Hydrodynamics of Liquid Crystals and Heat Flow of Harmonic Maps

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A significant part of this project is motivated by questions in physics, differential geometry, and material science. As such, the project is expected to have a positive impact on interdisciplinary research and applications to other fields of science. The list of concrete research projects includes the Ericksen-Leslie system, describing the dynamics of liquid crystals, and its mathematical and physical properties due to the nonlinear coupling between the Navier-Stokes fluid dynamics and the microscopic molecular orientation evolution. In particular, one is interested in the developments of singularities in both fluid flows and topological defects in the liquid crystal orientation as well as its long-time behavior. The project also investigates some geometric variational problems such as the classical problem of energy minimizing maps into spheres among continuous maps, the harmonic Ricci flows into surfaces. A fascinating mathematical fact is that these are somehow all connected with the research on liquid crystal dynamics. The project is an important and integral part of the principal investigator's training program for graduate students and postdoctoral researchers through topics courses, special lectures and thesis projects. The results obtained as part of the project are disseminated through publications in professional journals as well as through the lectures, seminars, and conferences. The principal investigator (PI) also organizes conferences and mentors graduate and undergraduate researchers on topics related to the work of the project.This project is aimed at solving several challenging problems from the theory of liquid crystals. Despite numerous efforts by various researchers and tremendous progress over the past three decades, the fundamental problem concerning the global existence of suitable weak solutions of the Ericksen-Leslie system in three dimensions remains a fascinating open problem. The project studies a new modifed model and explores the subtle underlying coupled nonlinear structure. Of related interest are the long time asymptotics and the partial regularity of suitable global weak solutions and the finite-time blow up in both two and three dimensions of initially smooth solutions for this new model. A main focus of the project is the defect and its dynamics. The PI also studies a list of concrete problems related to the heat flow of harmonic maps. When the target is a sphere, which is relevant in the study of liquid crystals, the PI is interested in the gradient flow of the so-called relaxed energy of maps. By using a minimizing movement scheme, one studies its connection to the generalized Brakke flow. A related problem for such coupled equations in geometry is to understand the blow-up mechanism of the harmonic Ricci flow. When the target is a non-positively curved Alexandroff space, the project studies, through a refined minimizing movement scheme, the existence of better weak solutions that satisfy, in addition to the Struwe monotonicity property, the monotonicity of Almgren's frequency. The latter has important consequences.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.