Coarsening Rates in Off-Critical Mixtures

Abstract. We study coarsening of a binary mixture within the Mullins-Sekerka evolution in the regime where one phase has small volume fraction φ ¿ 1. Heuristic arguments suggest that the energy density, which represents the inverse of a typical length scale, decreases as φt−1/3 as t → ∞. We prove rigorously a corresponding weak lower bound. Moreover, we establish a stronger result for the two-dimensional case, where we find a lower bound of the form φ(ln φ−1)1/3t−1/3. Our approach follows closely the analysis in [7], which exploits the relation between two suitable length scales. Our main contribution is an isoperimetric inequality in the two-dimensional case.