Dynamics, singularities and asymptotics of higher order PDEs

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

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.