A granular model for crowd motion and pedestrian flow

We study a granular model for congested crowd motion and pedestrian flow. Our approach is based on an approximation through a Hele-Shaw type equation involving a degenerate operator of $p$-Laplacian type and a linear drift, for which we prove existence and uniqueness using nonlinear semigroup methods and the doubling variables technique. Our main result shows that, as $p\to \mathrm{\infty }$, the weak solutions of the $p-$problem converge to a variational solution of the congested crowd motion problem.