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Relatively non-degenerate integrated decay estimates on subextremal Kerr de Sitter

Georgios Mavrogiannis

TL;DR

The paper proves a relatively non-degenerate integrated decay estimate for solutions to the Klein–Gordon equation on subextremal Kerr–de Sitter spacetimes, under a mode stability assumption on Carter’s radial ODE. The authors introduce a frequency-dependent pseudodifferential commutator \(\mathcal{G}\), regularized to \(\widetilde{\mathcal{G}}\) to enable Coifman–Meyer type estimates, and combine it with a subextremal Morawetz bound to control a bulk term that degenerates at trapped sets in a controlled way. The main result shows that axisymmetric data yield unconditional decay (and exponential convergence to a constant), while in the general non-axisymmetric case the decay follows from (MS) and the new commutation framework, with the very slowly rotating regime included. This work advances stability results in Kerr–de Sitter by blending spectral (Fourier) analysis with modern pseudodifferential commutators, paving the way for nonlinear stability results on such backgrounds.

Abstract

We study the Klein--Gordon equation $\Boxψ-μ^2_{\textit{KG}}ψ=0$ on subextremal Kerr de Sitter black hole backgrounds with parameters $(a,M,l)$, where $l^2=\frac{3}Λ$. We prove a "relatively non degenerate integrated" decay estimate assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results, in particular, apply unconditionally in the very slowly rotating case $|a|\ll M,l$, and in the case where $ψ$ is axisymmetric. Exponential decay for $ψ$ to a constant is a consequence of this estimate. To prove our result, we introduce a novel pseudodifferential commutation operator $\mathcal{G}$ that generalizes our previous purely physical space commutation \cite{mavrogiannis} and we use it in conjunction with the Morawetz estimate of our companion \cite{mavrogiannis4}. This pseudodifferential operator is defined using Fourier decomposition with respect to time frequencies $ω$ and azimuthal frequencies $m$, but does not require Carter's full separation.

Relatively non-degenerate integrated decay estimates on subextremal Kerr de Sitter

TL;DR

The paper proves a relatively non-degenerate integrated decay estimate for solutions to the Klein–Gordon equation on subextremal Kerr–de Sitter spacetimes, under a mode stability assumption on Carter’s radial ODE. The authors introduce a frequency-dependent pseudodifferential commutator , regularized to to enable Coifman–Meyer type estimates, and combine it with a subextremal Morawetz bound to control a bulk term that degenerates at trapped sets in a controlled way. The main result shows that axisymmetric data yield unconditional decay (and exponential convergence to a constant), while in the general non-axisymmetric case the decay follows from (MS) and the new commutation framework, with the very slowly rotating regime included. This work advances stability results in Kerr–de Sitter by blending spectral (Fourier) analysis with modern pseudodifferential commutators, paving the way for nonlinear stability results on such backgrounds.

Abstract

We study the Klein--Gordon equation on subextremal Kerr de Sitter black hole backgrounds with parameters , where . We prove a "relatively non degenerate integrated" decay estimate assuming an appropriate mode stability statement for real frequency solutions of Carter's radial ode. Our results, in particular, apply unconditionally in the very slowly rotating case , and in the case where is axisymmetric. Exponential decay for to a constant is a consequence of this estimate. To prove our result, we introduce a novel pseudodifferential commutation operator that generalizes our previous purely physical space commutation \cite{mavrogiannis} and we use it in conjunction with the Morawetz estimate of our companion \cite{mavrogiannis4}. This pseudodifferential operator is defined using Fourier decomposition with respect to time frequencies and azimuthal frequencies , but does not require Carter's full separation.

Paper Structure

This paper contains 81 sections, 35 theorems, 462 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Previous works
  3. The 'relatively non-degenerate' estimate of mavrogiannis on Schwarzschild--de Sitter
  4. A review of the results of our companion mavrogiannis4
  5. The g Fourier projections
  6. The Hawking--Reall vector fields
  7. The superradiant frequencies
  8. The radial ode in Kerr--de Sitter
  9. Energy identity and mode stability in Kerr--de Sitter
  10. The boundedness and Morawetz estimates of mavrogiannis4
  11. The operator g
  12. The function g graphically
  13. A bulk integral associated with g
  14. The time cut-offs and the energy density
  15. The rough version of Theorem \ref{['main theorem, relat. non-deg']}
  16. ...and 66 more sections

Key Result

Theorem \ref{main theorem, relat. non-deg}

Let $(a,M,l)$ be subextremal Kerr--de Sitter black hole parameters and let $\mu^2_{KG}\geq 0$. Let $T\geq 3$ be sufficiently large and let $\tilde{T}< \tau_1<\tau_1+T< \tau_2$, where for $\tilde{T}$ see eq: subsec: sec: intro, 3, eq 1.1. Assume that and let $\psi$ be a solution of the Klein--Gordon equation eq: kleingordon on the subextremal Kerr-de Sitter black hole with parameters $(a,M,l)$. Th

Figures (10)

  • Figure 1: The function $g_2(\omega,m,r)$ for the values \ref{['eq: subsec: sec: intro: subsec 3.1, eq 1']} respectively
  • Figure 2: The cut-offs \ref{['eq: subsec: sec: intro, 3, eq 1.2']}
  • Figure 3: The $\Delta$ polynomial with 4 roots
  • Figure 4: Carter-Penrose diagram of Kerr--de Sitter
  • Figure 5: The cut-off \ref{['eq: subsec: sec: carter separation, subsec 1, eq 2']}
  • ...and 5 more figures

Theorems & Definitions (106)

  • Remark 1.1
  • Definition 1.1
  • Definition \ref{def: sec: G, def 1}: rough version
  • Remark 1.2
  • Remark 1.3
  • Theorem \ref{main theorem, relat. non-deg}: rough version
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • ...and 96 more