  We consider a hydrodynamic system that models the Smectic-A liquid crystal
flow. The model consists of the Navier-Stokes equation for the fluid velocity
coupled with a fourth-order equation for the layer variable $\vp$, endowed with
periodic boundary conditions. We analyze the long-time behavior of the
solutions within the theory of infinite-dimensional dissipative dynamical
systems. We first prove that in 2D, the problem possesses a global attractor
$\mathcal{A}$ in certain phase space. Then we establish the existence of an
exponential attractor $\mathcal{M}$ which entails that the global attractor
$\mathcal{A}$ has finite fractal dimension. Moreover, we show that each
trajectory converges to a single equilibrium by means of a suitable
Lojasiewicz--Simon inequality. Corresponding results in 3D are also discussed.
