{
  "id": "2305.08761",
  "version": "v1",
  "published": "2023-05-15T16:20:15.000Z",
  "updated": "2023-05-15T16:20:15.000Z",
  "title": "Weak well-posedness by transport noise for a class of 2D fluid dynamical equations",
  "authors": [
    "Lucio Galeati",
    "Dejun Luo"
  ],
  "comment": "62 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "A fundamental open problem in fluid dynamics is whether solutions to $2$D Euler equations with $(L^1_x\\cap L^p_x)$-valued vorticity are unique, for some $p\\in [1,\\infty)$. A related question, more probabilistic in flavour, is whether one can find a physically relevant noise regularizing the PDE. We present some substantial advances towards a resolution of the latter, by establishing well-posedness in law for solutions with $(L^1_x\\cap L^2_x)$-valued vorticity and finite kinetic energy, for a general class of stochastic 2D fluid dynamical equations; the noise is spatially rough and of Kraichnan type and we allow the presence of a deterministic forcing $f$. This class includes as primary examples logarithmically regularized 2D Euler and hypodissipative 2D Navier-Stokes equations. In the first case, our result solves the open problem posed by Flandoli. In the latter case, for well-chosen forcing $f$, the corresponding deterministic PDE without noise has recently been shown by Albritton and Colombo to be ill-posed; consequently, the addition of noise truly improves the solution theory for such PDE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-15T16:20:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H50",
      "35Q35"
    ],
    "keywords": [
      "transport noise",
      "weak well-posedness",
      "examples logarithmically regularized 2d",
      "logarithmically regularized 2d euler",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 62,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}