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, Tome 1 (2014) , pp. 1-28.

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.

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
@article{JEP_2014__1__1_0,
     author = {Serre, Denis and Vasseur, Alexis F.},
     title = {$L^2$-type contraction for systems of conservation laws},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {1--28},
     publisher = {Ecole polytechnique},
     volume = {1},
     year = {2014},
     doi = {10.5802/jep.1},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.1/}
}
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://archive.numdam.org/articles/10.5802/jep.1/

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