1 What is a liquid crystal 2 1.1 Nematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Cholesterics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Smectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Columnar phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 Lyotropic liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Introduction to Liquid Crystals

『Introduction to PDEs and Waves for the Atmosphere and Ocean(Courant Lecture Notes, Volume 9)』, Andrew Majda著, American Mathematical Society刊, 2003年1月発行, 234頁, ペーパーバック, $40.95, ISBN:0-8218-2954-8

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Liquid Crystals: Nature's Delicate Phase of Matter

We consider a full Navier--Stokes and $Q$-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter $\xi$ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.

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Global Strong Solutions of the Full Navier-Stokes and Q-Tensor System for Nematic Liquid Crystal Flows in Two Dimensions

We derive a model describing the evolution of a nematic liquid–crystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen–Leslie system where the Leslie stress tensor and the Oseen–Frank energy density are considered in their general forms. The work postulate proposed by Ericksen–Leslie is traduced in terms of entropy production.We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

A global 2D well-posedness result on the order tensor liquid crystal theory

Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals

This paper is devoted to the full system of incompressible liquid crystals, as modeled in the Q-tensor framework. The main purpose is to establish the uniqueness of weak solutions in a two dimensional setting, without imposing an extra regularity on the solutions themselves. This result only requires the initial data to fulfill the features which allow the existence of a weak solution. Thus, we also present a revisit of the global existence result in dimension two and three.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2016/0014/0008/CMS-2016-0014-0008-a003.pdf

Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D

We derive a model describing the evolution of a nematic liquid-crystal material under the action of thermal effects. The first and second laws of thermodynamics lead to an extension of the general Ericksen-Leslie system where the Leslie stress tensor and the Oseen-Frank energy density are considered in their general forms. The work postulate proposed by Ericksen-Leslie is traduced in terms of entropy production. We finally analyze the global-in-time well-posedness of the system for small initial data in the framework of Besov spaces.

Francesco De Anna

Papers

Non-isothermal General Ericksen–Leslie System: Derivation, Analysis and Thermodynamic Consistency

A global 2D well-posedness result on the order tensor liquid crystal theory

Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals

Uniqueness of weak solutions of the full coupled Navier-Stokes and Q-tensor system in 2D

Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamics-consistency