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

A diffuse interface model for two-phase incompressible flows with nonlocal interactions and nonconstant mobility