Equilibria of an aggregation model with linear diffusion in domains with boundaries

TL;DR

This paper analyzes how linear diffusion and boundary interactions influence swarm equilibria, revealing domain-dependent energy properties, conditions for minimizer existence, and the necessity of external forces for confinement.

Contribution

It introduces a sharp domain-dependent criterion for the existence of global minimizers and highlights the role of external forces in confinement.

Findings

Energy unboundedness depends on domain volume filling.

Metastable mass translation occurs in asymmetric domains.

External forces are needed for confinement and minimizer existence.

Abstract

We investigate the effect of linear diffusion and interactions with the domain boundary on swarm equilibria by analyzing critical points of the associated energy functional. Through this process we uncover two properties of energy minimization that depend explicitly on the spatial domain: (i) unboundedness from below of the energy due to an imbalance between diffusive and aggregative forces depends explicitly on a certain volume filling property of the domain, and (ii) metastable mass translation occurs in domains without sufficient symmetry. From the first property, we present a sharp condition for existence (resp. non-existence) of global minimizers in a large class of domains, analogous to results in free space, and from the second property, we identify that external forces are necessary to confine the swarm and grant existence of global minimizers in general domains. We also introduce a numerical method for computing critical points of the energy and give examples to motivate further research.