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

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} $\text{bu}$ and the {\it director field} $\text{di}$, 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.