Revista Matemática Iberoamericana


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Volume 10, Issue 3, 1994, pp. 557–579
DOI: 10.4171/RMI/161

Published online: 1994-12-31

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

Sylvie Benzoni-Gavage[1] and Denis Serre[2]

(1) Université Claude Bernard Lyon 1, Villeurbanne, France
(2) École Normale Supérieure de Lyon, France

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 $(u^\epsilon)_{\epsilon>0}$ given either by the vanishing viscosity method or by the Godunov scheme converge to weak entropy solutions as $\epsilon$ 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 $p > 3$.

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Benzoni-Gavage Sylvie, Serre Denis: Compacité par compensation pour une classe de systèmes hyperboliques de $p ≥ 3$ lois de conservation. Rev. Mat. Iberoam. 10 (1994), 557-579. doi: 10.4171/RMI/161