{
  "id": "1008.1546",
  "version": "v2",
  "published": "2010-08-09T17:07:00.000Z",
  "updated": "2011-11-01T01:42:26.000Z",
  "title": "Kolmogorov's Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $\\R^3$",
  "authors": [
    "Gui-Qiang G. Chen",
    "James Glimm"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in $\\R^3$. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the $\\alpha^{th}$-order fractional derivative of the velocity for some $\\alpha>0$ in the space variables in $L^2$, which is independent of the viscosity $\\mu>0$. Then it is shown that this key observation yields the $L^2$-equicontinuity in the time and the uniform bound in $L^q$, for some $q>2$, of the velocity independent of $\\mu>0$. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in $\\R^3$. We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $\\mu>0$, that is in the high Reynolds number limit.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-01T01:42:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equations",
      "inviscid limit",
      "kolmogorovs theory",
      "euler equations",
      "turbulence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1008.1546C"
    }
  }
}