Boundedness and Morawetz estimates on subextremal Kerr de Sitter
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arXiv
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| Format: | Preprint |
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2025
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| author | Mavrogiannis, Georgios |
| author_facet | Mavrogiannis, Georgios |
| contents | We study the Klein--Gordon equation $\Box_{g_{a,M,l}}ψ-μ^2_{\textit{KG}}ψ=0$ on subextremal Kerr--de Sitter black hole backgrounds with parameters $(a,M,l)$, where $l^2=\frac{3}Λ$. We prove boundedness and Morawetz estimates assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results in particular apply in the very slowly rotating case $|a|\ll M,l$, and in the case where the solution~$ψ$ is axisymmetric. This generalizes the work of Dafermos--Rodnianski \cite{DR3} on Schwarzschild--de~Sitter.
The boundedness and Morawetz results of the present paper will be used in our companion \cite{mavrogiannis2} to prove a `relatively non-degenerate integrated estimate' for subextremal Kerr--de Sitter black holes~(and as a consequence exponential decay). In a forthcoming paper \cite{mavrogiannis3}, this will immediately yield nonlinear stability results for quasilinear wave equations on subextremal Kerr--de Sitter backgrounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22077 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundedness and Morawetz estimates on subextremal Kerr de Sitter Mavrogiannis, Georgios Analysis of PDEs General Relativity and Quantum Cosmology We study the Klein--Gordon equation $\Box_{g_{a,M,l}}ψ-μ^2_{\textit{KG}}ψ=0$ on subextremal Kerr--de Sitter black hole backgrounds with parameters $(a,M,l)$, where $l^2=\frac{3}Λ$. We prove boundedness and Morawetz estimates assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results in particular apply in the very slowly rotating case $|a|\ll M,l$, and in the case where the solution~$ψ$ is axisymmetric. This generalizes the work of Dafermos--Rodnianski \cite{DR3} on Schwarzschild--de~Sitter. The boundedness and Morawetz results of the present paper will be used in our companion \cite{mavrogiannis2} to prove a `relatively non-degenerate integrated estimate' for subextremal Kerr--de Sitter black holes~(and as a consequence exponential decay). In a forthcoming paper \cite{mavrogiannis3}, this will immediately yield nonlinear stability results for quasilinear wave equations on subextremal Kerr--de Sitter backgrounds. |
| title | Boundedness and Morawetz estimates on subextremal Kerr de Sitter |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2503.22077 |