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


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.

Research Article
Copyright © Cambridge University Press 1996

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