A Hele-Shaw limit without monotonicity

TL;DR

This paper investigates the incompressible limit of a porous medium equation with source or sink terms and boundary conditions, characterizing the limit pressure through an obstacle problem without requiring monotonicity assumptions.

Contribution

It provides a new characterization of the Hele-Shaw limit and the limit pressure for non-monotone motions in tumor and crowd dynamics, extending previous models.

Findings

Limit density solves a Hele-Shaw type free boundary problem.

Limit pressure characterized by an obstacle problem at each time.

Generalizes congestion models to non-monotone motions.

Abstract

We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (\cite{AKY}, \cite{PQV}, \cite{KP}, \cite{MPQ}). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressure, which solves an obstacle problem at each time in the evolution