Nonstationary models for liquid crystals: A fresh mathematical perspective

TL;DR

This paper reviews mathematical models for nonstationary, inhomogeneous liquid crystals driven out of equilibrium, focusing on solution existence, model linkages, and introducing relative energies for future research.

Contribution

It provides a comprehensive mathematical analysis of popular liquid crystal models and introduces the novel concept of relative energies for comparing solutions.

Findings

Mathematical issues like existence of solutions are addressed.

Connections between different liquid crystal models are established.

A new concept of relative energies is introduced for future studies.

Abstract

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 popular models which physicists often just solve numerically. 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.