Mathematical Analysis of Topics from Materials Science

这个项目致力于对两类重要的当代材料进行数学研究，这两类材料用于各种技术：液晶材料和嵌段共聚物。虽然液晶材料很常见，例如用于液晶显示器，但嵌段共聚材料不太常见。它们具有非凡的自组装成有序结构的能力。这个项目的目标是研究这些材料的几个数学模型，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性培训提供机会。本项目将开展一些非线性偏微分方程组(PDE)模拟液晶和嵌段共聚体系的分析和数值研究。主要的研究人员将在Q张量序参数的框架内讨论(1)静态和动态组态的几个解析问题，包括胶体粒子周围的缺陷组态，流体动力学系统中的物理条件及其数学作用，所有非共转流体动力学系统中初始物理性的损失，由奇异势产生的梯度流动动力学，(2)由分子理论导出的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)模拟嵌段共聚物在时间演化过程中自组装的梯度流动动力学全局适定性和稳定、高效的数值格式。这项研究涉及的主要内容和技术包括椭圆和抛物型偏微分方程的方法、稳定性分析、希尔伯特和米制空间中的梯度流理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.