Mean-Curvature Flow of Voronoi Diagrams

We study the evolution of grain boundary networks by the mean-curvature flow under the restriction that the networks are Voronoi diagrams for a set of points. For such evolution we prove a rigorous universal upper bound on the coarsening rate. The rate agrees with the rate predicted for the evolution by mean-curvature flow of the general grain boundary networks, namely that the typical grain area grows linearly in time. We perform numerical simulations which provide evidence that the dynamics achieves the rate of coarsening that agrees with the upper bound in terms of scaling.