  A well-known diffuse interface model consists of the Navier-Stokes equations
nonlinearly coupled with a convective Cahn-Hilliard type equation. This system
describes the evolution of an incompressible isothermal mixture of
binary-fluids and it has been investigated by many authors. Here we consider a
variant of this model where the standard Cahn-Hilliard equation is replaced by
its nonlocal version. More precisely, the gradient term in the free energy
functional is replaced by a spatial convolution operator acting on the order
parameter phi, while the potential F may have any polynomial growth. Therefore
the coupling with the Navier-Stokes equations is difficult to handle even in
two spatial dimensions because of the lack of regularity of phi. We establish
the global existence of a weak solution. In the two-dimensional case we also
prove that such a solution satisfies the energy identity and a dissipative
estimate, provided that F fulfills a suitable coercivity condition.
