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

• Published in: European journal of applied mathematics Vol. 32; no. 1; pp. 89 - 112
• Main Authors: PERTHAME, BENOÎT, POULAIN, ALEXANDRE
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 01.02.2021; Cambridge University Press (CUP)
• Subjects: Analysis of PDEs; Applied mathematics; Decomposition; Energy; Entropy; Mathematical models; Mathematics; Tissues; Relaxation method; Mathematical biology; Living tissues; Degenerate Cahn-Hilliard equation; Asymptotic analysis
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792520000054

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.