Morawetz estimates for the wave equation at low frequency

TL;DR

This paper establishes Morawetz estimates for low-frequency solutions to the wave equation on scattering manifolds, demonstrating decay properties that are largely independent of the manifold's compact geometry and introducing localized estimates within the light cone.

Contribution

It proves the persistence of Morawetz estimates at low frequencies regardless of the manifold's compact geometry and introduces a new localized Morawetz estimate inside the light cone.

Findings

Morawetz estimates hold for low-frequency solutions on scattering manifolds.

A new localized Morawetz estimate inside the light cone is established.

Achieves a 1/2 power decay in time for solutions in a specific spacetime region.

Abstract

We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a 1/2 power of $t$ decay relative to what would be dictated by energy estimates, in a small part of spacetime.