Discontinuous solutions of \(U_t + H(U_x) = 0\) versus measure-valued solutions of \(u_t + [H(u)]_x = 0\). (English) Zbl 1483.35073
Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 32, No. 3, 391-411 (2021).
Summary: Let \(H\) be a bounded Lipschitz continuous function. We discuss some recent results concerning discontinuous viscosity solutions of the Hamilton-Jacobi equation \(U_t + H(U_x) = 0\), signed Radon measure-valued entropy solutions of the conservation law \(u_t + [H(u)]_x = 0\), and their connection.
MSC:
| 35F21 | Hamilton-Jacobi equations |
| 35D40 | Viscosity solutions to PDEs |
| 35L65 | Hyperbolic conservation laws |
| 28A33 | Spaces of measures, convergence of measures |
| 28A50 | Integration and disintegration of measures |
Keywords:
Hamilton-Jacobi equations; scalar conservation laws; discontinuous viscosity solutions; Radon measure-valued solutions; compatibility conditionsReferences:
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