Pointwise semigroup methods and stability of viscous shock waves

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 semigroup 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.