On a regularized family of models for the full Ericksen-Leslie system

TL;DR

This paper introduces a broad family of regularized models for the full Ericksen-Leslie liquid crystal system, establishing existence, stability, regularity, and convergence properties on compact Riemannian manifolds.

Contribution

It develops a unified framework for analyzing various regularized Navier-Stokes models coupled with liquid crystal dynamics, proving key mathematical properties.

Findings

Existence of solutions for the regularized models

Existence of a finite-dimensional global attractor

Convergence of trajectories to equilibrium under certain conditions

Abstract

We consider a general family of regularized systems for the full Ericksen-Leslie model for the hydrodynamics of liquid crystals in $n$-dimensional compact Riemannian manifolds, $n$=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier Stokes-$\alpha $-like equation, the Leray-$\alpha $ equation, the Modified Leray-$\alpha $ equation, the Simplified Bardina model, the Navier Stokes-Voigt model and the Navier-Stokes equation) for the fluid velocity $u$ suitably coupled with a parabolic equation for the director field $d$. We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz-Simon inequality.