Comptes Rendus
no. 9-10
Partial Differential Equations
Hyperbolic conservation laws on manifolds with limited regularity
[Lois de conservation hyperboliques sur les variétés à faible régularité]

Présenté par : Philippe G. Ciarlet

Philippe G. LeFloch1 ; Baver Okutmustur1
1 Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, UPMC, Université de Paris 6, 4, place Jussieu, 75252 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 539-543.

Nous proposons une formulation du problème de Cauchy avec conditions aux limites pour les lois de conservation hyperboliques nonlinéaires posées sur une variété différentiable munie d'une forme volume, avec ou sans bord ; notre étude couvre, en particulier, le cas important des variété Lorentzienne. Nous supposons une régularité limitée sur la géometrie de la variété. Pour ce problème nous démontrons l'existence et l'unicité d'un semi-groupe L1 de solutions faibles satisfaisant à des conditions d'entropie et à des conditions aux limites convenablement définies.

We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.03.017
Philippe G. LeFloch 1 ; Baver Okutmustur 1

1 Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, UPMC, Université de Paris 6, 4, place Jussieu, 75252 Paris cedex 05, France 
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Philippe G. LeFloch; Baver Okutmustur. Hyperbolic conservation laws on manifolds with limited regularity. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 539-543. doi : 10.1016/j.crma.2008.03.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.03.017/

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[2] P. Amorim, P.G. LeFloch, B. Okutmustur, Finite volume schemes on Lorentzian manifolds, 2007, submitted for publication

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[6] C. Kondo; P.G. LeFloch Measure-valued solutions and well-posedness of multi-dimensional conservation laws in a bounded domain, Portugal. Math., Volume 58 (2001), pp. 171-194

[7] S.N. Kruzkov First-order quasilinear equations with several space variables, Math. USSR Sb., Volume 10 (1970), pp. 217-243

[8] P.G. LeFloch, Hyperbolic conservation laws and spacetimes with limited regularity, in: Proc. Inter. Conf. on Hyper. Problems: Theory, Numerics, and Application, July 2006, Lyon, France

[9] P.G. LeFloch, B. Okutmustur, in preparation

[10] E.Y. Panov On the Cauchy problem for a first-order quasilinear equation on a manifold, Differential Equations, Volume 33 (1997), pp. 257-266

[11] A. Szepessy Measure-valued solutions of scalar conservation laws with boundary conditions, Arch. Rational Mech. Anal., Volume 107 (1989), pp. 181-193

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