Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Abstract

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ${\partial }_{t}u+{\mathrm{div}}_{y}A(y,u)-{\Delta }_{y}u=0$, where $y\in {\mathbb{R}}^N$ and the flux $A$ is periodic in $y$. More specifically, we consider the case when the initial data is an $L^1$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between $u$ and the stationary solution behaves in $L^1$ norm like a self-similar profile for large times. The proof uses a time and space change of variables which is well-suited for the analysis of the long time behaviour of parabolic equations. Then, convergence in rescaled variables follows from arguments from dynamical systems theory. One crucial point is to obtain compactness in $L^1$ on the family of rescaled solutions; this is achieved by deriving uniform bounds in weighted $L^2$ spaces.