{
  "id": "1105.1526",
  "version": "v1",
  "published": "2011-05-08T15:03:54.000Z",
  "updated": "2011-05-08T15:03:54.000Z",
  "title": "Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations",
  "authors": [
    "Kyudong Choi",
    "Alexis F. Vasseur"
  ],
  "comment": "62 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study weak solutions of the 3D Navier-Stokes equations in whole space with $L^2$ initial data. It will be proved that $\\nabla^\\alpha u $ is locally integrable in space-time for any real $\\alpha$ such that $1< \\alpha <3$, which says that almost third derivative is locally integrable. Up to now, only second derivative $\\nabla^2 u$ has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{loc}^{4/(\\alpha+1)}$. These estimates depend only on the $L^2$ norm of initial data and integrating domains. Moreover, they are valid even for $\\alpha\\geq 3$ as long as $u$ is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-05-08T15:03:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D05",
      "35Q30"
    ],
    "keywords": [
      "fractional higher derivatives",
      "local study",
      "initial data",
      "3d navier-stokes equations",
      "locally integrable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1105.1526C"
    }
  }
}