{
  "id": "1607.06252",
  "version": "v1",
  "published": "2016-07-21T10:09:35.000Z",
  "updated": "2016-07-21T10:09:35.000Z",
  "title": "Strong solutions to the 3D primitive equations with only horizontal dissipation: near $H^1$ initial data",
  "authors": [
    "Chongsheng Cao",
    "Jinkai Li",
    "Edriss S. Titi"
  ],
  "categories": [
    "math.AP",
    "physics.ao-ph",
    "physics.flu-dyn",
    "physics.geo-ph"
  ],
  "abstract": "In this paper, we consider the initial-boundary value problem of the three-dimensional primitive equations for oceanic and atmospheric dynamics with only horizontal viscosity and horizontal diffusivity. We establish the local, in time, well-posedness of strong solutions, for any initial data $(v_0, T_0)\\in H^1$, by using the local, in space, type energy estimate. We also establish the global well-posedness of strong solutions for this system, with any initial data $(v_0, T_0)\\in H^1\\cap L^\\infty$, such that $\\partial_zv_0\\in L^m$, for some $m\\in(2,\\infty)$, by using the logarithmic type anisotropic Sobolev inequality and a logarithmic type Gronwall inequality. This paper improves the previous results obtained in [Cao, C.; Li, J.; Titi, E.S.: Global well-posedness of the 3D primitive equations with only horizontal viscosity and diffusivity, Comm. Pure Appl.Math., Vol. 69 (2016), 1492-1531.], where the initial data $(v_0, T_0)$ was assumed to have $H^2$ regularity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-21T10:09:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35Q35",
      "76D03",
      "86A10"
    ],
    "keywords": [
      "3d primitive equations",
      "initial data",
      "strong solutions",
      "horizontal dissipation",
      "logarithmic type anisotropic sobolev inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}