| 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. |
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1Karima Sbihi 1
