Beginner’s guide to aggregation-diffusion equations

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–Vázquez, McKean–Vlasov, Kuramoto, and one-layer neural networks. In Sect. 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.