Radon measure-valued solutions of first order hyperbolic conservation laws

Abstract

We study nonnegative solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0&\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ is a Radon measure and $\varphi:[0,\infty)\mapsto \mathbb{R}$ is a globally Lipschitz continuous function. We construct suitably defined entropy solutions in the space of Radon measures. Under some additional conditions on $\varphi$, we prove their uniqueness if the singular part of $u_0$ is a finite superposition of Dirac masses. In terms of the behaviour of $\varphi$ at infinity we give criteria to distinguish two cases: either all solutions are function-valued for positive times (an instantaneous regularizing effect), or the singular parts of certain solutions persist until some positive {\em waiting time} (in the linear case $\varphi(u)=u$ this happens for all times). In the latter case we describe the evolution of the singular parts.