  This paper deals with an initial and boundary value problem for a system
coupling equation and boundary condition both of Cahn-Hilliard type; an
additional convective term with a forced velocity field, which could act as a
control on the system, is also present in the equation. Either regular or
singular potentials are admitted in the bulk and on the boundary. Both the
viscous and pure Cahn-Hilliard cases are investigated, and a number of results
is proven about existence of solutions, uniqueness, regularity, continuous
dependence, uniform boundedness of solutions, strict separation property. A
complete approximation of the problem, based on the regularization of maximal
monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and
rigorously discussed.
