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Cahn–Hilliard equations on an evolving surface

Part of: Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  16 June 2021

D. CAETANO Open the ORCID record for D. CAETANO [Opens in a new window]  and
C. M. ELLIOTT
D. CAETANO
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK emails: Diogo.Caetano@warwick.ac.uk; C.M.Elliott@warwick.ac.uk
C. M. ELLIOTT
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK emails: Diogo.Caetano@warwick.ac.uk; C.M.Elliott@warwick.ac.uk
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Abstract

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.

Keywords

Cahn–Hilliardevolving surfaces

MSC classification

Primary: 35A01: Existence problems
Secondary: 35Q92: PDEs in connection with biology and other natural sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 937 - 1000
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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