{
  "id": "1608.06775",
  "version": "v1",
  "published": "2016-08-24T10:50:40.000Z",
  "updated": "2016-08-24T10:50:40.000Z",
  "title": "Periodic solutions with prescribed minimal period of the 2-vortex problem in domains",
  "authors": [
    "Thomas Bartsch",
    "Matteo Sacchet"
  ],
  "comment": "17 pages",
  "categories": [
    "math.DS"
  ],
  "abstract": "We consider the Hamiltonian system \\[ \\dot{z}_k = J \\nabla_{z_k} H_\\Omega(z_1,z_2), \\quad k=1,2, \\] for two point vortices $z_1,z_2\\in\\Omega$ in a domain $\\Omega\\subset\\mathbb{R}^2$. The Hamiltonian $H_\\Omega$ is of the form \\[ H_\\Omega(z_1,z_2) = -\\frac{1}{2\\pi} \\log |z_1-z_2| - 2g(z_1,z_2) - h(z_1) - h(z_2), \\] where $g:\\Omega\\times\\Omega\\to\\mathbb{R}$ is the regular part of a hydrodynamic Green's function in $\\Omega$, and $h:\\Omega\\to\\mathbb{R}$ is the Robin function: $h(z)=g(z,z)$. The system is singular and not integrable, except when $\\Omega$ is a disk or an annulus. We prove the existence of infinitely many periodic solutions with minimal period $T$ which are a superposition of a slow motion of the center of vorticity along a level line of $h$ and of a fast rotation of the two vortices around their center of vorticity. These vortices move in a prescribed subset $\\mathcal{A}\\subset\\Omega$ that has to satisfy a geometric condition. The minimal period can be any $T$ in an interval $I(\\mathcal{A})\\subset\\mathbb{R}$. Subsets $\\mathcal{A}$ to which our results apply can be found in any generic bounded domain. The proofs are based on a recent higher dimensional version of the Poincar\\'e-Birkhoff theorem due to Fonda and Ure\\~na.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-24T10:50:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J45",
      "34C25",
      "37E40",
      "37N10",
      "76B47"
    ],
    "keywords": [
      "prescribed minimal period",
      "periodic solutions",
      "higher dimensional version",
      "hydrodynamic greens function",
      "point vortices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}