LEAPS-MPS: Energy decay for the time dependent damped wave equation

Unwanted vibrations pose hazards to mechanical systems at a variety of scales. Examples include: earthquakes and high winds for buildings, engine and road vibration for vehicles, and self-induced vibrations for washing machines or centrifuges. Adding damping material attenuates these vibrations and protects such systems. The investigator studies the damped wave equation, which models these systems. Damped systems exhibit counter-intuitive phenomena, such as overdamping, where more damping actually causes slower energy decay. The mathematical model predicts and explains these phenomena, which are important to understand in order to properly design damping for a given application. This project provides research training opportunities for undergraduate and graduate students.

This project studies how energy decay rates for the damped wave equation are influenced by time-dependence, unboundedness, and regularity of the damping. One major objective is to fully generalize the classical theory for bounded autonomous damping to unbounded or time-dependent damping. In particular, for unbounded damping, the goal is to provide a dynamical hypothesis on the support of the damping that is equivalent to exponential energy decay. For time-dependent damping, the goal is to provide the weakest possible hypothesis that still guarantees energy decay at some rate. Techniques from microlocal analysis, semigroup theory, functional analysis, and spectral theory will be used.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2530465
Principal Investigator: Peter Kleinhenz
Funds Obligated: $225,697
State: IL