  We consider solutions of the Navier-Stokes equations in $3d$ with vortex
filament initial data of arbitrary circulation, that is, initial vorticity
given by a divergence-free vector-valued measure of arbitrary mass supported on
a smooth curve. First, we prove global well-posedness for perturbations of the
Oseen vortex column in scaling-critical spaces. Second, we prove local
well-posedness (in a sense to be made precise) when the filament is a smooth,
closed, non-self-intersecting curve. Besides their physical interest, these
results are the first to give well-posedness in a neighborhood of large
self-similar solutions of $3d$ Navier-Stokes, as well as solutions which are
locally approximately self-similar.
