Universal bounds on coarsening rates for mean-field models of phase transitions

Shibin Dai
Department of Mathematics
University of Maryland
College Park, MD 20742, USA
sdai@math.umd.edu

Robert L. Pego
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
rpego@cmu.edu



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), 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.

Get the paper in its entirety as