Dynamics, singularities and asymptotics of higher order PDEs

该奖项支持主要研究者（PI）与微机电系统（MEMS）应用科学领域相关的研究计划，MEMS是微观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其他一些领域）。 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。 这可能是具有挑战性的，因为宏观尺度上的过程的直觉不一定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，相关方程非常复杂。 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。 我们的目标是为从业者建立一套数学方法，可以为这些重要技术的工程提供信息。 这些方法也将是有用的许多数学研究人员研究问题，从各种相关领域，如流体力学，图案形成，和数学Biology.Mathematical建模的纳米技术需要耦合的多个物理理论，特别是那些弹性和静电。 这就产生了具有非线性和高阶（大于两个）偏微分方程耦合系统的模型。 这些方程的复杂性意味着许多现有的工具，特别是那些基于最大值和比较原理的工具，不适用。 如果没有这样的约束原则，这些系统可以表现出丰富的和意想不到的模式化行为。 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。 该项目的特别重点是研究这些系统的奇异解，主要是以有限时间奇点和尖锐界面的形式。 PI将专注于获得关于这些奇异解结构的位置，多重性和局部动态的正式和严格的结果。 该项目的分析臂将与PI及其学生开发的高精度和自适应数值模拟相结合。

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.