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The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis

Published online by Cambridge University Press:  16 July 2009

J. F. Blowey  and
C. M. Elliott
J. F. Blowey
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
C. M. Elliott
Affiliation:
School of Mathematical and Physical Sciences, University of Sussex, Brighton BN1 9QH, UK
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Abstract

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 2 , Issue 3 , September 1991 , pp. 233 - 280
Copyright
Copyright © Cambridge University Press 1991

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