Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Incompressible limit of a mechanical model for tissue growth with non-overlapping constraint

Pages: 1913 – 1932



Sophie Hecht (Francis Crick Institute, Kings Cross, London, United Kingdom; and Imperial College London, United Kingdom)

Nicolas Vauchelet (Institut Galilée, Université Paris 13, Villetaneuse, France)


A mathematical model for tissue growth is considered. This model describes the dynamics of the density of cells due to pressure forces and proliferation. It is known that some cell population models of this kind converge at the incompressible limit towards a Hele–Shaw type free boundary problem. The novelty of this work is to impose a non-overlapping constraint. This constraint is important to be satisfied in many applications. One way to guarantee this non-overlapping constraint is to choose a singular pressure law. The aim of this paper is to prove that, although the pressure law has a singularity, the incompressible limit leads to the same Hele–Shaw free boundary problem.


nonlinear parabolic equation, incompressible limit, free boundary problem, tissue growth modelin

2010 Mathematics Subject Classification

35K55, 76D27, 92C50

S.H. would like to thank Pierre Degond and Jean-Paul Vincent for stimulating discussion and acknowledges support from the Francis Crick Institute which receives its core funding from Cancer Research UK (FC001204), the UK Medical Research Council (FC001204), and the Wellcome Trust (FC001204). N.V. acknowledges partial support from the ANR blanche project Kibord No ANR-13-BS01-0004 funded by the French Ministry of Research. Part of this work has been done while N.V. was a CNRS fellow at Imperial College; he is really grateful to the CNRS and to Imperial College for the opportunity of this visit. The authors would like to express their sincere gratitude to Pierre Degond for his help and suggestions during this work.

Received 24 February 2017

Accepted 30 June 2017

Published 16 October 2017