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

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 5, page 1245-1288
- ISSN: 1435-9855

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topDalibard, 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 -

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