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Upper bounds on coarsening rates. (English) Zbl 1004.82011
Summary: 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 \(\ell(t) \sim t^{1/3}\) in the first case and \(\ell(t) \sim t^{1/4}\) 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.

MSC:
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
74A25 Molecular, statistical, and kinetic theories in solid mechanics
74N20 Dynamics of phase boundaries in solids
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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