Analysis of some L-infinity variational problems and Aronsson's equation, Ericksen-Leslie system modeling hydrodynamic flow of liquid crystals

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

The goal of this project is to advance the principal investigator's research on the analytic issues arising from two areas: (i) the L-infinity variational problem and the associated Aronsson equation, and (ii) the hydrodynamic flow of nematic liquid crystal materials. L-infinity variational problems study the minimization problem of appropriate cost functionals that are suprema of integrand functions. The area has become very active recently. Particular foci of the first part of the project are the following: to identify the most general conditions guaranteeing that the equivalence relationship between absolute minimizers of quasi-convex Hamiltonian functionals and viscosity solutions to the highly degenerate Aronsson equation holds; to explore uniqueness issues for general Aronsson equations with spatial dependence; to study the homogenization problem for L-infinity variational problems in the framework of gamma convergence; and to investigate L-infinity variational problems under Dirichlet energy constraints.The second part of this project deals with the Ericksen-Leslie system modeling hydrodynamic flows of nematic liquid crystals. This is a strongly coupled system relating the incompressible Navier-Stokes equation of the underlying fluid velocity field and the transported heat flow of harmonic maps for the director field of the nematic liquid crystal materials. The objective is to establish both existence and partial regularity for Leray-Hopf-type weak solutions in dimension three. The proposed problems not only are mathematically important but also have potential applications to other fields of mathematics, applied science, and materials engineering. The nonlinear partial differential equations or systems involved in the project either are highly degenerate elliptic problems or equations with super-critical nonlinearities whose resolutions will definitely contribute new ideas and techniques that will be useful in a variety of contexts. So-called L-infinity variational problems arise in a number of different areas such as the determination of optimal radiation treatments in chemotherapy, image analysis and recovery engineering, and the design of winning strategies in certain types of random games. In mechanical engineering, the Ericksen-Leslie system is among the most fundamental equations used to describe the dynamics of viscoelastic fluids, including nematic liquid crystals. Rigorous analysis of both the existence and the regularity of various solutions to such a system can predict the formation of singularities, allow researchers to gain insight into turbulent phenomena, and justify both computational and experimental studies made by applied scientists and engineers. This project will result in the publication of monographs and lecture notes, involve active training of advanced graduate students, and include the organization of specific conferences or workshops.