{
  "id": "math/0410380",
  "version": "v2",
  "published": "2004-10-17T22:36:53.000Z",
  "updated": "2004-10-19T19:36:27.000Z",
  "title": "On some dyadic models of the Euler equations",
  "authors": [
    "Fabian Waleffe"
  ],
  "comment": "10 pages, submitted to AMS Proc. v2: slight generalization of formula (31) and Theorem 1 (wavenumber mu=lambda>1 and mu=2, instead of mu=2 only). Clarification of a remark on Katz and Pavlovic's work on top of page 9. First 6 pages identical to v1",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP",
    "physics.flu-dyn"
  ],
  "abstract": "Katz and Pavlovic recently proposed a dyadic model of the Euler equations for which they proved finite time blow-up in the $H^{3/2+\\epsilon}$ Sobolev norm. It is shown that their model can be reduced to the dyadic inviscid Burgers equation where nonlinear interactions are restricted to dyadic wavenumbers. The inviscid Burgers equation exhibits finite time blow-up in $H^{\\alpha}$, for $\\alpha \\ge 1/2$, but its dyadic restriction is even more singular, exhibiting blow-up for any $\\alpha > 0$. Friedlander and Pavlovic developed a closely related model for which they also prove finite time blow-up in $H^{3/2+\\epsilon}$. Some inconsistent assumptions in the construction of their model are outlined. Finite time blow-up in the $H^{\\alpha}$ norm, with $\\alpha > 0$, is proven for a class of models that includes all those models. An alternative shell model of the Navier-Stokes equations is discussed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-10-19T19:36:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q30",
      "35Q35",
      "76B03"
    ],
    "keywords": [
      "finite time blow-up",
      "euler equations",
      "dyadic model",
      "dyadic inviscid burgers equation",
      "navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10380W"
    }
  }
}