## Explicit formula for weighted scalar nonlinear hyperbolic conservation laws

HTML articles powered by AMS MathViewer

- by Philippe LeFloch and Jean-Claude Nédélec PDF
- Trans. Amer. Math. Soc.
**308**(1988), 667-683 Request permission

## Abstract:

We prove a uniqueness and existence theorem for the entropy weak solution of nonlinear hyperbolic conservation laws of the form \[ \frac {\partial } {{\partial t}}(ru) + \frac {\partial } {{\partial x}}(rf(u)) = 0,\] with initial data and boundary condition. The scalar function $u = u(x, t)$, $x > 0$, $t > 0$, is the unknown, the function $f = f(u)$ is assumed to be strictly convex with inf $f( \cdot ) = 0$ and the weight function $r = r(x)$, $x > 0$, to be positive (for example, $r(x) = {x^\alpha }$, with an arbitrary real $\alpha$). We give an explicit formula, which generalizes a result of P. D. Lax. In particular, a free boundary problem for the flux $r( \cdot )f(u( \cdot , \cdot ))$ 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.## References

- C. Bardos, A. Y. le Roux, and J.-C. Nédélec,
*First order quasilinear equations with boundary conditions*, Comm. Partial Differential Equations**4**(1979), no. 9, 1017–1034. MR**542510**, DOI 10.1080/03605307908820117
S. N. Kruskov, - P. D. Lax,
*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**93653**, DOI 10.1002/cpa.3160100406 - 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** - O. A. Oleĭnik,
*Discontinuous solutions of non-linear differential equations*, Amer. Math. Soc. Transl. (2)**26**(1963), 95–172. MR**0151737**, DOI 10.1090/trans2/026/05 - Barbara Keyfitz Quinn,
*Solutions with shocks: An example of an $L_{1}$-contractive semigroup*, Comm. Pure Appl. Math.**24**(1971), 125–132. MR**271545**, DOI 10.1002/cpa.3160240203 - Maria Elena Schonbek,
*Existence of solutions to singular conservation laws*, SIAM J. Math. Anal.**15**(1984), no. 6, 1125–1139. MR**762969**, DOI 10.1137/0515088
J. A. Smoller, - G. B. Whitham,
*Linear and nonlinear waves*, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR**0483954**
Ph. Le Floch and J. C. Nedelec,

*First order quasilinear systems in several independant variables*, Math. USSR Sb.

**10**(1970), 217-243.

*Reaction-diffusion equations and shock waves*, vol. 258, Springer-Verlag, 1983.

*Explicit formula for weighted scalar conservation laws*, Centre Math. Appl. Ecole Polytechnique, preprint, January 1985. Ph. Le Floch,

*Contributions á l’étude théorique et á l’approximation numérique des systémes hyperboliques nonlinéaires*, Thése, École Polytéchnique, France.

## Additional Information

- © Copyright 1988 American Mathematical Society
- 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