  We consider the relationship between Hele-Shaw evolution with drift, the
porous medium equation with superharmonic drift, and a congested crowd motion
model originally proposed by [MRS]- [MRSV]. We first use viscosity solutions to
show that the porous medium equation solutions converge to the Hele- Shaw
solution as the exponent tends to infinity. Next, using of the gradient flow
structure of both the porous medium equation and the crowd motion model, we
prove that the porous medium equation solutions also converge to the congested
crowd motion. Combining these results lets us deduce that in the case where the
initial data to the crowd motion model is given by a patch, or characteristic
function, the solution evolves as a patch that is the unique solution to the
Hele-Shaw problem. While proving our main results we also obtain a comparison
principle for solutions to the minimizing movement scheme based on the
Wasserstein metric, of independent interest.
