The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations

Osher, Stanley

doi:10.1090/S0002-9939-1983-0718989-X

The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations
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by Stanley Osher

Proc. Amer. Math. Soc. 89 (1983), 641-646

DOI: https://doi.org/10.1090/S0002-9939-1983-0718989-X

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Abstract:

We present a closed form expression for the viscosity solution to the Riemann problem for any scalar nonconvex conservation law. We then define an analogous problem for any scalar nonconvex Hamilton-Jacobi equation and obtain its (even simpler) solution. Extensions to two (and by inference, higher) space dimensional problems, when the initial discontinuity lies on a hyperplane, are also given.

References

Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8

First order quasi-linear equations in several independent variables

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Peter D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. MR 0350216
Stanley Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), no. 2, 217–235. MR 736327, DOI 10.1137/0721016