{
  "id": "math/0508481",
  "version": "v2",
  "published": "2005-08-24T21:23:58.000Z",
  "updated": "2005-09-21T21:09:34.000Z",
  "title": "Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation",
  "authors": [
    "Jared C. Bronski",
    "Tom Gambill"
  ],
  "comment": "17 pages, 1 figure; typos corrected, references added; figure modified",
  "doi": "10.1088/0951-7715/19/9/002",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions \\phi the exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-09-21T21:09:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35G25",
      "35P15"
    ],
    "keywords": [
      "kuramoto-sivashinsky equation",
      "uncertainty estimates",
      "lyapunov function argument similar",
      "spatial dimension",
      "additional linear destabilizing term"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}