Aggregation-diffusion equations: dynamics, asymptotics, and singular limits

TL;DR

This paper reviews the mathematical analysis of aggregation-diffusion equations, exploring their dynamics, singular limits, and numerical methods, with a focus on recent advances in particle-based simulations and the blob method.

Contribution

It provides a comprehensive review of analytical results, singular limits, and numerical techniques, including the application of the blob method to aggregation-diffusion equations.

Findings

Rich dynamics including symmetrization and metastability

Singular limits reveal different regimes like constrained and localized aggregation

Recent advances in deterministic particle methods improve simulation accuracy

Abstract

Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, as well as localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -- the blob method for diffusion -- to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.