Existence and uniqueness theorems for Riemann problems
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- by Tai Ping Liu PDF
- Trans. Amer. Math. Soc. 212 (1975), 375-382 Request permission
Abstract:
In [2] the author proposed the entropy condition (E) and solved the Riemann problem for general $2 \times 2$ conservation laws ${u_t} + f{(u,v)_x} = 0,{v_t} + g{(u,v)_x} = 0$, under the assumptions that the system is hyperbolic, and ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$. The purpose of this paper is to extend the above results to a much wider class of $2 \times 2$ conservation laws. Instead of assuming that ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$, we assume that the characteristic speed is not equal to the shock speed of different family. This assumption is motivated by the works of Lax [1] and Smoller [4].References
- P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
- Tai Ping Liu, The Riemann problem for general $2\times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89–112. MR 367472, DOI 10.1090/S0002-9947-1974-0367472-1
- Tai Ping Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), no. 1, 78–88. MR 387830, DOI 10.1016/0022-247X(76)90146-3
- J. A. Smoller, On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems, Michigan Math. J. 16 (1969), 201–210. MR 247283
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 212 (1975), 375-382
- MSC: Primary 35L65
- DOI: https://doi.org/10.1090/S0002-9947-1975-0380135-2
- MathSciNet review: 0380135