Abstract
This dissertation deals with minimisation problems related to magnetohydrodynamics. The main part of this thesis is divided in 5 chapters. In the first chapter it is shown that the magnetic energy under the so called helicity constraint admits a global minimiser for each prescribed value of the helicity and that all such minimisers are Beltrami fields, i.e. eigenvector fields of the curl operator. This result is already known in the settings of compact manifolds without boundary and bounded domains with smooth boundary in $\mathbb{R}^3$. We generalise these results to the setting of abstract, compact manifolds with boundary and by this we filled a gap in the literature. In the second chapter we consider the problem of finding a domain of prescribed volume for which the minimal energy in a given helicity class becomes minimal among all other minimal energies, in the same helicity class, of domains of the same volume. This problem was considered in the literature in the setting where the ambient space is $\mathbb{R}^3$. We generalise the known results to the setting where the ambient space is any Riemannian manifold and derive a second variation inequality, which contains terms involving the geometry of the ambient space. The question of whether or not an optimal domain exists is still open and subject of current research. In the third chapter the focus lies on the field line dynamics and zero set structure of Beltrami fields, hence in particular of energy minimising vector fields. Our most important results are on the one hand the observation that the restriction of a Beltrami field to the boundary, assuming it is tangent to it, on a simply connected, compact manifold is always a gradient field, which provides us with a good understanding of the boundary field line behaviour. On the other hand we show that the Hausdorff dimension of the zero set is at most one. This upper bound is already known for the zero set in the interior of the manifold. We show that this upper bound stays valid if we include the zeros on the boundary. Such a result appears to be new in the literature. In the fourth chapter we consider again the minimisation problem from the first chapter, but add an additional symmetry constraint. Results concerning the existence of rotationally symmetric Beltrami fields on domains in $\mathbb{R}^3$ are known in the literature. We generalise these results to the setting of abstract manifolds and develop new arguments to deal with general Killing vector fields. The last chapter deals with minimisation problems on $\mathbb{R}^3$. It is easy to see that the original energy functional does not admit any global minimisers if we consider it on the whole 3-space. That is why we consider two related energies under the helicity constraint. For the first energy it is shown, in view of Lions' “concentration compactness principle”, that the only possible obstruction for the existence of global minimisers are dichotomy effects. For the second energy we derive necessary conditions for global as well as local minimisers.