)). Here, x, t ∈ R 1 , u, f ,f ∈ R n , v,g, q ∈ R r , and B ∈ R n×n , where typically n, r > 1; we make the standard assumptions of strict hyperbolicity of Df , and genuine nonlinearity of the principal characteristic eigenvalue (the one associated with the approximate shock direction), along with further standard hypotheses on B andf ,g, q, to be described later on: in particular, that systems (1.3) and (1.4) be dissipative in the sense of Kawashima and Zeng [Kaw, Ze]. These are generically satisfied for most models of physical interest; for discussions of applications, we refer the reader to the general surveys [Z.3,Z.6,MZ.1,N,Yo.4]. A recent development in the stability analysis of smooth shock profiles has been the successful adaptation of Evans function/dynamical system techniques to this nonstandard setting, with an associated explosion of new results; see, e.g., [GZ,ZH,BSZ,BMSZ,ZS,HZ,. For the origins of these methods in the setting of reaction diffusion and singular perturbation problems, see J,AGJ,PW]; see also the important analyses of [Sat,JGK] in the scalar case. In particular, it has now been established under extremely general circumstances that both linearized and nonlinear stability are implied by spectral stability of the linearized operator about the profile, a fact that was left in doubt by earlier "direct" analyses [MN,Go.1-3,KMN,SX.1-2,S,GoM,LZ.1-2,FreL,L.1-3] carried out on a case-by-case basis. This reduces the question of stability to an ODE issue, and raises the hope that at least small amplitude stability can be treated in a uniform way across dimension and type of regularization.
On the other hand, spectral stability has up to now been verified only by energy estimates resembling those of the earliest direct analyses of [MN,KMN]; see [Z.1,HuZ,Liu,Hu.1,Z.4] for examples in various contexts. And, these have proven difficult to generalize: in particular, one-dimensional stability of general small amplitude relaxation profiles, and multi-dimensional stability of small amplitude viscous shock profiles remain outstanding open problems in the theory. In this, and a companion paper [PZ] treating the multidimensional case, we remove this troublesome gap in the theory, showing that the same Evans function tools that were used to effect the reduction to spectral stability may be used as the basis of a general procedure to verify spectral stability in the zero-shock-amplitude limit, applicable in particular in the two open cases mentioned just above. This both unifies and completes existing stability theory, at least as regards the small amplitude case; stability of large amplitude shock profiles remains as the outstanding open question in this area.
The basic idea behind our approach is a natural one suggested by Gardner and Jones [GJ.3] in the context of the one-dimensional strictly parabolic case. In the weighted energy method of , the approach on which most subsequent extensions have been based, the strict transversality of characteristic fields other than the principal one is used to "project out" behavior in transverse modes, reducing the problem to an approximate scalar conservation law in the principal characteristic field. Gardner and Jones pointed out that transverse modes correspond in the associated eigenvalue ODE roughly to "fast" and "super-slow" modes, which on a large portion of the relevant spectral domain may be projected out using dynamical system techniques to leave an approximately scalar "slow" manifold flow as in the argument of Goodman, and proposed this observation as the basis of a dynamical systems argument alternative to the one-dimensional stability argument of Goodman. However, Gardner and Jones did not examine the crucial small-frequency regime where super-slow and slow modes intermingle, and therefore eliminated only the possibility of eigenvalues with real part greater than a small, but fixed constant θ > 0; by contrast, a standard Gärding-type energy estimate gives the much stronger result Re λ ≤ Cǫ 2 , Im λ ≤ Cǫ, where ǫ := |u + − u − | denotes shock amplitude. Thus, at a technical level, substantial issues still remain; indeed, virtually all of our effort will be directed to the understanding of the "mixing" regime complementary to that considered in [GJ.3].
In the remainder of the introduction, we give an overview of the analysis and main results, deferring technical aspects to the main body of the paper.
Spectral stability. Let us first make precise the notion of spectral stability that we consider here. Without loss of generality taking s = 0 in (1.1) (by shifting to an appropriate traveling frame), we have thatū(x) is a stationary solution of the nonlinear evolution system (1.7) u t = F (u) described by its associated regularized conservation law, of form (1.3) or (1.4).
Linearizing this system about the stationary solutionū, we obtain a linearized evolution system where L := dF (ū) is the formal derivative of operator F aboutū. Remark 1.2. It has been shown in a variety of contexts, in particular, all contexts considered here, that strong spectral stability together with the "hyperbolic" conditions of inviscid stability of the corresponding ideal shock (u − , u + ) of (1.2) (dynamical, or "outer" stability) and transversality of the traveling wave connection (structural, or "inner" stability) implies linearized and nonlinear stability; see [ZH,Z.2,Z.3,. In the small amplitude case, under reasonable assumptions, inviscid stability holds always; see [M.1-3] in the one-dimensional case, FM,FZ, in the multidimensional case. Likewise, under reasonable assumptions, connections are always transverse; see discussion of existence theory given below. Thus, strong spectral stability is sufficient to imply linearized and nonlinear stability.
The canonical model. In the one-dimensional genuinely nonlinear context, it is well known that Burgers equation, approximately describes small-amplitude viscous behavior in the principal characteristic mode. In particular, the family of exact solutions (1.11)ū ǫ (x) := −ǫ tanh(ǫx/2) give an asymptotic description of the structure of weak viscous shock profiles in the principal direction, in the limit as amplitude ǫ goes to zero. Note that (1.10) is invariant under the parabolic scaling x → ǫx, t → ǫ 2 t, u → u/ǫ, whence we may immediately conclude that any unstable spectrum of the linearized operator about the profileū ǫ must lie within a ball of radius Cǫ 2 . (By standard considerations, the unstable spectrum of a second order parabolic operator is at least bounded). Burgers profiles are strongly spectrally stable, as may be readily established by a number of different techniques: for example, by L 1 contraction [Kr], Sturm-Liouville considerations [Sat,He,Z.5], or direct energy estimates as in [MN,Z.3 Appendix A.6)]. We record this fact as: Proposition 1.3. Shock profiles (1.11) (of any amplitude ǫ, by scale-invariance) are strongly spectrally stable as solutions of (1.10).
The reduced profile equations. Existence/structure of small-amplitude profiles may be determined by center manifold reduction, as pointed out by Majda and Pego [MP] in their pioneering treatment of the general strictly parabolic case; extensions to general partially parabolic, or "real" viscosity, and relaxation systems have been carried out in [P], [Yo.Z,FZe,MZ.1], respectively. The result in all cases, is that the center manifold associated with the weak profile problem (appropriately chosen) is foliated by one-dimensional fibers, on which the reduced flow, in the genuinely nonlinear case, is approximately that of the profile equation for Burgers equation (1.10).
Proof. This follows using tangency of the profile to direction r p together with the fact [Sm] that ∇a p · r p = Λ.
The reduced eigenvalue equations. Following [GJ.3], we carry out here a reduction of the generalized eigenvalue equations similar in spirit to that carried out by Majda&Pego and others for the profile equations. However, whereas those anlayses involved center manifold reduction of an autonomous nonlinear system, we shall work directly with the linear, nonautonomous system given by the eigenvalue equation (L − λ)w = 0 associated with (1.8), written as an appropriate first-order system using the tracking lemma of [GZ,ZH] (specifically, a refined version introduced in [MZ.1]) to effect the reduction to a slow manifold; see Section 2. Here, (1.18) is dimension 2n in the strictly parabolic viscous case, n + RankB in the general viscous case, and n + r in the relaxation case; as above, " ′ " denotes ∂/∂x. For the efficient coordinatization of the first-order system (1.18) in the relaxation and general viscous case (important for practical computation), we follow the scheme of [Z.3]; see sections 3-4 for details.
