{
  "id": "1708.09360",
  "version": "v1",
  "published": "2017-08-30T16:49:33.000Z",
  "updated": "2017-08-30T16:49:33.000Z",
  "title": "Finite time blow up for a fluid mechanics model with nonlocal velocity",
  "authors": [
    "Changhui Tan"
  ],
  "comment": "14 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly positive smooth initial data, global regularity has been proved by Do, Kiselev, Ryzhik and Tan. We construct a family of non-negative smooth initial data so that solution loses regularity in finite time. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-30T16:49:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "35Q92"
    ],
    "keywords": [
      "finite time blow",
      "nonlocal velocity",
      "positive smooth initial data",
      "1d fluid mechanics model",
      "global regularity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}