  This paper considers a class of nonlinear, degenerate drift- diffusion
equations. We study well-posedness and regularity properties of the solutions,
with the goal to achieve uniform H\"{o}lder regularity in terms of $L^p$-bound
on the drift vector field. A formal scaling argument yields that the threshold
for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$.
On the other hand we are able to show by a series of examples that one needs
$p>d$ for such estimates, even for divergence free drift.