Of course the tracking lemma is itself an analytic formulation of classical invariant manifold techniques; see the original formulation in [GZ] in terms of projectivized flow/invariant sets of autonomous systems. The calculation of the reduced flow on the slow manifold can equally well be carried out by classical center manifold reduction; see the concurrent treatment of Plaza [Pl] in the one-dimensional case; see also the independent treatment of [FSz] (described further in the note following the introduction). Indeed, this may be preferable in the more complicated situations arising with various types of degeneracies in the underlying equations. However, there is some advantage in taking account of the linear nature of the underlying problem, in that the calculation of reduced equations simplifies considerably.
First reduction. A first application of the tracking lemma projects out "fast" transverse modes: n − 1 in the strictly parabolic viscous case, RankB − 1 in the general viscous case, and r−1 in the relaxation case, in each case reducing to an (n+ 1)-dimensional "slow manifold," of which n−1 correspond to "super slow" transverse modes ρ j , and the remaining 2 to "medium slow" Burgers modes which are roughly linearized versions of η, η ′ =: z in the limiting equations above. Rescaling by x → Λǫx/β, λ → βλ/Λ 2 ǫ 2 , and (balancing by) η ′ → βη ′ /Λǫ, we obtain a normal form for the reduced flow that has bounded (O(1)) coefficients for bounded λ and, modulo error terms Φ j (x, λ)W (W := (ρ, η, z)), becomes a block triangular system in ρ = (ρ + , ρ − ) and (η, z), of form where: θ > 0; andM 0 in the viscous case corresponds exactly to the coefficient of the eigenvalue equation for the standard Burgers equation (1.10)-(1.11), ǫ = 1, written as a first-order system. Here, E ± (·) is a smooth function satisfying the growth bounds Remark 1.6. The asymmetric bounds on E ± are a result of rescaling. The stronger bound in the upper righthand corner is actually important for our later analysis, specifically, in the large-frequency regime where we approximately undo the rescaling in the course of our argument.
Region III, the region considered by Gardner and Jones [GJ], we will denote as the "parabolic" regime, as this is the regime on which behavior is dominated by dissipativity of the underlying system (which yields "parabolic" behavior for small frequencies, as evidenced by the structure of the reduced system (1.19)-(1.20)), independent of any other structure of the traveling wave profile whatsoever. This regime may be treated by the standard "high-frequency" techniques of the usual Evans function theory (see [AGJ,GZ,ZH,MZ.1]), which are roughly equivalent to sectorial, or Gärding-type, energy estimates, to preclude altogether the possibility of eigenvalues.
Specifically, (η, z) may be resolved into decaying/growing modes η ± that have a uniform spectral gap both from each other and from ρ + and ρ − , the latter of which also have a uniform gap from one another, to an order sufficient that a second application of the tracking lemma, to the once-reduced equations (1.19)-(1.20) yields further reductions of the flow onto four transverse invariant manifolds tangent to each of the ρ ± , η ± directions. In particular, we find that the manifold of solutions decaying at +∞ is transverse to that of solutions decaying at −∞, yielding the result. In other words, the normal form for this regime is trivial, with growing and decaying modes decoupled.
Region II we denote as the "reduced parabolic" regime, since this is the regime for which similar tracking/energy estimates prohibit eigenvalues for the Burgers equation. Here also, we find that eigenvalues cannot exist, independent of the structure of the profile. However, the analysis is much more delicate; indeed, we regard this as the trickiest case in our argument. Here, we can again separate off decay/growth modes η ± (parabolic behavior of the Burgers part) as in the previous case, but ρ ± are not sufficiently separated to apply the tracking lemma to these coordinates. In this case, therefore, our ultimate reduction is to the (n − 1)-dimensional superslow equations (1.19), but with error terms now involving ρ alone and not (η, z). Note that the coefficients M ± are bounded on regime II, so that this is indeed a valid normal form.
Region I we denote as the "gap regime," in reference to the fact that on this regime we rely solely on the gap lemma of [GZ,KS,ZH,Z.3] (specifically a refined version given in [MéZ]) in our analysis of the slow flow, rather than using the tracking lemma to carry out a further reduction as in Regions II and III. The "gap" in "gap lemma" refers in fact to absence of spectral gap, as is the case in this regime; the lemma asserts that uniform exponential convergence of coefficients as x → ±∞ and continuous extension of the stable/unstable eigenspaces of the limiting coefficient matrices A ± (λ) at x → ±∞ can substitute in this situation for uniform spectral gap, to yield estimates valid on x ≷ 0. In this regime, which is the crucial one determining behavior, the normal form is the entire (n + 1)-dimensional system (1.19)-(1.20).
The gap lemma: convergence to limiting flows. We complete our reduction in each case by an application of the gap lemma (specifically, a refinement given in [MéZ]), showing that the flow of the normal forms we have obtained con-verges to that of the "limiting" normal forms obtained by dropping error terms ΘW , etc.: more precisely, the flows associated with the stable manifold at +∞ and the unstable manifold at −∞, or in some cases their analytic continuation into the essential spectrum. The basic approach is to: (i) conjugate approximate and limiting equations on the half-lines x ≷ 0 to their asymptotic systems at ±∞ by a linear change of variables uniformly close to the identity (an application of the gap lemma, made possible by the uniform exponential decay estimate (1.21)), then (ii) check by direct linear algebraic computation that the stable and unstable subspaces of the respective asymptotic coefficient matrices are uniformly close. In the present applications, (ii) is straightforward, since the asymptotic matrices of approximate and limiting flows agree for the ρ equation, while for the (decoupled) (η, z) equation, the stable and unstable subspaces for the limiting equations are, under our strategically chosen rescaling, uniformly spectrally separated on the region I where it comes into play.
Convergence of the stable flow at +∞ and unstable flow at −∞ then implies uniform convergence as ǫ → 0 of suitably chosen Evans functions associated with the approximate and limiting normal forms. The Evans function is a Wronskian measuring solid angle between the stable and unstable subspaces of a given eigenvalue ODE written as a first-order system, constructed so as to be analytic in the frequency λ on a "region of consistent splitting" (defined (A1), Section 2). In each of the contexts considered here, the region of consistent splitting includes the entire set of frequencies {Re λ ≥ 0} \ {0} of interest; see, e.g., [Z.3]. Moreover, both the Evans function and its component subspaces extend continuously along rays in frequency space, in both the one-and multidimensional case [ZS,Z.3] to the closure at the origin, making possible uniform estimates of transversality; indeed, in the one-dimensional case considered here, the Evans function may in fact be extended analytically through the origin, without restricting to rays [GZ]. (Note: in the multidimensional case, the Evans function may be holomorphically extended along rays [ZS,Z.3]; however, in certain "glancing" directions, there arise branch singularities at the origin.) On the region of consistent splitting, the zeroes of the Evans function correspond both in location and multiplicity with the eigenvalues of the associated linear operator; for details, see ZH].
The multiplicity of the generalized eigenvalue at the origin λ = 0 is directly calculable and related to hyperbolic stability, as shown in the one-and multi-dimensional case in [GZ,BSZ] and [ZS], respectively. In the present, small-amplitude case, it is readily seen to be the same as the multiplicity for the limiting normal form equations, namely one. In the one-dimensional, analytically extendable case, we may thus conclude immediately from uniform convergence to the limiting Evans function (by degree, i.e., winding number, considerations) that strong spectral stability of the original system is equivalent to strong spectral stability for the limiting normal forms. Alternatively, we may remove the zero of the Evans function at the origin by working instead with the "integrated" eigenvalue equations following [MN,Go.1] or with "flux variables" following [Go.2], to recover the same conclusion using only continuity at the origin of D.
