Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions

TL;DR

This paper establishes sharp linear and nonlinear stability estimates for multidimensional periodic traveling waves in viscous conservation laws, revealing complex convective and diffusive behaviors in various dimensions.

Contribution

It provides new sharp $L^p$ estimates and stability results for multidimensional periodic waves under spectral stability assumptions.

Findings

Linearized solution operator is stable in $L^1\cap L^p \to L^p$ for all $p\ge 2$

Nonlinear stability and asymptotic behavior are established in $L^1\cap H^s$ and $L^2$ norms

Behavior involves both wave-like and diffusive effects, depending on dimension

Abstract

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp $L^p$ estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized $L^1\cap L^p\to L^p$ stability for all $p \ge 2$ and dimensions $d \ge 1$ and nonlinear $L^1\cap H^s\to L^p\cap H^s$ stability and $L^2$-asymptotic behavior for $p\ge 2$ and $d\ge 3$. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and diffusive effects.