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

Serre, Denis; Vasseur, Alexis F.

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.

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