Scaling limits in Kinetic theory

Summer School - Lyon - 2019

**Talk by Didier Bresch** (Université de Savoie Mont Blanc)

*Quantitative estimates for a stochastic particle approximation of the Patlak-Keller-Segel equation*

In this talk, I will discuss how a modulated free energy which combines the methods developed by P.-E. Jabin & Z. Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math (2018), and references therein] allows to cover more general kernels when we are interested in
explicit rates of convergence with respect to the number of particles in mean field approximation theory. I will focus on the attractive kernel involved in the stochastic particle approximation of the Patlak-Keller-Segel system in the subcritical regime. This completes in some sense the works by P. Cattiaux & L. Pédèche and by N. Fournier & B. Jourdain, where existence, uniqueness, and convergence results have been obtained in some regimes. I will also briefly discuss the ideas to treat some singular potentials which combine large smooth part, small attractive singular part, and large repulsive singular part.

(This is a recent joint work with P.-E. Jabin and Z. Wang.)