Crossover of the coarsening rates in demixing of binary viscous liquids
Felix Otto, Christian Seis, and Dejan Slepčev
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Submission date: 05. Apr. 2012
published in: Communications in mathematical sciences, 11 (2013) 2, p. 441-464
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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 t1∕3, when ℓ denotes the average domains size and t denotes time. Furthermore, he argues 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. Siggias 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.