Strichartz estimates on Schwarzschild black hole backgrounds

TL;DR

This work develops a rigorous framework for global dispersion of waves on Schwarzschild black hole backgrounds by combining Morawetz-type energy methods with microlocal analysis to handle photon-sphere trapping, yielding local energy decay with a logarithmic loss and robust global Strichartz estimates. The authors introduce refined local energy spaces and photon-sphere microlocalization (via symbols like $a_{ps}$ and operators $A_{ps}$) to overcome trapping, and they construct a forward parametrix to propagate forcing terms. With these tools, they prove global-in-time Strichartz estimates and apply them to the energy-critical NLW, establishing small-data global well-posedness in the Schwarzschild exterior. The results contribute to the understanding of wave propagation in curved spacetimes and provide a solid foundation for nonlinear stability analyses near black hole geometries.

Abstract

We study dispersive properties for the wave equation in the Schwarzschild space-time. The first result we obtain is a local energy estimate. This is then used, following the spirit of earlier work of Metcalfe-Tataru, in order to establish global-in-time Strichartz estimates. A considerable part of the paper is devoted to a precise analysis of solutions near the trapping region, namely the photon sphere.

Figures (2)

• Figure 1: The Penrose diagram for the Kruskal extension of the Schwarzschild solution
• Figure 2: The Schwarzschild space partition represented on the Penrose diagram

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• proof : Proof of Lemma \ref{['ibp']}
• Theorem 2
• Proposition 3
• proof