Nonstationary models for liquid crystals: A fresh mathematical perspective

In this article, we discuss nonstationary models for inhomogeneous liquid crystals driven out of equilibrium by flow. Emphasis is put on those models which are used in the mathematics as well as in the physics literature, the overall goal being to illustrate the mathematical progress on these models to date. Our discussion includes the Doi-Hess model for the orientational distribution function, the Q-tensor model and the Ericksen-Leslie model which focuses on the director dynamics. We survey particularly the mathematical issues (such as existence of solutions) and linkages between these models. Moreover, we introduce the new concept of relative energies measuring the distance between solutions of equation systems with nonconvex energy functionals and discuss possible applications of this concept for future studies.