Dynamics, singularities and asymptotics of higher order PDEs

该奖项支持首席研究员（PI）与微机电系统（MEMS）应用科学领域相关的研究项目，微机电系统是微型传感器和执行器的重要技术（应用于汽车工业、消费电子、医疗和健康技术以及许多其他领域）。在微观尺度上操纵和预测动态过程行为的能力是现代纳米技术设计的必要条件。这可能具有挑战性，因为宏观尺度上的过程直觉不一定能转化为非常小的空间尺度。数学建模是弥合这一差距的重要工具；然而，相关方程是非常复杂的。在这个项目中，PI和他的学生将开发分析和计算工具来研究这些复杂的数学模型。目标是为实践者建立一套数学方法，可以告知这些重要技术的工程。这些方法对许多数学研究人员研究各种相关领域的问题也很有用，如流体力学、模式形成和数学生物学。纳米技术的数学建模需要多种物理理论的耦合，特别是弹性和静电理论。这就产生了以非线性和高阶（大于两阶）偏微分方程耦合系统为特征的模型。这些方程的复杂性质意味着许多现有的工具，特别是那些基于最大值和比较原则的工具不适用。如果没有这样的约束原则，这些系统可能会表现出丰富和意想不到的模式行为。为了研究这些行为及其对微器件工程的影响，PI将开发新的奇异摄动方法和计算技术。这个项目的重点是研究这些系统的奇异解，主要是有限时间奇点和尖锐界面的形式。PI将侧重于获得关于这些奇异解结构的位置、多样性和局部动力学的正式和严格的结果。这个项目的分析臂将与PI和他的学生开发的高度精确和自适应的数值模拟相结合。

This award supports the research program of the Principal Investigator (PI) related to the applied science field of Micro-Electro-Mechanical Systems (MEMS), an important technology for microscopic sensors and actuators (with applications in the automotive industry, consumer electronics, medical and health technologies, and a number of other areas). The ability to manipulate and predict the behavior of dynamic processes on microscopic scales is a necessity for the design of modern nanotechnology. This can be challenging because intuition of processes on the macro scale does not necessarily translate to very small spatial scales. Mathematical modeling is an essential tool for bridging this gap; however, the relevant equations are highly complex. In this project, the PI and his students will develop analytical and computational tools for studying these complex mathematical models. The goal is to establish a set of mathematical methodologies for practitioners that can inform the engineering of these important technologies. These methodologies will also be useful to many mathematical researchers studying problems from a variety of related fields, such as Fluid Mechanics, Pattern Formation, and Mathematical Biology.Mathematical modeling of nanotechnology requires a coupling of multiple physical theories, in particular those of elasticity and electrostatics. This gives rise to models that feature coupled systems of non-linear and higher order (greater than two) partial differential equations. The complex nature of these equations means that many existing tools, in particular those based on maximum and comparison principles, are not applicable. Without such constraining principles, these systems can exhibit rich and unexpected patterning behaviors. To study these behaviors and their implications for the engineering of micro-devices, the PI will develop novel singular perturbation methods and computational techniques. The particular focus of this project is studying singular solutions of these systems, mainly in the form of finite-time singularities and sharp interfaces. The PI will focus on obtaining both formal and rigorous results regarding the location, multiplicity, and local dynamics of these singular solution structures. The analytical arm of this project will be coupled to highly accurate and adaptive numerical simulations developed by the PI and his students.