Nonlocal interactions by repulsive-attractive potentials: radial ins/stability

TL;DR

This paper analyzes the stability of solutions to nonlocal particle interaction equations with repulsive-attractive potentials, establishing conditions for convergence to spherical or singular stationary states, supported by theoretical proofs and numerical simulations.

Contribution

It provides sharp conditions for radial stability and instability of solutions, extending understanding of nonlocal interaction models with repulsive-attractive potentials.

Findings

Radially symmetric solutions converge exponentially to spherical shells under certain conditions.

Radial solutions cannot weakly converge to spherical shells if conditions are not met.

Non-radial solutions can converge to singular stationary states supported on hypersurfaces.

Abstract

In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins/stability are sharp.