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

TL;DR

This paper establishes existence and uniqueness of measure-valued solutions for scalar hyperbolic conservation laws with initial data as Radon measures, introducing a compatibility condition that ensures uniqueness alongside entropy conditions.

Contribution

It introduces a new compatibility condition that, together with entropy conditions, guarantees the uniqueness of measure-valued solutions for scalar conservation laws.

Findings

Proves existence of Radon measure-valued solutions.

Establishes uniqueness under a new compatibility condition.

Handles initial data with singular parts as Dirac masses.

Abstract

We prove existence and uniqueness of Radon measure-valued 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 $u_0$ a positive Radon measure whose singular part is a finite superposition of Dirac masses, and $\varphi\in C^2([0,\infty))$ is bounded. The novelty of the paper is the introduction of a compatibility condition which, combined with standard entropy conditions, guarantees uniqueness.