Mean-field limit and quantitative estimates with singular attractive kernels

TL;DR

This paper establishes the mean field limit and quantitative estimates for systems with singular attractive kernels, notably deriving the Patlak-Keller-Segel model rigorously in subcritical regimes using a novel modulated free energy approach.

Contribution

It provides the first rigorous derivation with quantitative estimates of the Patlak-Keller-Segel model in subcritical regimes for systems with singular attractive interactions.

Findings

Proves mean field limit for singular attractive kernels.

Derives quantitative estimates for particle systems.

Provides a rigorous derivation of the Patlak-Keller-Segel model.

Abstract

This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the s{\'e}minaire Laurent Schwartz proceeding [https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full proof of results announced by the authors in [C. R. Acad. Sciences Section Maths (2019)].