$L^2$-type contraction for systems of conservation laws

Serre, Denis; Vasseur, Alexis F.

L^{2}

-type contraction for systems of conservation laws
[Contraction de type

L^{2}

pour des systèmes de lois de conservation]

Serre, Denis ; Vasseur, Alexis F.

Journal de l'École polytechnique - Mathématiques, Tome 1 (2014), p. 1-28 / Harvested from Numdam

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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.

The semi-group associated with the Cauchy problem for a scalar conservation law is known to be a contraction in $L^{1}$ . However it is not a contraction in $L^{p}$ for any $p > 1$ . Leger showed in [20] that for a convex flux, it is however a contraction in $L^{2}$ up to a suitable shift. We investigate in this paper whether such a contraction may happen for systems. The method is based on the relative entropy method. Our general analysis leads us to the new geometrical notion of Genuinely non-Temple systems. We treat in details two examples: – the Keyfitz–Kranzer system with rotationally invariant flux, for which the $L^{2}$ contraction holds true, – the Euler system of gas dynamics, for which it does not.

Publié le : 2014-01-01
DOI : https://doi.org/10.5802/jep.1
Classification: 35L65, 35L67, 35L40
Mots clés: Lois de conservation, entropie relative, stabilité des ondes de choc, systèmes de Temple

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     author = {Serre, Denis and Vasseur, Alexis F.},
     title = {$L^2$-type contraction for systems of conservation laws},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     volume = {1},
     year = {2014},
     pages = {1-28},
     doi = {10.5802/jep.1},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEP_2014__1__1_0}
}

Serre, Denis; Vasseur, Alexis F. $L^2$-type contraction for systems of conservation laws. Journal de l'École polytechnique - Mathématiques, Tome 1 (2014) pp. 1-28. doi : 10.5802/jep.1. http://gdmltest.u-ga.fr/item/JEP_2014__1__1_0/

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