Localized energy for wave equations with degenerate trapping
Robert Booth, Hans Christianson, Jason Metcalfe, Jacob Perry
TL;DR
This paper analyzes the wave equation on a stationary, asymptotically Minkowski warped-product manifold with degenerate trapping at x=0, focusing on how trapping deteriorates local energy estimates. It proves a lossy global-in-time local energy bound away from the trapped set, first for all frequencies, then sharpens it using WKB-inspired energy functionals to obtain a precise algebraic loss of regularity $\frac{m-1}{2(m+1)}$ for $m\ge2$. The authors then establish sharpness by constructing quasimodes that saturate the bound, showing that no power improvement of the loss is possible. The results extend the understanding of trapping-induced losses beyond nondegenerate hyperbolic trapping and provide explicit, sharp estimates for degenerate trapping in a concrete geometric model.
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 Schrödinger 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.