Tightness of radially-symmetric solutions to 2D aggregation-diffusion equations with weak interaction forces

TL;DR

This paper establishes the tightness of radially-symmetric solutions to 2D aggregation-diffusion equations with weakly confining interaction potentials, using Wasserstein gradient flow and symmetrization techniques, advancing understanding of solution behavior.

Contribution

It introduces the ERC property and proves tightness for a broad class of weakly confining potentials, a first in 2D aggregation-diffusion equations.

Findings

Proves tightness of solutions with weakly confining potentials.

Uses Wasserstein gradient flow and symmetrization methods.

First such result for general weakly confining potentials in 2D.

Abstract

We prove the tightness of radially-symmetric solutions to 2D aggregation-diffusion equations, where the pairwise attraction force is possibly degenerate at large distance. We first reduce the problem into the finiteness of a time integral in the density on a bounded region, by introducing a new assumption on the interaction potential called the essentially radially contractive (ERC) property. Then we prove this finiteness by using the 2-Wasserstein gradient flow structure, combining with the continuous Steiner symmetrization curves and the local clustering curve. This is the first tightness result on the 2D aggregation-diffusion equations for a general class of weakly confining potentials, i.e., those $W(r)$ with $\lim_{r\rightarrow\infty}W(r)<\infty$, and serves as an important step towards the study of equilibration.