On the existence of force-free magnetic fields with small nonconstant \(\alpha\) in exterior domains
This paper is mainly motivated by astrophysical phenomena, e.g. coronal loops or magnetic arcades of the Sun or solar flares which are modelled by magnetic flux tubes from the solar convection zone into the coronal space. Typically, in stellar atmospheres the magnetic fields have to be force-free. Force-free magnetic fields are characterized by the alignment of magnetic field to the electric current, \(Bj = \alpha B\), leading to vanishing Lorentz force. Here the authors show that for small (in the Hölder norm) \(\alpha\) which is constant along the magnetic flux lines but varies across the magnetic tube, there exist magnetic fields for which the magnetic tubes start and end on a simply connected compact surface. The proof generalizes and avoids some shortcomings of a previous analysis of \textit{M. Bineau} [Commun. Pure Appl. Math. 25, 77-84 (1972; Zbl 0226.35016)], and uses an iteration scheme. At each step the authors need the solutions of an intial value problem for linear first-order differential equations, and a Neumann boundary value problem for inhomogeneous harmonic vector fields.
linear first-order differential equations
scientific article; zbMATH DE number 1514629
scientific article