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Transactions of the American Mathematical Society
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
MathSciNet review: 0380135
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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].

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PII: S 0002-9947(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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