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arXiv · 1803.09999

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

Michiel Bertsch·Flavia Smarrazzo·Andrea Terracina·Alberto Tesei
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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.

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Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. 2018-03-27. A uniqueness criterion for measure-valued solutions of scalar hyperbolic conservation laws. https://arxiv.org/abs/1803.09999

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