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

• Published in: Communications in partial differential equations Vol. 47; no. 11; pp. 2217 - 2269
• Main Authors: Chen, Li, Holzinger, Alexandra, Jüngel, Ansgar, Zamponi, Nicola
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.11.2022; Taylor & Francis Ltd
• Subjects: Diffusion; Estimates; Existence analysis; fractional diffusion; Inequalities; interacting particle systems; Mathematical analysis; mean-field limit; nonlocal porous-medium equation; Porous media; propagation of chaos; Regularization; Transport equations
• ISSN: 0360-5302; 1532-4133
• DOI: 10.1080/03605302.2022.2118608

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 Oelschläger'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.