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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
14/2012

On the coarsening rates for attachment-limited kinetics

Luca Mugnai and Christian Seis

Abstract

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 $\ell$ of the solid islands grows according to the power law $\ell\sim t^{1/2}$, when $t$ denotes time.

In this paper, we prove a weak one-sided version of this coarsening rate, namely we prove that $\ell\lesssim t^{1/2}$ 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.

Received:
05.03.12
Published:
06.03.12

Related publications

inJournal
2013 Repository Open Access
Luca Mugnai and Christian Seis

On the coarsening rates for attachment-limited kinetics

In: SIAM journal on mathematical analysis, 45 (2013) 1, pp. 324-344