Scalar conservation laws with nonlinear boundary conditions

Boris Andreianov; Karima Sbihi

doi:10.1016/j.crma.2007.09.008

Partial Differential Equations

Scalar conservation laws with nonlinear boundary conditions
[Lois de conservation scalaires avec des conditions non linéaires au bord]

Présenté par : Pierre-Louis Lions

Boris Andreianov ¹ ; Karima Sbihi ¹

¹ UFR des sciences et techniques, département de mathématiques, 16, route de Gray, 25030 Besançon cedex, France

Comptes Rendus. Mathématique, Volume 345 (2007) no. 8, pp. 431-434.

Résumé
Abstract

Cette Note est dédiée aux résultats d'unicité des solutions du problème $u_{t} +$ div $φ (u) = f$ sur $(0, T) \times Ω$ avec la condition initiale $u (0, \cdot) = u_{0}$ sur Ω et les conditions non linéaires $φ (u) \cdot ν \in β (t, x, u)$ sur $(0, T) \times \partial Ω$ ; ici $β (t, x, \cdot)$ désigne un graphe maximal monotone sur $R$ . Nous proposons une interprétation de la condition formelle « $φ (u) \cdot ν \in β (t, x, u)$ » 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 $u_{t} + div φ (u) = f$ on $(0, T) \times Ω$ with initial condition $u (0, \cdot) = u_{0}$ on Ω and with (formal) nonlinear boundary conditions $φ (u) \cdot ν \in β (t, x, u)$ on $(0, T) \times \partial Ω$ , where $β (t, x, \cdot)$ 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 : 2007-06-27
Accepté le : 2007-09-12
Publié le : 2007-10-10

DOI : 10.1016/j.crma.2007.09.008

Affiliations des auteurs :

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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     title = {Scalar conservation laws with nonlinear boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {431--434},
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     number = {8},
     year = {2007},
     doi = {10.1016/j.crma.2007.09.008},
     language = {en},
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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/

Bibliographie
Cité par

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