Newtonian repulsion and radial confinement: convergence towards steady state

TL;DR

This paper analyzes the long-term behavior of multi-dimensional aggregation equations with Newtonian repulsion and radial confinement, establishing algebraic convergence rates towards unique steady states and exploring perturbed attraction potentials.

Contribution

It identifies a family of radial steady states and proves dimension-dependent decay rates, extending understanding of aggregation dynamics with Newtonian and quadratic potentials.

Findings

Quantified algebraic decay rate towards steady state.

Identified a one-parameter family of steady states.

Extended analysis to perturbed radial quadratic attraction.

Abstract

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate towards the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence towards steady repulsive-attractive solutions.