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

TL;DR

This paper extends pointwise stability analysis to periodic standing waves in viscous conservation laws using resolvent kernels and Bloch decomposition, demonstrating nonlinear stability under small perturbations.

Contribution

It introduces a novel approach to establish pointwise bounds and nonlinear stability for periodic standing waves in conservation laws using advanced spectral methods.

Findings

Established pointwise bounds for the Green function of the linearized operator.

Proved nonlinear stability of periodic standing waves under small perturbations.

Demonstrated decay estimates for modulated perturbations.

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 $E_0>0$.