Mathematical Sciences: Mathematical Theory of Liquid Crystals and Free Boundaries

9401546该奖项支持对非线性偏微分方程组、变分微积分和几何测度论问题的数学研究。在第一部分中，将对奇点进行分析，包括Reifenberg定理的推广版本及其应用，到Ginzburg-Landau顶点的p-调和映射方法，以及向列学中缺陷的动力学和形变研究。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进化。它被表示为一个非线性系统，它可以看作是调和映射流和N-S系统之间的非线性耦合。尤其令人感兴趣的是经典解或弱解的整体存在性、唯一性和稳定性的确定。还将通过探索弹性、优化设计和液晶液滴等方面的自由界面问题来研究这类系统的部分正则性。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

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. ***