an:03986853
Zbl 0609.76018
Amick, C. J.; Fraenkel, L. E.
The uniqueness of Hill's spherical vortex
EN
Arch. Ration. Mech. Anal. 92, 91-119 (1986).
00151927
1986
j
76B47 35J25 35R35
free boundary problem; asymptotics at infinity; Stokes stream function; Hill's problem; explicit solution; weak solution; local maximizers of functional
The authors study the free boundary problem
\[
r(\frac{1}{r}\psi_ r)_ r+\psi_{zz}= \begin{cases} -\lambda r^ 2f_ 0(\psi) &\text{ in \(A;\)} \\ 0 &\text{ in \(\Pi \setminus A,\)}\end{cases}
\]
\(\psi |_{r-0}=-k,\quad |_{\partial A}=0\) together with certain asymptotics at infinity.
Here \(\Pi =\{(r,z)|\) \(r>0\), \(z\in {\mathbb{R}}\}\), \(f_ 0\geq 0\), and \(\psi\) is a Stokes stream function in cylindrical co-ordinates (no dependence on \(\theta)\). The set \(A\subset \Pi\) is bounded and open, but a priori unknown. A special case of the problem is Hill's problem, in which an explicit solution is known. It is proven that any weak solution to the problem is the explicit solution modulo a translation in z. Such solutions may be obtained as local maximizers of functional.
G.Warnecke