Variational Analysis and Hydrodynamics of Liquid Crystals

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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.