Confinement for Repulsive-Attractive Kernels

TL;DR

This paper studies how solutions to the aggregation equation with repulsive-attractive potentials stay confined within a fixed region, depending on initial data and potential properties.

Contribution

It establishes confinement results for solutions in both measure and smooth function settings with specific repulsive-attractive potentials.

Findings

Solutions remain within a fixed large ball depending on initial data.

Results apply to mildly singular and smooth potentials.

Confinement holds for both probability measures and smooth solutions.

Abstract

We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a smooth suitably growing at infinity attractive potential.