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 [29], 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.
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Acknowledgement
The authors are grateful to M. Dafermos and I. Rodnianski for pointing out their novel way of taking advantage of the red shift effect in [14], and to N. Burq and M. Zworski for useful conversations concerning the analysis near trapped null geodesics.
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Communicated by S. Zelditch
The authors were supported in part by the NSF grants DMS0354539 and DMS0301122.
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Marzuola, J., Metcalfe, J., Tataru, D. et al. Strichartz Estimates on Schwarzschild Black Hole Backgrounds. Commun. Math. Phys. 293, 37–83 (2010). https://doi.org/10.1007/s00220-009-0940-z
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DOI: https://doi.org/10.1007/s00220-009-0940-z