Smooth and swirling steady vortex rings near the Hill--Norbury family
An investigation of local aspects of the space of axisymmetric traveling-wave solutions to the three-dimensional Euler equations near Hill's zero-flux spherical vortex and Norbury's small-flux vortex rings is the endeavor undertaken herein. The aforementioned family consists of idealized, swirl-free, and nonsmooth objects, their azimuthal relative vorticity being prescribed by a scaled characteristic function of the compact vortex core. We confront the desingularization question, both within the swirl-free solution subspace and upon admitting swirl, and prove that each member of the Hill--Norbury family of sufficiently small flux is the limit of traveling-wave solutions to the Euler equations that are smooth to infinite order and have compact toroidal vorticity support. In the latter swirling regime, however, a curious flexibility is revealed: total vorticity may diverge along sequences converging in a natural desingularization topology. Our construction, founded upon a five-dimensional reformulation of the Stokes stream function, is an amalgamation of delicate thin-domain analysis, singular-parameter Newton-type nonlinear inversion, and topological fixed-point arguments. To the best of the authors' knowledge, this constitutes the first verification that the small-flux Hill--Norbury family is nowhere isolated from the collection of smooth vortex rings.