Mathematical Problems Modeling Nematic Liquid Crystals: from Macroscopic to Microscopic Theories

液晶是一种特殊类型的软物质，广泛应用于当代技术，包括显示器、温度计、光学成像、记录以及生物系统。向列相是所有液晶相中最简单的。“向列”一词来自希腊语，意思是“线”，这是由于早期在显微镜下观察到相邻液晶分子之间的线状不连续。物理学家已经在不同的尺度上制定了各种模型来描述向列型液晶。该项目将对这些模型中产生的方程进行严格的数学研究，这将有助于深入了解这些材料的潜在性质，并最终有利于应用。该项目将为本科生和研究生提供研究培训机会。该项目针对非线性偏微分方程，范围从宏观理论到微观理论的向列液晶。更具体地说，该项目考虑宏观理论中Beris-Edwards系统的物理参数及其性质的分析和数值研究。该项目还旨在推导由具有奇异型势的自由能产生的梯度流的全局适定性和相分离性质，该梯度流被认为处于连续统理论和动力学理论的中间阶段。此外，该项目将探索微观理论中的动力学方程，并推进对其长期动力学的理解。该项目将利用和扩展偏微分方程理论和变分学中的数学工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Liquid crystals are a special type of soft matter widely used in contemporary technologies including displays, thermometers, optical imaging, and recording, as well as biological systems. The nematic phase is the simplest among all liquid crystal phases. The word nematic comes from Greek, which means thread, due to early experimental observation under microscopes of thread-like discontinuities between neighboring liquid crystal molecules. Physicists have formulated various models at different scales to describe nematic liquid crystals. This project will conduct a rigorous mathematical study of the equations arising from these models, which will provide insight into the underlying nature of these materials and ultimately benefit applications. The project will provide research training opportunities for undergraduate and graduate students.The project is targeted at nonlinear partial differential equations that range from the macroscopic theory to the microscopic theory for nematic liquid crystals. More specifically, the project considers analytic and numerical studies of the physical parameters and their properties in the Beris-Edwards system in macroscopic theory. The project also aims to derive global well-posedness and the phase separation property of a gradient flow generated by a free energy with a potential of singular type, which is considered to be in the intermediate stage of continuum and kinetic theories. Further, the project will explore a kinetic equation in microscopic theory and advance the understanding of its long-time dynamics. The project will utilize and extend mathematical tools from the theory of partial differential equations and calculus of variations.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.