  Nowadays a vast literature is available on the Hele-Shaw or incompressible
limit for nonlinear degenerate diffusion equations. This problem has attracted
a lot of attention due to its applications to tissue growth and crowd motion
modelling as it constitutes a way to link soft congestion (or compressible)
models to hard congestion (or incompressible) descriptions. In this paper, we
address the question of estimating the rate of this asymptotics in the presence
of external drifts. In particular, we provide improved results in the
2-Wasserstein distance which are global in time thanks to the contractivity
property that holds for strictly convex potentials.
