Variational Analysis and Hydrodynamics of Liquid Crystals

作为该项目的一部分，将研究的问题不仅在数学上具有极大的挑战性，而且与其他领域有着密切的联系和重要的应用，包括复杂的流体力学、材料科学和工程(例如，在光学显示设备的设计和控制方面)。对静态或动态液晶模型方程的某些类型的解的严格分析可以预测缺陷的形成和结构，使研究人员能够更好地理解湍流现象，并证明应用科学家的实验和计算研究是合理的。这些问题也将被整合到研究生的培训中，结果将通过一本针对该领域研究人员的研究专著传播。该项目的技术方面包括三个部分：1)向列相液晶流体动力学的Ericksen-Leslie系统模拟；2)各向同性相、向列相和液晶液滴的相变问题；以及3)流形之间的分数调和图热流。在第一部分，主要研究人员和他的合作者将研究(拉格朗日平均)Ericksen-Leslie系统，这是一个强耦合底层流体的强迫Navier-Stokes方程(具有拉格朗日平均)的耗散系统，以及液晶分子取向指向量场(传输的调和映射热流)的演化。其目的是对任意大的初始数据建立整体Leray-Hopf型解的存在性和部分正则性。在项目的第二部分，我们将利用Gamma收敛理论来研究具有不同取向程度的液晶的奇异摄动Ericksen泛函的极小器的尖锐界面极限问题；另一个目标是建立其相应梯度流的Gamma极限结果，即界面的平均曲率流与体区指向场的广义调和映象热流耦合。还将研究描述液晶液滴的能量泛函的最小组态的存在和唯一性。该项目的第三部分是关于流形之间地图的非局部能量泛函的梯度流动的研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The questions that will be studied as part of this project are not only extremely challenging mathematically but also have close connections to and important applications in other fields, including complex fluid mechanics, materials science, and engineering (for example, in both the design and control of optical display devices). The rigorous analysis of certain types of solutions to the model equations for liquid crystals in either the static or the dynamic regime can predict the formation and structure of defects, allow researchers to better understand turbulence phenomena, and justify both experimental and computational studies by applied scientists. The questions will also be integrated into the training of graduate students, and the results will be disseminated through a research monograph aimed at researchers in the field.The technical side of the project consists of three parts: 1) Ericksen-Leslie system modeling the hydrodynamics of nematic liquid crystals; 2) Phase transition problems on isotropic and nematic phases and liquid crystal droplets; and 3) Fractional harmonic map heat flows between manifolds. In the first part, the principal investigator and his collaborators will investigate the (Lagrangian-averaged) Ericksen-Leslie system, which is a dissipative system strongly coupling the forced Navier-Stokes equation (with Lagrangian-average) for the underlying fluid and the evolution for the orientation director fields for liquid crystal molecules (a transported harmonic map heat flow). The aim is to establish both existence and partial regularity of global Leray-Hopf type solutions for any arbitrarily large initial data. In the second part of the project, the Gamma-convergence theory will be used to study the sharp interface limit problem of minimizers to a singularly perturbed Ericksen functional of liquid crystals with variable degrees of orientation; another goal is to establish Gamma-limit results for their corresponding gradient flows, that is, the mean curvature flow of interfaces coupled with the generalized harmonic map heat flow for the director fields in the bulk regions. The existence and uniqueness of minimal configurations of energy functionals describing liquid crystal droplets will also be investigated. The third part of the project is concerned with the study of the gradient flow of nonlocal energy functionals of maps between manifolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.