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

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 ∼ t and the other with l ∼ t. 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), 375-395). The l ∼ t 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.