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

1998, Indiana University Mathematics Journal

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.