Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up

TL;DR

This paper studies a particle-based approximation of the 1D Keller-Segel equation, analyzing blow-up behavior, stability, and rigidity of the system, with detailed insights into the dynamics involving small numbers of particles.

Contribution

It introduces a particle system derived from the Keller-Segel equation, providing new stability and rigidity results for blow-up phenomena in one dimension.

Findings

Identification of stability basins for particle numbers

Weak rigidity results for rescaled dynamics

Detailed analysis of three-particle interactions

Abstract

We investigate a particle system which is a discrete and deterministic approximation of the one-dimensional Keller-Segel equation with a logarithmic potential. The particle system is derived from the gradient flow of the homogeneous free energy written in Lagrangian coordinates. We focus on the description of the blow-up of the particle system, namely: the number of particles involved in the first aggregate, and the limiting profile of the rescaled system. We exhibit basins of stability for which the number of particles is critical, and we prove a weak rigidity result concerning the rescaled dynamics. This work is complemented with a detailed analysis of the case where only three particles interact.