Mathematical Analysis of Nematic Liquid Crystals and L-infinity Variational Problems

这个项目要解决的问题不仅在数学上具有很大的挑战性，而且与流体力学和复杂流体、应用物理和材料科学以及控制工程等其他领域也有很强的联系和深刻的应用。对我们的模型的某些类型解的存在性和光滑性进行严格的分析可以预测奇点的形成，让研究人员能够洞察这些湍流现象，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。该项目中提出的问题也将作为培养研究生的工具。作为未来面向高级研究生和研究人员的研究专著的主要部分。本项目的技术方面是从三个部分研究分析问题：(I)向列相液晶的流体动力学流动，(Ii)液晶液滴和液晶中各向同性-向列相相变的变分问题，以及(Iii)L先生的无限变分问题。第一部分是Ericksen-Leslie系统，它模拟了向列相液晶的流体动力学，这是一个在基本流体速度场的不可压缩的Navier-Stokes方程和向列相液晶分子取向矢量场的调和映象的传输热流之间的强非线性耦合系统。目的是建立任意大初值情况下三维Leray-Hopf型弱解的存在性和部分正则性。第二部分利用单轴向列相液晶的变取向度模型，研究了液滴中可能的最佳构型的存在和分类，以及各向同性相和向列相之间尖锐界面的形成。第三部分是研究具有空间依赖性的哈密顿函数的阿隆森方程或L无限泛函的绝对极小化的唯一性，一般阿隆松方程粘性解的正则性，以及任意维无限调和函数的Liouville性质。这一奖项反映了美国国家科学基金会的法定使命，并已通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The problems to be solved in this project are not only very challenging mathematically but also have strong connections and profound applications to other fields such as fluid mechanics and complex fluids, applied physics and material sciences, and control engineering.  Rigorous analysis of both the existence and the smoothness of certain kinds solutions to our models can predict the formation of singularities, allow researchers to gain insights into the turbulence phenomena, and justify both computational and experimental studies made by applied scientists and engineers. The proposed problems in the project will also serve as tools to train graduate students, and constitute as main parts of future research monographs aimed at both advanced graduate students and researchers.The technical side of this project is to study analytic issues in the three parts: (i) the hydrodynamic flow of nematic liquid crystals, (ii) variational problems on both liquid crystal droplets and the isotropic-nematic phase transitions in liquid crystals, and (iii) the L-infinity variational problems. The first part  deals with the Ericksen-Leslie system modeling the hydrodynamics 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 for arbitrary large initial data. The second part investigates both existence and classification of possible optimal configuration in the liquid crystal droplets and the formation of sharp interface between the isotropic and nematic phases by employing the Ericksen's model of variable degree of orientations for uniaxial nematic liquid crystals. The third part is to study the uniqueness of Aronsson's equations or absolute minimizers of L-infinity functionals that involve Hamiltonian functions with spatial dependence, the regularity of viscosity solutions to general Aronsson's equations, and the Liouville property of infinity harmonic functions in any dimension. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.