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

这个项目的目标是推进主要研究者在两个领域的分析问题的研究：(i) l无穷变分问题和相关的Aronsson方程，以及（ii）向列液晶材料的流体动力学流动。l -无穷变分问题研究的是可积函数上的适当代价泛函的最小化问题。这个地区最近变得非常活跃。项目第一部分的特别重点是：确定保证准凸哈密顿泛函的绝对极小值与高度简并的Aronsson方程的粘度解之间的等价关系成立的最一般条件；探讨具有空间依赖性的一般Aronsson方程的唯一性问题；在收敛的框架下研究l -无穷变分问题的齐化问题；研究Dirichlet能量约束下的l -∞变分问题。这个项目的第二部分涉及Ericksen-Leslie系统模拟向列液晶的流体动力学流动。这是一个关于向列液晶材料导向场的不可压缩流体速度场的Navier-Stokes方程和输运热流调和映射的强耦合系统。目的是建立三维leray - hopf型弱解的存在性和部分正则性。所提出的问题不仅在数学上很重要，而且在数学、应用科学和材料工程的其他领域也有潜在的应用。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新技术。所谓的l无穷变分问题出现在许多不同的领域，如确定化疗中的最佳放射治疗，图像分析和恢复工程，以及在某些类型的随机游戏中获胜策略的设计。在机械工程中，Ericksen-Leslie系统是用来描述黏弹性流体动力学的最基本方程之一，包括向列液晶。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。该项目将出版专著和讲稿，积极培训高级研究生，并组织专门的会议或讲习班。

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.