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Well-posedness of general boundary-value problems for scalar conservation laws


Authors: Boris Andreianov and Karima Sbihi
Journal: Trans. Amer. Math. Soc. 367 (2015), 3763-3806
MSC (2010): Primary 35L65, 35L04; Secondary 35A01, 35A02
DOI: https://doi.org/10.1090/S0002-9947-2015-05988-1
Published electronically: February 12, 2015
MathSciNet review: 3324909
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Abstract: In this paper we investigate well-posedness for the problem $ u_t+ \operatorname {div} \varphi (u)=f$ on $ (0,T)\!\times \!\Omega $, $ \Omega \subset \mathbb{R}^N$, with initial condition $ u(0,\cdot )=u_0$ on $ \Omega $ and with general dissipative boundary conditions $ \varphi (u)\cdot \nu \in \beta _{(t,x)}(u)$ on $ (0,T)\!\times \!\partial \Omega $. Here for a.e. $ (t,x)\in (0,T)\!\times \!\partial \Omega $, $ \beta _{(t,x)}(\cdot )$ is a maximal monotone graph on $ \mathbb{R}$. This includes, as particular cases, Dirichlet, Neumann, Robin, obstacle boundary conditions and their piecewise combinations.

As for the well-studied case of the Dirichlet condition, one has to interpret the formal boundary condition given by $ \beta $ by replacing it with the adequate effective boundary condition. Such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation. We claim that the formal boundary condition given by $ \beta $ should be interpreted as the effective boundary condition given by another monotone graph $ \tilde \beta $, which is defined from $ \beta $ by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with $ \tilde \beta $ (and thus also with $ \beta $).

For the notion of solution defined in this way, we prove existence, uniqueness and $ L^1$ contraction, monotone and continuous dependence on the graph $ \beta $. Convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results.


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Additional Information

Boris Andreianov
Affiliation: Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
Email: boris.andreianov@univ-fcomte.fr

Karima Sbihi
Affiliation: Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
Email: sbihi_k@yahoo.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-05988-1
Received by editor(s): June 16, 2012
Received by editor(s) in revised form: November 5, 2012
Published electronically: February 12, 2015
Additional Notes: The work of the first author was partially supported by the French ANR project CoToCoLa
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.