Hydrodynamics of Liquid Crystals and Extremum Problems for Eigenvalues

这一建议是对一类复杂的流体进行深入的理论研究，包括在显示设备中广泛应用的液晶和在现代医学和生命科学中极其重要的带电生物流体。令人特别感兴趣的是它们有趣而复杂的动力学现象以及图案和奇点的形成。理解这些复杂流体所涉及的建模、分析和模拟在理论上是非常重要和具有挑战性的。它需要新的思想和方法，因此它将促进我们的知识，这些知识也适用于许多其他科学问题。更具体地说，该建议包括两个部分。第一部分是研究描述液晶流体动力学的偏微分方程组和相关的复杂流体模型。这部分研究的重点将是研究Ericksen-Leslie理论中液晶流的适当弱解的整体存在性；不可压缩粘弹性流体的Oldroyd B模型解的整体存在性；无粘不可压缩磁流体力学系统以及一般情况下流体与其他几何对象的耦合非线性动力学。第二部分研究了一大类椭圆型特征值的极值问题。这样的问题也出现在优化设计、图案形成以及材料科学和凝聚态物理的其他应用中。PI计划研究的偏微分方程(PDE)涉及描述输运、相场、映射或几何对象的演化的方程与Navier-Stokes方程之间的非线性耦合。它们既可以是抛物线的，也可以是双曲型的，并且具有奇点或多个尺度。特征值和特征函数的变分问题是经典的和基本的。这些都是令人着迷和具有挑战性的问题，需要新的想法和方法，这可能会导致偏微分方程和变分分析的研究或程序的新方向。拟议的研究活动是国际和平研究所本科生、研究生和博士后培训计划的重要组成部分。

This proposal is for a deep theoretical study on a class of complex fluids including both the liquid crystals which have been widely used in display devices and charged biological fluids which are extremely important in modern medicine and life sciences. Of particular interests are their intriguing and complex dynamical phenomena as well as formations of patterns and singularities. The modeling, analysis and simulations involved in understanding these complex fluids are theoretically very important and challenging. It needs new ideas and methods, and hence it would advance our knowledge which would be applicable to many other scientific problems as well.More specifically, the proposal consists of two parts. The first part is to study partial differential equations that describe the hydrodynamics of liquid crystals and related complex fluid models. The main focus of this part of the research will be to study global existence of suitable weak solutions of liquid crystal flows in the Ericksen-Leslie theory; the global existence of solutions of Oldroyd B-model of incompressible visco-elastic fluids; the inviscid incompressible magneto-hydrodynamic system and, in general, coupled nonlinear dynamics of fluids with other geometric objects. The second part is to study a large class of extremum problems of elliptic eigenvalues. Such problems also arise in optimal designs, pattern formations and other applications in material sciences and condense matter physics. The partial differential equations (PDE) that the PI plans to study involve nonlinear couplings between equations that describe transport, phase-field, mapping or geometric object's evolutions and that of Navier Stokes equations. They may be of both parabolic and hyperbolic nature and possess singularities or multiple scales. The variational problems for eigenvalues and eigenfunctions that linked with underlying domains are classical and fundamental. These are fascinating and challenging problems that require new ideas and methods, which could lead to new directions of research or programs in the analysis of PDE and calculus of variations. The proposed research activity is an important and integral part of the PI's training program of under-graduate, graduate, and post-doctoral students.