Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3-manifolds with boundary i
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Existence and characterisation of magnetic energy minimisers on oriented, compact Riemannian 3‑manifolds with boundary in arbitrary helicity classes Wadim Gerner1 Received: 11 March 2020 / Accepted: 29 June 2020 © The Author(s) 2020
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. Keywords Beltrami fields · Helicity · Magnetohydrodynamics · Volume-preserving diffeomorphisms Mathematics Subject Classification 53Z05 · 53C80 · 76W05
1 Introduction 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). Most notably in the ideal case is the fact that the electric and magnetic fields are perpendicular to one another, which gives rise to a conserved quantity, the so-called helicity. More precisely, if we consider a simply connected, bounded domain 𝛺 ⊂ ℝ3 with smooth boundary and impose the boundary condition that the magnetic field B is always tangent to the boundary, then one formally checks that the quantity H(B) ∶= ∫𝛺 A ⋅ Bd3 x , where A is any vector potential of B , is in fact constant in time. The conservation of helicity was first observed by Woltjer [1] and a physical interpretation was, for instance, given by Moffatt [2]. The helicity may be regarded as a measure of linkage of distinct field lines of the underlying magnetic field. A similar interpretation of the helicity on closed 3-manifolds with
* Wadim Gerner [email protected]‑aachen.de 1
Lehrstuhl I für Mathematik, RWTH Aachen University, Turmstraße 46, 52064 Aachen, Germany
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Vol.:(0123456789)
Annals of Global Analysis and Geometry
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. Helicity has been widely studied in mathematics and physics, see, for example, [1–3, 5–8] to name a few. More recent works include [9], where the authors generalise the notion of helicity to higher dimensions and provide a characterisation of diffeomorphisms under which helicity is preserved, and [10], where it is shown that helicity is essentially the only regular integral invariant of exact vector fields which is preserved under the action of volume-preserving diffeomorphisms.
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