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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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DOI: https://doi.org/10.1090/S0002-9947-1989-0930070-3
Article copyright: © Copyright 1989 American Mathematical Society