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Relaxation of the Cahn–Hilliard equation with singular single-well potential and degenerate mobility

Part of: General higher-order equations and systems Partial differential equations Equations of mathematical physics and other areas of application Physiological, cellular and medical topics

Published online by Cambridge University Press:  24 March 2020

BENOÎT PERTHAME Open the ORCID record for BENOÎT PERTHAME [Opens in a new window]  and
ALEXANDRE POULAIN Open the ORCID record for ALEXANDRE POULAIN [Opens in a new window]
BENOÎT PERTHAME
Affiliation:
Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions, F-75005 Paris, France emails: Benoit.Perthame@sorbonne-universite.fr; poulain@ljll.math.upmc.fr
ALEXANDRE POULAIN
Affiliation:
Sorbonne Université, CNRS, Université de Paris, Inria, Laboratoire Jacques-Louis Lions, F-75005 Paris, France emails: Benoit.Perthame@sorbonne-universite.fr; poulain@ljll.math.upmc.fr
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Abstract

The degenerate Cahn–Hilliard equation is a standard model to describe living tissues. It takes into account cell populations undergoing short-range attraction and long-range repulsion effects. In this framework, we consider the usual Cahn–Hilliard equation with a singular single-well potential and degenerate mobility. These degeneracy and singularity induce numerous difficulties, in particular for its numerical simulation. To overcome these issues, we propose a relaxation system formed of two second-order equations which can be solved with standard packages. This system is endowed with an energy and an entropy structure compatible with the limiting equation. Here, we study the theoretical properties of this system: global existence and convergence of the relaxed system to the degenerate Cahn–Hilliard equation. We also study the long-time asymptotics which interest relies on the numerous possible steady states with given mass.

Keywords

Degenerate Cahn–Hilliard equationrelaxation methodasymptotic analysisliving tissues

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35G20: Nonlinear higher-order equations 35Q92: PDEs in connection with biology and other natural sciences 92C10: Biomechanics

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 89 - 112
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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Footnotes

†

The authors have received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740623).

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