Chen, L., Holzinger, A., Jüngel, A., & Zamponi, N. (2022). Analysis and mean-field derivation of a porous-medium equation with fractional diffusion. Communications in Partial Differential Equations, 47(11), 2217–2269. https://doi.org/10.1080/03605302.2022.2118608
existence analysis; fractional diffusion; interacting particle systems; mean-field limit; propagation of chaos; nonlocal porous- medium equation
en
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 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.