{
  "id": "1011.0171",
  "version": "v1",
  "published": "2010-10-31T15:36:21.000Z",
  "updated": "2010-10-31T15:36:21.000Z",
  "title": "Dissipative models generalizing the 2D Navier-Stokes and the surface quasi-geostrophic equations",
  "authors": [
    "Dongho Chae",
    "Peter Constantin",
    "Jiahong Wu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $\\theta$ through $\\mathcal{R} \\Lambda^{-1} P(\\Lambda) \\theta$ where $\\mathcal{R}$ denotes a Riesz transform, $\\Lambda=(-\\Delta)^{1/2}$ and $P(\\Lambda)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(\\Lambda)=I$ while the surface quasi-geostrophic (SQG) equation to $P(\\Lambda) =\\Lambda$. We obtain the global regularity for a class of equations for which $P(\\Lambda)$ and the fractional power of the dissipative Laplacian are required to satisfy an explicit condition. In particular, the active scalar equations with any fractional dissipation and with $P(\\Lambda)= (\\log(I-\\Delta))^\\gamma$ for any $\\gamma>0$ are globally regular.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-31T15:36:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76D03"
    ],
    "keywords": [
      "surface quasi-geostrophic equations",
      "dissipative models generalizing",
      "active scalar equations",
      "2d navier-stokes vorticity equations correspond",
      "fractional dissipation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1011.0171C"
    }
  }
}