Discontinuous Solutions of Hamilton–Jacobi Equations Versus Radon Measure-Valued Solutions of Scalar Conservation Laws: Disappearance of Singularities

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