{
  "id": "1310.2141",
  "version": "v2",
  "published": "2013-10-08T14:09:50.000Z",
  "updated": "2013-10-31T00:47:50.000Z",
  "title": "Analyticity for the (generalized) Navier-Stokes equations with rough initial data",
  "authors": [
    "Chunyan Huang",
    "Baoxiang Wang"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \\begin{align} u_t+(-\\Delta)^{\\alpha}u+u\\cdot \\nabla u +\\nabla p=0, \\ \\ {\\rm div} u=0, \\ \\ u(0,x)= u_0. \\nonumber \\end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($\\alpha=1$) with any initial data in critical Besov spaces $\\dot{B}^{n/p-1}_{p,q}(\\mathbb{R}^n)$ with $1< p<\\infty, \\ 1\\le q\\le \\infty $ and the solution is global if $u_0$ is sufficiently small in $\\dot{B}^{n/p-1}_{p,q}(\\mathbb{R}^n)$. In the case $p=\\infty$, the analyticity for the local solutions of the Navier-Stokes equation ($\\alpha=1$) with any initial data in modulation space $M^{-1}_{\\infty,1}(\\mathbb{R}^n)$ is obtained. We prove the global well-posedness for a fractional Navier-stokes equation ($\\alpha=1/2$) with small data in critical Besov spaces $\\dot{B}^{n/p}_{p,1}(\\mathbb{R}^n) \\ (1\\leq p\\leq\\infty)$ and show the analyticity of solutions with small initial data either in $\\dot{B}^{n/p}_{p,1}(\\mathbb{R}^n) \\ (1\\leq p<\\infty)$ or in $\\dot{B}^0_{\\infty,1} (\\mathbb{R}^n)\\cap {M}^0_{\\infty,1}(\\mathbb{R}^n)$. Similar results also hold for all $\\alpha\\in (1/2,1)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-10-31T00:47:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "35K55"
    ],
    "keywords": [
      "rough initial data",
      "analyticity",
      "critical besov spaces",
      "local solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.2141H"
    }
  }
}