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The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature

Published online by Cambridge University Press:  26 September 2008

J. W. Cahn
Affiliation:
Materials Science and Engineering Laboratory, NIST, Gaithersburg, MD 20899, USA
C. M. Elliott
Affiliation:
Centre for Mathematical Analysis and its Applications, University of Sussex, Brighton BN1 9QH, UK
A. Novick-Cohen
Affiliation:
Department of Mathematics, Technion-IIT, Haifa, Israel 32000

Abstract

We show by using formal asymptotics that the zero level set of the solution to the Cahn–Hilliard equation with a concentration dependent mobility approximates to lowest order in ɛ. an interface evolving according to the geometric motion,

(where V is the normal velocity, Δ8 is the surface Laplacian and κ is the mean curvature of the interface), both in the deep quench limit and when the temperature θ is where є2 is the coefficient of gradient energy. Equation (0.1) may be viewed as motion by surface diffusion, and as a higher-order analogue of motion by mean curvature predicted by the bistable reaction-diffusion equation.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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