Scalar conservation laws with nonlinear boundary conditions
Résumé
This Note deals with uniqueness and continuous dependence of solutions to the problem u_t + div ϕ(u) = f on (0,T ) ×Ω with initial condition u(0, ·) = u_0 on Ω and with (formal) nonlinear boundary conditions ϕ(u) · ν ∈ β(t,x,u) on (0,T ) × ∂Ω, where β(t,x, ·) stands for a maximal monotone graph on R. We suggest an interpretation of the formal boundary condition which generalizes the Bardos–LeRoux–Nédélec 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.