Abstract:
We consider two standard models of surface-energy-driven coarsening: a constant-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to Mullins-Sekerka dynamics; and a degenerate-mobility Cahn-Hilliard equation, whose large-time behavior corresponds to motion by surface diffusion. Arguments based on scaling suggest that the typical length scale should behave as in the first case and in the second. We prove a weak, one-sided version of this assertion – showing, roughly speaking, that no solution can coarsen faster than the expected rate. Our result constrains the behavior in a time-averaged sense rather than pointwise in time, and it constrains not the physical length scale but rather the perimeter per unit volume. The argument is simple and robust, combining the basic dissipation relations with an interpolation inequality and an ODE argument.
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Received: 20 September 2001 / Accepted: 5 February 2002 Published online: 12 August 2002
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Kohn, R., Otto, F. Upper Bounds on Coarsening Rates. Commun. Math. Phys. 229, 375–395 (2002). https://doi.org/10.1007/s00220-002-0693-4
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DOI: https://doi.org/10.1007/s00220-002-0693-4