Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux
Anne-Laure Dalibard · 2009
This article investigates the long-time behaviour of parabolic scalar conservation laws of the type ∂tu + divyA(y,u) − ∆yu = 0, where y ∈ 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.