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

本项目的目标是继续首席研究员对四个子领域的分析问题的研究：(i)向列液晶材料的流体动力学流动，（ii）两个流形之间的高维相变问题，（iii）协维两个表面的守恒几何运动，以及（iv） l -无穷变分问题。本项目第一部分研究了Ericksen-Leslie系统对向列液晶流体动力流动的建模，该系统是向列液晶分子定向场的调和映射传递热流与底层流体速度场的不可压缩Navier-Stokes方程之间的强非线性耦合系统。目的是建立三维Leray-Hopf型弱解的存在性和部分正则性。第二个项目研究了奇异摄动泛函在伽玛收敛意义下的能量渐近性，并解决了基于新边界条件下调和映射热流和尖锐界面平均曲率流的动力学Keller-Rubinstein-Sternberg问题。第三个课题是建立一般初始曲面的守恒平均曲率流的局部适定性。第四个课题是研究具有空间依赖性的哈密顿函数的一般Aronsson方程的唯一性以及一致凸哈密顿函数对应的无穷调和函数和Aronsson方程的正则性。这些领域提出的问题不仅在数学上具有挑战性，而且与生物学、化学工程、物理学、流体力学和材料科学等其他领域有着密切的联系和深刻的应用。在数学上，该项目涉及的非线性偏微分方程或系统要么是高度退化的椭圆问题，要么是具有超临界非线性的方程，其解决方案肯定会为各种环境提供有用的新思想和技术。向列液晶的流体动力学流动方程是描述粘弹性流体动力学的最基本方程之一，它起源于LCD设计和工程。几何运动守恒与玻色凝聚物理有着密切的联系。l -∞变分问题在最优控制、图像恢复工程、化疗中最佳放射治疗的确定以及随机博弈论中获胜策略的设计等方面都有其应用。对这样一个系统的各种解的存在性和规律性进行严格分析，可以预测奇点的形成，使研究人员能够深入了解湍流现象，并证明应用科学家和工程师所做的计算和实验研究是正确的。该项目将为研究人员和研究生出版国际暑期学校的专著和演讲笔记，包括积极培养高级博士生，并包括组织特定会议，如俄亥俄河分析会议，AMS和SIAM特别会议，AIM或BIRS研讨会。

The goal of this project is to continue the principal investigator's research on analytic issues arising from four subareas: (i) the hydrodynamic flow of nematic liquid crystal materials, (ii) high dimensional phase-transition problem between two manifolds, (iii) conserved geometric motion of co-dimension two surfaces, and (iv) L-infinity variational problems. The first part of this project deals with the Ericksen-Leslie system modeling hydrodynamic flow of nematic liquid crystals, which is a strongly nonlinear-coupled system between the incompressible Navier-Stokes equation of the underlying fluid velocity field and the transported heat flow of harmonic maps for the orientation director field of the nematic liquid crystal molecules. The objective is to establish existence and partial regularity for Leray-Hopf type weak solutions in dimension three. The second project investigates the energy asymptotic of a singularly perturbed functional in the sense of Gamma-convergence and resolve the Keller-Rubinstein-Sternberg problem on the dynamics in terms of harmonic map heat flow under new boundary conditions and mean curvature flow of the sharp interface. The third project is to establish the local well-posedness of such a conserved mean curvature flow for generic initial surfaces. The fourth project is to study the uniqueness of general Aronsson's equations for Hamiltonian functions with spatial dependence and the regularity of infinity harmonic functions and Aronsson's equations corresponding to uniformly convex Hamiltonians.The proposed problems in these areas are not only very challenging mathematically but also have strong connections and profound applications to other fields such as biology, chemical engineering, physics, fluid mechanics and material sciences. Mathematically, the nonlinear partial differential equations or systems involved in the project either are either 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. The hydrodynamic flow of nematic liquid crystals is among the most fundamental equations describing the dynamics of viscoelastic fluids and has its origination in LCD design and engineering. The conserved geometric motion has close connection with the Bose condensate physics. The L-infinity variational problems has found its applications in the optimal control, the image recovery engineering arise, the determination of optimal radiation treatments in chemotherapy, and the design of winning strategies for random game theories. 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 from international summer schools for both researchers and graduate students, involve active training of advanced PhD students, and include the organization of specific conferences such as Ohio River Analysis Meetings, AMS and SIAM special sessions, and AIM or BIRS workshops.