Crossover of the coarsening rates in demixing of binary viscous liquids

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 [24] predicts that at an initial stage, coarsening proceeds mainly by diffusion, which leads to the well-known evaporation-recondensation growth law ` ∼ t1/3, 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 ` ∼ 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 [15], 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.