A HELE-SHAW LIMIT WITH A VARIABLE UPPER BOUND AND DRIFT

We investigate a generalized Hele--Shaw equation with a source and drift terms where the density is constrained by an upper-bound density constraint that varies in space and time. By using a generalized porous medium equation approximation, we are able to construct a weak solution to the generalized Hele-Shaw equations under mild assumptions. Then we establish uniqueness of weak solutions to the generalized Hele-Shaw equations. Our next main result is a pointwise characterization of the density variable in the generalized Hele--Shaw equations when the system is in the congestion case. To obtain such a characterization for the congestion case, we derive new uniform lower bounds on the time derivative pressure of the generalized porous medium equation via a refined Aronson--Benilan estimate that implies monotonicity properties of the density and pressure.