JournalsjemsVol. 13, No. 5pp. 1245–1288

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

  • Anne-Laure Dalibard

    Ecole Normale Superieure, Paris, France
Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux cover
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Abstract

This article investigates the long-time behaviour of parabolic scalar conservation laws of the type tu+divyA(y,u)Δyu=0\partial_t u + \mathrm{div}_yA(y,u) - \Delta_y u=0, where yRNy\in\mathbb R^N and the flux AA is periodic in yy. More specifically, we consider the case when the initial data is an L1L^1 disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between uu and the stationary solution behaves in L1L^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 L1L^1 on the family of rescaled solutions; this is achieved by deriving uniform bounds in weighted L2L^2 spaces.

Cite this article

Anne-Laure Dalibard, Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux. J. Eur. Math. Soc. 13 (2011), no. 5, pp. 1245–1288

DOI 10.4171/JEMS/280