Compacité par compensation pour une classe de systèmes hyperboliques de $p\ge 3$ lois de conservation

Abstract

We are concerned with a strictly hyperbolic system of conservation laws $u_t+f(u{)}_x=0$, where $u$ runs in a region $\Omega$ of ${\mathbb{R}}^p$, such that two of the characteristic fields are genuinely non-linear whereas the other ones are of Blake Temple's type. We begin with the case $p=3$ and show, under some more or less technical assumptions, that the approximate solutions $(u^{ϵ}{)}_{ϵ>0}$ given either by the vanishing viscosity method or by the Godunov scheme converge to weak entropy solutions as $ϵ$ goes to 0. The first step consists in using techniques from the Blake Temple systems lying in the separate works of Leveque-Temple and Serre. Then we apply a compensated compactness method and the theory of Di Perna on 2 x 2 genuinely non-linear systems. Eventually the proof is extended to the general case $p>3$.