{
  "id": "2105.09068",
  "version": "v1",
  "published": "2021-05-19T11:33:17.000Z",
  "updated": "2021-05-19T11:33:17.000Z",
  "title": "Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms",
  "authors": [
    "Patrik Knopf",
    "Andrea Signori"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We investigate a multiphase Cahn-Hilliard model for tumor growth with general source terms. The multiphase approach allows us to consider multiple cell types and multiple chemical species (oxygen and/or nutrients) that are consumed by the tumor. Compared to classical two-phase tumor growth models, the multiphase model can be used to describe a stratified tumor exhibiting several layers of tissue (e.g., proliferating, quiescent and necrotic tissue) more precisely. Our model consists of a convective Cahn-Hilliard type equation to describe the tumor evolution, a velocity equation for the associated volume-averaged velocity field, and a convective reaction-diffusion type equation to describe the density of the chemical species. The velocity equation is either represented by Darcy's law or by the Brinkman equation. We first construct a global weak solution of the multiphase Cahn-Hilliard-Brinkman model. After that, we show that such weak solutions of the system converge to a weak solution of the multiphase Cahn-Hilliard-Darcy system as the viscosities tend to zero in some suitable sense. This means that the existence of a global weak solution to the Cahn-Hilliard-Darcy system is also established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-19T11:33:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "35K35",
      "35K86",
      "35Q92",
      "76D07",
      "92C17",
      "92C50"
    ],
    "keywords": [
      "general source terms",
      "stratified tumor growth",
      "multiphase cahn-hilliard-darcy",
      "cahn-hilliard-brinkman model",
      "two-phase tumor growth models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}