Partial Differential Equations
Hyperbolic conservation laws on manifolds with limited regularity
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 539-543.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.03.017
LeFloch, Philippe G. 1; Okutmustur, Baver 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
@article{CRMATH_2008__346_9-10_539_0,
     author = {LeFloch, Philippe G. and Okutmustur, Baver},
     title = {Hyperbolic conservation laws on manifolds with limited regularity},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {539--543},
     publisher = {Elsevier},
     volume = {346},
     number = {9-10},
     year = {2008},
     doi = {10.1016/j.crma.2008.03.017},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.03.017/}
}
TY  - JOUR
AU  - LeFloch, Philippe G.
AU  - Okutmustur, Baver
TI  - Hyperbolic conservation laws on manifolds with limited regularity
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 539
EP  - 543
VL  - 346
IS  - 9-10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.03.017/
DO  - 10.1016/j.crma.2008.03.017
LA  - en
ID  - CRMATH_2008__346_9-10_539_0
ER  - 
%0 Journal Article
%A LeFloch, Philippe G.
%A Okutmustur, Baver
%T Hyperbolic conservation laws on manifolds with limited regularity
%J Comptes Rendus. Mathématique
%D 2008
%P 539-543
%V 346
%N 9-10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.03.017/
%R 10.1016/j.crma.2008.03.017
%G en
%F CRMATH_2008__346_9-10_539_0
LeFloch, Philippe G.; Okutmustur, Baver. 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. http://www.numdam.org/articles/10.1016/j.crma.2008.03.017/

[1] Amorim, P.; Ben-Artzi, M.; LeFloch, P.G. Hyperbolic conservation laws on manifolds: total variation estimates and the finite volume method, Meth. Appl. Anal., Volume 12 (2005), pp. 291-324

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

[3] Ben-Artzi, M.; LeFloch, P.G. The well-posedness theory for geometry compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Nonlinéaire, Volume 24 (2007), pp. 989-1008

[4] M. Ben-Artzi, J. Falcovitz, P.G. LeFloch, Hyperbolic conservation laws on the sphere. A geometry compatible finite volume scheme, in preparation

[5] DiPerna, R.J. Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal., Volume 88 (1985), pp. 223-270

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

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

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

Cited by Sources: