{
  "id": "math-ph/0409012",
  "version": "v1",
  "published": "2004-09-03T21:58:43.000Z",
  "updated": "2004-09-03T21:58:43.000Z",
  "title": "Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane",
  "authors": [
    "James P. Kelliher"
  ],
  "journal": "SIAM Journal on Mathematical Analysis, Vol 38(1) 2006 p. 210-232",
  "categories": [
    "math-ph",
    "math.AP",
    "math.MP"
  ],
  "abstract": "We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikeli\\'{c}, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas under the assumption that the initial vorticity lies in L^p for some p > 2. We extend the results of these authors to non-simply connected domains. Assuming a particular bound on the growth of the L^p-norms of the initial vorticity with p, and also assuming that the boundary and the function A have fractionally greater smoothness, we obtain a bound on the rate of convergence in L^2 uniform over any finite time interval to the solution of the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H^3 and the boundary is C^3, then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L^2 uniformly over any finite time interval to the solution to the Navier-Stokes equations with the usual no-slip boundary conditions as we let the function A grow large uniformly on the boundary.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-03T21:58:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "76D03"
    ],
    "keywords": [
      "navier-stokes equations",
      "bounded domain",
      "finite time interval",
      "usual no-slip boundary conditions",
      "navier boundary conditions converge"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.ph...9012K"
    }
  }
}