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Variational principles for Hill's spherical vortex and nearly spherical vortices


Author: Yieh Hei Wan
Journal: Trans. Amer. Math. Soc. 308 (1988), 299-312
MSC: Primary 76C05; Secondary 35J20
DOI: https://doi.org/10.1090/S0002-9947-1988-0946444-X
MathSciNet review: 946444
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Abstract: In this paper, vortex rings are regarded as axisymmetric motions without swirl of an incompressible inviscid fluid in space, with vorticity confined to their finite cores. The main results of this paper are (H) Hill's spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, among vortex rings. (N) Norbury's nearly spherical vortex is a "nondegenerate" local maximum of the energy function subject to a fixed impulse, and a fixed circulation. Estimates are established to overcome the discontinuity of vorticity distributions, and the singular behavior of Stoke's stream functions near the axis of symmetry. The spectral analysis involves the use of Legendre's functions.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0946444-X
Keywords: Vortex rings, Hill's spectral vortex, impulse, circulations, energy-Casimir method, spherical analysis, Legendre functions
Article copyright: © Copyright 1988 American Mathematical Society

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