{
  "id": "0706.0758",
  "version": "v1",
  "published": "2007-06-06T02:56:50.000Z",
  "updated": "2007-06-06T02:56:50.000Z",
  "title": "Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations",
  "authors": [
    "Bin Cheng",
    "Eitan Tadmor"
  ],
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [H. Liu and E. Tadmor, Phys. D 188 (2004), no. 3-4, 262-276] we have shown that the pressureless version of these equations admit global smooth solution for a large set of sub-critical initial configurations. In the present work we prove that when rotational force dominates the pressure, it \\emph{prolongs} the life-span of smooth solutions for t < ln(1/d); here d << 1 is the ratio of the pressure gradient measured by the inverse squared Froude number, relative to the dominant rotational forces measured by the inverse Rossby number. Our study reveals a ``nearby'' periodic-in-time approximate solution in the small d-regime, upon which hinges the long time existence of the exact smooth solution. These results are in agreement with the close-to periodic dynamics observed in the ``near inertial oscillation'' (NIO) regime which follows oceanic storms. Indeed, our results indicate the existence of smooth, ``approximate periodic'' solution for a time period of \\emph{days}, which is the relevant time period found in NIO obesrvations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-06-06T02:56:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76U05",
      "76E07",
      "76N10"
    ],
    "keywords": [
      "long time existence",
      "rapidly rotating shallow-water",
      "euler equations",
      "equations admit global smooth solution",
      "rotational force"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0706.0758C"
    }
  }
}