{
  "id": "math/0601074",
  "version": "v2",
  "published": "2006-01-04T17:54:47.000Z",
  "updated": "2006-06-08T14:15:36.000Z",
  "title": "Blow-up in finite time for the dyadic model of the Navier-Stokes equations",
  "authors": [
    "Alexey Cheskidov"
  ],
  "comment": "19 pages, corrected typos",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovi\\'c. They showed a finite time blow-up in the case where the dissipation degree $\\alpha$ is less than 1/4. In this paper we prove the existence of weak solutions for all $\\alpha$, energy inequality for every weak solution with nonnegative initial datum starting from any time, local regularity for $\\alpha > 1/3$, and global regularity for $\\alpha \\geq 1/2$. In addition, we prove a finite time blow-up in the case where $\\alpha<1/3$. It is remarkable that the model with $\\alpha=1/3$ enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for all $\\alpha$ and becomes a strong global attractor for $\\alpha \\geq 1/2$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-06-08T14:15:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "76D03",
      "76D05"
    ],
    "keywords": [
      "dyadic model",
      "finite time blow-up",
      "weak solution",
      "strong global attractor",
      "4d navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1074C"
    }
  }
}