On a mean field model for 1D thin film droplet coarsening

A thin liquid film coating a solid substrate is unstable and the late stage morphology is essentially quasiequilibrium droplets connected by an ultra thin film. Droplets exchange mass and coarsening occurs-the total number of droplets N(t) decreases while the average size of droplets increases. It is predicted that N(t) obeys a power law N(t) similar to ct(-2/5) in the 1D case. We study a mean field model proposed by Gratton and Witelski for the one-dimensional thin film coarsening and prove a universal one-sided bound on N(t) that partially justifies the power law. A corresponding bound on the average drop size is also obtained. Then we study the relation of this model and the mean field model for Ostwald ripening and show that they are equivalent after a nonlinear rescaling of time. Finally we present some further estimates on the coarsening rates for the Ostwald ripening.