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

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.