  We investigate the stability properties for a family of equations introduced
by Moffatt to model magnetic relaxation. These models preserve the topology of
magnetic streamlines, contain a cubic nonlinearity, and yet have a favorable
$L^2$ energy structure. We consider the local and global in time well-posedness
of these models and establish a difference between the behavior as $t\to
\infty$ with respect to weak and strong norms.
