Mathematical Sciences: Mathematical Theory of Liquid Crystals and Free Boundaries

9401546林本奖项支持在非线性偏微分方程、变分微积分和几何测度理论等方面的数学研究。在第一部分中，将分析奇异点，包括广义赖芬贝格定理及其应用，金兹堡-朗道顶点的p调和映射方法，以及向列数学中缺陷的动态和变形的研究。此外，还将对双轴液晶中的缺陷进行研究。该项目的第二部分涉及液晶的演变。它可以用一个非线性系统来表示，这个非线性系统可以看作是谐波映射流与Navier-Stokes系统之间的非线性耦合。特别令人感兴趣的是确定经典解或弱解的整体存在性、唯一性和稳定性。通过探索弹性、优化设计和液晶液滴中出现的自由界面问题，研究该系统的部分规律性。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创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***