Finite--dimensional global attractor for a system modeling the 2D nematic liquid crystal flow

TL;DR

This paper proves that a 2D model of nematic liquid crystal flow, based on coupled Navier-Stokes and reaction-diffusion equations, possesses a finite-dimensional global attractor, indicating long-term predictability and stability.

Contribution

It establishes the existence of a smooth, finite-dimensional global attractor for a 2D nematic liquid crystal flow model with periodic boundary conditions, extending previous well-posedness results.

Findings

Existence of a dissipative dynamical system

Presence of a smooth global attractor

Finite fractal dimension of the attractor

Abstract

We consider a 2D system that models the nematic liquid crystal flow through the Navier--Stokes equations suitably coupled with a transport-reaction-diffusion equation for the averaged molecular orientations. This system has been proposed as a reasonable approximation of the well-known Ericksen--Leslie system. Taking advantage of previous well-posedness results and proving suitable dissipative estimates, here we show that the system endowed with periodic boundary conditions is a dissipative dynamical system with a smooth global attractor of finite fractal dimension.