A Hele-Shaw Limit Without Monotonicity

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 (Alexander et al. in Nonlinearity 27(4):823-858, 2014; Perthame et al. in Arch Ration Mech Anal 212(1):93-127, 2014; Kim and Pozar in Trans Am Math Soc 370(2):873-909, 2018; Mellet et al. in J Funct Anal 273(10):3061-3093, 2017). 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.