Congested aggregation via Newtonian interaction

TL;DR

This paper introduces a novel congested aggregation model with Newtonian attraction and a height constraint, analyzing its dynamics and long-term behavior using a blend of gradient flow and viscosity solution techniques.

Contribution

It develops a new mathematical framework for a nonlocal aggregation model with height constraints, addressing nonconvexity and nonlocality challenges.

Findings

Characterizes patch solutions via a Hele-Shaw free boundary problem.

Proves convergence of patch solutions to a disk in long-time limit.

Combines energy methods with viscosity solutions for analysis.

Abstract

We consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard height constraint. This provides a counterpoint to existing literature on repulsive-attractive nonlocal interaction models, where the repulsive effects instead arise from an interaction kernel or the addition of diffusion. We formulate our model as the Wasserstein gradient flow of an interaction energy, with a penalization to enforce the constraint on the height of the density. From this perspective, the problem can be seen as a singular limit of the Keller-Segel equation with degenerate diffusion. Two key properties distinguish our problem from previous work on height constrained equations: nonconvexity of the interaction kernel (which places the model outside the scope of classical gradient flow theory) and nonlocal dependence of the velocity field on the density (which causes the problem to lack a comparison principle). To overcome these obstacles, we combine recent results on gradient flows of nonconvex energies with viscosity solution theory. We characterize the dynamics of patch solutions in terms of a Hele-Shaw type free boundary problem and, using this characterization, show that in two dimensions patch solutions converge to a characteristic function of a disk in the long-time limit, with explicit rate of convergence. We believe that a key contribution of the present work is our blended approach, combining energy methods with viscosity solution theory.