{
  "id": "1706.02022",
  "version": "v1",
  "published": "2017-06-07T01:58:31.000Z",
  "updated": "2017-06-07T01:58:31.000Z",
  "title": "Global existence to a $3D$ chemotaxis-Navier-stokes system with nonlinear diffusion and rotation",
  "authors": [
    "Jiashan Zheng",
    "Yanyan Li",
    "Xinhua Zou",
    "Dongfang Zhang",
    "Weifang Yan"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \\left\\{ \\begin{array}{l} n_t+u\\cdot\\nabla n=\\Delta n^m-\\nabla\\cdot(nS(x,n,c)\\cdot\\nabla c),\\quad x\\in \\Omega, t>0, c_t+u\\cdot\\nabla c=\\Delta c-nc,\\quad x\\in \\Omega, t>0,\\\\ u_t+\\kappa(u \\cdot \\nabla)u+\\nabla P=\\Delta u+n\\nabla \\phi ,\\quad x\\in \\Omega, t>0,\\\\ \\nabla\\cdot u=0,\\quad x\\in \\Omega, t>0 \\end{array}\\right.\\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $\\Omega\\subseteq \\mathbb{R}^3$ with smooth boundary, which describes the motion of oxygen-driven bacteria in a fluid. Here % $\\Omega\\subseteq \\mathbb{R}^3$ is a , $\\kappa\\in \\mathbb{R}$ and $S$ denotes the strength of nonlinear fluid convection and a given tensor-valued function, respectively. Assume $m>\\frac{10}{9}$ and $S$ fulfills $|S(x,n,c)| \\leq S_0(c)$ for all $(x,n,c)\\in \\bar{\\Omega} \\times [0, \\infty)\\times[0, \\infty)$ with $S_0(c)$ nondecreasing on $[0,\\infty)$, then for any reasonably regular initial data, the corresponding initial-boundary problem $(CNF)$ admits at least one global weak solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-07T01:58:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear diffusion",
      "global existence",
      "no-flux boundary conditions",
      "dirichlet boundary condition",
      "global weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}