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.References
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Additional Information
- Inwon Kim
- Affiliation: Department of Mathematics, University of California Los Angeles, 520 Portola Plaza, Los Angeles, California 90095
- MR Author ID: 684869
- Email: ikim@math.ucla.edu
- Norbert Požár
- Affiliation: Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Japan
- MR Author ID: 923481
- Email: npozar@se.kanazawa-u.ac.jp
- 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 - © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3729490