Uniqueness of Hill's Spherical Vortex

摘要

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 Hill's 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. (Author)