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

Abstract

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 long-time 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 $\mathcal{A}$ in a 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 Łojasiewicz–Simon inequality. Corresponding results in three dimensions are also discussed.