Title: Pointwise semigroup methods and stability of viscous shock waves

Authors: Kevin Zumbrun and Peter Howard

Issue: Volume 47 (1998), Issue 3, 741-872

Abstract: Considered as rest points of ODE on $L^p$, stationary viscous shock waves present a critical case for which standard semigroup methods do not suffice to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact that precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semiroup setting by Sattinger's method of weighted norms. We overcome this difficulty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard [H.1], Kapitula [K.1-2], and Zeng [Ze,LZe]. These techniques allow us to do 'hard' analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, sufficient for the analysis of linear and nonlinear stability. The method is general, and should find applications also in other situations of sensitive stability. Central to our analysis is a notion of 'effective' point spectrum that can be extended to regions of essential spectrum. This turns out to be intimately related to the Evans function, a well-known tool for the spectral analysis of traveling waves. Indeed, crucial to our whole analysis is the 'Gap Lemma' of [GZ,KS], a technical result developed originally in the context of Evans function theory. Using these new tools, we can treat general over- and undercompressive, and even strong shock waves for systems within the same framework used for standard weak (i.e. slowly varying) Lax waves. In all cases, we show that stability is determined by the simple and numerically computable condition that the number of zeroes of the Evans function in the right complex half-plane be equal to the dimension of the stationary manifold of nearby traveling wave solutions. Interpreting this criterion in the conservation law setting, we quickly recover all known analytic stability results, obtaining several new results as well.