{
  "id": "1803.01988",
  "version": "v1",
  "published": "2018-03-06T02:23:37.000Z",
  "updated": "2018-03-06T02:23:37.000Z",
  "title": "Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion",
  "authors": [
    "Weirun Tao",
    "Yuxiang Li"
  ],
  "comment": "22pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper investigates an incompressible chemotaxis-Navier-Stokes system with slow $p$-Laplacian diffusion \\begin{eqnarray} \\left\\{\\begin{array}{lll} n_t+u\\cdot\\nabla n=\\nabla\\cdot(|\\nabla n|^{p-2}\\nabla n)-\\nabla\\cdot(n\\chi(c)\\nabla c),& x\\in\\Omega,\\ t>0, c_t+u\\cdot\\nabla c=\\Delta c-nf(c),& x\\in\\Omega,\\ t>0, u_t+(u\\cdot\\nabla) u=\\Delta u+\\nabla P+n\\nabla\\Phi,& x\\in\\Omega,\\ t>0, \\nabla\\cdot u=0,& x\\in\\Omega,\\ t>0 \\end{array}\\right. \\end{eqnarray} under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a bounded convex domain $\\Omega\\subset \\mathbb{R}^3$ with smooth boundary. Here, $\\Phi\\in W^{1,\\infty}(\\Omega)$, $0<\\chi\\in C^2([0,\\infty))$ and $0\\leq f\\in C^1([0,\\infty))$ with $f(0)=0$. It is proved that if $p>\\frac{32}{15}$ and under appropriate structural assumptions on $f$ and $\\chi$, for all sufficiently smooth initial data $(n_0,c_0,u_0)$ the model possesses at least one global weak solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-06T02:23:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q92",
      "35K55",
      "35Q35",
      "76S05",
      "92C17"
    ],
    "keywords": [
      "global weak solution",
      "three-dimensional chemotaxis-navier-stokes system",
      "laplacian diffusion",
      "sufficiently smooth initial data",
      "appropriate structural assumptions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}