Beginner’s guide to aggregation-diffusion equations
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David Gómez-Castro [1]
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[1] Universidad Autónoma de Madrid
Universidad Autónoma de Madrid
Madrid, España
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- Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 81, Nº. 4, 2024, págs. 531-587
- Idioma: inglés
- DOI: 10.1007/s40324-024-00350-y
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Resumen
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.
