A Hele-Shaw Limit With A Variable Upper Bound and Drift

TL;DR

This paper extends the classical Hele-Shaw model to include a variable upper bound and drift, providing existence, uniqueness, and a pointwise density characterization using advanced PDE techniques.

Contribution

It introduces a generalized Hele-Shaw model with variable constraints and drift, establishing weak solution existence, uniqueness, and a novel density characterization in congestion scenarios.

Findings

Constructed weak solutions under mild assumptions

Proved uniqueness of solutions

Derived uniform lower bounds on pressure derivatives

Abstract

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 a new uniform lower bounds on the time derivative pressure of the generalized porous medium equation via a refined Aronson-Benilan estimate that implies monotonicity on the density and pressure.