{
  "id": "1809.09490",
  "version": "v1",
  "published": "2018-09-23T22:30:18.000Z",
  "updated": "2018-09-23T22:30:18.000Z",
  "title": "Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in $R^3$",
  "authors": [
    "Gui-Qiang G. Chen",
    "James Glimm"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "physics.flu-dyn"
  ],
  "abstract": "We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for isentropic compressible fluids in $R^3$. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for compressible flow, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in $L^2$, which is independent of the viscosity coefficient $\\mu>0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^\\gamma$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $\\mu>0$, for some fixed $q_1>\\gamma$ and $q_2>2$, where $\\gamma>1$ is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations for isentropic compressible fluids in $R^3$. 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": "v1",
      "updated": "2018-09-23T22:30:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76N10",
      "76F02",
      "76N17",
      "65M12",
      "35L65"
    ],
    "keywords": [
      "navier-stokes equations",
      "inviscid limit",
      "kolmogorov-type theory",
      "compressible turbulence",
      "isentropic compressible fluids"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}