Analysis and mean-field derivation of a porous-medium equation with fractional diffusion

TL;DR

This paper derives a nonlocal porous-medium equation with fractional diffusion from particle systems, providing existence results and analyzing the propagation of chaos, thus connecting stochastic particle models with fractional PDEs.

Contribution

It introduces a mean-field derivation of a fractional porous-medium equation from particle systems with Riesz potential, including existence analysis and propagation of chaos.

Findings

Derivation of a nonlocal porous-medium equation with fractional Laplacian

Establishment of existence results for the fractional porous-medium equation

Proof of propagation of chaos for the particle system

Abstract

A mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential is performed, leading to nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The proof is based on Oelschl\"ager's approach and a priori estimates for the associated diffusion equations, coming from energy-type and entropy inequalities as well as parabolic regularity. An existence analysis of the fractional porous-medium equation is also provided, based on a careful regularization procedure, new variants of fractional Gagliardo--Nirenberg inequalities, and the div-curl lemma. A consequence of the mean-field limit estimates is the propagation of chaos property.