{
  "id": "0711.4183",
  "version": "v2",
  "published": "2007-11-27T05:48:56.000Z",
  "updated": "2007-11-28T05:06:39.000Z",
  "title": "Existence and Stability of Steady-State Solutions with Finite Energy for the Navier-Stokes equation in the Whole Space",
  "authors": [
    "Clayton Bjorland",
    "Maria E. Schonbek"
  ],
  "comment": "22 pages, submitted",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the steady-state Navier-Stokes equation in the whole space $\\mathbb{R}^3$ driven by a forcing function $f$. The class of source functions $f$ under consideration yield the existence of at least one solution with finite Dirichlet integral ($\\|\\nabla U\\|_2<\\infty$). Under the additional assumptions that $f$ is absent of low modes and the ratio of $f$ to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite $L^2$ norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If $U$ is such a solution then any viscous, incompressible flow in the whole space, driven by $f$ and starting with finite energy, will return to $U$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2007-11-28T05:06:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35Q30",
      "76D05"
    ],
    "keywords": [
      "finite energy",
      "steady-state solutions",
      "finite dirichlet integral",
      "steady-state navier-stokes equation",
      "source functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0711.4183B"
    }
  }
}