Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Local energy estimate on Kerr black hole backgrounds

Daniel Tataru, Mihai Tohaneanu

TL;DR

This paper establishes uniform energy bounds and local energy decay for the wave equation on Kerr spacetimes with small angular momentum, by viewing Kerr as a small Schwarzschild perturbation and developing a microlocal local energy norm that accounts for trapping near the photon sphere. The authors extend the Schwarzschild local energy framework using a pseudodifferential multiplier near $r\approx 3M$, constructing symbol-based weights and corrections to prove a Kerr analogue of the Schwarzschild estimate. Key contributions include the definition of Kerr-specific local smoothing norms $LE_K^1$, a corresponding dual $LE_K^*$, and high-order extensions, all valid for small $a$; these yield uniform energy bounds and local decay in the exterior region and a neighborhood of the horizon. The results pave the way for further analysis (e.g., Strichartz estimates) for Kerr backgrounds and highlight the role of trapping as a source of degeneracy that can be mitigated by tailored microlocal techniques.

Abstract

We study dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of thispaper is to establish uniform energy bounds and local energy decay for such backgrounds.

Local energy estimate on Kerr black hole backgrounds

TL;DR

This paper establishes uniform energy bounds and local energy decay for the wave equation on Kerr spacetimes with small angular momentum, by viewing Kerr as a small Schwarzschild perturbation and developing a microlocal local energy norm that accounts for trapping near the photon sphere. The authors extend the Schwarzschild local energy framework using a pseudodifferential multiplier near , constructing symbol-based weights and corrections to prove a Kerr analogue of the Schwarzschild estimate. Key contributions include the definition of Kerr-specific local smoothing norms , a corresponding dual , and high-order extensions, all valid for small ; these yield uniform energy bounds and local decay in the exterior region and a neighborhood of the horizon. The results pave the way for further analysis (e.g., Strichartz estimates) for Kerr backgrounds and highlight the role of trapping as a source of degeneracy that can be mitigated by tailored microlocal techniques.

Abstract

We study dispersive properties for the wave equation in the Kerr space-time with small angular momentum. The main result of thispaper is to establish uniform energy bounds and local energy decay for such backgrounds.

Paper Structure

This paper contains 4 sections, 7 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Local energy decay in the Minkowski space-time
  3. Local energy decay in the Schwarzschild space-time
  4. Local energy decay in the Kerr space-time

Key Result

Theorem 1

MT Let $g$ be a Lorenzian metric in ${\mathbb R}^{3+1}$ which satisfies gm with $\epsilon$ small enough. Then the solution $u$ to the inhomogeneous problem satisfies the estimate gm.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 1
  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Corollary 5