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

摘要

We consider a hydrodynamic system that models the Smectic-A liquid crystalflow. The model consists of the Navier-Stokes equation for the fluid velocitycoupled with a fourth-order equation for the layer variable $p$, endowed withperiodic boundary conditions. We analyze the long-time behavior of thesolutions within the theory of infinite-dimensional dissipative dynamicalsystems. We first prove that in 2D, the problem possesses a global attractor$mathcal{A}$ in certain phase space. Then we establish the existence of anexponential attractor $mathcal{M}$ which entails that the global attractor$mathcal{A}$ has finite fractal dimension. Moreover, we show that eachtrajectory converges to a single equilibrium by means of a suitableLojasiewicz--Simon inequality. Corresponding results in 3D are also discussed.