{
  "id": "1403.0980",
  "version": "v2",
  "published": "2014-03-05T01:08:28.000Z",
  "updated": "2014-11-03T23:29:56.000Z",
  "title": "Uniform Regularity for free-boundary navier-stokes equations with surface tension",
  "authors": [
    "Tarek Elgindi",
    "Donghyun Lee"
  ],
  "comment": "45 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the zero-viscosity limit of free boundary Navier-Stokes equations with surface tension in $\\mathbb{R}^3$ thus extending the work of Masmoudi and Rousset [1] to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev co-normal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. One of the main differences between this work and the work [1] is our use of time-derivative estimates and certain properties of the Dirichlet-Neumann operator. In a forthcoming work, we discuss how we can take the simultaneous limit of zero viscosity and surface tension [].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-05T01:08:28.000Z",
      "abstract": "We study the zero-viscosity limit of free boundary Navier-Stokes equations with surface tension in $\\mathbb{R}^3$ thus extending the work of Masmoudi and Rousset to take surface tension into account. Due to the presence of boundary layers, we are unable to pass to the zero-viscosity limit in the usual Sobolev spaces. Indeed, as viscosity tends to zero, normal derivatives at the boundary should blow-up. To deal with this problem, we solve the free boundary problem in the so-called Sobolev conormal spaces (after fixing the boundary via a coordinate transformation). We prove estimates which are uniform in the viscosity. And after inviscid limit process, we get the local existence of free-boundary Euler equation with surface tension. In a forthcoming work, we discuss how we can take the simultaneous limit of zero viscosity and surface tension [].",
      "comment": "43 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-03T23:29:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "surface tension",
      "free-boundary navier-stokes equations",
      "uniform regularity",
      "zero-viscosity limit",
      "free boundary navier-stokes equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.0980E"
    }
  }
}