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

In this paper we investigate well-posedness for the problem ut + divφ(u) = f on (0, T )×Ω, Ω ⊂ R , with initial condition u(0, ·) = u0 on Ω and with general dissipative boundary conditions φ(u) · ν ∈ β(t,x)(u) on (0, T )×∂Ω. Here for a.e. (t, x) ∈ (0, T )×∂Ω, β(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 interprete the formal boundary condition given by β 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 β should be interpreted as the effective boundary condition given by another monotone graph β̃, which is defined from β by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with β̃ (and thus also with β). For the notion of solution defined in this way, we prove existence, uniqueness and L1 contraction, monotone and continuous dependence on the graph β. Convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results.