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


Authors: Inwon Kim and Norbert Požár
Journal: Trans. Amer. Math. Soc. 370 (2018), 873-909
MSC (2010): Primary 92C50; Secondary 35Q35, 35D40, 76D27
DOI: https://doi.org/10.1090/tran/6969
Published electronically: October 5, 2017
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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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Additional Information

Inwon Kim
Affiliation: Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
Email: ikim@math.ucla.edu

Norbert Požár
Affiliation: Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Japan
Email: npozar@se.kanazawa-u.ac.jp

DOI: https://doi.org/10.1090/tran/6969
Received by editor(s): October 22, 2015
Received by editor(s) in revised form: April 7, 2016, and May 2, 2016
Published electronically: October 5, 2017
Additional Notes: The first author was partially supported by NSF DMS-1300445
The second author was partially supported by JSPS KAKENHI Grant Number 26800068
Article copyright: © Copyright 2017 American Mathematical Society

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