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Transactions of the American Mathematical Society

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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. M. Gel′fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 87–158 (Russian). MR 0110868
  • [2] Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
  • [3] S. N. Kružkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
  • [4] David H. Wagner, The Riemann problem in two space dimensions for a single conservation law, SIAM J. Math. Anal. 14 (1983), no. 3, 534–559. MR 697528,
  • [5] W. B. Lindquist, Construction of solutions for two-dimensional Riemann problems, Comput. Math. Appl. Part A 12 (1986), no. 4-5, 615–630. Hyperbolic partial differential equations, III. MR 841991
  • [6] John Guckenheimer, Shocks and rarefactions in two space dimensions, Arch. Rational Mech. Anal. 59 (1975), no. 3, 281–291. MR 0387829,
  • [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] Tong Chang and Gui Qiang Chen, Some fundamental concepts about system of two spatial dimensional conservation laws, Acta Math. Sci. (English Ed.) 6 (1986), no. 4, 463–474. MR 924036

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Article copyright: © Copyright 1989 American Mathematical Society