The Riemann Problem in Two Space Dimensions for a Single Conservation Law

Source abstract

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.