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The uniqueness of a family of steady vortex rings. (English) Zbl 0694.76011
We work at some length to prove uniqueness of a particular, one-parameter family of solutions in the general theory of steady vortex rings. We believe such labour to be worthwhile; uniqueness seems precious in this context not only for its own sake, but also because it connects the results of the diverse formulations, variational principles and existence theorems that have appeared in recent years.
Norbury solved the following problem for small \(k>0:\) find \(\psi \in C^ 1({\bar \Pi})\cap C^ 2(\Pi \setminus \partial A)\), where \(A=A(\psi,k)\), such that \[ \begin{cases} L_{\psi}\equiv \left\{ r\frac{\partial}{\partial r}\left( \frac{1}{r}\frac{\partial}{\partial r} \right)+ \frac{\partial^ 2}{\partial z^ 2} \right\} \psi =\begin{cases} -10r^ 2 \quad&\text{ in} A(\psi,k), \\ 0\quad& \text{ in} \Pi \setminus \bar A(\psi,k),\\ &\psi |_{r=0}=0, \end{cases} \\ \psi(r,z)\to 0\text{ and } \frac{1}{r}| \nabla \psi(r,z)| \to 0\text{ as } r^ 2+z^ 2\to \infty \text{ in }{\bar \Pi}, \end{cases} \] where \({\bar \Pi}\) is the closure of the half-plane \(\Pi =\{(r,z)/r>0\), \(z\in {\mathbb{R}}\}.\)
We shall consider weak solutions of the problem. The Hilbert space H(\(\Pi)\) is the completion of the set \(C_ 0^{\infty}(\Pi)\), of real- valued functions having derivatives of every order and compact support in \(\Pi\), in the norm \(\| \cdot \|\) corresponding to the inner product \(<\phi,\chi >=\int_{\Pi}1/r^ 2(\phi_ r\chi_ r+\phi_ z\chi_ z)d\tau\), where \(d\tau =r dr dz\). Thus (for fluid of unit density) \(\pi \| \phi \|^ 2\) is the kinetic energy of the motion with stream function \(\phi\) ; also \(<\phi,\chi >=-\int_{\Pi}1/r^ 2\phi L_{\chi}d\tau\) if \(\phi,\chi \in C_ 0^{\infty}(\Pi)\). We shall say that \(\phi\) is a weak solution of Norbury’s problem, for given \(k>0\), if \[ \psi \in H(\Pi)\setminus \{0\}\quad and\quad <\phi,\psi >=10\int_{A(\psi,k)}\phi d\tau \quad for\quad all\quad \phi \in H(\Pi). \] Note that any (pointwise or weak) solution of Norbury’s problem remains a solution under arbitrary translation in the z-direction. However, we shall show that any given solution can be centred by a unique translation that makes it an even function of z.

MSC:
76B47 Vortex flows for incompressible inviscid fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
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