{
  "id": "1010.1506",
  "version": "v1",
  "published": "2010-10-07T18:31:07.000Z",
  "updated": "2010-10-07T18:31:07.000Z",
  "title": "Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations",
  "authors": [
    "Dongho Chae",
    "Peter Constantin",
    "Jiahong Wu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $\\theta$ through $u_j =\\mathcal{R} \\Lambda^{-1} P(\\Lambda) \\theta$ where $\\mathcal{R}$ is a Riesz transform and $\\Lambda=(-\\Delta)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(\\Lambda)=I$ while the SQG equation to the case $P(\\Lambda) =\\Lambda$. We develop tools to bound $\\|\\nabla u||_{L^\\infty}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(\\Lambda)= (\\log(I+\\log(I-\\Delta)))^\\gamma$ with $0\\le \\gamma\\le 1$. In addition, a regularity criterion for the model corresponding to $P(\\Lambda)=\\Lambda^\\beta$ with $0\\le \\beta\\le 1$ is also obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-07T18:31:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76D03"
    ],
    "keywords": [
      "surface quasi-geostrophic equations",
      "inviscid models generalizing",
      "2d euler vorticity equation corresponds",
      "regularity",
      "paper studies solutions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00205-011-0411-5",
      "journal": "Archive for Rational Mechanics and Analysis",
      "year": 2011,
      "month": "Oct",
      "volume": 202,
      "number": 1,
      "pages": 35
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011ArRMA.202...35C"
    }
  }
}