  In this paper, we study the longtime asymptotic behavior of a phase
separation process occurring in a three-dimensional domain containing a fluid
flow of given velocity. This process is modeled by a viscous convective
Cahn-Hilliard system, which consists of two nonlinearly coupled second-order
partial differential equations for the unknown quantities, the chemical
potential and an order parameter representing the scaled density of one of the
phases. In contrast to other contributions, in which zero Neumann boundary
conditions were are assumed for both the chemical potential and the order
parameter, we consider the case of dynamic boundary conditions, which model the
situation when another phase transition takes place on the boundary. The phase
transition processes in the bulk and on the boundary are driven by free
energies functionals that may be nondifferentiable and have derivatives only in
the sense of (possibly set-valued) subdifferentials. For the resulting
initial-boundary value system of Cahn-Hilliard type, general well-posedness
results have been established in a recent contribution by the same authors. In
the present paper, we investigate the asymptotic behavior of the solutions as
times approaches infinity. More precisely, we study the $\omega$-limit (in a
suitable topology) of every solution trajectory. Under the assumptions that the
viscosity coefficients are strictly positive and that at least one of the
underlying free energies is differentiable, we prove that the $\omega$-limit is
meaningful and that all of its elements are solutions to the corresponding
stationary system, where the component representing the chemical potential is a
constant.
