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\(N\)-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler equations. (English) Zbl 1195.35250
Commun. Math. Phys. 297, No. 3, 653-686 (2010); erratum ibid. 333, No. 2, 1107 (2015).
Summary: 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 \(\Omega \) 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 \(\Omega \) with Dirichlet boundary conditions for \(\rho > 0\) small. The stream function \({\psi_\rho}\) induces a stationary velocity field \({v_\rho}\) solving the Euler equation in \(\Omega \). On an arbitrary bounded domain we obtain velocitiy fields having three or four counter-rotating vortices. If \(\Omega \) 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 \(\Omega \) with \(p\) large.

MSC:
35Q35 PDEs in connection with fluid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q31 Euler equations
35J25 Boundary value problems for second-order elliptic equations
76D17 Viscous vortex flows
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