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Transactions of the American Mathematical Society

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



The Riemann problem for general $ 2\times 2$ conservation laws

Author: Tai Ping Liu
Journal: Trans. Amer. Math. Soc. 199 (1974), 89-112
MSC: Primary 35L65
MathSciNet review: 0367472
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Abstract: The Riemann Problem for a system of hyperbolic conservation laws of form

$\displaystyle (1)\quad \begin{array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} \\ {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \\ \end{array} $

with arbitrary initial constant states

$\displaystyle (2)\quad ({u_0}(x),{v_0}(x)) = \left\{ {\begin{array}{*{20}{c}} {... ...},{v_l}),\quad x < 0,} \\ {({u_r},{v_r}),\quad x > 0,} \\ \end{array} } \right.$

is considered. We assume that $ {f_\upsilon } < 0,{g_u} < 0$. Let $ {l_i}({r_i})$ be the left (right) eigenvectors of $ dF \equiv d(f,g)$ for eigenvalues $ {\lambda _1} < {\lambda _2}$. Instead of assuming the usual convexity condition $ d{\lambda _i}({r_i}) \ne 0,i = 1,2$ we assume that $ d{\lambda _i}({r_i}) = 0$ on disjoint union of $ 1$-dim manifolds in the $ (u,\upsilon )$ plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case $ d{\lambda _i}({r_i}) \ne 0$, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).

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Keywords: Conservation laws, shocks $ S$, rarefaction waves $ R$, contact discontinuities, Oleinik condition (E), Lax shock inequalities (L), shock speed
Article copyright: © Copyright 1974 American Mathematical Society