Comptes Rendus
no. 11-12
Partial Differential Equations
A remark on duality solutions for some weakly nonlinear scalar conservation laws
[Remarque sur les solutions en dualité pour une loi de conservation scalaire faiblement non-linéaire]

Présenté par : the Editorial Board

François James1, 2 ; Nicolas Vauchelet3, 4, 5
1 Université dʼOrléans, mathématiques, applications et physique mathématique dʼOrléans (MAPMO), CNRS UMR 6628, 45067 Orléans cedex 2, France
2 Fédération Denis-Poisson, CNRS FR 2964, 45067 Orléans cedex 2, France
3 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
4 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
5 INRIA Paris-Rocquencourt, Equipe BANG, 75005 Paris, France
Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 657-661.

Nous considérons lʼexistence et lʼunicité de solutions en dualité pour une loi de conservation scalaire avec un noyau dʼinteraction non-local. En suivant Bouchut et James (1999) [3], une notion de solution en dualité pour un tel système non-linéaire est proposée pour laquelle nous nʼavons cependant pas dʼunicité. Dans ce travail nous prouvons alors quʼen sélectionnant le flux, nous retrouvons un résultat dʼexistence et dʼunicité des solutions mesures de notre système.

We investigate existence and uniqueness of duality solutions for a scalar conservation law with a nonlocal interaction kernel. Following Bouchut and James (1999) [3], a notion of duality solution for such a nonlinear system is proposed, for which we do not have uniqueness. However we prove that a natural definition of the flux allows to select a solution for which uniqueness holds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.05.004

François James 1, 2 ; Nicolas Vauchelet 3, 4, 5

1 Université dʼOrléans, mathématiques, applications et physique mathématique dʼOrléans (MAPMO), CNRS UMR 6628, 45067 Orléans cedex 2, France
2 Fédération Denis-Poisson, CNRS FR 2964, 45067 Orléans cedex 2, France
3 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
4 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France
5 INRIA Paris-Rocquencourt, Equipe BANG, 75005 Paris, France 
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François James; Nicolas Vauchelet. A remark on duality solutions for some weakly nonlinear scalar conservation laws. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 657-661. doi : 10.1016/j.crma.2011.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.004/

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