IUMJ

Title: Long time behavior of parabolic scalar conservation laws with space periodic flux

Authors: Anne-Laure Dalibard

Issue: Volume 59 (2010), Issue 1, 257-300

Abstract:

This paper is concerned with the stability of stationary solutions of the conservation law $\partial_{t}u + \mathrm{div}_{y} A(y,u) - \Delta_{y} u = 0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviors are studied here: the case when the equation is set on $\mathbb{R}$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of standing shocks which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^{1}$, provided the initial disturbance satisfies some appropriate boundedness conditions. Furthermore, a recent result enables us to extend this stability property to arbitrary initial data. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^{\infty}$ bounds on the solution of the conservation law, and on sub- and super-solution techniques.