Research

My research is on partial differential equations, using the tools of microlocal analysis. Most of my work has focused on the damped wave equation, which describes how vibrating objects, like a guitar string, drum head or building, come to rest. I am particularly interested in understanding how regularity properties of the damping function, like how many derivatives it has or what powers of it are integrable, determine the how quickly the energy decays.  I am also interested in understanding  how classical decay results change when the damping is allowed to depend on time, have negative values or depend on the direction waves are moving (pseudodifferential or anisotropic damping).

Papers

Exponential damping on the torus (in preparation)

Optimal backward uniqueness and polynomial stability of second order equations with unbounded damping,, with Ruoyu Wang

Non-exponential uniform stabilization for the time dependent damped wave equation (2023)

Polynomial decay rates for the damped wave equation with singular damping, with Ruoyu Wang, submitted (2022).

Energy decay for the time dependent damped wave equation submitted, (2022)

Decay rates for the damped wave equation with finite regularity damping in Math Research Letters Vol 29, no.  4 (2022).

Sharp Exponential Decay Rates for Anisotropically Damped Waves, in Annales Henri Poincaré with Blake Keeler (2022).

Decay Rates for the Damped Wave Equation on the Torus (2020). This is my thesis, it is a synthesis of my previous two papers and a new result.  It also includes expository proofs for exponential decay in the presence of the Geometric Control Condition and logarithmic decay for damping supported on an open set. If you know or are learning microlocal analysis this is a good introduction to the DWE on compact manifolds.

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping in Proceedings of the AMS 148, with Kiril Datchev (2020).

Stabilization rates for the damped wave equation with Hölder-regular damping  Communications in Mathematical Physics 369 (2019).

On a variational problem on hyperbolic space with Chenjie Fan (2016).