[Contraction de type pour des systèmes de lois de conservation]
On sait que le semi-groupe associé au Problème de Cauchy pour une loi de conservation scalaire est contractant dans , mais qu’il ne l’est pas dans si . Leger a montré dans [20], pour un flux convexe, une propriété de contraction dans 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 . However it is not a contraction in for any . Leger showed in [20] that for a convex flux, it is however a contraction in 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 contraction holds true, – the Euler system of gas dynamics, for which it does not.
Keywords: Conservation laws, relative entropy, shock stability, Temple systems
Mot 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://www.numdam.org/articles/10.5802/jep.1/}
}
TY - JOUR AU - Serre, Denis AU - Vasseur, Alexis F. TI - $L^2$-type contraction for systems of conservation laws JO - Journal de l’École polytechnique — Mathématiques PY - 2014 SP - 1 EP - 28 VL - 1 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.1/ DO - 10.5802/jep.1 LA - en ID - JEP_2014__1__1_0 ER -
%0 Journal Article %A Serre, Denis %A Vasseur, Alexis F. %T $L^2$-type contraction for systems of conservation laws %J Journal de l’École polytechnique — Mathématiques %D 2014 %P 1-28 %V 1 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.1/ %R 10.5802/jep.1 %G en %F 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://www.numdam.org/articles/10.5802/jep.1/
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