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Crossover in coarsening rates for the monopole approximation of the Mullins–Sekerka model with kinetic drag

Published online by Cambridge University Press:  21 May 2010

Shibin Dai
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA (
Barbara Niethammer
Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK (
Robert L. Pego
Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA 15213, USA (


The Mullins–Sekerka sharp-interface model for phase transitions interpolates between attachment-limited and diffusion-limited kinetics if kinetic drag is included in the Gibbs–Thomson interface condition. Heuristics suggest that the typical length-scale of patterns may exhibit a crossover in coarsening rate from l(t) ˜ t1/2 at short times to l(t) ˜ t1/3 at long times. We establish rigorous, universal one-sided bounds on energy decay that partially justify this understanding in the monopole approximation and in the associated Lifshitz–Slyozov–Wagner mean-field model. Numerical simulations for the Lifshitz–Slyozov–Wagner model illustrate the crossover behaviour. The proofs are based on a method for estimating coarsening rates introduced by Kohn and Otto, and make use of a gradient-flow structure that the monopole approximation inherits from the Mullins–Sekerka model by restricting particle geometry to spheres.

Research Article
Copyright © Royal Society of Edinburgh 2010

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