TBA

We give examples of solutions to the magnetohydrodynamic (MHD) equation with positive resistivity for which the topology of the magnetic lines changes under the flow. By Alfvén's theorem this is known to be impossible in the ideal case (resistivity = 0). This is a joint work with Pedro Caro and Gennaro Ciampa.

In view of numerical evidence, ideal MHD should possess weak solutions that 1) arise at the ideal (inviscid, non-resistive) limit and 2) conserve magnetic helicity but 3) dissipate total energy. Faraco, L., MacTaggart and Valli have shown that magnetic helicity is, indeed, conserved at the ideal limit. It is a very challenging problem to prove that some of these solutions dissipate energy. As a more tractable problem, one might first try and construct dissipative solutions of ideal MHD in an integrability class where magnetic helicity is automatically conserved, without considering any regularisation such as the ideal limit. (Magnetic helicity is conserved as long as the solution is \(L^3\)-integrable.)

Faraco, L. and Székelyhidi have constructed bounded (in particular, \(L^3\)) energy dissipative solutions via the method of convex integration, by elaborating on the ideas of the 2009 Annals paper of De Lellis and Székelyhidi. Another natural conjecture, posed by Buckmaster and Vicol, is that \(L^3\) is an optimal integrability class for magnetic helicity conservation. Beekie, Buckmaster and Vicol had proved a first result in this direction by constructing solutions that lie slightly above \(L^2\) and do not conserve magnetic helicity. Faraco, L. and Székelyhidi have proved the full conjecture by using so-called staircase laminates invented by Faraco. In the talk I describe some of the main points of the proofs of these two results with Faraco and Székelyhidi.

We consider a solid of revolution in 3D. Our eventual goal is to describe the eigenvectors and multiplicity of eigenvalues as explicitly as possible. Eigenvectors of the curl (of non-zero eigenvalue) in the MHD community are better-known as strong Beltrami fields. Our starting point has been to leverage the continuous symmetry of rotation to decompose the eigenspaces of the curl on smooth vector fields tangent to the boundary into further eigenspaces. This gives rise to an equivalence between being a eigenvector field and solving a certain vector PDE on the cross section. We will discuss some implications of this PDE including a geometric lower bound on the eigenvalues which support non-rotationally symmetric eigenvectors. This has been joint work with David Pfefferlé. I will also speak about related problems suggested to me by Daniel Peralta-Salas.

We will discuss the existence of stationary solutions of the two-component Vlasov-Poisson system on a domain \(\Omega\subset\mathbb{R}^3\) describing a high-temperature plasma which due to the influence of an external magnetic field is spatially confined to a subregion of \(\Omega\). As specific examples we consider cylindrical domains and domains with toroidal symmetry corresponding to Mirror trap and Tokamak devices. The presented existence result is based on a joint work with Yulia O. Belyaeva and Alexander L. Skubachevskii.

Magnetic helicity is an invariant of magnetohydrodynamics that describes some fascinating topological properties of magnetic fields. In many practical applications of magnetic helicity, such as in solar physics or fusion studies, the classical version of magnetic helicity is not strictly applicable as the magnetic field is not everywhere tangent to the boundary. To resolve this, a related quantity, known as relative magnetic helicity, is required. In this talk, we will extend the topological properties of classical helicity to relative helicity. In particular, we will describe how relative helicity can be defined in domains of arbitrary topology and argue that a 'natural choice' for the definition of relative helicity leads to a description of the magnetic field that is both interesting theoretically and of substantial practical use, particularly for understanding solar active regions.