Partial differential equations arising from physical and biological sciences: Singularity and Nonlinearity

本计画致力于对物理与生物科学中的非线性偏微分方程（PDE）进行数学研究。 首席研究员计划解决一个单一的激活剂-抑制剂系统模拟生物学中的再生现象。该项目旨在了解模式形成的机制，因此将分析解决方案的长期行为，稳定性和分叉。首席研究员还将研究各种自由边界问题，并将其应用于薄膜动力学，液晶配置和晶体凝固过程。这里的重点是奇异集的结构和动力学以及所涉及的自由边界的存在性、唯一性和正则性。由于问题本身的奇异性和非线性，需要用到非线性椭圆和抛物偏微分方程理论、调和分析、变分法、几何测度理论和几何分析等分析工具，偏微分方程已被广泛应用于物理、生物、化学和金融等领域。当方程涉及奇点时，许多数学上具有挑战性的问题就会出现。奇点的例子包括薄膜的干点，液体/固体界面的尖角，以及无界的物理量。这个项目将发展新的想法和方法来研究相关的非线性偏微分方程中的奇异性，从而对潜在的物理和生物过程产生新的理解。 该项目还将对人力资源开发产生影响：它将为研究生课程提供教材，并为本科生和研究生提供高级培训，从而使他们能够参与该项目的各个方面。

This project is devoted to a mathematical study of some nonlinear partial differential equations (PDE) arising from physical and biological sciences. The principal investigator plans to address a singular activator-inhibitor system modeling the regeneration phenomena in biology. The project seeks to understand the mechanism of pattern formation, so the long-time behavior, stability, and bifurcation of the solutions will be analyzed. The principal investigator will also study various free boundary problems with applications to thin film dynamics, liquid crystal configurations, and crystal solidification processes. Here the focus is on the structure and dynamics of the singular set and the existence, uniqueness, and regularity of the free boundaries involved. Due to the inherent singularity and nonlinearity of the problems, it is necessary to use analytical tools from nonlinear elliptic and parabolic PDE theory, harmonic analysis, the calculus of variation, geometric measure theory, and geometric analysis.Partial differential equations have been broadly used to model a wide variety of phenomena in physics, biology, chemistry, and finance. Many mathematically challenging problems arise when the equations involve singularities. Examples of singularities include dry spots of thin films, sharp corners in liquid/solid interfaces, and unbounded physical quantities. This project will develop novel ideas and methods to study singularities in the relevant nonlinear partial differential equations and hence yield new understanding of the underlying physical and biological processes. The project will also have an impact on the development of human resources: it will supply teaching materials in graduate courses and provide advanced training for undergraduate and graduate students, thereby allowing them to participate in various aspects of the project.