Swarming in domains with boundaries: approximation and regularization by nonlinear diffusion

TL;DR

This paper studies an aggregation model with nonlinear diffusion in bounded domains, analyzing the zero diffusion limit, convergence of solutions, and how small diffusion improves stability by preventing unstable equilibria.

Contribution

It provides a rigorous analysis of the zero diffusion limit and demonstrates numerically that small nonlinear diffusion stabilizes the model in bounded domains.

Findings

Weak solutions converge as diffusion vanishes

Energy minimizers also converge in the limit

Small nonlinear diffusion prevents unstable equilibria

Abstract

We consider an aggregation model with nonlinear diffusion in domains with boundaries and investigate the zero diffusion limit of its solutions. We establish the convergence of weak solutions for fixed times, as well as the convergence of energy minimizers in this limit. Numerical simulations that support the analytical results are presented. A second key scope of the numerical studies is to demonstrate that adding small nonlinear diffusion rectifies a flaw of the plain aggregation model in domains with boundaries, which is to evolve into unstable equilibria (non-minimizers of the energy).