Swarm Equilibria in Domains with Boundaries

We study equilibria in domains with boundaries for a first-order aggregation model that includes social interactions and exogenous forces. Such equilibrium solutions can be connected or disconnected, the latter consisting in a delta concentration on the boundary and a free swarm component in the interior of the domain. Equilibria are stationary points of an energy functional, and stable configurations are local minimizers of this functional. We find a one-parameter family of disconnected equilibrium configurations which are not energy minimizers; the only stable equilibria are the connected states. Nevertheless, we demonstrate that in certain cases the dynamical evolution, along the gradient flow of the energy functional, tends to overwhelmingly favor the formation of (unstable) disconnected equilibria.