{
  "id": "2101.08586",
  "version": "v1",
  "published": "2021-01-21T12:58:58.000Z",
  "updated": "2021-01-21T12:58:58.000Z",
  "title": "Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces",
  "authors": [
    "Stan Palasek"
  ],
  "comment": "48 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\\|r^{1-\\frac3q}u\\|_{L_t^\\infty L_x^q}<\\infty$ where $r=\\sqrt{x_1^2+x_2^2}$ and either $q\\in(3,\\infty)$, or $u$ is axisymmetric and $q\\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We use some new tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-21T12:58:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76D05"
    ],
    "keywords": [
      "critical spaces",
      "double logarithmic lower bound",
      "three-dimensional navier-stokes equations satisfying",
      "blowup rate",
      "quantitative regularity theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 48,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}