We consider a diffuse interface model for the evolution of an
iso-thermal incompressible two-phase flow in a two-dimensional bounded
domain. The model consists of the Navier-Stokes equation for the fluid
velocity u coupled with a convective Allen-Cahn equation for
the order (phase) parameter $\phi$, both endowed with suitable boundary
conditions. We analyze the asymptotic behavior of the solutions within the
theory of infinite-dimensional dissipative dynamical systems. We first prove
that the initial and boundary value problem generates a strongly continuous
semigroup on a suitable phase space which possesses the global attractor $
\mathcal{A}$. Then we establish the existence of an exponential attractor $
\mathcal{E}$ which entails that $\mathcal{A}$ has finite fractal dimension.
This dimension is then estimated in terms of some model parameters.
Moreover, assuming the potential to be real analytic, we demonstrate that,
in absence of external forces, each trajectory converges to a single
equilibrium by means of a Łojasiewicz-Simon inequality. We also obtain a
convergence rate estimate. Finally, we discuss the case where $\phi $ is
forced to take values in a bounded interval, e.g., by a so-called singular
potential.