Universal Bounds on Coarsening Rates for Mean-Field Models of Phase Transitions

Abstract

We prove one-sided universal bounds on coarsening rates for two kinds of mean-field models of phase transitions, one with a coarsening rate $l \sim t^{1/3}$ and the other with $l\sim t^{1/2}$. Here l is a characteristic length scale. These bounds are both proved by following a strategy developed by Kohn and Otto [ Comm. Math. Phys., 229 (2002), pp. 375--395]. The $l\sim t^{1/2}$ rate is proved using a new dissipation relation which extends the Kohn--Otto method. In both cases, the dissipation relations are subtle and their proofs are based on a residual lemma (Lagrange identity) for the Cauchy--Schwarz inequality.