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

Kim, Inwon; Požár, Norbert

doi:10.1090/tran/6969

Porous medium equation to Hele-Shaw flow with general initial density
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by Inwon Kim and Norbert Požár PDF

Trans. Amer. Math. Soc. 370 (2018), 873-909 Request permission

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 B. Perthame, F. Quirós, and J. L. Vázquez [The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Ration. Mech. Anal. 212 (2014), 93–127] describes the growth of a tumor zone with a restriction on the maximal cell density. In a general context, this extends previous results of Gil–Quirós 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.

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