Coarsening Rates for a Droplet Model: : Rigorous Upper Bounds
Pages 503 - 529
Abstract
Certain liquids on solid substrates form a configuration of droplets connected by a precursor layer. This configuration coarsens: The average droplet size grows while the number of droplets decreases and the characteristic distance between them increases. We study this type of coarsening behavior in a model given by an evolution equation for the film height on an n-dimensional substrate. Heuristic arguments based on the asymptotic analysis of Glasner and Witelski [Phys. Rev. E, 67 (2003), p. 016302, Phys. D., 209 (2005), pp. 80-104] and numerical simulations suggest a statistically self-similar behavior characterized by a single exponent which determines the coarsening rate. In this paper, we establish rigorously an upper bound on the coarsening rate in a time-averaged sense. We use the fact that the evolution is a gradient flow, i.e., a steepest descent in an energy landscape. Coarse information on the geometry of the energy landscape serves to obtain coarse information on the dynamics. This robust method was proposed in [R. V. Kohn and F. Otto, Comm. Math. Phys., 229 (2002), pp. 375-395]. Our main analytical contribution is an interpolation inequality involving the Wasserstein distance, which characterizes the coarse shape of the energy landscape. The upper bound we obtain is in agreement with heuristic arguments and numerical simulations.
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Published: 01 January 2006
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