  We study an initial-boundary value problem for the incompressible
Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels,
Garcke and Gr\"{u}n in 2012. This model arises in the diffuse interface theory
for binary mixtures of viscous incompressible fluids. This system is a
generalization of the well-known model H in the case of fluids with unmatched
densities. In three dimensions, we prove that any global weak solution (for
which uniqueness is not known) exhibits a propagation of regularity in time and
stabilizes towards an equilibrium state as $t \rightarrow \infty$. More
precisely, the concentration function $\phi$ is a strong solution of the
Cahn-Hilliard equation for (arbitrary) positive times, whereas the velocity
field $\mathbf{u}$ becomes a strong solution of the momentum equation for large
times. Our analysis hinges upon the following key points: a novel global
regularity result (with explicit bounds) for the Cahn-Hilliard equation with
divergence-free velocity belonging only to $L^2(0,\infty;
\mathbf{H}^1_{0,\sigma}(\Omega))$, the energy dissipation of the system, the
separation property for large times, a weak-strong uniqueness type result, and
the Lojasiewicz-Simon inequality. Additionally, in two dimensions, we show the
existence and uniqueness of global strong solutions for the full system.
Finally, we discuss the existence of global weak solutions for the case of the
double obstacle potential.
