The uniqueness of a family of steady vortex rings
type: Item
instance of: scholarly article
title: The uniqueness of a family of steady vortex rings
zbMATH Open document ID: 0694.76011
DOI: 10.1007/BF00251515
author: L. Edward Fraenkel
author: Charles J. Amick
published in: Archive for Rational Mechanics and Analysis
publication date: 1988-01-01T00:00:00Z
review text: 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 rem…
Mathematics Subject Classification ID: 76B47
Mathematics Subject Classification ID: 35Q99
zbMATH DE Number: 4137421
zbMATH Keywords: family of steady vortex rings
zbMATH Keywords: one-parameter family of solutions
zbMATH Keywords: Norbury's problem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00251515
OpenAlex ID: W2006611544
description: scientific article; zbMATH DE number 4137421