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We investigate the nonlocal version of the Abels-Garcke-Gr\"{u}n (AGG)system, which describes the motion of a mixture of two viscous incompressiblefluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard systemcharacterized by concentration-dependent density and viscosity, and anadditional flux term due to interface diffusion. In particular, theCahn-Hilliard dynamics of the concentration (phase-field) is governed by theaggregation/diffusion competition of the nonlocal Helmholtz free energy withsingular (logarithmic) potential and constant mobility. We first prove theexistence of global strong solutions in general two-dimensional bounded domainsand their uniqueness when the initial datum is strictly separated from the purephases. The key points are a novel well-posedness result of strong solutions tothe nonlocal convective Cahn-Hilliard equation with singular potential andconstant mobility under minimal integral assumption on the incompressiblevelocity field, and a new two-dimensional interpolation estimate for the$L^4(\Omega)$ control of the pressure in the stationary Stokes problem.Secondly, we show that any weak solution, whose existence was already known, isglobally defined, enjoys the propagation of regularity and converges towards anequilibrium (i.e., a stationary solution) as $t\rightarrow \infty$.Furthermore, we demonstrate the uniqueness of strong solutions and theircontinuous dependence with respect to general (not necessarily separated)initial data in the case of matched densities and unmatched viscosities (i.e.,the nonlocal model H with variable viscosity, singular potential and constantmobility). Finally, we provide a stability estimate between the strongsolutions to the nonlocal AGG model and the nonlocal Model H in terms of thedifference of densities.

Global well-posedness and convergence to equilibrium for the Abels-Garcke-Gr\"{u}n model with nonlocal free energy