{
  "id": "2003.04431",
  "version": "v1",
  "published": "2020-03-09T22:09:58.000Z",
  "updated": "2020-03-09T22:09:58.000Z",
  "title": "Statistical solutions to the barotropic Navier-Stokes system",
  "authors": [
    "Francesco Fanelli",
    "Eduard Feireisl"
  ],
  "comment": "Submitted",
  "categories": [
    "math.AP"
  ],
  "abstract": "We introduce a new concept of statistical solution in the framework of weak solutions to the barotropic Navier--Stokes system with inhomogeneous boundary conditions. Statistical solution is a family $\\{ M_t \\}_{t \\geq 0}$ of Markov operators on the set of probability measures $\\mathfrak{P}[\\mathcal{D}]$ on the data space $\\mathcal{D}$ containing the initial data $[\\varrho_0, \\mathbf{m}_0]$ and the boundary data $\\mathbf{d}_B$. (1) $\\{ M_t \\}_{t \\geq 0}$ possesses a.a. semigroup property, $ M_{t + s}(\\nu) = M_t \\circ M_s(\\nu)$ for any $t \\geq 0$, a.a. $s \\geq 0$, and any $\\nu \\in \\mathfrak{P}[\\mathcal{D}]$. (2) $\\{ M_t \\}_{t \\geq 0}$ is deterministic when restricted to deterministic data, specifically $ M_t(\\delta_{[\\varrho_0, \\mathbf{m}_0, \\mathbf{d}_B]}) = \\delta_{[\\varrho(t, \\cdot), \\mathbf{m}(t, \\cdot), \\mathbf{d}_B]}, $ where $[\\varrho, \\mathbf{m}]$ is a finite energy weak solution of the Navier--Stokes system corresponding to the data $[\\varrho_0, \\mathbf{m}_0, \\mathbf{d}_B] \\in \\mathcal{D}$. (3) $M_t: \\mathfrak{P}[\\mathcal{D}] \\to \\mathfrak{P}[\\mathcal{D}]$ is continuous in a suitable Bregman--Wasserstein metric at measures supported by the data giving rise to regular solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-09T22:09:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "barotropic navier-stokes system",
      "statistical solution",
      "finite energy weak solution",
      "boundary conditions",
      "regular solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}