On the Topology of the Magnetic Lines of Solutions of the MHD Equations

We construct examples of smooth periodic solutions to the Magnetohydrodynamic equations in dimension 2 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). In the resistive case the reconnection of the magnetic lines is known to occur and has deep physical implications, being responsible for many dynamic phenomena in astrophysics. The construction is a simplified proof of Caro, Ciampa, and Lucà (2022) and in addition we consider the case of the forced system.