N-Vortex Equilibria for Ideal Fluids in Bounded Planar Domains and New Nodal Solutions of the sinh-Poisson and the Lane-Emden-Fowler Equations

Abstract

We prove the existence of equilibria of the N-vortex Hamiltonian in a bounded domain Omega subset of 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 is an element of 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 = vertical bar u vertical bar(p-1)u in Omega with p large

Deformation retracts to the fat diagonal and applications to the existence of peak solutions of nonlinear elliptic equations

“…This is a higher-dimensional analogue of results in [Bartsch et al 2010] for special values of parameters. We follow the proof there, mainly pointing out the differences, though we do modify it a little to simplify it in our particular case.…”

Planar vortex patch problem in incompressible steady flow

“…When all the constants κ j are positive, a typical domain Ω in which W has a non-degenerate critical point is a dumb-bell shaped domain with long thin handles. The readers can refer to [6] for other results in this aspect.…”

Symmetric equilibria for the N-vortex problem

“…The case p = 1 is the one discussed in [4], and we give a slight sharpening of their theorem here. The other case is a simple application of Theorem 3.6, which we state first.…”

“…This is a small sharpening of [4,Theorem 3.3], since now we are allowed to break the pattern of alternating vorticities from one connected component to another.…”