液晶分子模型中若干偏微分方程问题的研究

The liquid crystal state is a distinct phase of matter that is between those of ordinary liquid and solid crystal. They may flow as a liquid while the molecules are oriented in a crystal-like way. A classification of liquid crystals based on their structural properties was first proposed by G. Friedel in 1922 and they are generally divided into three main classes, the nematic phase, the cholesteric phase, and the smectic phase. This project will be only concerned with the continuum model for the nematic case which can generally be described by three theories. Among them, the most intuitive one is the vector theory and the representative model is Oseen-Frank model in the static case and the Ericksen-Leslie model in the dynamic. The second theory is the molecular theory. This is a microscopic theory which uses a number density function f(x;m) to characterize, at each spatial point x, the number density of molecules whose orientations are parallel to a certain direction. On this field, the most frequently used model is the Maier-Saupe-Onsager theory in the static case and the Doi-Onsager model in the dynamics. The last continuum theory for nematic liquid crystal is called Landau-de Gennes theory. In this framework, the order parameter is a 3 by 3 traceless symmetric matrix-valued function Q(x) characterizing the orientation of molecules near spatial point x. As a phenomenological theory, it was derived based on the thermo-dynamical consideration of the Gibbs free energy of the system and it gives a phenomenological description of the nematic-isotropic phase transition of liquid crystals. .The main theme of this project is, under the framework of weak solutions of partial differential solutions, to understand the relationships between the Doi-Onsager equation and the Ericksen-Leslie equation: In the first stage, we shall prove that, when the re-scaled length of the molecule tends to 0, the solutions to the Doi-Onsager equation with very general initial data will converges weakly to a weak solution of the harmonic map heat flow equation. Based on this, we shall study the small Deborah number limit of the coupling between Doi-Onsager equations with incompressible viscous hydrodynamics. Under certain assumptions on the initial data of such a system, we shall recover the general Ericksen-Leslie equation in the small Deborah number limit. The novelty brought in with respect to existing literature lies on that, our results will take into account the possible defects, which is a key feature of liquid crystal.

丝状液晶的连续介质模型基本分为三类：分子模型，以Doi-Onsager模型为代表，它用概率密度函数来描述液晶分子的指向，是从统计力学观点出发得到的微观理论；向量模型，以描述稳态情形的Oseen-Frank理论和描述动力学性质的Ericksen-Leslie理论为基础，是使用最广泛的液晶宏观模型。张量理论，一种介于分子模型与向量模型之间的唯象理论，以Landau-De Gennes模型为代表。本科研项目将重点围绕分子模型对应的偏微分方程的弱解适定性理论以及它与向量模型的关系展开：一方面，本项目将研究液晶分子模型的Doi-Onsager方程的弱解适定性及其大时间性态问题。另一方面，我们将要考察它如何在极限状态下转化为向量模型的Ericksen-Leslie方程。

在项目执行期内，我们取得了显著的进展。其中最重要的突破是在弱解框架下建立了液晶的Doi-Onsager理论与经典Oseen-Frank弹性理论的关系。我们从Doi-Onsager方程的弱解出发，利用相对熵方法证明当系统的Deborah数趋于0时，方程的解趋于某个不变测度并且它的二阶矩是由调和映射热流的解描述的。之前的结果都局限于形式推导或者给予极限方程的光滑解，从而不能囊括物理上最有意义的情况，即带有缺陷的液晶，对应于方程的不连续解。我们的证明不需要对极限方程有任何了解。