Nonlocal approximation of nonlinear diffusion equations

TL;DR

This paper demonstrates that degenerate nonlinear diffusion equations can be approximated as limits of nonlocal PDEs derived from gradient flows, extending previous results and providing a particle approximation method.

Contribution

It introduces a novel approach to approximate nonlinear diffusion equations via nonlocal PDEs using gradient flow techniques, covering the porous medium equation.

Findings

Established asymptotic connection between nonlocal PDEs and nonlinear diffusion equations.

Constructed weak solutions using the Jordan-Kinderlehrer-Otto scheme.

Provided a qualitative particle approximation for the equations.

Abstract

We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of $2$-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.