Localized energy for wave equations with degenerate trapping

TL;DR

This paper investigates localized energy estimates for wave equations on a specific warped product manifold, revealing a sharp algebraic loss of regularity due to degenerate trapping, with implications for understanding wave behavior in complex geometries.

Contribution

It provides the first explicit example of an energy estimate with a sharp algebraic loss of regularity caused by degenerate trapping in a warped product manifold.

Findings

Established an initial sub-optimal energy estimate considering low frequency contributions.

Improved the estimate using energy functionals inspired by WKB analysis.

Proved the sharpness of the algebraic loss by constructing a saturating quasimode.

Abstract

Localized energy estimates have become a fundamental tool when studying wave equations in the presence of asymptotically at background geometry. Trapped rays necessitate a loss when compared to the estimate on Minkowski space. A loss of regularity is a common way to incorporate such. When trapping is sufficiently weak, a logarithmic loss of regularity suffices. Here, by studying a warped product manifold introduced by Christianson and Wunsch, we encounter the first explicit example of a situation where an estimate with an algebraic loss of regularity exists and this loss is sharp. Due to the global-in-time nature of the estimate for the wave equation, the situation is more complicated than for the Schr\"{o}dinger equation. An initial estimate with sub-optimal loss is first obtained, where extra care is required due to the low frequency contributions. An improved estimate is then established using energy functionals that are inspired by WKB analysis. Finally, it is shown that the loss cannot be improved by any power by saturating the estimate with a quasimode.