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Transactions of the American Mathematical Society

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



Two-dimensional Riemann problem for a single conservation law

Authors: Tong Zhang and Yu Xi Zheng
Journal: Trans. Amer. Math. Soc. 312 (1989), 589-619
MSC: Primary 35L65; Secondary 35L67
MathSciNet review: 930070
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Abstract: The entropy solutions to the partial differential equation

$\displaystyle (\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,$

with initial data constant in each quadrant of the $ (x,y)$ plane, have been constructed and are piecewise smooth under the condition $ f''(u) \ne 0, g''(u) \ne 0, (f''(u)/g''(u))\prime \ne 0$. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.

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  • [1] I. Gel'fand, Some problems in the theory of quasilinear equations, Uspekhi Mat. Nauk 14 (1959), 87-158; English transl., Amer. Math. Soc. Transl. (2) (1963), 295-381. MR 0110868 (22:1736)
  • [2] J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, 1983. MR 688146 (84d:35002)
  • [3] S. Kruzkov, First order quasilinear equations with several space variables, Mat. Sb. 123 (1970), 228-255; English transl., Math. USSR-Sb. 10 (1970), 217-243. MR 0267257 (42:2159)
  • [4] D. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), 534-559. MR 697528 (84f:35092)
  • [5] W. B. Lindquist, Construction of solutions to two-dimensional Riemann problem, and the scalar Riemann problem in one and two spatial dimensions: piecewise smooth solutions, New York Univ. Preprints, 1, 2, 1984. MR 841991 (87j:35065)
  • [6] J. Guckenheimer, Shocks and rarefactions in two space dimensions, Arch. Rational Mech. Anal. 59 (1975), 281-291. MR 0387829 (52:8668)
  • [7] Y. Val'ka, Discontinuous solutions of a multidimensional quasilinear equation (numerical experiments), U.S.S.R. Comput. Math. and Math. Phys. 8 (1968), 257-264.
  • [8] Zhang Tong and Chen Gui-qiang, Some fundamental concepts about system of two spatial dimensional conservation laws, Acta Math. Sci. 6 (1986), 463-474. MR 924036 (89a:35136)

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Article copyright: © Copyright 1989 American Mathematical Society

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