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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics