On the long-time behavior of some mathematical models for nematic liquid crystals

TL;DR

This paper studies the long-term dynamics of a mathematical model for nematic liquid crystals, showing that solutions tend to stationary states and providing conditions for unique equilibrium convergence.

Contribution

It improves and generalizes existing results on the long-time behavior of nematic liquid crystal models, especially regarding the structure of omega-limit sets.

Findings

Solutions have non-empty omega-limit sets containing only stationary solutions.

Under certain conditions, the omega-limit set reduces to a single point.

The approach extends previous results on the asymptotic behavior of liquid crystal models.

Abstract

A model describing the evolution of a liquid crystal substance in the nematic phase is investigated in terms of two basic state variables: the {\it velocity field} $\bu$ and the {\it director field} $\di$, representing the preferred orientation of molecules in a neighborhood of any point in a reference domain. After recalling a known existence result, we investigate the long-time behavior of weak solutions. In particular, we show that any solution trajectory admits a non-empty $\omega$-limit set containing only stationary solutions. Moreover, we give a number of sufficient conditions in order that the $\omega$-limit set contains a single point. Our approach improves and generalizes existing results on the same problem.