Phase segregation in binary fluid mixtures

We consider a binary mixture of fluids interacting with a repulsive long range force between different species. Masses, total momentum and energy are conserved. The time evolution is described by a couple of Boltzmann equations for the phase space densities of the two species. Under appropriate space-time scalings hydrodynamic equations of Euler and Navier-Stokes type are obtained, with additional self-consistent forces. The equilibria are described on all the scales as minimizers of an appropriate free energy. The minimizers, corresponding to segregated phases at low temperature, are determined, in the kinetic scale, under the assumption of monotonicity of the potential, via rearrangement inequalities and a local analysis. A formal asymptotic expansion for the evolution of a sharp interface is obtained.