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Comptes Rendus Mathématique
Volume 345, n° 8
pages 431-434 (octobre 2007)
Doi : 10.1016/j.crma.2007.09.008
Received : 27 June 2007 ;  accepted : 12 September 2007
Scalar conservation laws with nonlinear boundary conditions
Lois de conservation scalaires avec des conditions non linéaires au bord
 

Boris Andreianov , Karima Sbihi
UFR des sciences et techniques, département de mathématiques, 16, route de Gray, 25030 Besançon cedex, France 

Abstract

This Note deals with uniqueness and continuous dependence of solutions to the problem   on   with initial condition   on and with (formal) nonlinear boundary conditions   on  , where   stands for a maximal monotone graph on  . We suggest an interpretation of the formal boundary condition which generalizes the Bardos-LeRoux-Nédélec condition, and introduce the corresponding notions of entropy and entropy process solutions using the strong trace framework of E.Yu. Panov. We prove uniqueness and provide some support for our interpretation of the boundary condition. To cite this article: B. Andreianov, K. Sbihi, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

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Résumé

Cette Note est dédiée aux résultats dʼunicité des solutions du problème   div   sur   avec la condition initiale   sur et les conditions non linéaires   sur   ; ici   désigne un graphe maximal monotone sur  . Nous proposons une interprétation de la condition formelle «   » qui généralise celle de Bardos-LeRoux-Nédélec ; nous introduisons les notions de solutions entropiques et solutions processus entropiques. Nous montrons lʼunicité et argumentons en faveur de notre interprétation de la condition au bord. Pour citer cet article : B. Andreianov, K. Sbihi, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

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