Pointwise semigroup methods and stability of viscous shock waves Indiana Univ (original) (raw)

1998, Indiana Univ Math J

Considered as rest points of ODE on L p , stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce 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 which 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 semigroup setting by Sattinger's method of weighted norms. We o v ercome this di culty 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, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications also in other situations of sensitive stability. Central to our analysis is a notion of e ective" point spectrum which 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 w a v es 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. TABLE OF CONTENTS Section 1. Introduction. Part I. Preliminaries Section 2. The Asymptotic Eigenvalue Equations. Section 3. Asymptotic Behavior of ODE. Thanks to Cidney Bever for her e cient, skillfull, and tireless typesetting of this paper, and to David Ho for his encouragement and for generously reviewing both this and an earlier draft. We gratefully acknowledge the contribution of Heinrich F reist uhler in pointing out the result of Fries quoted in section 1.2.4, and its application to overcompressive shocks in MHD. Finally, special thanks to Robert Gardner and Chris Jones, for their generosity i n i n troducing us to the Evans function and its uses, and for their continued encouragement, example, and collaboration in the dynamical systems approach to stability of traveling waves. Research o f both authors was supported in part by the ONR under Grant No. N00014-94-1-0456 and by the NSF under Grants No. DMS-9107990 and DMS9706842.