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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Existence and uniqueness theorems for Riemann problems


Author: Tai Ping Liu
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
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537-566. MR 20 #176. MR 0093653 (20:176)
  • [2] T.-P. Liu, The Riemann problem for general $ 2 \times 2$ conservation laws, Trans. Amer. Math. Soc. 199 (1974), 89-112. MR 0367472 (51:3714)
  • [3] -, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. (to appear). MR 0387830 (52:8669)
  • [4] 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 40 #552. MR 0247283 (40:552)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0380135-2
Keywords: Conservation laws, shocks, rarefaction waves, contact discontinuities, Lax shock inequalities (L), entropy condition (E)
Article copyright: © Copyright 1975 American Mathematical Society

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