On the coarsening rates for attachment-limited kinetics
Luca Mugnai and Christian Seis
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Submission date: 05. Mar. 2012
published in: SIAM journal on mathematical analysis, 45 (2013) 1, p. 324-344
DOI number (of the published article): 10.1137/120865197
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We study the coarsening rates for attachment-limited kinetics which is modeled by nonlocal mean-curvature flow. Attachment-limited kinetics is observed during solidification processes, in which the system is divided into two domains of the two pure phases, more precisely islands of a solid phase surrounded by an undercooled liquid phase, and the relaxation process is due to material redistribution form high to low interfacial curvature regions. The interfacial area between the phases decreases in time while the volume of each phase is preserved. Consequently, the domain morphology coarsens. Experiments, heuristics and numerics suggest that the typical domain size ℓ of the solid islands grows according to the power law ℓ t1∕2, when t denotes time.
In this paper, we prove a weak one-sided version of this coarsening rate, namely we prove that ℓ ≲ t1∕2 in time average. The bound on the coarsening rate is uniform in the initial configuration but requires some control on collisions of different domains. Our approach is based on a method, introduced by Kohn and Otto [?], relying on the gradient flow structure of the dynamics.