{
  "id": "0907.4357",
  "version": "v2",
  "published": "2009-07-24T18:41:00.000Z",
  "updated": "2009-10-16T23:32:32.000Z",
  "title": "A Solvability criterion for Navier-Stokes equations in high dimensions",
  "authors": [
    "T. M. Viswanathan",
    "G. M. Viswanathan"
  ],
  "comment": "Fixed references, priority etc",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "We define the Ladyzhenskaya-Lions exponent $\\alpha_{\\rm {\\tiny \\sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-\\Delta)^{\\alpha}$ in ${\\Bbb R}^n$, for all $n\\geq 2$. We review the proof of strong global solvability when $\\alpha\\geq \\alpha_{\\rm {\\tiny \\sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1\\over 2} \\|\\nabla u \\|^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $\\alpha<\\alpha_{\\rm {\\tiny \\sc l}} (n)$. The energy is critical under scale transformations only for $\\alpha=\\alpha_{\\rm {\\tiny \\sc l}} (n)$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-10-16T23:32:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "navier-stokes equations",
      "high dimensions",
      "solvability criterion",
      "smooth initial data",
      "strong global solvability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.4357V"
    }
  }
}