FINITE DIMENSIONAL REDUCTION AND CONVERGENCE TO EQUILIBRIUM FOR INCOMPRESSIBLE SMECTIC-A LIQUID CRYSTAL FLOWS

We consider a hydrodynamic system that models 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, endowed with periodic boundary conditions. We analyze the longtime behavior of the solutions within the theory of infinite-dimensional dissipative dynamical systems. We first prove that in two dimensions, the problem possesses a global attractor A in a certain phase space. Then we establish the existence of an exponential attractor M, which entails that the global attractor 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 three dimensions are also discussed.