Decay of linear waves on higher dimensional Schwarzschild black holes
Dec, 201093 pages
Published in:
- Anal.Part.Diff.Eq. 6 (2013) 3, 515-600
- Published: Jul 11, 2013
e-Print:
- 1012.5963 [gr-qc]
DOI:
- 10.2140/apde.2013.6.515 (publication)
View in:
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Abstract: (MSP)
We consider solutions to the linear wave equation on higher dimensionalSchwarzschild black hole spacetimes and prove robust nondegenerate energydecay estimates that are in principle required in a nonlinear stabilityproblem. More precisely, it is shown that for solutions to the wave equation□gϕ= 0 onthe domain of outer communications of the Schwarzschild spacetime manifold(ℳmn,g) (wheren≥ 3 is the spatialdimension, and m> 0is the mass of the black hole) the associated energy fluxE[ϕ](Στ) through a foliationof hypersurfaces Στ(terminating at future null infinity and to the future of the bifurcation sphere) decays,E[ϕ](Στ)≤CD∕τ2, whereC is a constantdepending on nand m, andD<∞ is a suitable higher-orderinitial energy on Σ0;moreover we improve the decay rate for the first-order energy toE[∂tϕ](ΣτR)≤CDδ∕τ4−2δ for anyδ> 0, whereΣτR denotes the hypersurfaceΣτ truncated at an arbitrarilylarge fixed radius R<∞ providedthe higher-order energy Dδon Σ0 is finite.We conclude our paper by interpolating between these two results to obtain the pointwiseestimate |ϕ|ΣτR≤CDδ′∕τ32−δ.In this work we follow the new physical-space approach to decay for the waveequation of Dafermos and Rodnianski (2010).Note:
- 93 pages, 7 figures
- decay
- wave equation
- Schwarzschild black hole
- spacetime,higher dimensions
- mathematical general relativity
- black hole: Schwarzschild
- energy: decay
- energy: flux
- stability: nonlinear
- black hole: higher-dimensional
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