  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.
