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

TL;DR

This paper explores the relationship between discontinuous viscosity solutions of Hamilton-Jacobi equations and Radon measure-valued entropy solutions of scalar conservation laws, focusing on the disappearance of singularities over time.

Contribution

It establishes a precise relation between solutions of Hamilton-Jacobi and conservation laws and provides estimates for when singularities vanish.

Findings

Proves the formal relation $U_x=u$ between solutions.

Provides estimates for the times when singularities disappear.

Analyzes the nature of singularities in both equations.

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.