WELL-POSEDNESS OF GENERAL BOUNDARY-VALUE PROBLEMS FOR SCALAR CONSERVATION LAWS

In this paper we investigate well-posedness for the problem u(t) + div phi(u) = f on (0, T) x Omega, Omega subset of R-N, with initial condition u(0, .) = u(0) on Omega and with general dissipative boundary conditions phi(u) . nu is an element of beta((t, x))(u) on (0, T) x partial derivative Omega. Here for a.e. (t, x) is an element of (0, T) x partial derivative Omega, beta((t, x))(.) 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 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 (beta) over tilde, which is defined from beta by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with (beta) over tilde (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.