Nonlinear Stability of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Dimensions One and Two

Abstract

Extending results of Oh and Zumbrun in dimensions $d\geq3$, we establish nonlinear stability and asymptotic behavior of spatially periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions $d=1,2$, under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions $d\geq3$ are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the nonlinear iteration in the Duhamel representation of a modulated periodic wave.