Analysis of nematic liquid crystal flows, high dimensional phase transtion, conserved geometric motion, and L-infintiy variational problems

The PI plans to investigate a number of important issues in nonlinear partial differential equations and variational problems. This proposal is focused on Summary. 1) Analysis of a simplified Ericksen-Leslie system modeling hydrodynamic flow of nematic liquid crystals. The goal is to establish the global existence of Leray-Hopf type weak solutions in dimensions three, partial regularity of suitable weak solutions, and regularity and uniqueness in the end point case of Serrinfs classes; 2) Analysis of high dimensional phase transitions between two manifolds. The goal is to study the energy asymptotic in the sense of ƒ¡-convergence, and resolve the dynamics of the phase transition problem or the well known Keller-Rubinstein- Sternberg problem. More precisely, we want to show that the sharp interface moves by mean curvature, while the limiting map moves by the heat flow of harmonic maps under intriguing partially constrained boundary conditions on the sharp interface; 3) Conserved geometric motion of codimension two surfaces. The goal is to establish the local well-posedness of the bi-normal mean curvature motion for initial surface in the W2,p-class; and 4) Analysis of some L‡-variational problems and Aronssonfs equations. Here we plan to study the regularity issue of infinity harmonic functions and viscosity solutions of Aronssonfs equation, and the uniqueness of general Aronssonfs equations. Intellectual Merit. The simplified Ericksen-Leslie system strongly couples the nonhomogeneous Navier-Stokes equation for the fluid and the transported heat flow of harmonic maps for the liquid crystal molecule. Since the equation is only dissipative with super-critical nonlinearities, the advances require development of new ideas and methods that may have potential impacts to other complex fluid and flow problems. The Keller-Rubinstein-Sternberg problem is a long outstanding problem in the phase transitions of high dimensional wells. Establishing the dynamic law for both the sharp interface and the map part rigorously is very challenging and will have significant impact to the study of problems involving different dimensional objects. The bi-normal mean curvature flow of codimension two surfaces is a new direction of research for geometric evolution equations. While the L‡-variational problem involves highly singular and unconventional PDEs, resolving the uniqueness of general Aronssonfs equations and their regularities is one of the most important problems that will certainly bring many new ideas to the field of nonlinear PDEs. Broader Impact. The proposed problems in these areas have strong connections and profound applications to other fields such as biology, chemical engineering, physics, and fluid mechanics and material sciences. For example, the L‡-variational problem has found its applications in optimal control and the image recovery engineering, and the nematic liquid crystal flow has its origination in engineering of LCD. The proposed problems are fascinating and challenging mathematically. They either involve highly degenerate elliptic PDEs or system of PDEs with critical nonlinearities. Their resolutions will contribute new ideas and techniques that shall be useful for many other problems. The proposed research activity is an important and integral part of the PIfs training program for undergraduate and graduate students. The PI is actively engaged in training PhD students to work in these areas. The PI, joint with Dr. Y. Yu, is currently preparing a research monograph on calculus of variations in L‡, which will include research findings from this proposal. The results obtained will be disseminated by publications in journals, development of topic courses for graduate students, mini-courses in national and international graduate student summer schools. The PI has devoted a lot of efforts to organize special workshops and regional conferences, and plans to continue such activities by coorganizing Ohio River Analysis Meetings (ORAM III), AMS special session meetings, SIAM mini-symposiums, and proposes to submit workshop proposals to AIM and BIRS on topics related to development in L‡-variational problems and nematic liquid crystals.