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

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.