Abstract
We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain \({\Omega\subset\mathbb{R}^2}\) , which is not necessarily simply connected. On an arbitrary bounded domain we obtain new equilibria for N = 3 or N = 4. If Ω has an axial symmetry we obtain a symmetric equilibrium for each \({N\in\mathbb{N}}\) . We also obtain new stream functions solving the sinh-Poisson equation \({-\Delta\psi=\rho\sinh\psi}\) in Ω with Dirichlet boundary conditions for ρ > 0 small. The stream function \({\psi_\rho}\) induces a stationary velocity field \({v_\rho}\) solving the Euler equation in Ω. On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If Ω has an axial symmetry we obtain for each N a velocity field \({v_\rho}\) that has a chain of N counter-rotating vortices, analogous to the Mallier-Maslowe row of counter-rotating vortices in the plane. Our methods also yield new nodal solutions for other semilinear Dirichlet problems, in particular for the Lane-Emden-Fowler equation \({-\Delta u=|u|^{p-1}u}\) in Ω with p large.
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Communicated by P. Constantin
Supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.
An erratum to this article is available at http://dx.doi.org/10.1007/s00220-014-2230-7.
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Bartsch, T., Pistoia, A. & Weth, T. N-Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations. Commun. Math. Phys. 297, 653–686 (2010). https://doi.org/10.1007/s00220-010-1053-4
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DOI: https://doi.org/10.1007/s00220-010-1053-4