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

Author:
Stanley Osher

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

MSC:
Primary 35L65

MathSciNet review:
718989

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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.

**[1]**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**, 10.1090/S0002-9947-1983-0690039-8**[2]**S. N. Kružkov,*First order quasi-linear equations in several independent variables*, Math. USSR Sb.**10**(1970), 217-243.**[3]**Peter D. Lax,*Hyperbolic systems of conservation laws and the mathematical theory of shock waves*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. MR**0350216****[4]**Stanley Osher,*Riemann solvers, the entropy condition, and difference approximations*, SIAM J. Numer. Anal.**21**(1984), no. 2, 217–235. MR**736327**, 10.1137/0721016

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Additional Information

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

Keywords:
Conservation law,
Hamilton-Jacobi equations,
Riemann problem

Article copyright:
© Copyright 1983
American Mathematical Society