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

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_t u + \mathrm{div}_yA(y,u) - \Delta_y u=0$, 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 vanishes for large times in $L^1$ norm. The proof uses a self-similar change of variables which is well-suited for the analysis of the long time behaviour of parabolic equations. Then, convergence in self-similar 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.