Abstract
AbstractThe 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.
Funder
Ministerio de Ciencia e Innovación
Spanish Government
Universidad Autónoma de Madrid
Publisher
Springer Science and Business Media LLC
Reference180 articles.
1. Amann, H.: Dynamic theory of quasilinear parabolic systems—III. Global existence. Math. Zeitschrift 205(1), 331 (1990)
2. Ambrosio, L., Brué, E., Semola, D.: Lectures on Optimal Transport. Springer International Publishing, Berlin (2021)
3. Ambrosio, L., Gigli, N., Savare, G.: Gradient Flows. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, pp 1–27 (2005)
4. Ambrosio, L., Mainini, E., Serfaty, S.: Gradient flow of the Chapman–Rubinstein–Schatzman model for signed vortices. Annales de l’I.H.P.Analyse non linéaire 28(2), 217–246 (2011)
5. Ambrosio, L., Serfaty, S.: A gradient flow approach to an evolution problem arising in superconductivity. Commun. Pure Appl. Math. 61(11), 1495–1539 (2008)