{
  "id": "1007.4412",
  "version": "v3",
  "published": "2010-07-26T09:46:41.000Z",
  "updated": "2012-09-05T16:05:54.000Z",
  "title": "On the constants in a basic inequality for the Euler and Navier-Stokes equations",
  "authors": [
    "Carlo Morosi",
    "Livio Pizzocchero"
  ],
  "comment": "LaTeX, 36 pages. The numerical values of the upper bounds K^{+}_{5} and K^{+}_{10} for d=3 have been corrected. Some references have been updated. arXiv admin note: text overlap with arXiv:1009.2051 by the same authors, not concerning the main results",
  "journal": "an abridged version published in Applied Mathematics Letters 26 (2013), 277-284",
  "doi": "10.1016/j.aml.2012.09.007",
  "categories": [
    "math.AP",
    "math-ph",
    "math.FA",
    "math.MP"
  ],
  "abstract": "We consider the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w : T^d -> R^d into v . D w, and also involves the Leray projection L onto the space of divergence free vector fields. We derive upper and lower bounds for the constants in some inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants K_{n d} = K_n in the basic inequality || L(v . D w)||_n <= K_n || v ||_n || w ||_{n+1}, where n in (d/2, + 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. Some practical motivations are indicated for an accurate analysis of the constants K_n.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-09-05T16:05:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D05",
      "26D10",
      "46E35"
    ],
    "keywords": [
      "basic inequality",
      "divergence free vector fields",
      "navier-stokes equations",
      "quadratic term",
      "lower bounds"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.4412M"
    }
  }
}