Motion by curvature in generalized Cahn-Allen models
type: Item
instance of: scholarly article
title: Motion by curvature in generalized Cahn-Allen models
zbMATH Open document ID: 0844.35042
DOI: 10.1007/BF02186837
published in: Journal of Statistical Physics
publication date: 1996-03-19T00:00:00Z
review text: The authors generalize the Cahn-Allen model for the motion of phase-antiphase boundaries by allowing the kinetic coefficient and the coefficient of the gradient free to depend on the order parameter \(u\). The resulting equation is \[ \varepsilon^2 u_t= \alpha(u) \{\varepsilon^2\sqrt{\kappa(u)} \nabla\cdot (\sqrt{\kappa(u)} \nabla u)- f(u)\}, \] where \(\alpha\) and \(\kappa\) are physical quantities and \(f\) is bistable. This reduces to the classical Cahn-Allen equation if \(\alpha\) and \(\kappa\) are constants. Grain boundaries in a solid material, away from triple junctions, are modelled by solutions of the above equation with internal layers. The authors show that the classical motion by mean curvature law for internal layers remains valid, the only difference being the value of the coefficient entering into the relation. They also extend the analysis to the case that \(\alpha\) is infinite for some values of \(u\).
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 82C24
zbMATH DE Number: 843814
zbMATH Keywords: internal layers
zbMATH Keywords: grain boundaries
zbMATH Keywords: Cahn-Allen equation
zbMATH Keywords: motion by mean curvature
author: Paul C. Fife
author: A. A. Lacey
reviewed by: John Urbas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 843814