Analysis of coupled PDE systems modelling micro-electro-mechanical systems

Abstract

This thesis studies some mathematical models for a Micro-Electro-Mechanical System (MEMS) capacitor, consisting of a fixed plate and a flexible plate separated by a fluid. It investigates the wellposedness of solutions to the resulting quasilinear coupled systems, as well as the finite-time blow-up (quenching) of solutions. The models considered include a parabolic-dispersive system modelling the fluid flow under an elastic plate, a parabolic-hyperbolic system for a thin membrane, as well as an elliptic-dispersive system for quasistatic fluid flow under an elastic plate. Short-time existence, uniqueness and smoothness are obtained by combining wellposedness results for a single equation with an abstract semigroup approach for the system. Quenching is shown to occur, if the solution ceases to exist after a finite time. The thesis concludes with a study of self-similar quenching solutions and their stability for a simple hyperbolic membrane model for a MEMS capacitor.