  We consider a diffuse interface model for incompressible isothermal mixtures
of two immiscible fluids with matched constant densities. This model consists
of the Navier-Stokes system coupled with a convective nonlocal Cahn-Hilliard
equation with non-constant mobility. We first prove the existence of a global
weak solution in the case of non-degenerate mobilities and regular potentials
of polynomial growth. Then we extend the result to degenerate mobilities and
singular (e.g. logarithmic) potentials. In the latter case we also establish
the existence of the global attractor in dimension two. Using a similar
technique, we show that there is a global attractor for the convective nonlocal
Cahn-Hilliard equation with degenerate mobility and singular potential in
dimension three.
