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Denis Serre; Alexis F. Vasseur
$L^2$-type contraction for systems of conservation laws
(Contraction de type $L^2$ pour des systèmes de lois de conservation)
Journal de l'École polytechnique — Mathématiques, 1 (2014), p. 1-28, doi: 10.5802/jep.1
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Class. Math.: 35L65, 35L67, 35L40
Mots clés: Lois de conservation, entropie relative, stabilité des ondes de choc, systèmes de Temple

Résumé - Abstract

On sait que le semi-groupe associé au Problème de Cauchy pour une loi de conservation scalaire est contractant dans $L^1$, mais qu’il ne l’est pas dans $L^p$ si $p>1$. Leger a montré dans [20], pour un flux convexe, une propriété de contraction dans $L^2$ moyennant une translation. Nous examinons ici la possibilité d’une telle propriété pour les systèmes. Notre analyse nous conduit à la notion géométrique de système Vraiment pas Temple. Nous traitons en détail deux exemples : – le système de Keyfitz et Kranzer avec flux isotrope, pour lequel la contraction a lieu, – le système de la dynamique des gaz, où ce n’est pas le cas.

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