  We present regularity results for nonlinear drift-diffusion equations of
porous medium type (together with their incompressible limit). We relax the
assumptions imposed on the drift term with respect to previous results and
additionally study the effect of linear diffusion on our regularity result (a
scenario of particular interest in the incompressible case, for it represents
the motion of particles driven by a Brownian motion subject to a density
constraint). Specifically, this work concerns the $L^4$-summability of the
pressure gradient in porous medium flows with drifts that is stable with
respect to the exponent of the nonlinearity, and $L^2$-estimates on the
pressure Hessian (in particular, in the incompressible case with linear
diffusion we prove that the pressure is the positive part of an
$H^2$-function).
