Existence of solutions to degenerate parabolic equations via the Monge-Kantorovich theory

Abstract

We obtain solutions of the nonlinear degenerate parabolic equation \[ \frac{\partial \rho}{\partial t}= \mbox{div} \Big\{ \rho \nabla c^\star \left[ \nabla \left(F^\prime(\rho)+V\right) \right] \Big\} \] as a steepest descent of an energy with respect to a convex cost functional. The method used here is variational. It requires fewer uniform convexity assumptions than those imposed by Alt and Luckhaus in their pioneering work [4]. In fact, their assumptions may fail in our equation. This class of equations includes the Fokker-Planck equation, the porous-medium equation, the fast diffusion equation and the parabolic p-Laplacian equation.