Finite Dimensional Reduction and Convergence to Equilibrium for Incompressible Smectic-A Liquid Crystal Flows

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 φ, 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 A in 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 3D are also discussed.