Pointwise semigroup methods and stability of viscous shock waves

Considered as rest points of ODE on L-p, stationary viscous shock waves pre sent a critical case for which standard semigroup methods do not suffice to determine stability. More precisely, there is no spectral gap between stat ionary modes and essential spectrum of the linearized operator about the wa ve, a fact that precludes the usual analysis by decomposition into invarian t subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or tota lly 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, p ointwise semigroup techniques, generalizing earlier work of Howard [H.1], K apitula [K.1-2], and Zeng [Ze,LZe]. These techniques allow us to do "hard" analysis in PDE within the dynamical systems/semigroup framework: in partic ular, to obtain sharp, global pointwise bounds on the Green's function of t he linearized operator around the wave, sufficient for the analysis of line ar and nonlinear stability. The method is general, and should find applicat ions 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 intimate ly related to the Evans function, a well-known tool for the spectral analys is of traveling waves. Indeed, crucial to our whole analysis is the "Gap Le mma" of [GZ,KS], a technical result developed originally in the context of Evans function theory. Using these new tools, we can treat general over- an d undercompressive, and even strong shock waves for systems within the same framework used for standard weak (i.e. slowly varying) Lax waves. In all c ases, we show that stability is determined by the simple and numerically co mputable condition that the number of zeroes of the Evans function in the r ight complex halfplane be equal to the dimension of the stationary manifold of nearby traveling wave solutions. Interpreting this criterion in the con servation law setting, we quickly recover all known analytic stability resu lts, obtaining several new results as well.