  In this paper we address the regularity issues of drift-diffusion equation
with nonlocal diffusion, where the diffusion operator is in the realm of
stable-type L\'evy operator and the velocity field is defined from the
considered quantity by a zero-order pseudo-differential operator. Through using
the method of nonlocal maximum principle in a unified way, we prove the
eventual regularity result in the supercritical type cases and the global
regularity at some logarithmically supercritical cases. The feature of these
results is that the time after which the solution is smoothly regular in the
supercritical type cases can be evaluated explicitly.
