Physical-space estimates for axisymmetric waves on extremal Kerr spacetime

Abstract

We study axisymmetric solutions to the wave equation on extremal Kerr backgrounds and obtain integrated local energy decay (or Morawetz estimates) through an analysis \textit{exclusively in physical-space}. Boundedness of the energy and Morawetz estimates for axisymmetric waves in extremal Kerr were first obtained by Aretakis through the construction of frequency-localized currents used in particular to express the trapping degeneracy. Here we extend to extremal Kerr a method introduced by Stogin in the sub-extremal case, simplifying Aretakis' derivation of Morawetz estimates through purely classical currents.

Added appendix with weighted estimates on the horizon, version accepted for publication

References in corpus (20)

• Global existence for the Einstein vacuum equations in wave coordinates
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations I
• The linear stability of the Schwarzschild solution to gravitational perturbations
• The global non-linear stability of the Kerr-de Sitter family of black holes
• Stability and Instability of Extreme Reissner-Nordström Black Hole Spacetimes for Linear Scalar Perturbations II
• A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds
• Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
• Strichartz estimates on Schwarzschild black hole backgrounds
• Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime
• Late-time asymptotics for the wave equation on extremal Reissner-Nordström backgrounds
• Non-linear stability of the Kerr-Newman-de Sitter family of charged black holes
• Stability of Minkowski space and polyhomogeneity of the metric
• Uniform energy bound and Morawetz estimate for extreme components of spin fields in the exterior of a slowly rotating Kerr black hole II: linearized gravity
• Mode stability for the Teukolsky equation on extremal and subextremal Kerr spacetimes
• The linear stability of Reissner-Nordström spacetime: the full subextremal range
• The linear stability of the Schwarzschild solution to gravitational perturbations in the generalised wave gauge
• The linear stability of Reissner-Nordström spacetime for small charge
• The trapping effect on degenerate horizons
• Boundedness and decay for the Teukolsky equation of spin $\pm 1$ on Reissner-Nordström spacetime: the $\ell =1$ spherical mode
• Boundedness and decay for the Teukolsky system of spin $\pm 2$ on Reissner-Nordström spacetime: the case $|Q|\ll M$