In this paper, we follow the latter strategy exclusively. In the multidimensional analysis of [PZ], we use a modification of the same strategy, specifically, an interesting variation intermediate to flux and integrated variable methods, neither of which themselves directly generalize to multidimensions in a useful way, to achieve the same result; this "balanced flux form" is quite similar to that introduced in the treatment of relaxation systems in Section 4 of the present paper. The former strategy is perhaps viable as well, using computation of winding number on Riemann surfaces as suggested in [GZ]; however, this would involve extra bookkeeping (specifically, tracking of branch points in approximate vs. limiting systems) that we prefer to avoid.
Conclusions/summary of the main results. From this point, we may immediately deduce our main results. For, in each of the (nontrivial) limiting normal forms arising in regions I-II, the equations for ρ ± completely decouple, and clearly support no "decaying" solutions other than the trivial, zero solution (in general, "decaying" is defined as lying in ρ + direction at +∞ and ρ − direction at −∞; away from the essential spectrum, this is equivalent to actual decay at ±∞). Thus, we may conclude that ρ ≡ 0 for any generalized eigensolution, leaving a decoupled (η, z) equation in region I, agreeing with the generalized eigenvalue equations for canonical model (1.10), and the trivial flow in region II. Recall that the reduced flow was already trivial in region III. Likewise, ν ≡ 0 in each of these regions, since fast growing and fast decaying modes completely decouple.
Noting that the canonical model, being among the class considered, is subject to the same reduction in regions II-III, we may conclude, finally, our main result: that strong spectral stability is equivalent to strong spectral stability of the canonical model, at least modulo the degenerate case that the canonical model yields nontrivial nonstable eigenvalues lying precisely on the imaginary axis; see Proposition 3.2 and Remark 3.3 below. As a corollary, recalling the result of Proposition 1.3, we obtain the stated results of strong spectral stability of small-amplitude profiles, across the general class of models considered in (1.3) and (1.4). More precisely, in this paper, we carry out the details of the one-dimensional case for general strictly parabolic viscosities, and for general relaxation systems. The case of general real viscosity may be carried out similarly as the relaxation case, following the dual framework set out in Appendix A.2 of [Z.3]; we omit these calculations, as satisfactory onedimensional results already exist [HuZ].
Plan of the paper. In Section 2, we recall and slightly extend the basic Evans function tools we will need. In Section 3, we carry out the one-dimensional, strictly parabolic viscous case. For clarity, we first carry out in its entirety the simpler identity viscosity case, afterward indicating the necessary adjustments in the general case. In Section 4, we carry out the general relaxation case, for clarity first treating the 2 × 2 case by reduction to the scalar viscous case; we note that this gives an independent proof of stability in this case, different from that of Liu [L.2]. Finally, in Section 5, we briefly indicate the extension of our analysis to the multidimensional viscous case; details will be given in [PZ].
Note: Shortly before the completion of this analysis (precisely: after the completion of the one-dimensional viscous case and before the completion of the relaxation case), we have learned of closely related work of H. Freistühler and P. Szmolyan [FreS] in which they establish a reduction method similar (indeed, perhaps equivalent) to ours, but proceeding entirely from the point of view of geometric singular perturbation theory. In the cited work, the first in a planned series of three, they carry out a complete analysis of the one-dimensional identity viscosity case, announcing the intention to treat one-dimensional relaxation/real viscosity and multidimensional stability in papers two and three, respectively.
Section 2. Evans function framework. We begin our analysis by recalling, and in some cases slightly extending, the Evans function tools we will need. For later reference, we carry out the analysis in generality sufficient for the multidimensional case as well.
2.1. The gap lemma and convergence of approximate flows. Consider a family of first-order systems of unstable-neutrally stable frequencies, ǫ varies within some given open set V, and x varies within R 1 . Equations (2.1) are to be thought of as generalized eigenvalue equations, with parameter λ representing a Laplace transform frequency in time, andξ representing a Fourier transform frequency in directions of spatial symmetry; in the present, one-dimensional case,ξ ≡ 0. In the applications of this paper and of [PZ], ǫ will be just the shock strength |u + − u − |.
We make the following basic assumptions.
These are satisfied in all the contexts considered in both this paper and [PZ]; see [Z.3]. Condition (A0) is induced, at least for ǫ bounded from zero, by the origins of (2.1) through the linearization about smooth shock profiles of (1.3) or (1.4). Uniform approach (2.3) as ǫ → 0 will be recovered through reduction/rescaling. Condition (A1) may be recognized as the standard hypothesis of "consistent splitting," as introduced in [AGJ], and follows by the assumption of dissipativity of systems (1.3) and (1.4). Condition (A2) is an extension suitable for multidimensions of the "geometric separation" hypothesis of [GZ] and is a consequence of the hyperbolic structure of (1.2). (Note: separation was not needed for the arguments of [GZ], and in fact does not hold in multidimensions; see discussion of Appendix The gap lemma. Under these circumstances, the "gap lemma" established in various degrees of generality in [GZ,KS,ZH,Z.3] asserts that behavior of the variablecoefficient equation (2.1) may be related to that of its constant-coefficient limiting systems on x ≷ 0, while maintaining the assumed regularity in all parameters. This has recently been greatly improved in [MéZ], in the form of the following conjugation lemma, a version that is particularly convenient in applications.
(ii) The change of coordinates W := P ǫ ± Z reduces (2.1) on x ≥ 0 and x ≤ 0, respectively, to the asymptotic constant-coefficient equations Proof. As described in [MéZ], this is a straighforward corollary of the gap lemma as stated in [Z.3], applied to the "lifted" matrix-valued equations for the conjugating matrices P ǫ ± . (A1), chosen with the same regularity assumed on S ǫ + , U ǫ − (note: that such a choice is possible is a consequence of standard matrix perturbation theory [Kat]), and P ǫ ± be transformations defined as above on some neighborhood of (ξ, λ, ǫ ∈ Ω × V. Then, the local Evans function for (2.1) associated with these choices is defined as Remark 2.3. Evidently, the choice of local Evans function is highly nonunique. Though we shall not need it here, it can be shown using the original formulation of the gap lemma in [GZ,KS] that a globally analytic choice is possible in both the one-and multidimensional case; see [GZ] and [ZS,Z.3], respectively.
Combining (2.5) and (2.6), we immediately obtain the following result sufficient for our needs. 1 Proof. Evidently, the first K columns of the matrix on the righthand side of (2.6) are a basis for the stable manifold of (2.1) at x → +∞, while the final N − K columns are a basis for the unstable manifold at x → −∞. Thus, its determinant vanishes if and only if these manifolds have nontrivial intersection, and the result follows.
Convergence of approximate flows. We now turn to the crucial question: Under what circumstances does the Evans function of (2.1) converge as ǫ → 0 + to the Evans function of the limiting equations at ǫ = 0? The following proposition gives a simple and sharp answer.
