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

TL;DR

This paper establishes a quantitative link between a stochastic particle system with branching and the Keller--Segel PDE with logistic damping, showing convergence and suppression of blow-up in certain conditions.

Contribution

It introduces a microscopic particle system model with branching and singular interactions that approximates the Keller--Segel PDE, providing convergence rates and insights into blow-up suppression.

Findings

Empirical measure of particles converges to the PDE solution at rate N^{-1/(2(d+1))}.

The model demonstrates suppression of finite-time blow-up under certain damping conditions.

Provides a stochastic particle framework for analyzing chemotaxis models with logistic damping.

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, a logistic damping term is added in this PDE and local well-posedness of mild solutions is proven. When the space dimension is $2$ 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 $N^{-\frac{1}{2(d+1)}}$.