A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole

Luk, Jonathan

doi:10.2140/apde.2012.5.553

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A vector field method approach to improved decay for solutions to the wave equation on a slowly rotating Kerr black hole

Jonathan Luk

Vol. 5 (2012), No. 3, 553–625

DOI: 10.2140/apde.2012.5.553

Abstract

We prove that sufficiently regular solutions to the wave equation $□_{g_{K}} Φ = 0$ on the exterior of a sufficiently slowly rotating Kerr black hole obey the estimates $| Φ | \leq C {(t^{*})}^{- 3 ∕ 2 + η}$ on a compact region of $r$ . This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski and Schwarzschild spacetime. This result improves the known robust decay rates that are proved using the vector field method in the region of finite $r$ and along the event horizon.

Keywords

Kerr spacetime, wave equation, decay estimates

Mathematical Subject Classification 2010

Primary: 83C99, 35R01

Milestones

Received: 5 October 2010

Revised: 25 March 2011

Accepted: 21 April 2011

Published: 15 October 2012

Authors

Jonathan Luk
Department of Mathematics Princeteon University Fine Hall Washington Road Princeton, NJ 08544 United States

Vol. 5, No. 3, 2012