Discontinuous solutions of Hamilton-Jacobi equations versus Radon measure-valued solutions of scalar conservation laws: Disappearance of singularities

Let \(H\) be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the fo...

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Bibliographic Details
Published in:arXiv.org 2020-07
Main Authors: Bertsch, M, Smarrazzo, F, Terracina, A, Tesei, A
Format: Article
Language:English
Subjects:
Conservation laws
Continuity (mathematics)
Discontinuity
Hamilton-Jacobi equation
Linear equations
Radon
Singularities
Online Access:Get full text
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Summary:Let \(H\) be a bounded and Lipschitz continuous function. We consider discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_{t}+H(U_x)=0\) and signed Radon measure valued entropy solutions of the conservation law \(u_{t}+[H(u)]_x=0\). After having proved a precise statement of the formal relation \(U_x=u\), we establish estimates for the (strictly positive!) times at which singularities of the solutions disappear. Here singularities are jump discontinuities in case of the Hamilton-Jacobi equation and signed singular measures in case of the conservation law.
ISSN:2331-8422