  This paper studies the Cauchy problem for a helical vortex filament evolving
by the 3D incompressible Navier-Stokes equations. We prove global-in-time
well-posedness and smoothing of solutions with initial vorticity concentrated
on a helix. We provide a local-in-time well-posedness result for vortex
filaments periodic in one spatial direction, and show that solutions with
helical initial data preserve this symmetry. We follow the approach of [4],
where the analogue local-in-time result has been obtained for closed vortex
filaments in $\mathbb{R}^3$. Next, we apply local energy weak solutions theory
with a novel estimate for helical functions in non-helical domains to uniquely
extend the solutions globally in time. This is the first global-in-time
well-posedness result for a vortex filament without size restriction and
without vanishing swirl assumptions.
