Two-dimensional riemann problem for a single conservation law

The entropy solutions to the partial differential equation (∂/∂t)u(t, x, y) + (∂/∂x)f(u(t, x, y)) + (∂/∂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) ≠ 0, g"(u) ≠ 0, (f"(u)lg"(u))' ≠ 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.