Mathematical Sciences: Mathematical Theory of Liquid Crystals and Free Boundaries

数学科学：液晶和自由边界的数学理论

• 批准号: 9401546
• 负责人: Fang-Hua Lin
• 金额: $ 11.38万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-06-15 至 1997-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401546&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Theory; Liquid; Crystals

9401546 Lin This award supports mathematical research on problems in nonlinear partial differential equations, variational calculus and geometric measure theory. In the first part, work will be done analyzing singularities which contain the generalized version of Reifenberg's theorem and its applications, the p- harmonic mapping approach to Ginzburg-Landau vertices and the study of dynamic and deformation of defects in nematics. In addition, an investigation defects in biaxial liquid crystals will be initiated. The second part of the project concerns the evolution of liquid crystals. It is represented by a nonlinear system which can be viewed as a nonlinear coupling between flow of harmonic maps and a Navier-Stokes system. Of particular interest is the determination of global existence, uniqueness and stability of classical or weak solutions. Work will also be done in studying the partial regularity of such systems by exploring free interface problems arising in elasticity, optimal design and liquid crystal droplets. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***

小行星9401546 该奖项支持非线性偏微分方程，变分学和几何测量理论问题的数学研究。 在第一部分中，我们将分析奇异性，其中包括Reifenberg定理的推广版本及其应用，Ginzburg-Landau顶点的p-调和映射方法以及向列型中缺陷的动力学和变形的研究。 此外，双轴向列晶缺陷之研究也将开始。 该项目的第二部分涉及液晶的演变。 它是由一个非线性系统，可以看作是一个非线性耦合的调和映射流和Navier-Stokes系统。 特别感兴趣的是确定的整体存在性，唯一性和稳定性的经典或弱解。 通过探索弹性、最佳设计和液晶液滴中产生的自由界面问题，还将研究此类系统的部分规律性。 偏微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。 ***