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## Upper Bounds on the Coarsening Rates for Some Non-Conserving Equations

##### Date

2012-08-02##### Author

Jiang, Nan

##### Type of Degree

dissertation##### Department

Mathematics and Statistics##### Metadata

Show full item record##### Abstract

In this thesis, we prove one-sided bounds on the coarsening rates for two models of
non-conserved curvature driven dynamics by following a strategy developed by Kohn and
Otto in [20].
In the rst part, we analyze the Allen-Cahn equation in one and two dimensions, with
di erent choices of length scales. The analysis follows the framework of Kohn and Yan in
[24]. In the one-dimensional domain, by choosing an H^{-1}-type length scale, our analysis
supports the assertion that the coarsening occurs at the rate t^{1/3}. In the two-dimensional
domain, we consider two types of length scales. First, we obtain the coarsening rate of t^{1/3}
using an H^{-1}-type length scale, and then, using another L^2-type length scale yields that the
energy decays no faster than the rate t^{-1/6}. In all the cases, among the main ingredients, the
interpolation inequality requires the most delicate analysis, and the dissipation inequalities
are based on basic calculations using H older's inequality. An ODE argument is adapted to
combine these two components in each case. The well-posedness of the Allen-Cahn equation
obtained using xed point method is presented in the appendix.
For the Swift-Hohenberg equation, we again consider an L^2-type length scale in a twodimensional
domain. The coarsening rate of t^{1/3} rate is established using an interpolation
inequality which extends Kohn and Otto's method. This rate is consistent with numerical
results as an upper bound on coarsening rates.

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