{
  "id": "1903.03833",
  "version": "v1",
  "published": "2019-03-09T17:40:38.000Z",
  "updated": "2019-03-09T17:40:38.000Z",
  "title": "A Regularity Criterion for Solutions to the 3D NSE in `Dynamically Restricted' Local Morrey Spaces",
  "authors": [
    "Zoran Grujic",
    "Liaosha Xu"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "It is shown that a local-in-time strong solution $u$ to the 3D Navier-Stokes equations remains regular on an interval $(0,T)$ provided a smallness $\\epsilon_0$-condition on $u$ in a lower time-restricted local Morrey space is stipulated; more precisely, $$\\sup_{t\\in(0,T)} \\ \\sup_{x \\in \\mathbb{R}^3, \\ \\eta(t) \\le r \\le 1} \\ \\frac{1}{r^\\alpha} \\int_{B_r(x)} |u(y,t)|^p dy \\le \\epsilon_0$$ where $\\eta$ is a dynamic dissipation scale consistent with the turbulence phenomenology and $\\alpha$ and $p$ are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at $T$ under the framework of $Z_\\alpha$-sparseness classes introduced in [Bradshaw, Farhat and Grujic, ARMA, 2018].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-09T17:40:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularity criterion",
      "3d nse",
      "3d navier-stokes equations remains regular",
      "lower time-restricted local morrey space",
      "dynamic dissipation scale consistent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}