Phase segregation and interface dynamics in kinetic systems

We consider a kinetic model of two species of particles interacting with a reservoir at fixed temperature, described by two coupled Vlasov-Fokker-Planck equations. We prove that in the diffusive limit the evolution is described by a macroscopic equation in the form of the gradient flux of the macroscopic free energy functional. Moreover, we study the sharp interface limit and find by formal Hilbert expansions that the interface motion is given in terms of a quasi-stationary problem for the chemical potentials. The velocity of the interface is the sum of two contributions: the velocity of the Mullins-Sekerka motion for the difference of the chemical potentials and the velocity of a Hele-Shaw motion for a linear combination of the two potentials. These equations are identical to the ones in Otto and E (1997 J. Chem. Phys. 107 10177-84) modelling the motion of a sharp interface for a polymer blend.