Abstract
In this paper we deal with the existence, regularity and Beltrami field property of magnetic energy minimisers under a helicity constraint. We in particular tackle the problem of characterising local as well as global minimisers of the given minimisation problem. Further we generalise Arnold’s results concerning the problem of finding the minimum magnetic energy in an orbit of the group of volume-preserving diffeomorphisms to the setting of abstract manifolds with boundary.
Highlights
Magnetohydrodynamics is concerned with the dynamics of electrically conducting fluids under the influence of an external electromagnetic field
Of particular interest is the special case of an ideal fluid, that is, a perfectly electrically conducting, incompressible, Newtonian fluid of constant viscosity. The dynamics in this case are governed by the equations of ideal magnetohydrodynamics (IMHD)
We prove in Theorem 2.1 that solutions of the problem (MP1) are Beltrami fields, i.e. they are eigenfields of the curl operator corresponding to nonzero eigenvalues
Summary
Magnetohydrodynamics is concerned with the dynamics of electrically conducting fluids under the influence of an external electromagnetic field. Annals of Global Analysis and Geometry (2020) 58:267–285 vanishing first and second de Rham cohomology groups was derived by Arnold [3] and Vogel [4] In particular they prove that the helicity of a smooth divergence-free vector field coincides with the average linking number of the field lines of the considered vector field. Local minimisers of the magnetic energy in fixed helicity classes are potential terminal static configurations for suitable initial magnetic field configurations in IMHD. These type of vector fields have been studied in recent years on R3 [16] and on open domains [17]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
