Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system

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, while the potential 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. 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.