The Uniqueness of Hill's Spherical Vortex.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The only explicit exact solution of the problem of steady vortex rings is that found, for a particular case, by Hill in 1984 it solves a semilinear elliptic equation, of order two, involving a Stokes stream function psi r,z and a non-linearity sub H psi that has a simple discontinuity at psi 0. This paper proves that a any weak solution of the corresponding boundary-value problem is Hills solution, modulo translation along the axis of symmetry r 0, b any solution of the isoperimetric variational problem in another paper is a weak solution, indeed, any local maximizer is a weak solution. The result b is not immediate because f sub H is discontinuous consequently, the functional that is maximized is not Frechet differentiable on the whole Hilbert space in question. Additional keywords Fluid velocity TransformationsMathematics and Variational principles. Author
- Theoretical Mathematics