{
  "id": "2410.10040",
  "version": "v1",
  "published": "2024-10-13T23:04:49.000Z",
  "updated": "2024-10-13T23:04:49.000Z",
  "title": "Aggregation-diffusion equations with saturation",
  "authors": [
    "José Antonio Carrillo",
    "Alejandro Fernández-Jiménez",
    "David Gómez-Castro"
  ],
  "comment": "51 pages, 7 figures",
  "categories": [
    "math.AP",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $\\rho$ with a continuous and compactly supported nonlinear mobility $\\mathrm{m}(\\rho)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-13T23:04:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35K55",
      "35K65",
      "35B40",
      "65M08",
      "35Q70",
      "35Q92",
      "47H20"
    ],
    "keywords": [
      "aggregation-diffusion equations",
      "saturation",
      "structure preserving implicit finite-volume scheme",
      "free energy",
      "long-time behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}