Hele–Shaw Limit for a System of Two Reaction-(Cross-)Diffusion Equations for Living Tissues

  • Federica Bubba
  • Benoît Perthame
  • Camille PoucholEmail author
  • Markus Schmidtchen


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 phase-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 led, 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 \(L^1\) version in place of the standard upper bound.



F.B. and B.P. have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant Agreement No. 740623). M.S. acknowledges the kind invitation to LJLL funded by the previous grant. Furthermore, M.S. received funding for two research visits from the Doris Chen Mobility Award awarded by Imperial College London. C.P. acknowledges support from the Swedish Foundation of Strategic Research Grant AM13-004.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Inria MAMBA Team, Laboratoire Jacques-Louis LionsSorbonne Université, CNRS, Université Paris-Diderot SPCParisFrance
  2. 2.Department of MathematicsKTH - Royal institute of TechnologyStockholmSweden
  3. 3.Department of MathematicsImperial College LondonLondonUK

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