Quantitative approximation of a Keller–Segel PDE by a branching moderately interacting particle system and suppression of blow-up

Abstract The Keller–Segel PDE is a model for chemotaxis known to exhibit possible finite-time blow-up. Following a seminal work by Tello and Winkler (2007 Commun. Partial Differ. Equ. 32 849–77), a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is two or when the damping is strong enough, the solution is global in time. In the second part of this work, a microscopic description of this model is introduced in terms of a system of stochastic moderately interacting particles. This system features two main characteristics: the interaction between particles happens through a singular (Coulomb-type) kernel which is attractive; and the particles are subject to demographic events, birth and death due to local competition with other particles. The latter induces a branching structure of the particle system. Then the main result of this work is the convergence of the empirical measure of the particle system towards the Keller–Segel PDE with logistic damping, with a rate of order N − 1 2 ( d + 1 ) .