Beginner's guide to Aggregation-Diffusion Equations

TL;DR

This survey introduces key techniques, models, and analytical frameworks in Aggregation-Diffusion Equations, covering historical context, well-posedness, asymptotic analysis, and numerical methods for a broad audience.

Contribution

It provides a comprehensive overview of the field, including modeling, key equations, well-posedness, asymptotic behavior, and numerical approaches, serving as an accessible introduction.

Findings

Overview of main models like Heat, Fokker-Planck, Keller-Segel

Discussion of well-posedness in Sobolev and Wasserstein spaces

Presentation of numerical methods for simulation

Abstract

The aim of this survey is to serve as an introduction to the different techniques available in the broad field of Aggregation-Diffusion Equations. We aim to provide historical context, key literature, and main ideas in the field. We start by discussing the modelling and famous particular cases: Heat equation, Fokker-Plank, Porous medium, Keller-Segel, Chapman-Rubinstein-Schatzman, Newtonian vortex, Caffarelli-V\'azquez, McKean-Vlasov, Kuramoto, and one-layer neural networks. In Section 4 we present the well-posedness frameworks given as PDEs in Sobolev spaces, and gradient-flow in Wasserstein. Then we discuss the asymptotic behaviour in time, for which we need to understand minimisers of a free energy. We then present some numerical methods which have been developed. We conclude the paper mentioning some related problems.