{
  "id": "2002.06350",
  "version": "v1",
  "published": "2020-02-15T09:30:33.000Z",
  "updated": "2020-02-15T09:30:33.000Z",
  "title": "Navier-Stokes equations in a curved thin domain, Part III: thin-film limit",
  "authors": [
    "Tatsu-Hiko Miura"
  ],
  "comment": "118 pages. This paper is the last part of the divided and revised version of arXiv:1811.09816",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a given closed surface. Under suitable assumptions we show that the average in the thin direction of a strong solution to the bulk Navier-Stokes equations converges weakly in appropriate function spaces on the limit surface as the thickness of the thin domain tends to zero. Moreover, we characterize the limit as a weak solution to limit equations, which are the damped and weighted Navier-Stokes equations on the limit surface. We also prove the strong convergence of the average of a strong solution to the bulk equations towards a weak solution to the limit equations by showing estimates for the difference between them. In some special case our limit equations agree with the Navier-Stokes equations on a Riemannian manifold in which the viscous term contains the Ricci curvature. This is the first result on a rigorous derivation of the surface Navier-Stokes equations on a general closed surface by the thin-film limit.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-15T09:30:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B25",
      "35Q30",
      "76D05",
      "35R01",
      "76A20"
    ],
    "keywords": [
      "curved thin domain",
      "thin-film limit",
      "limit equations",
      "bulk navier-stokes equations converges",
      "strong solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 118,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}