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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
sdai@wpi.edu

Barbara Niethammer
Mathematical Institute
University of Oxford
Oxford, OX1 3LB, UK
niethammer@maths.ox.ac.uk

Robert L. Pego
Department of Mathematical Sciences
and Center for Nonlinear Analysis
Carnegie Mellon University
Pittsburgh, PA 15213, USA
rpego@cmu.edu

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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