Mathematical Analysis of Topics from Materials Science

材料科学主题的数学分析

• 批准号: 2007157
• 负责人: Xiang Xu
• 金额: $ 14.1万
• 依托单位: Old Dominion University Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2007157&HistoricalAwards=false
• 关键词: Mathematical; Analysis; Topics; Materials; Science

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.

该项目致力于对用于各种技术的两类重要的当代材料进行数学研究：液晶材料和嵌段共聚物。虽然液晶材料非常常见并且被用于例如液晶显示器，但嵌段共聚物材料却不太熟悉。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培训提供机会。该项目将对一些非线性偏微分方程（PDE）建模液晶和嵌段共聚物系统进行分析和数值研究。首席研究员将解决（1）Q张量阶参数框架中静态和动态配置的几个分析问题，包括胶体颗粒周围的缺陷配置、物理条件及其在流体动力系统中的数学作用、所有非同向旋转流体动力系统中初始物理性的损失、奇异势产生的梯度流动动力学； (2) Doi-Onsager 方程的经典解的渐近行为和平衡趋势，该方程是在分子理论中导出的，用于模拟刚性棒状聚合物分子； (3) 梯度流动动力学的全局适定性和稳定、高效的数值方案，模拟时间演化过程中嵌段共聚物的自组装。该研究涉及的主要成分和技术包括椭圆偏微分方程和抛物线偏微分方程、稳定性分析、希尔伯特梯度流理论和度量空间。该奖项反映了 NSF 的法定使命，并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。

项目成果

Regularity of a Gradient Flow Generated by the Anisotropic Landau--de Gennes Energy with a Singular Potential

奇异势各向异性Landau-de Gennes能量产生的梯度流的规律性

• DOI: 10.1137/20m1386499
• 发表时间: 2021
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Liu, Yuning;Lu, Xin Yang;Xu, Xiang
• 通讯作者: Xu, Xiang

Blowup Rate Estimates of a Singular Potential and Its Gradient in the Landau-de Gennes Theory

• DOI: 10.1007/s00332-021-09761-x
• 发表时间: 2021-11
• 期刊: Journal of Nonlinear Science
• 影响因子: 3
• 作者: X. Lu;Xiang Xu;Wujun Zhang