{
  "id": "1009.2051",
  "version": "v1",
  "published": "2010-09-10T16:52:39.000Z",
  "updated": "2010-09-10T16:52:39.000Z",
  "title": "On the constants in a Kato inequality for the Euler and Navier-Stokes equations",
  "authors": [
    "Carlo Morosi",
    "Livio Pizzocchero"
  ],
  "comment": "LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412 by the same authors, not concerning the main results",
  "journal": "Commun. Pure Appl. Anal. 11(2)(2012), 557-586",
  "doi": "10.3934/cpaa.2012.11.557",
  "categories": [
    "math.AP",
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-09-10T16:52:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D05",
      "26D10",
      "46E35"
    ],
    "keywords": [
      "kato inequality",
      "navier-stokes equations",
      "bilinear map",
      "lower bounds",
      "divergence free vector fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 39,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.2051M"
    }
  }
}