The ideal magnetohydrodynamic system in three space dimensions consists of the incompressible Euler equations coupled to the Faraday system via Ohm’s law. This system has a wealth of interesting structure, including three conserved quantities : the total energy, cross-helicity and magnetic helicity. Whilst the former two are analogous to the total kinetic energy for the Euler system, magnetic helicity is known to be more robust and of a different nature. In particular, when studying weak solutions, Onsager-type conditions for all three quantities are known, and are basically on the same level of 1/3-differentiability as the kinetic energy in the ideal hydrodynamic case for the former two. In contrast, magnetic helicity does not require any differentiability, only $L^3$ integrability. In the talk we present and compare some recent constructions of weak solutions and along the way highlight some of the hidden structures in the ideal magnetohydrodynamic system.
Co-author: Matteo Giardi
De Stephen Griffies
De Catherine Sulem
De Richard Bamler
De Richard Bamler
De Richard Bamler
De Richard Bamler
De Kenji Nakanishi
