Mathematical Analysis of Topics from Materials Science

该项目致力于对两种重要的当代材料进行数学研究，这些材料用于各种技术：液晶材料和嵌段共聚物。虽然液晶材料是相当普遍和使用，例如，用于液晶显示器，嵌段共聚物材料是不太熟悉的。它们具有自组装成有序结构的非凡能力。该项目的目标是研究这些材料的几种数学模型，以更好地了解它们的结构、缺陷和行为。该项目将为本科生和研究生的研究训练提供机会。本项目将发展一些非线性偏微分方程（PDE）模拟液晶和嵌段共聚物系统的分析和数值研究。主要研究者将解决(1)在q张量序参数框架下的静态和动态配置的几个分析问题，包括胶体粒子周围的缺陷配置，流体动力系统中的物理条件及其数学作用，所有非共旋流体动力系统的初始物理性损失，由奇异势产生的梯度流动动力学；(2)在分子理论中推导出用于模拟刚性棒状聚合物分子的Doi-Onsager方程经典解的渐近行为和趋于平衡的趋势；(3)模拟嵌段共聚物自组装过程的梯度流动动力学的全局适定性和稳定高效的数值格式。研究涉及的主要成分和技术包括椭圆和抛物线偏微分方程的方法、稳定性分析、希尔伯特梯度流动理论和度量空间。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is devoted to the mathematical study of two important classes of contemporary materials that are used in a variety of technologies: liquid crystal materials and block copolymers. While liquid crystal materials are quite common and are used, for example, for liquid crystal displays, the block copolymer materials are less familiar. They possess a remarkable capacity for self-assembly into ordered structures. The goal of this project is to investigate several mathematical models of these materials to better understand their structure, defects, and behavior. The project will provide opportunities for the research training of undergraduate and graduate students. This project will develop analytical and numerical study of some nonlinear partial differential equations (PDE) modeling liquid crystals and block copolymer systems. The principal investigator will address (1) several analytic problems in both static and dynamic configurations in the framework of Q-tensor order parameters, that include defect configurations around a colloid particle, a physical condition and its mathematical role in the hydrodynamic system, loss of initial physicality in all non-co-rotational hydrodynamic system, a gradient flow dynamics generated by a singular potential; (2) the asymptotic behavior and trend to equilibrium of classical solutions of the Doi-Onsager equation that is derived in molecular theory to model rigid rod-like polymer molecules; (3) the global well-posedness and stable, efficient numeric schemes of a gradient flow dynamics modeling the self-assembly of block copolymers during time evolution. The main ingredients and techniques involved in the study include methods from elliptic and parabolic PDE, stability analysis, gradient flow theory in Hilbert and metric spaces.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.