Appears in collection : 2026 - T2 - WS2 - Instabilities and transitions in geophysical flows

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