Proposition 2.5. Suppose that, in an Ω-neighborhood of some (ξ, λ): (i) As ǫ → 0 + , the asymptotic subspaces S ǫ + , U ǫ − converge uniformly in angle to S 0 + , U 0 − , with rate δ(ǫ): equivalently, for 0 < ǫ ≤ ǫ 0 , their spanning bases satisfy (ii) The coefficient matrices A ǫ converge uniformly exponentially to their limits, in the sense that, for 0 < ǫ ≤ ǫ 0 , Then, on some (possibly smaller) Ω-neighborhood of (ξ, λ), the local Evans function D ǫ defined as in Definition 2.2 converges uniformly as ǫ → 0 + to D 0 , with (More precisely, the columns of the defining matrix on the righthand side of (2.6) converge uniformly, with the same rate CC 1 C 2 η(ǫ).) Proof. Clearly, it is sufficient to prove the stronger parenthetical assertion of convergence of individual columns, and for this we may restrict without loss of generality to the + columns, and the right half-line x ≥ 0. Applying the conjugating transformation W → (P 0 + ) −1 W for the ǫ = 0 equations, we may reduce to the case that A 0 is constant, and P 0 − ≡ 0. In this case, (2.8) becomes just and we obtain directly from the conjugation lemma, Lemma 2.1, the estimate and in particular The result now follows, from (2.6), (2.7), and (2.10).
Remark. The same argument shows that the proposition is sharp, since easy examples show that the conjugation lemma itself is sharp.
2.2. The tracking lemma and reduction of the flow. Next, consider complementary situation of a family of equations of form (2.1) on an ǫ-neighborhood for which the coefficient A ǫ does not exhibit uniform exponential decay to its asymptotic limits, but instead is slowly varying. This occurs quite generally for rescaled eigenvalue equations arising in the study of the large-frequency regime; see, e.g., [GZ,ZH,MZ.1,Z.3]. In the present context, it arises for the unrescaled equations in the small-shock strength limit ǫ → 0.
In this situation, it frequently occurs that not only A ǫ but also certain of its invariant (group) eigenspaces are slowly varying with x, i.e., there exist matrices where " ′ " as usual denotes ∂/∂x. In this case, making the change of coordinates W ǫ = R ǫ Z, we may reduce (2.1) to the approximately block-diagonal equation where M ǫ is as in (2.12), Θ ǫ (x) are uniformly bounded matrices, and δ ǫ (x) ≤ δ(ǫ) is a (relatively) small scalar. A sometimes crucial improvement may be obtained by arranging that error Θ vanish on diagonal blocks, i.e., that L j R ′ j ≡ 0; see [Go.1-2,MZ.1]. Here, we shall make use of this observation only in the principal, 2 × 2 "Burgers" block.
Let us assume that such a procedure has been successfully carried out, and, moreover, that there exists a uniform spectral gap, in the strong sense that (2.14) Re In the "standard" case that M ǫ are uniformly bounded and η(ǫ) may be taken independent of ǫ, that (2.14) may be arranged by a suitable coordinate transformation provided that there holds the weaker condition for someη > η. Then, there holds the following reduction lemma, a refinement established in [MZ.1] of the "tracking lemma" given in varying degrees of generality in [GZ,ZH,Z.3]. The new feature of the reduction lemma is that it asserts the existence of smooth invariant manifolds in the vicinity of the first and second block coordinates, whereas the tracking lemma(s) assert only the existence of forward and backward attracting cones in the same vicinity and do not give a reduction in the usual dynamical systems sense.
From Proposition 2.6, we obtain reduced flows on the two invariant manifolds described. Should we arrange that Θ 11 ≡ 0 or Θ 22 ≡ 0, then the error terms would become O(δ ǫ (x)δ(ǫ)/η(ǫ))Z ǫ 1 or O(δ ǫ (x)δ(ǫ)/η(ǫ))Z ǫ 2 , respectively; that is, enforcing vanishing of Θ on a diagonal block reduces the error term by factor δ/η. Remark 2.7. As pointed out in [MZ.1], the gap condition (2.14) may be allowed to fail by order α on an interval of length order 1/α, for any fixed α. For the applications of this paper and of [PZ], as for those of [MZ.1], this means that the gap condition need only be checked at x = ±∞.
Remark 2.8. As described in [MZ.1], provided that M 1 and M 2 are (uniformly) bounded and spectrally separated, one can in fact repeat the procedure used to construct (2.13) to obtain an approximate block-diagonalization to arbitrarily high order: i.e., an asymptotic expansion in powers of δ(ǫ). In general, e.g., in the large-amplitude situation considered in [MZ.1], these higher order terms are not explicitly computable. However, we point out that, in the present small-amplitude situation, they may in principle be determined exactly to any desired order, using the error expansion for the traveling wave profile to approximate L ′ ; indeed, this can be recognized as just a variation of the usual center manifold calculations used to approximate center manifold flow. In the calculations of this paper and of [PZ], we shall not need to perform any such higher-order corrections: a single iteration will be enough.
2.3. Reduction to small frequency. As a first application of Proposition 2.6, we obtain the following preliminary result analogous to that of [GJ], reducing the small-amplitude stability problem to the small frequency, or "diffusive," regime |λ| << 1.
Proof. Equivalently, we show, for arbitrary r > 0, that |λ| ≤ r for ǫ sufficiently small. For |λ| > r and A ǫ as in (2.1) denoting the coefficient of the generalized eigenvalue problem (1.8) written as a first-order system, there is a uniform spectral gap between stable and unstable subpaces of A ǫ (x), independent of 0 < ǫ ≤ ǫ 0 and x; as discussed in [Z.3], this is a straightforward consequence of the assumed dissipativity of the systems (in the sense of Kawashima-Zeng [Kaw,Ze]). By standard matrix perturbation theory [Kat] plus boundedness (compactness) of A ǫ , we may therefore deduce the existence of well-conditioned bases (2.11) for these subspaces varying smoothly with A ǫ , and therefore satisfying by Proposition 1.4. Moreover, by a proper choice of basis, we may arrange (2.14) with η > 0 uniformly bounded, independent of 0 < ǫ ≤ ǫ 0 and x; for details, see Appendix A4, [Z.3].
Applying Proposition 2.6, we obtain a pair of reduced systems (2.16) and (2.17) for which the coefficent matrices (with O(ǫ 2 ) error term taken into account) are, respectively, uniformly positive and negative definite, and solutions therefore uniformly exponentially grow and decay, provided that ǫ is sufficiently small. It follows that the equations have no solutions bounded at plus and minus spatial infinity save for the trivial solution W ≡ 0.
For |λ| ≥ R, R > 0 sufficiently large, on the other hand, the nonexistence of eigenvalues may be established independent of the size of ǫ by a rescaling followed by a similar reduction argument; see [ZH,. for a multidimensional version, see Lemma 4.38, [Z.3].
Remark 2.10. The same argument applies word for word in the multidimensional case, to yield |(ξ, λ)| << 1, whereξ denotes the Fourier transform frequency in directions parallel to the shock front, and λ as here denotes the Laplace transform frequency with respect to time.
(H4) a p = 0 is a simple eigenvalue of Df (u 0 ) with left and right eigenvectors l p and r p , and l t p D 2 f (r p , r p ) = 0 (genuine nonlinearity of the principal characteristic field).
More precisely, we shall establish the following proposition, from which, together with Proposition 1.3, Theorem 3.1 follows as an immediate corollary.
Proposition 3.2. Under assumptions (H0)-(H4), profilesū ǫ are strongly spectrally stable (and therefore linearly and nonlinearly orbitally stable [ZH,Z.2]) for ǫ sufficiently small, if the standard Burgers profile (1.11) is strongly spectrally stable for ǫ = 1 and only if the Burgers profile is not strongly spectrally unstable for ǫ = 1 in the sense that there exists an eigenvalue Re λ > 0.
