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Variational principles for Hill’s spherical vortex and nearly spherical vortices. (English) Zbl 0661.76017
In this paper, Vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in the space $${\mathbb{R}}^ 3$$. A “steady” vortex ring moves without change of shape and propagates at constant speed along its axis of symmetry. Hill’s spherical vortex (H) is steady and possesses a solid sphere as its core. Norbury found a family of nearly spherical steady vortex rings (N) which connect Helmholtz’s vortex ring to the ring H [J. Norburg, e.g. Proc. Cambridge Philos. Soc. 72, 253-284 (1972; Zbl 0256.76016); J. Fluid Mech. 57, 417-431 (1973; Zbl 0254.76018)]. Author shows (i) that H is a “nondegenerate” local maximum of the energy function among vortex rings and (ii) that N also possess this property. In both cases there are constraints consisting in fixed values of impulse (H,N) or circulation (N) or a quantity related to translations (N).