Porous medium equation to Hele-Shaw flow with general initial density

Understand

In this paper we study the "stiff pressure limit" of the porous medium equation, where the initial density is a bounded, integrable function with a sufficient decay at infinity.

• Our particular model, introduced by Perthame-Quiros-Vazquez, describes the growth of a tumor zone with a restriction on the maximal cell density.
• In a general context, this extends previous results of Caffarelli-Vazquez and Kim who restrict the initial data to be the characteristic function of a compact set.
• In the limit a Hele-Shaw type problem is obtained, where the interface motion law reflects the acceleration effect of the presence of a positive cell density on the expansion of the maximal density (tumor) zone.