Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics

TL;DR

This paper investigates the conditions under which nonlinear aggregation-diffusion equations reach radially symmetric equilibrium states, demonstrating existence, symmetry, and long-term convergence of solutions, especially for Newtonian interactions in two dimensions.

Contribution

It establishes the symmetry, existence, and uniqueness of equilibrium solutions, and proves convergence of solutions to these equilibria over time, particularly for Newtonian interactions in 2D.

Findings

Radially symmetric equilibrium configurations exist for all masses.

All stationary states with suitable regularity are radially symmetric.

Solutions converge to a unique equilibrium profile up to translation in 2D Newtonian case.

Abstract

We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses. All stationary states with suitable regularity are shown to be radially symmetric by means of continuous Steiner symmetrization techniques. Calculus of variations tools allow us to show the existence of global minimizers among these equilibria. Finally, in the particular case of Newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation-diffusion equations towards this unique equilibrium profile up to translations as $t\to\infty$.