The Riemann Problem in Two Space Dimensions for a Single Conservation Law
The Riemann Problem in Two Space Dimensions for a Single Conservation Law
复制标题
DOI：
10.1137/0514045
复制
发表时间：
1983-05
影响因子：
2
通讯作者：
David Wagner
中科院分区：
文献类型：
--
作者：
David Wagner
给出了偏微分方程${\partial /{\partial t}}u(t,x,y) + {\partial /{\partial x}}f(u(t,x,y)) + {\partial / {\partial y}}g(u(t,x,y)) = 0$的解，在$(x,y)$平面的每个象限中初始数据为常数。这个问题推广了一维空间方程的黎曼问题。虽然解的存在性和唯一性是已知的，但关于解的定性行为所知甚少。当f和g为凸且$f \equiv g$时，则解满足Kruzkov和Vol 'pert给出的唯一性或熵条件。在f和g的某些附加条件下，当f和g是凸且足够接近时，我们的解满足熵条件。给出了一个反例来说明f和g上这些附加条件的必要性。这个反例的正确熵解显示出新的有趣的现象。
Solutions are given for the partial differential equation ${\partial /{\partial t}}u(t,x,y) + {\partial /{\partial x}}f(u(t,x,y)) + {\partial / {\partial y}}g(u(t,x,y)) = 0$, with initial data constant in each quadrant of the $(x,y)$ plane. This problem generalizes the Riemann problem for equations in one space dimension. Although existence and uniqueness of solutions are known, little is known concerning the qualitative behavior of solutions.When f and g are convex and $f \equiv g$, then our solutions satisfy the uniqueness, or entropy condition given by Kruzkov and Vol’pert. Under certain extra conditions on f and g, our solutions satisfy the entropy condition if f and g are convex and sufficiently close. A counterexample is given to show the necessity of these extra conditions on f and g. The correct entropy solution for this counterexample exhibits new and interesting phenomena.