Gradient flows of modified Wasserstein distances and porous medium equations with nonlocal pressure

TL;DR

This paper investigates porous medium equations with nonlocal pressure, constructing weak solutions using modified Wasserstein distances and analyzing their regularization and decay properties.

Contribution

It introduces a novel approach using JKO schemes with modified Wasserstein distances to solve nonlocal porous medium equations.

Findings

Existence of weak solutions via JKO schemes.

Regularization effects for solutions.

Decay estimates for $L^p$ norms.

Abstract

We study families of porous medium equation with nonlocal pressure. We construct their weak solutions via JKO schemes for modified Wasserstein distances. We also establish the regularization effect and decay estimates for the $L^p$ norms.