Discontinuous solutions of $U_t + H(U_x) = 0$ versus measure-valued solutions of $u_t + [H(u)]_x = 0$

Abstract

Let $H$ be a bounded Lipschitz continuous function. We discuss some recent results concerning discontinuous viscosity solutions of the Hamilton–Jacobi equation $U_t + H(U_x) = 0$ 0, signed Radon measure-valued entropy solutions of the conservation law $u_t + H(u)\_x = 0$, and their connection.