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

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Bertsch, Michiel, Smarrazzo, Flavia, Terracina, Andrea, Tesei, Alberto
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.03.2018
Subjects
Cauchy problems
Conservation laws
Radon
Superposition (mathematics)
Uniqueness
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Summary: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.
ISSN:2331-8422