Blow-up dynamics of self-attracting diffusive particles driven by competing convexities

TL;DR

This paper investigates the blow-up behavior of self-attracting diffusive particles modeled by gradient flows, revealing how competition between entropy and interaction energy influences particle dynamics and blow-up conditions.

Contribution

It provides the first rigorous analysis of blow-up criteria for a particle system approximating a non-local drift-diffusion equation in one dimension.

Findings

Proves global existence of particle trajectories under certain initial conditions.

Establishes two criteria for finite-time blow-up of solutions.

Shows consistency with the continuous Keller-Segel model in high dimensions.

Abstract

In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is a consistent approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel in high dimensions.