{
  "id": "2406.09267",
  "version": "v1",
  "published": "2024-06-13T16:06:42.000Z",
  "updated": "2024-06-13T16:06:42.000Z",
  "title": "Global smooth solutions by transport noise of 3D Navier-Stokes equations with small hyperviscosity",
  "authors": [
    "Antonio Agresti"
  ],
  "comment": "36 pages",
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "The existence of global smooth solutions to the Navier-Stokes equations (NSEs) with hyperviscosity $(-\\Delta)^{\\gamma}$ is open unless $\\gamma $ is close to the J.-L. Lions exponent $ \\frac{5}{4}$ at which the energy balance is strong enough to prevent singularity formation. If $1<\\gamma \\ll \\frac{5}{4}$, then the global well-posedness of the hyperviscous NSEs is widely open as for the usual NSEs. In this paper, for all $\\gamma>1$, we show the existence of a transport noise for which global smooth solutions to the stochastic hyperviscous NSEs on the three-dimensional torus exist with high probability. In particular, a suitable transport noise considerably improves the known well-posedness results in the deterministic setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-13T16:06:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H50",
      "60H15",
      "76M35",
      "35Q35"
    ],
    "keywords": [
      "global smooth solutions",
      "transport noise",
      "3d navier-stokes equations",
      "small hyperviscosity",
      "prevent singularity formation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}