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

向列液晶流、高维相变、守恒几何运动和L-无穷变分问题的分析

• 批准号: 1265574
• 负责人: Changyou Wang
• 金额: $ 16.8万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2015-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265574&HistoricalAwards=false
• 关键词: Analysis; nematic; liquid; crystal; flows

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.

该项目的目标是继续主要研究者对四个子领域所产生的分析问题的研究：（i）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）余维两个表面的守恒几何运动，以及（iv）L-无穷变分问题。本计画的第一部份是以Ericksen-Leslie系统模拟液晶分子的流体力学流动，这是一个强非线性的耦合系统，介于流体速度场的不可压缩Navier-Stokes方程与液晶分子取向指向矢场的谐波映射的热传导之间。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Gamma收敛意义下的能量渐近性，并利用调和映射热流和锐界面的平均曲率流解决了动力学上的Keller-Rubinstein-斯滕贝格问题。第三个项目是建立这样一个保守的平均曲率流的一般初始曲面的局部适定性。第四个项目是研究具有空间依赖性的Hamilton函数的一般Aronsson方程的唯一性、无穷调和函数的正则性以及一致凸Hamilton函数的Aronsson方程，这些领域的问题不仅在数学上具有很大的挑战性，而且在生物学、化学工程、物理学、流体力学和材料科学。在数学上，该项目中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，其解决方案肯定会带来新的想法和技术，这些想法和技术在各种情况下都是有用的。 粘弹性液晶的流体动力学流动是描述粘弹性流体动力学的最基本方程之一，其起源于LCD设计和工程。守恒几何运动与玻色凝聚物理有着密切的联系。L-无穷变分问题在最优控制、图像恢复工程的出现、化疗中最优放疗方案的确定以及随机博弈中获胜策略的设计等方面都有着广泛的应用。严格分析这种系统的各种解的存在性和规律性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。该项目将导致出版专着和国际暑期学校的研究人员和研究生的讲义，涉及先进的博士生的积极培训，并包括具体会议的组织，如俄亥俄州河流分析会议，AMS和SIAM特别会议，和AIM或BIRS研讨会。