Comptes Rendus
Partial Differential Equations
Scalar conservation laws with nonlinear boundary conditions
[Lois de conservation scalaires avec des conditions non linéaires au bord]
Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 431-434.

Cette Note est dédiée aux résultats d'unicité des solutions du problème ut+ div φ(u)=f sur (0,T)×Ω avec la condition initiale u(0,)=u0 sur Ω et les conditions non linéaires φ(u)νβ(t,x,u) sur (0,T)×Ω ; ici β(t,x,) désigne un graphe maximal monotone sur R. Nous proposons une interprétation de la condition formelle « φ(u)νβ(t,x,u) » qui généralise celle de Bardos–LeRoux–Nédélec ; nous introduisons les notions de solutions entropiques et solutions processus entropiques. Nous montrons l'unicité et argumentons en faveur de notre interprétation de la condition au bord.

This Note deals with uniqueness and continuous dependence of solutions to the problem ut+divφ(u)=f on (0,T)×Ω with initial condition u(0,)=u0 on Ω and with (formal) nonlinear boundary conditions φ(u)νβ(t,x,u) on (0,T)×Ω, where β(t,x,) stands for a maximal monotone graph on R. 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.09.008

Boris Andreianov 1 ; Karima Sbihi 1

1 UFR des sciences et techniques, département de mathématiques, 16, route de Gray, 25030 Besançon cedex, France
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Boris Andreianov; Karima Sbihi. Scalar conservation laws with nonlinear boundary conditions. Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 431-434. doi : 10.1016/j.crma.2007.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.09.008/

[1] B. Andreianov, K. Sbihi, Strong boundary traces and well-posedness for scalar conservation laws with dissipative boundary conditions, in: Proceedings of the XIth International Conference on Hyperbolic Problems, Lyon 2006

[2] C. Bardos; A.Y. Le Roux; J.C. Nédélec First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, Volume 4 (1979), pp. 1017-1034

[3] R. Bürger; H. Frid; K.H. Karlsen On the well-posedness of entropy solutions to conservation laws with a zero-flux boundary condition, J. Math. Anal. Appl., Volume 326 (2007), pp. 108-120

[4] J. Carrillo Entropy solutions for nonlinear degenerate problems, Arch. Ration. Mech. Anal., Volume 147 (1999), pp. 269-361

[5] G.-Q. Chen; H. Frid Divergence-Measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., Volume 147 (1999), pp. 89-118

[6] R. Eymard; R. Gallouët; R. Herbin Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Ann. Math. Ser. B, Volume 16 (1995), pp. 1-14

[7] E.Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., in press

[8] K. Sbihi, Etude de quelques E.D.P. non linéaires dans L1 avec des conditions générales sur le bord, Thesis, University of Strasbourg, France, 2006

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