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

2007/09/30 by Dafermos, Mihalis, Rodnianski, Igor
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc)

paper · doi:10.48550/arxiv.0710.0171

Abstract

In recent work, we have proven uniform decay bounds for solutions of the wave equation \Box_gϕ=0 on a Schwarzschild exterior, in particular, the uniform pointwise estimate |ϕ|≤ Cv₊^-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.

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