Remark 3.3. Of course, the "only if" part of Proposition 3.2 follows vacuously in the case at hand; the point is that we obtain this information by our method of proof, independent of any knowledge of behavior of the the limiting system. This distinction may be important in future applications to more general situations.
Remark 3.4. Theorem 3.1 recovers the spectral stability result obtained by energy methods in [Go.1-2], and slightly extends it from the class of A, B such that LBR > 0 for some diagonalizing bases L, R = L −1 of A to the class of all "stable" pairs A, B in the sense of Majda&Pego [MP]. (In the case that A, B are simultaneously symmetrizable, these two classes are equivalent; see [MP].) It is straightforward with some additional bookkeeping (see [Z.3,HuZ]) to extend our results to the case, to which the nonlinear analyses of [ZH,Z.2] also apply, that (H2) is relaxed to: (H2') σ(Df ) real, semisimple.
3.1. The case B ≡ I. To illustrate the method, we first carry out the proof of Proposition 3.2 and Theorem 3.1 in the setting of identity viscosity B ≡ I, for which the associated linear algebra is particularly simple.
Writing (3.4) as the family of first-order systems we are ready to begin our analysis.
Lemma 3.6. The eigenvalues µ and associated left and right eigenvectors L, R of A ǫ (x) in (3.6) may be denoted as where a j (x) denote the eigenvalues of A ǫ (x) and l(x), r(x) associated left and right eigenvectors, normalized so that l j r k = δ j k (here, we have suppressed the ǫ dependence for the sake of readability).
By (H2) and (H4), we may order the (real) eigenvalues of A ǫ as (3.9) a 1 , . . . , a p−1 ≤ −θ < 0, and (3.11) 0 < θ ≤ a p+1 , . . . , a n , where Λ and β denote the genuine nonlinearity and effective diffusion coefficients described in the introduction, andη the standard Burgers profile (1.10). Like the principal eigenvalue a p , the transverse eigenvalues a j , j = p vary within an O(ǫ) neighborhood of the corresponding eigenvalues of Df (u 0 ). Evidently, for the indices j = p such that a j is bounded from zero, the eigenvalues µ ± j are jointly analytic in a j and thus A ǫ and λ within neighborhoods of Df (u 0 ), 0; a brief calculation shows that L ± j and R ± j are bounded, hence analytic in these parameters as well. The two-dimensional group eigenspace spanned by L ± p and R ± p likewise varies analytically as subspaces, as A ǫ and λ are varied; however, the individual eigenvectors that make it up do not (indeed, blow up for a p = 0 and λ → 0). We therefore do not attempt to resolve these subspaces, but just choose convenient analytically varying spanning bases Summing up, we have: Lemma 3.7. In the transverse fields j = p, there hold expansions for j < p and (3.15) µ + j = a j + . . . , µ − j = −λ/a j + λ 2 /a 3 j + . . .
for j > p, analytic in λ and smooth (indeed, analytic) in a j , or, through a j , in A ǫ , on sufficiently small neighborhoods of λ = 0 and a j = a j (u 0 ) or A ǫ = Df (u 0 ). Likewise, the eigenvectors L ± j , R ± j , j = p possess the same regularity, as do bases L η , L z and R η , R z (defined in (3.12)-(3.13)) for their complementary two-dimensional invariant subspaces.

Proof. Taylor expansion/direct calculation.
Approximate block-diagonalization. Following the procedure outlined in Section 2.2, define now ). This is just such an approximately block-diagonalizing transformation as described in Section 2.2: in particular, because L ǫ , R ǫ depend smoothly on A ǫ , we have that where η := l p (u − )·(ū ǫ −(u − +u + )/2) as in the introduction. Thus, in Z coordinates, where, for Re λ ≥ 0:
We could carry out a similar normalization at the L j , R j level (with α j now matrixvalued) to annihilate all diagonal blocks Θ jj ; see Lemma 4.9 of [MZ.1]. However, we do not require this for our argument. (In fact, we only require vanishing in the lower lefthand corner Θ z,η = 0 of the η, z block, which is automatic; however, we wish to point out early on this more general procedure.) First reduction. Observing that coefficients M ν − and M ν + of the "fast transverse modes" ν − and ν + are separated by a uniform spectral gap both from each other, and from the "slow" modes (ρ − , η, z, ρ + ) t , we may apply Proposition 2.6 twice to reduce (2.13) to decoupled flows on three invariant manifolds, associated respectively with ν − , ν + , and (ρ − , η, z, ρ + ) t , appearing as where, by (3.25), is small on the principal block: (3.39)Θ ǫ (η,z),(η,z) = O(δ(ǫ)) = O(ǫ 2 ).
Observing that the flow is uniformly exponentially decreasing for (3.36) and increasing for (3.37), we find that the only possible decaying solution are the trivial ones ν ± ≡ 0, and so we may discard these "fast transverse" equations, leaving us with (3.38). Rescaling now by x → Λǫx, λ → λ/Λ 2 ǫ 2 , z → z/Λǫ, we obtain the block-triangular system of equations (1.19)-(1.24) described in the introduction, modulo coefficient errors withΘ ǫ uniformly bounded and O(ǫ) on the principal, (η, z) block), and E ǫ ≡ 0 in the ρ ± equations and for the (η, z) equations is by Corollary 1.5. Rearranging, we have the error bounds asserted in the introduction. Note that the forcing term Further reductions/Normal forms. Loosely following the introduction, we now examine separately each of the three regimes (with respect to the rescaled variable λ) I. |λ| ≤ C, II. C ≤ |λ| ≤ Cǫ −1 : and III. Cǫ −1 ≤ |λ| ≤ C −1 ǫ −2 , where C > 0 is a sufficiently large constant to be determined later. (Recall: In Proposition 2.9 we have already disposed of the case |λ| ≥ C −1 ǫ −2 .) Region I. On region I, the rescaled ρ ± , (η, z) equations evidently have bounded coefficients satisfying condition (ii) of Lemma 2.5, with η(ǫ) = O(ǫ). Moreover, the asymptotic coefficients at plus and minus spatial infinity are block-diagonal, with M + and M − positive and negative definite, respectively, and M 0 =M 0 + O(ǫ), whereM at x = ±∞ have uniform spectral gap between stable and unstable eigenvalues for all Re λ > 0; it follows by standard matrix perturbation theory that the stable and unstable subspaces for the asymptotic coefficient matrices of the ǫ-approximate system are within O(ǫ) of those of the limiting, ǫ = 0 system, verifying condition (ii) of Lemma 2.5, with η(ǫ) = O(ǫ). Thus, we may conclude from the Lemma that the approximate system has a local Evans function D ǫ that is O(ǫ) close to the local Evans function (with some chosen basis) D 0 for the limiting system with error terms dropped. As described in the introduction, the limiting ρ ± equations decouple, and are readily seen to admit only trivial L 2 solutions, and the limiting (η, z) equation with ρ ± set to zero is just the Burgers eigenvalue equation. Since the latter is known to admit no decaying solutions for λ = 0, we thus have that the limiting Evans function D 0 has no zeroes on Region I if the Burgers equation is strongly stable on Region I, and has a zero of strictly positive real part if the Burgers equation is strongly unstable on Region I (possesses a positive real part eigenvalue). By uniform convergence/analyticity in λ, we may conclude the same regarding the zeroes of the approximate Evans function D ǫ , and thus also the eigenvalues of the original operator about the wave.
