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 φ, 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 φ. 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.