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

TL;DR

This paper investigates the stiff pressure limit of the porous medium equation with general initial densities, leading to a Hele-Shaw type problem that models tumor growth with variable initial conditions.

Contribution

It extends previous models by allowing general initial densities, not just characteristic functions, and derives the resulting Hele-Shaw limit with an acceleration effect.

Findings

Derived the Hele-Shaw limit for general initial densities.

Extended previous results to broader initial conditions.

Revealed the influence of initial density on interface acceleration.

Abstract

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.