From resolvent estimates to damped waves

TL;DR

This paper links resolvent estimates to decay rates for damped wave equations on compact manifolds, showing how complex absorbing potentials can be used to derive decay estimates from resolvent bounds.

Contribution

It establishes a method to derive decay estimates for damped waves using resolvent bounds from complex absorbing potentials, connecting geometric control and spectral analysis.

Findings

Polynomial resolvent bounds imply decay estimates for damped waves

Complex absorbing potentials can be used to obtain resolvent estimates

Gluing techniques from non-compact models are applicable

Abstract

In this paper we show how to obtain decay estimates for the damped wave equation on a compact manifold without geometric control via knowledge of the dynamics near the un-damped set. We show that if replacing the damping term with a higher-order \emph{complex absorbing potential} gives an operator enjoying polynomial resolvent bounds on the real axis, then the "resolvent" associated to our damped problem enjoys bounds of the same order. It is known that the necessary estimates with complex absorbing potential can also be obtained via gluing from estimates for corresponding non-compact models.