{
  "id": "1109.4043",
  "version": "v2",
  "published": "2011-09-19T14:44:21.000Z",
  "updated": "2013-02-22T07:33:24.000Z",
  "title": "On the stability in weak topology of the set of global solutions to the Navier-Stokes equations",
  "authors": [
    "Hajer Bahouri",
    "Isabelle Gallagher"
  ],
  "comment": "To appear in Archive for Rational and Mechanical Analysis",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $X$ be a suitable function space and let $\\cG \\subset X$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three dimensional Navier-Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of $\\cG$ belongs to $\\cG$ if $n$ is large enough, provided the convergence holds \"anisotropically\" in frequency space. Typically that excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier-Stokes equations; it is also shown that initial data which does not belong to $\\cG$ (hence which produces a solution blowing up in finite time) cannot have a strong anisotropy in its frequency support.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-02-22T07:33:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equations",
      "global solutions",
      "divergence free vector fields",
      "weak topology",
      "free vector fields converging"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.4043B"
    }
  }
}