Cahn–Hilliard equations on an evolving surface

• Published in: European journal of applied mathematics Vol. 32; no. 5; pp. 937 - 1000
• Main Authors: CAETANO, D., ELLIOTT, C. M.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2021
• Subjects: Applied mathematics; Energy; Evolution; Initial conditions; Integrals; Order parameters; Well posed problems; evolving surfaces; Cahn–Hilliard; 35A01; 35Q92
• ISSN: 0956-7925; 1469-4425
• DOI: 10.1017/S0956792521000176

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.