L 2 -type contraction for systems of conservation laws
Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 1-28.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.1
Classification: 35L65, 35L67, 35L40
Keywords: Conservation laws, relative entropy, shock stability, Temple systems
Mot clés : Lois de conservation, entropie relative, stabilité des ondes de choc, systèmes de Temple
Denis Serre 1; Alexis F. Vasseur 2

1 UMPA, ENS-Lyon 46 allée d’Italie, 69364 Lyon Cedex 07, France
2 University of Texas at Austin 1 University Station C1200, Austin, TX 78712-0257, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Denis Serre; Alexis F. Vasseur. $L^2$-type contraction for systems of conservation laws. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 1-28. doi : 10.5802/jep.1. https://jep.centre-mersenne.org/articles/10.5802/jep.1/

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