{
  "id": "2309.13713",
  "version": "v1",
  "published": "2023-09-24T18:10:39.000Z",
  "updated": "2023-09-24T18:10:39.000Z",
  "title": "Beginner's guide to Aggregation-Diffusion Equations",
  "authors": [
    "David Gómez-Castro"
  ],
  "categories": [
    "math.AP",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "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-V\\'azquez, 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-24T18:10:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "aggregation-diffusion equations",
      "beginners guide",
      "one-layer neural networks",
      "main ideas",
      "heat equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}