Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

Authors:
Philippe LeFloch and Jean-Claude Nédélec

Journal:
Trans. Amer. Math. Soc. **308** (1988), 667-683

MSC:
Primary 35L65

DOI:
https://doi.org/10.1090/S0002-9947-1988-0951622-X

MathSciNet review:
951622

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form

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.

**[1]**C. Bardos, A. Y. Leroux and J. C. Nedelec,*First order quasilinear erquations with boundary conditions*, Comm. Partial Differential Equations**4**(1979), 1017-1034. MR**542510 (81b:35052)****[2]**S. N. Kruskov,*First order quasilinear systems in several independant variables*, Math. USSR Sb.**10**(1970), 217-243.**[3]**P. D. Lax,*Hyperbolic systems of conservation laws*, Comm. Pure Appl. Math.**10**(1957), 537-566. 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)****[5]**O. A. Oleinik,*Discontinuous solutions of nonlinear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95-172. MR**0151737 (27:1721)****[6]**B. L. Keyfitz,*Solutions with shocks, an example of**contractive semi-groups*, Comm. Pure Appl. Math.**24**(1971), 125-132. MR**0271545 (42:6428)****[7]**M. E. Schonbek,*Existence of solutions to singular conservation laws*, SIAM J. Math. Anal.**15**(1984). MR**762969 (86c:35099)****[8]**J. A. Smoller,*Reaction-diffusion equations and shock waves*, vol. 258, Springer-Verlag, 1983.**[9]**G. B. Whitham,*Linear and nonlinear waves*, Wiley-Interscience, New York, 1974. MR**0483954 (58:3905)****[10]**Ph. Le Floch and J. C. Nedelec,*Explicit formula for weighted scalar conservation laws*, Centre Math. Appl. Ecole Polytechnique, preprint, January 1985.**[11]**Ph. Le Floch,*Contributions á l'étude théorique et á l'approximation numérique des systémes hyperboliques nonlinéaires*, Thése, École Polytéchnique, France.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35L65

Retrieve articles in all journals with MSC: 35L65

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0951622-X

Article copyright:
© Copyright 1988
American Mathematical Society