Dissipation for a non-convex gradient flow problem of a Patlack-Keller-Segel type for densities on $\mathbb{R}^n$, $n\geq 3$

TL;DR

This paper analyzes a non-convex gradient flow evolution equation in higher dimensions, revealing conditions under which diffusion dominates and mass disperses at a quantifiable polynomial rate.

Contribution

It extends the study of Patlack-Keller-Segel type systems to higher dimensions, identifying parameter regimes where diffusion overcomes aggregation.

Findings

Diffusion dominates in certain parameter ranges.

Mass disperses at an explicit polynomial rate.

The model generalizes Patlack-Keller-Segel dynamics to $ abla^2$-based flows.

Abstract

We study an evolution equation that is the gradient flow in the $2$-Wasserstien metric of a non-convex functional for densities in $\mathbb{R}^n$ with $n \geq 3$. Like the Patlack-Keller-Segel system on $\mathbb{R}^2$, this evolution equation features a competition between the dispersive effects of diffusion, and the accretive effects of a concentrating drift. We determine a parameter range in which the diffusion dominates, and all mass leaves any fixed compact subset of $\mathbb{R}^n$ at an explicit polynomial rate.