Communications in Mathematical Sciences

Volume 20 (2022)

Number 7

Degenerate Cahn–Hilliard and incompressible limit of a Keller–Segel model

Pages: 1901 – 1926



Charles Elbar (Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL), Paris, France)

Benoît Perthame (Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions (LJLL), Paris, France)

Alexandre Poulain (Department of Numerical Analysis and Scientific Computing, Simula Research Laboratory, Oslo, Norway)


The Keller–Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller–Segel to the degenerate Cahn–Hilliard system. This analysis is made possible from the observation that the Keller–Segel system is equivalent to a relaxed version of the Cahn–Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn–Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy’s law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller–Segel system to the Cahn–Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller–Segel system and the relaxed degenerate Cahn–Hilliard system.


degenerate Cahn–Hilliard equation, asymptotic analysis, Keller–Segel system, incompressible limit, Hele–Shaw equations, surface tension

2010 Mathematics Subject Classification

35B40, 35B45, 35G20, 35Q92

Received 18 January 2022

Accepted 29 January 2022

Published 21 October 2022