Region II. On Region II, the eigenvalues of the (η, z) ("Burgers") blockM 0 satisfy with also |μ ± p | = O(|λ| 1/2 ). Thus, "balancing" the reduced equations by the rescaling z → z/|λ| 1/2 , we obtain well-conditioned basis vectors for this block. At the same time, the forcing matrix N of (1.23) becomes now O(|λ| −1/2 ) (note: we have partially undone the original rescaling z → z/ǫ), and may be viewed as an error term of order O(C −1/2 ): uniformly small, but not vanishing as ǫ → 0. Indeed, reviewing the origins of various error terms, we find that this is the dominant error term within both of Regions II and III, with error terms in the ρ ± equations retaining their originally stated form.
DiagonalizingM 0 by a well-conditioned change of basis, we thus obtain (in rescaled coordinates) an approximately block-diagonalized system where η ± p denote the now-separated Burgers modes. Observing that theμ ± p are separated by uniform spectral gap η|λ| 1/2 both from each other and from the O(ǫ|λ|) blocks M ± on all of Region II (indeed, on Region III as well), we may therefore apply Lemma 2.6 as second time to further reduce to decoupled equations on three invariant manifolds, associated respectively with η − p , η + p , and (ρ − , ρ + ), appearing as whereδ ǫ is still O(ǫe −θ|x| ) as in the original rescaling: note that we are here using in a crucial way the fact that the error terms in ρ ± coordinates alone are better than the O(|λ| −1/2 error terms in the η ± p blocks, which were the ones (recorded in the uniform description (3.46)) that led to the error/gap ratio |λ| −1/2 /|λ| 1/2 ∼ |λ| −1 determining the tracking angle in Proposition 2.6.
Noting as usual that η ± p flows are uniformly exponentially growing/decaying, so support only trivial decaying solutions, we may discard the first two equations, leaving us with the (ρ − , ρ + ) equation alone. Noting that coefficients M ± = O(ǫλ) for these"superslow" modes are bounded on Region II, we may then apply Lemma 2.5 to obtain that suitably chosen local Evans functions for this superslow flow converge uniformly (as O(ǫ)) to that for the limiting equations with error terms omitted. But, just as in the treatment of Region I, the limiting ρ ± equations decouple into uniformly exponentially growing/decaying flows, which evidently support no nontrivial decaying solutions. Thus, we may conclude, finally, that there exist no decaying solutions of the reduced equations, and therefore no eigenvalues for the original equations, for λ within Region II, and ǫ sufficiently small.
Region III. The treatment of Region III is similar to but somewhat simpler than the treatment of Region II. For, as already pointed out, all steps in the reduction to ρ ± equations (3.49) remain valid on this region as well. Noting that, by (1.24), M + and M − have uniform spectral gap of order θC 2 ǫ on this region, θ > 0 fixed and C > 0 arbitrarily large, whileδ ǫ remains uniformly O(ǫ), so we may in this case apply Lemma 2.6 to obtain the further reduction to decoupled equations on two invariant manifolds associated with modes ρ − and ρ + , appearing as But, these, having negative and positive definite coefficients are uniformly exponentially decaying and growing, respectively, so support no nontrivial decaying solutions. We may thus conclude that there exist no eigenvalues in Region III, for ǫ sufficiently small.

Conclusions.
Collecting the results for each regime, we obtain the result of Proposition 3.2, and therefore of Theorem 3.1. This completes the analysis of the case B ≡ I.
3.2. The general strictly parabolic case. The general case follows quite similarly as in the case B ≡ I, in terms of overall argument structure. However, there is an important new technical aspect introduced here, in the way that we carry out the linear algebra underlying the initial reduction to "slow" coordinates (ρ ± , η, z).
For general B, the linearized eigenvalue equations (3.2) become and the associated integrated equations become We express these as a family of first-order systems Note that A ǫ lies within O(|(ū ǫ ) ′ |) of dF (ū ǫ ), hence is likewise strictly hyperbolic, with eigenvalues a j and eigenvectors l j and r j lying within O(|(ū ǫ ) ′ |) of those of dF (ū ǫ ). For purposes of our argument, therefore, A ǫ and Df (ū ǫ ) are indistinguishable; in particular, a j ≤ −η < 0 for j < p, a j ≥ η > 0 for j > p, and the principle eigenvalue a p after rescaling approaches to the Burgers profileη with the same rate predicted in Proposition 1.5. (This last is the main point, and holds also in the more general setting alluded to in Remark 3.4, with (H2) replaced by (H2').) For simplicity of exposition, we make the provisional hypothesis: This concerns only the transverse modes j = p, since the argument of [MP] shows that zero is a simple eigenvalue under assumptions (H1), (H3), and is easily removed at the expense of further bookkeeping. The next lemma furnishes the linear algebra necessary to carry out the reduction to slow coordinates in the general case; for similar arguments, see [ZH,Z.3,MZ.1].

and
(3.66) where (3.67) l j r * = γ j l j s p , l * s j = −l p s j /γ j , for j = p and (3.68) l p r * + l * s p = 0.
Proof. By standard matrix perturbation theory, we have analytic dependence in (A ǫ , B ǫ ) and ξ of the eigenvalues λ j and eigenvectorsl j ,r j of the symbol (3.70) λ j = −iξa j − ξ 2 β j + . . . , l j = l j + . . . , r j = r j + . . . , about (Df (u 0 ), B(u 0 ), 0, where β j := l j B ǫ r j . For each j = p, a j is bounded from zero, and so (3.70) may be inverted, by the Analytic Implicit Function Theorem, to yield analytic series for µ := iξ and thereforel j ,r j in terms of λ, (A ǫ , B ǫ ). This verifies expansions (3.57), (3.63) for the "slow" transverse modes; the corresponding eigenvector expansions (3.58), (3.64) then follow by the relations betweenl j ,r j and L j , R j ; for similar arguments.
Expansions (3.59), (3.60) and (3.61), (3.62) for the "fast" transverse modes follow in more straightforward fashion by (3.56) and the analytic dependence of isolated eigenvalues/eigenvectors of A ǫ at λ = 0. Finally, analytic dependence of the complementary two-dimensional invariant subspaces follows by duality, whence (see [Kat]) we obtain the existence of analytic bases (L η , L z ), (R η , R z ). By inspection, the choices described are valid representatives at λ = 0. Now, similarly as in the case B ≡ I, we make the definitions (3.16)-(3.22), to obtain an approximately block-diagonal system satisfying (3.23)-(3.31), but with (3.32) now replaced by (3.71) where, in the crucial lower lefthand corner of the final (error) matrix, we have used the special properties to obtain the stated bound. The O(·) terms in (3.71) may be expressed more precisely as with coefficients O j analytic in all parameters. Finally, we observe as in [MZ.1] that we may achieve L 0 R ′ 0 ≡ 0 by a renormalization L 0 → α −1 L 0 , R 0 → R 0 α, where α is a 2 × 2 matrix defined by ODE (3.74) α ′ = −L 0 R ′ 0 α, α| x=0 = I. At λ = 0, L 0 R ′ 0 is initially upper-triangular, hence α is as well. In other words, the form of L 0 , R 0 are preserved, and (3.74) amounts to the pair of vector normalizations l p r ′ p ≡s p s ′ p ≡ 0 together with the condition l p r ′ * + l * s ′ p ≡ 0 restricting choices of l * , r * . This does not affect any of the previously stated equations or bounds, but only accomplishes again that error Θ vanish on the η, z block: (3.75) Θ (η,z),(η,z) ≡ 0.
