Kerr 扰动中的 Teukolsky 方程
Teukolsky equations in perturbations of Kerr
AI总结：
本文作为 Kerr 稳定性猜想证明的配套论文，给出了 Kerr 扰动中 Teukolsky 波-输运系统的推导及其能量-Morawetz 估计的完整证明。
AI中文摘要：
Kerr 稳定性猜想已在慢旋转情形（即 $|a|\ll m$）下被证明，这一证明见于 Sergiu Klainerman 与作者的一系列工作 \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr}、Elena Giorgi、Sergiu Klainerman 与作者的 \cite{GKS22}，以及 Dawei Shen 的 \cite{Shen}；作者在 \cite{Sze} 中将其推广至完整的次极端范围 $|a|<m$，从而完成了 Kerr 稳定性猜想的证明。\cite{Sze} 中的证明关键依赖于 Siyuan Ma 与作者的两篇配套论文 \cite{MaSz24} \cite{MaSz26}，其中我们分别证明了在 $|a|<m$ 的 Kerr 扰动上标量波动方程与 Teukolsky 方程的能量-Morawetz 估计。此外，\cite{Sze} 中关于 Kerr 扰动中 Teukolsky 波-输运系统的推导以及 \cite{Sze} 设定下 Teukolsky 方程的能量-Morawetz 估计这两个结果，在文中仅陈述而未给出证明。本文作为 \cite{Sze} 的配套论文，旨在提供这两个结果的证明。
英文摘要：
The Kerr stability conjecture has been proved in the slowly rotating case, i.e., $|a|\ll m$, in the sequence of works \cite{KS-GCM1} \cite{KS-GCM2} \cite{KS:Kerr} by Sergiu Klainerman and the author, \cite{GKS22} by Elena Giorgi, Sergiu Klainerman and the author, and \cite{Shen} by Dawei Shen, and extended in \cite{Sze}, by the author, to the full subextremal range $|a|<m$, thereby completing the proof of the Kerr stability conjecture. The proof in \cite{Sze} crucially relies on the two companion papers \cite{MaSz24} \cite{MaSz26} by Siyuan Ma and the author, in which we prove energy-Morawetz estimates respectively for the scalar wave equation and for Teukolsky equations on perturbations of Kerr with $|a|<m$. In addition, two results in \cite{Sze}, concerning the derivation of the Teukolsky wave-transport system in perturbations of Kerr and energy-Morawetz estimates for Teukolsky in the setting of \cite{Sze}, are stated without proofs. The goal of the present paper companion paper to \cite{Sze} is to provide the proof of these two results.
