{
  "id": "1304.2972",
  "version": "v2",
  "published": "2013-04-10T14:22:05.000Z",
  "updated": "2013-09-04T09:06:47.000Z",
  "title": "On the Reynolds number expansion for the Navier-Stokes equations",
  "authors": [
    "Carlo Morosi",
    "Livio Pizzocchero"
  ],
  "comment": "Author's note. To appear in Nonlinear Analysis TMA. In the preliminaries and in the final paragraph there is some overlap with previous works of ours (arXiv:1203.6865, arXiv:0709.1670, arXiv:0909.3707, arXiv:1009.2051, arXiv:1104.3832, arXiv:1007.4412); these overlaps aim to make the paper self-contained, and do not involve the main results about the Reynolds expansion",
  "journal": "Nonlinear Analysis 95 (2014) 156-174",
  "doi": "10.1016/j.na.2013.08.029",
  "categories": [
    "math.AP"
  ],
  "abstract": "In a previous paper of ours [Nonlinear Anal. 2012] we have considered the incompressible Navier-Stokes (NS) equations on a d-dimensional torus T^d, in the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields (n > d/2+1). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution u_a, this allows to infer a lower bound T_c on the time of existence of the exact solution u and to construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for t in [0,T_c). In certain cases it is T_c = + infinity, so global existence is granted for u. In the present paper the framework of [Nonlinear Anal., 2012] is applied using as an approximate solution an expansion u^N(t) = Sum_{j=0}^N R^j u_j(t), where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R_{*} (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N=1,2,5 in dimension d=3, with the initial datum of Behr, Necas and Wu [M2AN, 2001]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in [Nonlinear Anal. 2012], where a Galerkin approximate solution was employed in place of the Reynolds expansion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-04T09:06:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D03",
      "76D05"
    ],
    "keywords": [
      "reynolds number expansion",
      "navier-stokes equations",
      "nonlinear anal",
      "initial datum",
      "zero mean vector fields"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2972M"
    }
  }
}