A review on mathematical analysis for nematic and smectic-A liquid crystal models

Abstract

We review the mathematical analysis of some uniaxial, liquid crystal phases. First, we state the models for the two di fferent studied phases: nematic and smectic-A liquid crystals. The spatial and temporal pro les of the liquid crystal con gurations will be described by means of strongly nonlinear parabolic partial di erential systems, which are presented at the same time. Then, we will state some results about existence, regularity, time-periodicity and stability of solutions at in nite time for both models. It is our aim to show that, although nematic and smectic-A phases have di fferent physical properties and are modeled by di erent nonlinear parabolic problems, there exists a common mathematical machinery to rewrite the models and to obtain the analytical results.