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

Anne-Laure Dalibard

Journal of the European Mathematical Society, Tome 13 (2011) no. 5, pp. 1245-1288

Résumé

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ∂t​u+divy​A(y,u)−Δy​u=0, where y∈RN and the flux A is periodic in y. More specifically, we consider the case when the initial data is an L1 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 L1 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 L1 on the family of rescaled solutions; this is achieved by deriving uniform bounds in weighted L2 spaces.

Détail

DOI : 10.4171/jems/280

Classification : 35-XX, 00-XX
Keywords: Long time asymptotics, parabolic scalar conservation law, asymptotic expansion, moment estimates, homogenization