Finite dimensional reduction and convergence to equilibrium for incompressible Smectic-A liquid crystal flows

TL;DR

This paper studies the long-term behavior of Smectic-A liquid crystal flows, proving the existence of finite-dimensional attractors and convergence to equilibrium in both 2D and 3D cases.

Contribution

It establishes the existence of global and exponential attractors with finite fractal dimension for the model, and proves convergence to equilibrium using a Lojasiewicz--Simon inequality.

Findings

Existence of a global attractor in 2D

Finite fractal dimension of the attractor

Convergence of solutions to equilibrium

Abstract

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.