N-Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations |
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Authors: | Thomas Bartsch Angela Pistoia Tobias Weth |
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Institution: | (1) Mathematical Institute, Tohoku university, Sendai 980-8578, Japan;(2) Present address: Osaka City University, Osaka 558-8585, Japan |
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Abstract: | We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain ${\Omega\subset\mathbb{R}^2}We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain
W ì \mathbbR2{\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 ? \mathbbN{N\in\mathbb{N}} . We also obtain new stream functions solving the sinh-Poisson equation -Dy = rsinhy{-\Delta\psi=\rho\sinh\psi} in Ω with Dirichlet boundary conditions for ρ > 0 small. The stream function yr{\psi_\rho} induces a stationary velocity field vr{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 vr{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 -Du=|u|p-1u{-\Delta u=|u|^{p-1}u} in Ω with p large. |
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