posted on 2023-06-09, 00:16authored byDavid S McCormick, James C Robinson, Jose L Rodrigo
We prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in 2D, and existence of weak solutions in 3D; we consider the standard equations of magnetohydrodynamics with the advective terms removed from the velocity equation. Despite the apparent simplicity of the model, the proof in 2D requires results that are at the limit of what is available, including elliptic regularity in $L^{1}$ and a strengthened form of the Ladyzhenskaya inequality () which we derive using the theory of interpolation. The model potentially has applications to the method of magnetic relaxation introduced by Moffatt (J. Fluid. Mech. 159, 359–378, 1985) to construct stationary Euler flows with non-trivial topology.