Lemma 3.9. Matrix M 0 defined in (3.71) satisfies and (3.77) Proof. Taylor expanding about a p = 0, we have (up to scalar multipliers) and with O j analytic. From these expansions, together with the fact that A, B, l p , l * , r p , r * ,s p , s p , and a p , as well as coefficients O j in (3.73), all approach their asymptotic values as x → ±∞ at rate proportional to |ū ǫ − u ± | ∼ ǫe −θǫ|x| , the result readily follows.
From this point on, the argument goes almost exactly as in the case B ≡ I. Carrying out the first reduction eliminating fast transverse modes, and rescaling by x → Λǫx/β, λ → βλ/Λ 2 ǫ 2 , z → βz/Λǫ, we obtain again the block-triangular system (1.19)-(1.24) described in the introduction, with the errors (1.26) there described. The latter errors are slightly larger than occurred in case B ≡ I; however, they are sufficient that the arguments in each of Regions I-III carry through exactly as before. This completes the proof of one-dimensional spectral stability in the general strictly parabolic case.
Remark 3.10. The crucial estimate of the lower lefthand entry in the final (error) matrix of (3.71) is a generalization of the standard observation, in the case of a one-parameter bifurcation in λ from an isolated semisimple block L(0), R(0) of matrix A(λ), L(0) and R(0 consisting of genuine eigenvectors with a single common eigenvalue µ(0), that In the case at hand, which is essentially a two-parameter bifurcation in λ and a p from a Jordan block at λ = 0, a p = 0, the computation (3.80) of course breaks down in general, but does remain valid for the single, lower lefthand entry of the block that corresponds to the pairing of genuine left and right eigenvectors L z and R η , respectively, to give the stated result of validity to order O(|λ| 2 + |λ||a p |). This explains the success of the calculation. Indeed, the entire calculation could have been done by expansion about λ = 0, a p = 0, giving slightly degraded error terms O(|λ| + |a p |) in other entries, since these would in fact have been sufficient for the analysis; however, we prefer to carry a p within the computation, since it is no more difficult and gives sharper estimates.
(H4) a p = 0 is a simple eigenvalue of Df (u 0 ) with left and right eigenvectors l p and r p , and l t p D 2 f (r p , r p ) = 0 (genuine nonlinearity for the equilibrium system of the principal characteristic field).
As in the previous section, ǫ > 0 denotes shock strength |u ǫ + − u ǫ − |, and profiles (ū ǫ ,v ǫ )(·) are assumed to converge as ǫ → 0 to (u 0 , v 0 ) (where necessarily v 0 = v * (u 0 )), under which assumptions the center-manifold argument of [MZ.1] verifies the assertion of Proposition 1.4, yielding convergence after rescaling ofū ǫ to the standard Burgers profile (1.11); see also related analyses of [Yo.Z,FZe]. In the rest of this section we establish the following analogs of Theorem 3.1 and Proposition 3.2.
Proposition 4.2. Under assumptions (H0)-(H4), profilesū ǫ are strongly spectrally stable (and therefore linearly and nonlinearly orbitally stable [MZ.1]) for ǫ sufficiently small, if the standard Burgers profile (1.11) is strongly spectrally stable for ǫ = 1 and only if the Burgers profile is not strongly spectrally unstable for ǫ = 1 in the sense that there exists an eigenvalue Re λ > 0. Remark 4.3. The "real," or partially parabolic case is closely related to the relaxation case, and may be treated by quite similar analysis, following the framework set up in Appendix A2 of [Z.3]; see [Z.6]. In the real viscous case, satisfactory small-amplitude stability results have already been obtained by energy methods [HuZ].
4.1. The 2 × 2 case. For clarity, we first carry out the 2 × 2 casef ,g, q, f ∈ R 1 , which was treated by energy methods in [L.3]. The linearized eigenvalue equations about profile (ū ǫ ,v ǫ ) are Following [Z.3], we make the (nonsingular, by (H1)) change of variables to obtain a first-order system So far, this is analogous to the "flux transform" of [Go.2]. If we were now to convert to a single, second-order equation forf , then the analogy would be exact, since the equations would then clearly not support any L 2 (decaying) solutions at λ = 0. Instead, we take a different approach, keeping the convenient variablesf ,g, and rescaling as (4.6)f →f /λ,g →g, to obtain in place of (4.5). Sincef ′ = λu in the original equations (4.2), we see that the rescaledf variable is, in the case of decaying solutions, given byf (x) = x −∞ u(z)dz, i.e., is exactly the integrated variable of standard energy analyses [L.3,Liu,HuZ]; in particular, the rescaled equations (4.7) have the favorable property that they do not support L 2 (decaying) solutions at λ = 0.
Remark 4.4. Equations (4.7) are neither pure flux nor pure integrated form, but a new form intermediate to the two, which we might be called "balanced flux form." This form will prove useful again in the multidimensional case treated in [PZ]. (All three forms (flux, integrated, and balanced flux) coincide in the one-dimensional viscous case, so this distinction is not apparent, or necessary, here.)
Lemma 4.5. There hold det A = −f v gu +g v a 1 = −f v g u + O(ǫ) and In particular, det A < 0 and sgnf v = sgn g u = 0, for ǫ sufficiently small.

Proof.
A column operation reduces det A to det f ufv g ugv , from which the first assertion follows. Combining with (4.8), we obtain the second assertion.
Remark 4.6. Condition det A < 0 for a 1 near zero may be recognized as the subcharacteristic condition, which in the 2 × 2 case is equivalent to (H3). Conditioñ f v = 0 is the "genuine coupling" assumption of Liu [L.3]; as we have just shown, it is in fact a consequence of (H3), and not a separate assumption.

Now, setting
β := β 1 A/q v (0), rescaling x → Λǫx/β, λ → βλ/Λ 2 ǫ 2 , and balancingf → −g u βf /Λǫ, we obtain the normal form f g where, by Lemma 4.5, error term Φ satisfies bounds (1.22)-(1.26) of the introduction: that is, just an approximate Burgers eigenvalue equation. From this form we may immediately conclude the stability results, by a simplified (because transverse, ρ terms are absent) version of the arguments of the previous section. This completes the analysis of the 2 × 2 case.
