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

Dalibard, Anne-Laure. "Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux." Journal of the European Mathematical Society 013.5 (2011): 1245-1288. <http://eudml.org/doc/277295>.

@article{Dalibard2011,
abstract = {This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial _tu+\operatorname\{div\}_yA(y,u)-\Delta _yu=0$, 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 behaves in $L^1$ 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 $L^1$ on the family of rescaled solutions; this is achieved by deriving uniform bounds in weighted $L^2$ spaces.},
author = {Dalibard, Anne-Laure},
journal = {Journal of the European Mathematical Society},
keywords = {long time asymptotics; parabolic scalar conservation law; asymptotic expansion; moment estimates; homogenization; scalar parabolic conservation law; moment estimates; periodic flux},
language = {eng},
number = {5},
pages = {1245-1288},
publisher = {European Mathematical Society Publishing House},
title = {Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux},
url = {http://eudml.org/doc/277295},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Dalibard, Anne-Laure
TI - Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 5
SP - 1245
EP - 1288
AB - This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial _tu+\operatorname{div}_yA(y,u)-\Delta _yu=0$, 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 behaves in $L^1$ 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 $L^1$ on the family of rescaled solutions; this is achieved by deriving uniform bounds in weighted $L^2$ spaces.
LA - eng
KW - long time asymptotics; parabolic scalar conservation law; asymptotic expansion; moment estimates; homogenization; scalar parabolic conservation law; moment estimates; periodic flux
UR - http://eudml.org/doc/277295
ER -