Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Pointwise Bounds on the Green Function

Abstract

We investigate stability of periodic traveling-wave solutions of systems of conservation laws with viscosity within the abstract Evans function framework established by R. A. Gardner. Our main result, generalizing the work of Zumbrun and Howard in the traveling-front or -pulse setting, is to establish sharp pointwise bounds on the Green function for the linearized evolution equations, provided that an appropriate Evans function condition applies to the linearized operator about the wave. This condition is equivalent to a spectral stability criterion introduced by Schneider in the context of periodic reaction-diffusion waves. An immediate consequence is that strong spectral stability (in the sense of Schneider) implies linearizedL1 → Lpasymptotic stability for all p > 1. On the other hand, we show that the strict version of Schneider's condition generically fails in the conservation law setting, leading to complicated new “metastable” behavior reminiscent of that seen for degenerate, neutrally stable families in the traveling-front or -pulse case. Our results apply also to the reaction-diffusion setting, sharpening (at the linearized level) results obtained by Schneider using weighted-norm and Bloch-decomposition methods.