4.2. The general case. Now, consider the case of general n, r ≥ 1, starting as in the previous subsection with the linearized eigenvalue equation written in the "balanced flux variables" introduced in the previous subsection: (4.10) again, this is easily removed at the expense of further bookkeeping. A consequence of dissipativity, (H3), is that on a sufficiently small neighborhood of the base point u 0 , H(u) has a single eigenvalue (4.12) with the (r − 1) remaining eigenvalues uniformly bounded from zero as ǫ → 0: (4.13) γ 1 , . . . , γ k−1 ≤ −η < 0 < η ≤ γ k+1 , . . . , γ r ; we denote the associated left and right eigenvectors bys j and s j , respectively. At λ = 0, A ǫ reduces to where γ j ,s j , s jsj are eigenvalues and associated left and right eigenvectors of H, and E is as defined in (4.15). Likewise, associated with each of the transverse fields j = p of A ǫ , there exist "slow" eigenvalues µ s j and associated left and right eigenvectors L s j and R s j of A ǫ , analytic in λ and (A ǫ , Q ǫ ) on sufficiently small neighborhoods of λ = 0 and (A ǫ , Q ǫ )(u 0 , v 0 ), of λ = 0 and (A, Q)(u 0 , v 0 )), with expansions (4.20) µ s j = −λ/a j + λ 2 β 2 j /a 3 j + . . . , where β j := l j B * r j , with B * the effective, "Chapman-Enskog" viscosity defined in (1.15) of the introduction, andt j satisfies Dual to the right-and left-invariant subspaces Span j =k {R f j } ⊕ Span j =p {R s j } and Span j =k {L f j } ⊕ Span j =p {L s j } are two-dimensional left-and right-invariant subspaces spanned by vectors L η , L z and R η , R z analytic in λ and (A ǫ , Q ǫ ) on the same neighborhoods of λ = 0 and (A ǫ , Q ǫ )(u 0 , v 0 ), with expansions (4.23) L η = l p , l * + . . . , L z = 0,s p + . . . ,
Proof. This follows similarly as in the proof of Lemma 3.8, namely, analytic variation of the fast transverse modes follows by their spectral separation, while analytic variation of slow transverse modes follows by inversion (and appropriate conjugation) of the associated dispersion relations (4.27) λ j = −iξa j − ξ 2 β j + . . . , L j = L j + . . . , As in the treatment of the viscous case, we make the definitions (3.16)-(3.22), to obtain an approximately block-diagonal system satisfying (3.23)-(3.31), with (3.32) replaced by the following analog of (3.71): (4.28) where all O(·) terms are analytic in λ, A, Q. Here, as in the viscous case, we have used (3.72) to improve the crucial bound in the lower lefthand corner of the final, error matrix; more precisely, we have description (3.73) for this error bound, with coefficients O j analytic in all parameters. As in the viscous case, we may achieve L 0 R ′ 0 ≡ 0 by a renormalization leaving the form of the equations otherwise unchanged; thus, error Θ vanishes on the η, z block: (4.29) Θ (η,z),(η,z) ≡ 0. and (4.31) for x ≷ 0, where (4.32)M 0 := 0 1 λ/β a p /β , β := β p (0), and as usual β j := l j B * r j with B * as defined in (1.15).
Remark. Note that the natural choice of basis vectors L z and R z given in Lemma 4.7 automatically accomplishes the desired rescaling to the viscous case, so that we need not do this "by hand" as in the treatment of the 2 × 2 case above.
From this point, the argument goes through exactly as in the general viscous case. Carrying out the first reduction eliminating the r − 1 fast transverse modes, and rescaling by x → Λǫx/β, λ → βλ/Λ 2 ǫ 2 , z → βz/Λǫ, we obtain as in the general viscous case the block-triangular system (1.19)-(1.24) described in the introduction, with errors (1.26), and so the remaining arguments go through as before. This completes the proof of one-dimensional spectral stability in the general relaxation case.
Section 5. Multidimensional stability for parabolic systems.
Finally, we briefly discuss the extension of the above analysis to the multidimensional viscous case; details will be given in [PZ]. Consider a sequence of planar stationary profilesū ǫ (x 1 ) of a strictly parabolic viscous conservation law of form lying in a neighborhood U of a particular state u 0 , where x ∈ R d , u, f j ∈ R n , and B jk ∈ R n×n . We make the assumptions: (H0) f j , B jk ∈ C 2 (regularity). (H1) Re σ( B jk (ū)ξ j ξ k ) > 0 for ξ ∈ R d (strict parabolicity). (H2) There exists A 0 (·), symmetric positive definite and smoothly depending on u, such that A 0 Df j (u) is symmetric for all 1 ≤ j ≤ d. (simultaneous symmetrizability, ⇒ nonstrict hyperbolicity).
(H4) (i) a p = 0 is a simple eigenvalue of Df 1 (u 0 ) with left and right eigenvectors l p and r p , and l t p D 2 f 1 (r p , r p ) = 0 (genuine nonlinearity of the principal characteristic field in the normal spatial direction x 1 ).
(ii) r p is not an eigenvector of any Dfξ := j =1 ξ j Df j forξ ∈ R d−1 = 0, and, in the intermediate case 1 < p < n: Here, ǫ > 0 as usual denotes shock strength |u ǫ + − u ǫ − |, and profilesū ǫ (·) are assumed to converge as ǫ → 0 to u 0 . Since the profile problem is restricted to the normal direction, the one-dimensional analysis of Majda&Pego [MP] again yields Proposition 1.4, with convergence after rescaling ofū ǫ to the standard Burgers profile (1.11). Linearizing aboutū ǫ , we obtain a linearized evolution system (1.8) with coefficients depending only on x 1 ; taking the Fourier transform in directions x := (x 2 , . . . , x d ), we then obtain for each fixed ǫ a family of one-dimensional equations Under hypotheses (H0)-(H4), the results of [Z.3] show that strong spectral spectral stability implies linearized and nonlinear stability with sharp rates of decay (as discussed in [PZ], our conditions (H0)-(H4) imply conditions (H0)-(H7) assumed in the reference). In [PZ] we establish: Theorem 5.2. Under assumptions (H0)-(H4), profilesū ǫ are strongly spectrally stable (and therefore linearly and nonlinearly orbitally stable [ZH,Z.2]) for ǫ sufficiently small. Remark 5.3. As in the inviscid small amplitude case FM,FZ], the condition of simultaneous symmetrizability (H2) may be relaxed under appropriate alternative structural conditions; however, so far there has been identified no such conditions of simple and general application. Likewise, the nonresonance condition (H4)(ii) might possibly be relaxed; however, the degenerate case that this condition fails would appear to require a considerably more detailed analysis than the one carried out here. In any case, these conditions are satisfied for most systems arising in physical applications; see [Mé.5,Z.3,. In the extreme shock case, (H4)(ii) is equivalent to a simpler and easily verified condition identified by : that the principal eigenvalue a p of the full symbol d j=1 ξ j Df j be a strictly convex, resp. concave, function of ξ = (ξ 1 , . . . , ξ d ), according as p = 1 or n; see [FZ] or (implicitly) [Mé.1]. Note that (5.2) is automatically satisfied in the extreme shock case. The case of "real," or partially parabolic, viscosity, or relaxation may be treated similarly, following the model of the present, one-dimensional analysis.
Similarly as in the one-dimensional case, Theorem 5.2 is established by showing that stability is equivalent, modulo pure imaginary eigenvalues as in Theorems 3.2 and 4.2, to stability of an appropriate canonical system. However, this system is not a 2 × 2 system as in the scalar multidimensional case, but a 3 × 3 system corresponding to the full slow flow of an appropriate 2 × 2 viscous conservation law, namely, the coupled Burgers-linear degenerate equations, considered with the family of profiles where a > 0 and B jk are constant, diagonal, and satisfy the uniform parabolicity condition (5.7) jk ξ j ξ k B jk ≥ θ|ξ| 2 , θ > 0, for all ξ := (ξ 1 , ξ 2 ) ∈ R 2 , with B 11 11 = 1. That is, in multiple dimensions, small amplitude behavior is not captured by any scalar model, but generically requires a 2 × 2 model for its expression. This fundamental observation was made in the inviscid setting by FM] in his study of the weak shock limit, and is due to the new phenomenon of "glancing modes" arising in the multi-dimensional case; see also FZ] for related discussions in the inviscid case. Accordingly, the analysis of the reduced equations on the "slow-superslow" manifold becomes considerably more complicated in multiple dimensions; in particular, we can no longer rely on simple Sturm-Liouville, or maximum principles to conclude stability of the reduced system. The following proposition, though it does not (because the equations (5.5) are not scale-invariant) directly imply stability of the reduced system, nonetheless indicates at an intuitive level why the analysis can still be carried out, to yield a full stability result and not only a reduction. Note that the reduction to constant coefficients a, A 2 , B jk (achieved as in the one-dimensional analysis by an application of Lemma 2.5) is crucial in the proof.