Propagation of chaos for the Keller–Segel equation with a logarithmic cut-off

We consider an $N$-particle interacting system with the Newtonian potential aggregation and Brownian motions. Assuming that the initial data are independent and identically distributed (i.i.d.) with a common probability density function $\rho_0 \in L^{\infty} (\mathbb{R}^d) \cap L^1 (\mathbb{R}^d, (1 + \lvert x \rvert) dx)$. We rigorously prove the propagation of chaos for this interacting system with a cut-off parameter $\varepsilon \sim (\operatorname{ln} N)^{-\frac{1}{d}}\: $: when $N \to \infty$, the empirical measure of the particle system converges in law to a probability measure and this measure possesses a density which is a weak solution to the mean-field Keller–Segel (KS) equation. More precisely, as $N \to \infty$, each particle path is approximated by a strong solution to a mean-field self-consistent stochastic differential equation (SDE). The global existence and uniqueness of strong solution to this SDE is proved and consequently we also prove the uniqueness of weak solution to the KS equation.

For $d = 2$, if $8 \pi \nu \gt 1$, the propagation of chaos is valid globally in time. On the other hand, if $8 \pi \nu \lt 1$, we show that the expectation of the collision time for the interacting particles system is bounded by $\frac{2 \pi \operatorname{Var} \lbrace \rho_0 \rbrace}{1 - 8 \pi \nu}$. For $d \geq 3$, if ${\lVert \rho_0 \rVert}_{L^\frac{d}{2} (\mathbb{R}^d)}$ is bounded by a universal constant depending only on $\nu$ and $d$, then the propagation of chaos is also valid globally in time.

Keywords

Newtonian potential aggregation, self-gravitating Brownian particles system, meanfield limit, $L^{\infty}$ bound, $\log$-Lipschitz continuity, uniqueness of weak solution, stability in Wasserstein metric

2010 Mathematics Subject Classification

35K55, 35Q92, 60H30, 65M75

Full Text (PDF format)

Received 18 April 2019

Accepted 9 August 2019

Published 13 May 2020