Problems on Hydrodynamics and Phase Transitions from Nematic Liquid Crystal Materials

Purdue University, West Lafayette IN

Investigators

Abstract

This project is aimed at problems in the subject of nematic liquid crystals that are not only important and challenging mathematically, but also have close connections to other fields, such as material sciences and fluid mechanics, and have found profound applications to the liquid crystal device (LCD) industry. The rigorous analysis of certain types of solutions to the governing equations for nematic liquid crystals in either static or dynamic situations can predict the formation and structure of defects in the materials, allow better understandings of turbulence phenomena, and justify both experimental and computational studies by applied scientists. This, in turn, can advance the design and control of optical features in display devices. The sharp phase transition problems in the vectorial valued setting have their origins in applied sciences and have already found important applications in many chemical reaction-diffusion problems and isotropic-nematic transition phenomena in liquid crystal materials. The questions will be integrated into the training of Ph.D. students, and the findings will be disseminated through a research monograph and other means. The project consists of three parts: 1) The hydrodynamics of nematic liquid crystals modeled by the Ericksen-Leslie system; 2) Phase transition problems between isotropic-nematic phases and the structure of line defects in the framework of Landau - de Gennes Q-tensor theory; and 3) the heat flow of s-harmonic maps into manifolds. The goal of the first part is to establish the existence of global Leray-Hopf weak solutions in dimension three for any finite energy initial data; in particular, the investigator is focused on the existence of global axisymmetric solutions with swirls, the existence of forward self-similar solutions with sole singularity at the origin, and the Liouville problem for steady Ericksen-Leslie system in three spaces. The second part of the project is devoted to continuing the study of the sharp interface limit problem of minimizers to a singular perturbed Landau - de Gennes energy functional by the Gamma convergence theory and the geometric description of the defect set of Q-tensor minimizers around its negative uniaxial set. In the third part of the project, the investigator aims to establish the global existence of partially smooth solutions of the heat flow of s-harmonic maps. 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.