Pointwise stability estimates for periodic traveling wave solutions of systems of viscous conservation laws

Abstract

In the previous paper \cite{J1}, we established pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves $\bar u$ of a system of reaction diffusion equations, and also obtained pointwise nonlinear stability and behavior of $\bar u$ under small perturbations. In this paper, using periodic resolvent kernels and the Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with periodic standing waves $\bar u$ of a system of conservation laws. We also show pointwise nonlinear stability of $\bar u$ by estimating decay of modulated perturbation $v$ of $\bar u$ under small perturbation $|v_0| \leq E_0(1+|x|)^{-3/2}$ for sufficiently small $E_0>0$.