{
  "id": "0705.1687",
  "version": "v1",
  "published": "2007-05-11T16:18:08.000Z",
  "updated": "2007-05-11T16:18:08.000Z",
  "title": "Existence results for mean field equations with turbulence",
  "authors": [
    "Cheikh Birahim Ndiaye"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we consider the following form of the so-called Mean field equation arising from the statistical mechanics description of two dimensional turbulence \\begin{equation}\\label{eq:study} - \\D_g u = \\rho_1 (\\frac{e^{u}}{\\int_\\Sig e^{u} dV_g}-1)-\\rho_2 (\\frac{e^{-u}}{\\int_\\Sig e^{-u} dV_g} - 1) \\end{equation} on a given closed orientable Riemannian surface ($\\Sigma, g$) with volume 1, where $\\rho_1, \\rho_2$ are real parameters. Exploiting the variational structure of the problem and running a min-max scheme introduced by Djadli and Malchiodi, we prove that if $k$ is a positive integer, $\\rho_1$ and $\\rho_2$ two real numbers such that $\\rho_1\\in (8k\\pi, 8(k+1)\\pi)$ and $\\rho_2<4\\pi$ then $\\eqref{eq:study}$ is solvable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-05-11T16:18:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "existence results",
      "mean field equation arising",
      "real numbers",
      "variational structure",
      "min-max scheme"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0705.1687B"
    }
  }
}