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

Abstract

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