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

TL;DR

This paper proves the existence of solutions for a class of degenerate parabolic equations using a variational approach based on Monge-Kantorovich optimal transport theory, relaxing previous convexity assumptions.

Contribution

It introduces a variational method to establish solutions for degenerate parabolic equations with weaker convexity conditions than prior approaches.

Findings

Solutions exist for a broad class of degenerate parabolic equations.

The method applies to equations like Fokker-Planck, porous-medium, and p-Laplacian.

Less restrictive convexity assumptions are sufficient for existence.

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.