Beginner's guide to Aggregation-Diffusion Equations
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910325191737344 |
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| author | Gómez-Castro, David |
| author_facet | Gómez-Castro, David |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13713 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Beginner's guide to Aggregation-Diffusion Equations Gómez-Castro, David Analysis of PDEs Numerical Analysis 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 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. |
| title | Beginner's guide to Aggregation-Diffusion Equations |
| topic | Analysis of PDEs Numerical Analysis |
| url | https://arxiv.org/abs/2309.13713 |