Stability of the magnetic Couette-Taylor flow

In this paper we consider the magnetic Couette-Taylor problem, that is, a conducting fluid between two infinite rotating cylinders, subject to a magnetic field parallel to the rotation axis. This configuration admits an equilibrium solution of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ (0,ar + br^{{ - 1}} ,0,0,0,\alpha + \beta \log r). $ \end{document} It is shown that this equilibrium is Ljapounov stable under small perturbations in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{L}^{2} (\Gamma ), $ \end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \Gamma = \{ (r,\varphi ,z)/r_{1} < r < r_{2} ,\varphi \in [0,2\pi ],z \in \mathbb{R}\} , $ \end{document} provided that the parameters a, b, α, β are small. The methods of proof are a combination of an energy method, based on Bloch space analysis and small data techniques.