  We investigate the nonlocal version of the Abels-Garcke-Gr\"{u}n (AGG)
system, which describes the motion of a mixture of two viscous incompressible
fluids. This consists of the incompressible Navier-Stokes-Cahn-Hilliard system
characterized by concentration-dependent density and viscosity, and an
additional flux term due to interface diffusion. In particular, the
Cahn-Hilliard dynamics of the concentration (phase-field) is governed by the
aggregation/diffusion competition of the nonlocal Helmholtz free energy with
singular (logarithmic) potential and constant mobility. We first prove the
existence of global strong solutions in general two-dimensional bounded domains
and their uniqueness when the initial datum is strictly separated from the pure
phases. The key points are a novel well-posedness result of strong solutions to
the nonlocal convective Cahn-Hilliard equation with singular potential and
constant mobility under minimal integral assumption on the incompressible
velocity 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, is
globally defined, enjoys the propagation of regularity and converges towards an
equilibrium (i.e., a stationary solution) as $t\rightarrow \infty$.
Furthermore, we demonstrate the uniqueness of strong solutions and their
continuous 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 constant
mobility). Finally, we provide a stability estimate between the strong
solutions to the nonlocal AGG model and the nonlocal Model H in terms of the
difference of densities.
