{
  "id": "1207.0957",
  "version": "v1",
  "published": "2012-07-04T12:39:06.000Z",
  "updated": "2012-07-04T12:39:06.000Z",
  "title": "On a one-dimensional α-patch model with nonlocal drift and fractional dissipation",
  "authors": [
    "Hongjie Dong",
    "Dong Li"
  ],
  "comment": "21 pages, submitted",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a one-dimensional nonlocal nonlinear equation of the form: $\\partial_t u = (\\Lambda^{-\\alpha} u)\\partial_x u - \\nu \\Lambda^{\\beta}u$ where $\\Lambda =(-\\partial_{xx})^{\\frac 12}$ is the fractional Laplacian and $\\nu\\ge 0$ is the viscosity coefficient. We consider primarily the regime $0<\\alpha<1$ and $0\\le \\beta \\le 2$ for which the model has nonlocal drift, fractional dissipation, and captures essential features of the 2D $\\alpha$-patch models. In the critical and subcritical range $1-\\alpha\\le \\beta \\le 2$, we prove global wellposedness for arbitrarily large initial data in Sobolev spaces. In the full supercritical range $0 \\le \\beta<1-\\alpha$, we prove formation of singularities in finite time for a class of smooth initial data. Our proof is based on a novel nonlocal weighted inequality which can be of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-04T12:39:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional dissipation",
      "nonlocal drift",
      "one-dimensional nonlocal nonlinear equation",
      "novel nonlocal weighted inequality",
      "arbitrarily large initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.0957D"
    }
  }
}