  We establish regularity and, under suitable assumptions, convergence to
stationary states for weak solutions of a parabolic equation with a non-linear
non-local drift term; this equation was derived from a model of active Brownian
particles with repulsive interactions in a previous work, which incorporates
advection-diffusion processes both in particle position and orientation. We
apply De Giorgi's method and differentiate the equation with respect to the
time variable iteratively to show that weak solutions become smooth away from
the initial time. This strategy requires that we obtain improved integrability
estimates in order to cater for the presence of the non-local drift. The
instantaneous smoothing effect observed for weak solutions is shown to also
hold for very weak solutions arising from distributional initial data; the
proof of this result relies on a uniqueness theorem in the style of M.~Pierre
for low-regularity solutions. The convergence to stationary states is proved
under a smallness assumption on the drift term.
