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A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr

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Abstract

In this article, we prove a local energy estimate for the linear wave equation on metrics with slow decay to a Kerr metric with small angular momentum. As an application, we study the quasilinear wave equation \(\Box _{g(u, t, x)} u = 0\) where the metric g(u, t, x) is close (and asymptotically equal) to a Kerr metric with small angular momentum g(0, t, x). Under suitable assumptions on the metric coefficients, and assuming that the initial data for u is small enough, we prove global existence and decay of the solution u.

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Acknowledgements

H.L. was supported in part by NSF grant DMS-1500925 and Simons Collaboration Grant 638955. M.T. was supported in part by the NSF Ggrant DMS–1636435 and Simons collaboration Grant 586051. We would also like to thank the Mittag Leffler Institute for their hospitality during the Fall 2019 program in Geometry and Relativity.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Hans Lindblad

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA

    Mihai Tohaneanu

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  1. Hans Lindblad
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  2. Mihai Tohaneanu
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Correspondence to Mihai Tohaneanu.

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Communicated by Mihalis Dafermos.

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Lindblad, H., Tohaneanu, M. A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr. Ann. Henri Poincaré 21, 3659–3726 (2020). https://doi.org/10.1007/s00023-020-00950-0

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