{
  "id": "1402.4560",
  "version": "v3",
  "published": "2014-02-19T05:55:39.000Z",
  "updated": "2014-10-11T01:32:11.000Z",
  "title": "Remark on Luo-Hou's ansatz for a self-similar solution to the 3D Euler equations",
  "authors": [
    "Dongho Chae",
    "Tai-Peng Tsai"
  ],
  "comment": "9 pages(to appear in the Journal of Nonlinear Science)",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under suitable decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution. These results show that the self-similar ansatz may be valid either only in a time-dependent region which shrinks to the boundary circle at the self-similar rate, or under different boundary conditions at spatial infinity of the self-similar profile.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-06T07:31:25.000Z",
      "abstract": "In this note we show that Luo-Hou's ansatz for the self-similar solution to the axisymmetric solution to the 3D Euler equations leads to triviality of the solution under a mild decay condition of the blow-up profile. The equations for the blow-up profile reduces to an over-determined system of partial differential equations, whose only solution with decay is the trivial solution. We also propose a natural generalization of Luo-Hou's ansatz. Using the vanishing of the normal velocity at the boundary, we show that this generalized self-similar ansatz also leads to a trivial solution.",
      "comment": "8 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-11T01:32:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q31",
      "76B03"
    ],
    "keywords": [
      "3d euler equations",
      "luo-hous ansatz",
      "self-similar solution",
      "trivial solution",
      "mild decay condition"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00332-014-9225-6",
      "journal": "Journal of NonLinear Science",
      "year": 2015,
      "month": "Feb",
      "volume": 25,
      "number": 1,
      "pages": 193
    },
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JNS....25..193C"
    }
  }
}