Date of Original Version
copyright © by International Press of Boston
Abstract or Description
We consider a model for phase separation in binary viscous liquids that allows for material transport due to cross-diffusion of unlike particles and convection by the hydrodynamic bulk flow. Typically, during the evolution, the average size of domains of the pure phases increases with time — a phenomenon called coarsening. Siggia  predicts that at an initial stage, coarsening proceeds mainly by diffusion, which leads to the well-known evaporation-recondensation growth law ℓ ∼ t 1/3 , when ` denotes the average domains size and t denotes time. Furthermore, he argued that at a later stage, convection by the bulk flow becomes the dominant transport mechanism, leading to a crossover in the coarsening rates to ℓ ∼ t. Siggia’s predictions have been confirmed by experiments and numerical simulations.
In this work, we prove the crossover in the coarsening rates in terms of time-averaged lower bounds on the energy, which scales like an inverse length. We use a method proposed by Kohn and the first author , which exploits the gradient flow structure of the dynamics. Our adaption uses techniques from optimal transportation. Our main ingredient is a dissipation inequality. It measures how the optimal transportation distance changes under the effects of convective and diffusive transport.
Communications in Mathematical Sciences, 11, 2, 441-464.