A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws

Abstract

We prove existence and uniqueness of Radon measure-valued solutions of the Cauchy problem $$\{\begin{array}{ll}{u}_t+[\phi (u){]}_x=0& \text{in }\mathbb{R}\times (0,T)\\ u=u_0\ge 0& \text{in }\mathbb{R}\times \{0\},\end{array}$$ where $u_0$ a positive Radon measure whose singular part is a finite superposition of Dirac masses, and $\phi \in C^2([0,\infty ))$ is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.