{
  "id": "0710.1604",
  "version": "v5",
  "published": "2007-10-08T17:58:11.000Z",
  "updated": "2009-05-21T18:01:19.000Z",
  "title": "A quantitative formulation of the global regularity problem for the periodic Navier-Stokes equation",
  "authors": [
    "Terence Tao"
  ],
  "comment": "12 pages, no figures. More minor corrections (not appearing in the published version)",
  "journal": "Dynamics of PDE 4 (2007), 293--302",
  "categories": [
    "math.AP"
  ],
  "abstract": "The global regularity problem for the periodic Navier-Stokes system asks whether to every smooth divergence-free initial datum $u_0: (\\R/\\Z)^3 \\to \\R^3$ there exists a global smooth solution u. In this note we observe (using a simple compactness argument) that this qualitative question is equivalent to the more quantitative assertion that there exists a non-decreasing function $F: \\R^+ \\to \\R^+$ for which one has a local-in-time \\emph{a priori} bound $$ \\| u(T) \\|_{H^1_x((\\R/\\Z)^3)} \\leq F(\\|u_0\\|_{H^1_x((\\R/\\Z)^3)})$$ for all $0 < T \\leq 1$ and all smooth solutions $u: [0,T] \\times (\\R/\\Z)^3 \\to \\R^3$ to the Navier-Stokes system. We also show that this local-in-time bound is equivalent to the corresponding global-in-time bound.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2009-05-21T18:01:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30"
    ],
    "keywords": [
      "global regularity problem",
      "periodic navier-stokes equation",
      "quantitative formulation",
      "periodic navier-stokes system asks",
      "smooth divergence-free initial datum"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.1604T"
    }
  }
}