{
  "id": "1811.09816",
  "version": "v1",
  "published": "2018-11-24T11:43:19.000Z",
  "updated": "2018-11-24T11:43:19.000Z",
  "title": "Navier--Stokes equations in a curved thin domain",
  "authors": [
    "Tatsu-Hiko Miura"
  ],
  "comment": "149 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the three-dimensional incompressible Navier--Stokes equations in a curved thin domain with Navier's slip boundary conditions. The curved thin domain is defined as a region between two closed surfaces which are very close to each other and degenerates into a given closed surface as its width tends to zero. We establish the global-in-time existence and uniform estimates of a strong solution for large data when the width of the thin domain is very small. Moreover, we study a singular limit problem as the thickness of the thin domain tends to zero and rigorously derive limit equations on the limit surface, which are the damped and weighted Navier--Stokes equations on a surface with viscous term involving the Gaussian curvature of the surface. We prove the weak convergence of the average in the thin direction of a strong solution to the bulk Navier--Stokes equations and characterize the weak limit as a weak solution to the limit equations as well as provide estimates for the difference between solutions to the bulk and limit equations. To deal with the weighted surface divergence-free condition of the limit equations we also derive the weighted Helmholtz--Leray decomposition of a tangential vector field on a closed surface.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-24T11:43:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B25",
      "35Q30",
      "76D03",
      "76D05",
      "35R01",
      "76A20"
    ],
    "keywords": [
      "curved thin domain",
      "limit equations",
      "closed surface",
      "strong solution",
      "naviers slip boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 149,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}