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 $|v_0|\le E_0(1+|x|{)}^{-3/2}$ for sufficiently small $E_0>0$.