A note on energy currents and decay for the wave equation on a Schwarzschild background

Abstract

In recent work, we have proven uniform decay bounds for solutions of the wave equation ${\square }_g\varphi =0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|\varphi |\le C{v}_{+}^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_{+}=\max \{v,1\}$, and $C$ is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition.

10 pages

References in corpus (2)

• The wave equation on Schwarzschild-de Sitter spacetimes
• Stability of two soliton collision for nonintegrable gKdV equations

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