A gradient flow approach to the porous medium equation with fractional pressure

TL;DR

This paper introduces a gradient flow framework for fractional porous media equations, establishing existence, regularity, decay properties, and connecting to classical models through limits.

Contribution

It constructs weak solutions as Wasserstein gradient flows of fractional Sobolev norms, providing new insights into fractional porous media equations.

Findings

Energy dissipation inequality proved

Regularizing effects demonstrated

Decay estimates for $L^p$ norms established

Abstract

We consider a family of fractional porous media equations, recently studied by Caffarelli and V\'azquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the $L^p$ norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.