Existence and uniqueness for a coupled parabolic-elliptic model with applications to magnetic relaxation

We prove existence, uniqueness and regularity of weak solutions of a coupled parabolic-elliptic model in two dimensions; 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 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 $$\| f \|_{L^{4}} \leq c \| f \|_{L^{2,\infty}}^{1/2} \| \nabla f \|_{L^{2}}^{1/2},$$ which we derive using the theory of interpolation. The model 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.