Scalar conservation laws with nonlinear boundary conditions

This Note deals with uniqueness and continuous dependence of solutions to the problem u(t) + div phi (u) = f on (0, T) x Omega with initial condition u(0, .) = u(0) on Omega and with (formal) nonlinear boundary conditions phi(u).nu is an element of beta(t, x, u) on (0, T) x partial derivative Q, where, beta(t, x, .) stands for a maximal monotone graph on R. We suggest an interpretation of the formal boundary condition which generalizes the Bardos-LeRoux-Nedelec condition, and introduce the corresponding notions of entropy and entropy process solutions using the strong trace framework of E.Yu. Panov. We prove uniqueness and provide some support for our interpretation of the boundary condition.