A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr
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Annales Henri Poincar´ e
A Local Energy Estimate for Wave Equations on Metrics Asymptotically Close to Kerr Hans Lindblad and Mihai Tohaneanu 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 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.
Contents 1.
Introduction 1.1. Statement of the Results and History 1.2. The Heuristics 1.3. The Proof 2. Local Energy Estimates on Kerr Backgrounds 2.1. The Setup and Statement 2.2. The Schwarzschild Case 2.3. The Kerr Case 2.4. Vanishing of Symbols on the Trapped Set 2.5. The Operator Quadratic Form 3. Norm Estimates for the Pseudodifferential Operators and Commutators 3.1. Estimates for Operators with Low Regularity Symbols 3.2. Estimates for the Kerr Operators 3.3. Commutator Estimates 4. Local Energy Estimates Close to the Trapped Set 4.1. Conditions on the Metric
H. Lindblad, M. Tohaneanu
Ann. Henri Poincar´e
4.2. Additional Control of Dx −1 Dt 4.3. Proof of the Main Theorem 5. Quasilinear Wave Equations Close to Kerr 6. Price’s Law for Perturbations of Kerr Acknowledgements References
1. Introduction In this paper, we consider energy estimates for the wave operator g on the background of a metric g that is close to the Kerr black hole metric gK with small angular momentum a expressed in the modified Boyer–Lindquist coordinates that are smooth over the event horizon, see Sect. 2.1.1. We show that if the metric decays slowly in time toward Kerr then we have local energy decay estimates. We also use our estimate to prove global existence of solutions to quasilinear wave equations close to Kerr, as well as pointwise decay for solutions to the linear problem. The paper is structured as follows. In Sect. 1, we discuss the estimate and present some heuristics of why it should hold. Section 2 contains a detailed proof of the estimate in the case of the Kerr metric. In Sect. 3, we prove some estimates for pseudodifferential operators with limited regularity. Section 4 contains our main linear estimate for perturbations of the Kerr metric. Section 5 deals with how to apply the estimate to obtain global well-posedness for the quasilinear problem following [41]. Section 6 applies the estimate to obtain pointwise decay for solutions to the linear problem, following [46]. 1.1. Statement of the Results and History 1.1.1. Assumptions on the Metric. We consider metrics that are perturbations of a Kerr metric (gK , M, a) with mass M and angular momentum per unit mass 0 < a M . In Boyer–Lindquist coordinates, √ the Kerr metric has a singularity at r = 0, the event horizon r = M + M 2 − a2 and the Cauchy horizon + √ r− = M − M 2 − a2 . By a suitable change of coordina
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