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 \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), 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.