Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues

Abstract : Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Bénilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound.
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https://hal.archives-ouvertes.fr/hal-01970313
Contributor : Federica Bubba <>
Submitted on : Tuesday, January 22, 2019 - 11:57:06 AM
Last modification on : Tuesday, December 10, 2019 - 3:08:21 PM
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  • HAL Id : hal-01970313, version 2
  • ARXIV : 1901.01692

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Federica Bubba, Benoît Perthame, Camille Pouchol, Markus Schmidtchen. Hele-Shaw limit for a system of two reaction-(cross-)diffusion equations for living tissues. 2019. ⟨hal-01970313v2⟩

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