{
  "id": "1708.08888",
  "version": "v1",
  "published": "2017-08-29T17:11:29.000Z",
  "updated": "2017-08-29T17:11:29.000Z",
  "title": "Periodic solutions for the N-vortex problem via a superposition principle",
  "authors": [
    "Björn Gebhard"
  ],
  "comment": "27 pages, 1 figure",
  "categories": [
    "math.DS",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We examine the $N$-vortex problem on general domains $\\Omega\\subset\\mathbb{R}^2$ concerning the existence of nonstationary collision-free periodic solutions. The problem in question is a first order Hamiltonian system of the form $$ \\Gamma_k\\dot{z}_k=J\\nabla_{z_k}H(z_1,\\ldots,z_N),\\quad k=1,\\ldots,N, $$ where $\\Gamma_k\\in\\mathbb{R}\\setminus\\{0\\}$ is the strength of the $k$th vortex at position $z_k(t)\\in\\Omega$, $J\\in\\mathbb{R}^{2\\times 2}$ is the standard symplectic matrix and $$ H(z_1,\\ldots,z_N)=-\\frac{1}{2\\pi}\\sum_{\\underset{k\\neq j}{k,j=1}}^N\\Gamma_j\\Gamma_k\\log|z_k-z_j|-\\sum_{k,j=1}^N\\Gamma_j\\Gamma_k g(z_k,z_j) $$ with some regular and symmetric, but in general not explicitely known function $g:\\Omega\\times\\Omega\\rightarrow \\mathbb{R}$. The investigation relies on the idea to superpose a stationary solution of a system of less than $N$ vortices and several clusters of vortices that are close to rigidly rotating configurations of the whole-plane system. We establish general conditions on both, the stationary solution and the configurations, under which multiple $T$-periodic solutions are shown to exist for every $T>0$ small enough. The crucial condition holds in generic bounded domains and is explicitely verified for an example in the unit disc $\\Omega=B_1(0)$. In particular we therefore obtain various examples of periodic solutions in $B_1(0)$ that are not rigidly rotating configurations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-29T17:11:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37J45",
      "37N10",
      "76B47"
    ],
    "keywords": [
      "superposition principle",
      "n-vortex problem",
      "nonstationary collision-free periodic solutions",
      "stationary solution",
      "first order hamiltonian system"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}