Description

• Title:
  Equilibria of vortex type Hamiltonians on closed surfaces
• Creator:
  Ahmedou, Mohameden;
  Bartsch, Thomas;
  Fiernkranz, Tim
• Description:
  We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1}\atop{i\ne j}} ^N \Gamma_i\Gamma_jG(p_i,p_j)+\Psi(p_1,\dots,p_N) \] on a closed Riemannian surface $(\Sigma,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $\Sigma$, $\Psi\colon \Sigma^N\to\mathbb{R}$ may be any function of class ${\mathcal C}^1$, and $\Gamma_1,\dots,\Gamma_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $\Psi(p) = -\sum\limits_{i=1}^N \Gamma_i^2h(p_i,p_i)$ where $h\colon \Sigma\times\Sigma\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(\Gamma_1,\dots,\Gamma_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2023 Mohameden Ahmedou, Thomas Bartsch, Tim Fiernkranz;
  https://creativecommons.org/licenses/by-nd/4.0
• Date:
  2023.02.26
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 61, No 1 (March 2023); 239 - 256;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/42828/34998 ;https://apcz.umk.pl/TMNA/article/view/42828/34999
• Subject:
  Point vortex Hamiltonian;
  point vortex equilibria;
  counter-rotating vortices;
  mean field equations;
  sinh-Poisson equation;
  blow-up solutions;
  37J12, 35J15, 35J25, 35J60, 35R01, 76B47
• Identifier: https://apcz.umk.pl/TMNA/article/view/42828
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other