Integrated Local Energy Decay for the Damped Wave Equation on Stationary Space-Times

TL;DR

This paper proves integrated local energy decay for the damped wave equation on stationary, asymptotically flat space-times, extending previous Euclidean results to Lorentzian geometries and handling trapped trajectories effectively.

Contribution

It generalizes local energy decay results from Euclidean to Lorentzian space-times, utilizing geometric control to manage trapping and applying frequency-specific estimates.

Findings

Established integrated local energy decay on stationary space-times.

Handled trapped trajectories without loss using geometric control.

Extended Euclidean results to Lorentzian geometries.

Abstract

We prove integrated local energy decay for the damped wave equation on stationary, asymptotically flat space-times in (1 + 3) dimensions. Local energy decay constitutes a powerful tool in the study of dispersive partial differential equations on such geometric backgrounds. By utilizing the geometric control condition to handle trapped trajectories, we are able to recover high frequency estimates without any loss. We may then apply known estimates from the work of Metcalfe, Sterbenz, and Tataru in the medium and low frequency regimes in order to establish local energy decay. This generalizes the integrated version of results established by Bouclet and Royer from the setting of asymptotically Euclidean manifolds to the full Lorentzian case.