Explicit formula for weighted scalar nonlinear hyperbolic conservation laws
Authors:
Philippe LeFloch and JeanClaude Nédélec
Journal:
Trans. Amer. Math. Soc. 308 (1988), 667683
MSC:
Primary 35L65
MathSciNet review:
951622
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Abstract: We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form with initial data and boundary condition. The scalar function , , , is the unknown, the function is assumed to be strictly convex with inf and the weight function , , to be positive (for example, , with an arbitrary real ). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux at the boundary is solved by introducing a variational inequality. The uniqueness result is obtained by extending a semigroup property due to B. L. Keyfitz.
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Ph. Le Floch, Contributions á l'étude théorique et á l'approximation numérique des systémes hyperboliques nonlinéaires, Thése, École Polytéchnique, France.
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 C. Bardos, A. Y. Leroux and J. C. Nedelec, First order quasilinear erquations with boundary conditions, Comm. Partial Differential Equations 4 (1979), 10171034. MR 542510 (81b:35052)
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 S. N. Kruskov, First order quasilinear systems in several independant variables, Math. USSR Sb. 10 (1970), 217243.
 [3]
 P. D. Lax, Hyperbolic systems of conservation laws, Comm. Pure Appl. Math. 10 (1957), 537566. MR 0093653 (20:176)
 [4]
 P. D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM, Philadelphia, Pa., 1973. MR 0350216 (50:2709)
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 O. A. Oleinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95172. MR 0151737 (27:1721)
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 B. L. Keyfitz, Solutions with shocks, an example of contractive semigroups, Comm. Pure Appl. Math. 24 (1971), 125132. MR 0271545 (42:6428)
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 M. E. Schonbek, Existence of solutions to singular conservation laws, SIAM J. Math. Anal. 15 (1984). MR 762969 (86c:35099)
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 J. A. Smoller, Reactiondiffusion equations and shock waves, vol. 258, SpringerVerlag, 1983.
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 G. B. Whitham, Linear and nonlinear waves, WileyInterscience, New York, 1974. MR 0483954 (58:3905)
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 Ph. Le Floch and J. C. Nedelec, Explicit formula for weighted scalar conservation laws, Centre Math. Appl. Ecole Polytechnique, preprint, January 1985.
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 Ph. Le Floch, Contributions á l'étude théorique et á l'approximation numérique des systémes hyperboliques nonlinéaires, Thése, École Polytéchnique, France.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719880951622X
PII:
S 00029947(1988)0951622X
Article copyright:
© Copyright 1988
American Mathematical Society
