Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities

Abstract

Let H be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation Ut + H(Ux) = 0 and signed Radon measure valued entropy solutions of the conservation law ut + [H(u)]x = 0. After having proved a precise statement of the formal relation Ux = u, we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.