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Publication 09-CNA-03

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

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

Abstract: 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 $l(t)\sim t^{1/2}$ at short times to $l(t)\sim t^{1/3}$ 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 LSW mean-field model. Numerical simulations for the LSW model illustrate the crossover behavior. 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.

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