Aggregation-Diffusion to Constrained Interaction: Minimizers & Gradient Flows in the Slow Diffusion Limit

TL;DR

This paper demonstrates that constrained interaction energies can be derived as the slow diffusion limit of aggregation-diffusion energies, establishing convergence of minimizers and gradient flows, and extends well-posedness theory for these equations.

Contribution

It proves the slow diffusion limit for a broad class of interaction energies and extends the theory to nonconvex potentials, with applications to numerical methods.

Findings

Minimizers of aggregation-diffusion energies converge to constrained interaction energy minimizers.

Gradient flows of aggregation-diffusion energies converge to gradient flows of constrained energies.

Extended well-posedness theory for aggregation-diffusion equations with nonconvex potentials.

Abstract

Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.