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Article Dans Une Revue Indiana University Mathematics Journal Année : 2010

Long time behaviour of viscous scalar conservation laws

Résumé

This paper is concerned with the stability of stationary solutions of the conservation law $\partial_t u + \mathrm{div}_y A(y,u) -\Delta_y u=0$, where the flux $A$ is periodic with respect to its first variable. Essentially two kinds of asymptotic behaviours are studied here: the case when the equation is set on $\R$, and the case when it is endowed with periodic boundary conditions. In the whole space case, we first prove the existence of viscous stationary shocks - also called standing shocks - which connect two different periodic stationary solutions to one another. We prove that standing shocks are stable in $L^1$, provided the initial disturbance satisfies some appropriate boundedness conditions. We also extend this result to arbitrary initial data, but with some restrictions on the flux $A$. In the periodic case, we prove that periodic stationary solutions are always stable. The proof of this result relies on the derivation of uniform $L^\infty$ bounds on the solution of the conservation law, and on sub- and super-solution techniques.
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Dates et versions

hal-00345324 , version 1 (08-12-2008)

Identifiants

Citer

Anne-Laure Dalibard. Long time behaviour of viscous scalar conservation laws. Indiana University Mathematics Journal, 2010, 59 (1), pp.257-300. ⟨10.1512/iumj.2010.59.3874⟩. ⟨hal-00345324⟩
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