Analysis and mean-field derivation of a porous-medium equation with fractional diffusion
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.
Keywords
Cite
@article{arxiv.2109.08598,
title = {Analysis and mean-field derivation of a porous-medium equation with fractional diffusion},
author = {Li Chen and Alexandra Holzinger and Ansgar Jüngel and Nicola Zamponi},
journal= {arXiv preprint arXiv:2109.08598},
year = {2021}
}