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...
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| Formato: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Recursos: | Universidad Autónoma de Madrid |
| Repositorio: | Biblos-e Archivo. Repositorio Institucional de la UAM |
| Idioma: | inglés |
| OAI Identifier: | oai:repositorio.uam.es:10486/711615 |
| Acesso em linha: | http://hdl.handle.net/10486/711615 https://dx.doi.org/10.1007/s40324-024-00350-y |
| Access Level: | acceso abierto |
| Palavra-chave: | Aggregation-Diffusion 35K55 Nonlinear Parabolic Equations 35K65 Degenerate Parabolic Equations 35K67 Singular Parabolic Equations Nonlinear Diffusion 35B40 - 35Q84 Fokker-Planck Equations Matemáticas |
| Resumo: | 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 |
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