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 Available versions: v1 (2012-06-17) v2 (2012-10-31)
 Well-posedness of general boundary-value problems for scalar conservation laws
 Boris Andreianov ( ) 1, Karima Sbihi 1
 (2012-06-16)
 In this paper we investigate well-posedness for the problem $u_t+ \div \ph(u)=f$ on $(0,T)\!\times\!\Om$, $\Om \subset \R^N$, with initial condition $u(0,\cdot)=u_0$ on $\Om$ and with general dissipative boundary conditions $\varphi(u)\cdot \nu \in \beta_{(t,x)}(u)$ on $(0,T)\!\times\!\ptl\Om$. Here for a.e. $(t,x)\in(0,T)\!\times\!\ptl\Om$, $\beta_{(t,x)}(\cdot)$ is a maximal monotone graph on $\R$. 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 interprete the {\it formal boundary condition} given by $\beta$ by replacing it with the adequate {\it 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.
 1: Laboratoire de Mathématiques (LM-Besançon) CNRS : UMR6623 – Université de Franche-Comté
 Subject : Mathematics/Analysis of PDEs
 Keyword(s): scalar conservation law – boundary-value problem – entropy solution – vanishing viscosity limit – formal boundary condition – effective boundary condition – maximal monotone graph – strong boundary trace – $L^1$ contraction – well-posedness – convergence of approximations
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 hal-00708973, version 1 http://hal.archives-ouvertes.fr/hal-00708973 oai:hal.archives-ouvertes.fr:hal-00708973 From: Boris Andreianov <> Submitted on: Saturday, 16 June 2012 14:20:39 Updated on: Sunday, 17 June 2012 15:45:03