Aggregation-diffusion phenomena in domains with boundaries

This thesis is concerned with a class of mathematical models for the collective behaviour of autonomous agents, or particles, in general spatial domains, where particles exhibit pairwise interactions and may be subject to environmental forces. Such models have been shown to exhibit non-trivial behaviour due to interactions with the boundary of the domain. More specifically, when there is a boundary, it has been observed that the swarm of particles readily evolves into unstable states. Given this behaviour, we investigate the regularizing effect of adding noise to the system in the form of Brownian motion at the particle level, which produces linear diffusion in the continuum limit. To investigate the effect of linear diffusion and interactions with spatial boundaries on swarm equilibria, we analyze critical points of the associated energy functional, establishing conditions under which global minimizers exist. Through this process we uncover a new metastability phenomenon which necessitates the use of external forces to confine the swarm. We then introduce numerical methods for computing critical points of the energy, along with examples to motivate further research. Finally, we consider the short-time dynamics of the stochastic particle system as diffusion approaches zero. We verify that the analytical convergence rate in the zero diffusion limit is represented in numerics, which we believe validates and motivates the use of stochastic particle simulations for further exploration of the regularizing effect of Brownian motion on aggregation phenomena in domains with boundaries.