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

这个项目的目的是促进主要研究人员对两个领域的分析问题的研究：(I)L无穷大变分问题和相关的Aronsson方程，(Ii)向列相液晶材料的流体动力学流动。L-无限变分问题研究作为被积函数上界的适当代价泛函的最小化问题。这一地区最近变得非常活跃。本项目第一部分的重点是：确定保证拟凸哈密顿泛函的绝对极小值与高度退化的Aronsson方程的粘性解之间的等价关系的最一般条件；探索具有空间相关性的一般Aronsson方程的唯一性问题；在Gamma收敛的框架下研究L-无穷大变分问题；以及在Dirichlet能量约束下研究L-无穷大变分问题。本项目的第二部分涉及Ericksen-Leslie系统模拟向列相液晶流体动力学流动。这是一个强耦合系统，它将基本流体速度场的不可压缩Navier-Stokes方程和向列相液晶材料指向矢场的调和映射的传输热流联系在一起。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具有重要意义，而且在数学、应用科学和材料工程等领域具有潜在的应用价值。该项目所涉及的非线性偏微分方程组或者是高度退化的椭圆问题，或者是具有超临界非线性的方程，它们的解肯定会提供新的思想和技术，在各种背景下都是有用的。所谓的L无限变分问题出现在许多不同的领域，例如在化疗、图像分析和恢复工程中确定最佳放射治疗，以及在某些类型的随机博弈中设计获胜策略。在机械工程中，Ericksen-Leslie系统是描述包括向列相液晶在内的粘弹性流体动力学的最基本方程之一。对这样一个系统的各种解的存在性和规律性的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合理的。该项目将出版专著和讲稿，积极培训高级研究生，并组织具体的会议或讲习班。

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.