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Nonlinear Wave Dynamics in Black Hole Spacetimes

John Stogin

Abstract

This manuscript is a lightly reformatted version of my 2017 PhD thesis. I am posting it on arXiv at the request of my advisor, Sergiu Klainerman, who noted that it has been useful to some students. The content largely reflects the thesis in its original form. This thesis details a method for proving global boundedness and decay results for nonlinear wave equations on black hole spacetimes. The method is applied to five example problems of increasing difficulty. The first problem, which addresses the semilinear wave equation on Minkowski space, is quite simple and should be accessible to a reader who is still new to the field of partial differential equations. The final problem, which was posed by Ionescu and Klainerman in [IK14], constitutes a step toward proving stability for slowly rotating Kerr black holes. The remaining intermediate problems are: a semilinear wave equation on the Schwarzschild spacetime, a semilinear wave equation on any subextremal Kerr spacetime with the additional assumption of axisymmetry, and a restriction of the final problem to the Schwarzschild case. The method used in this thesis is based on a few particular developments that may be useful for other related problems. These include: a new method for constructing Morawetz-type estimates that is fairly robust (insofar as it may be successfully applied to all five problems), a strategy based on a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces using, in particular, a notion of weak decay, and a technique for handling certain terms with factors that are singular on an axis of symmetry.

Nonlinear Wave Dynamics in Black Hole Spacetimes

Abstract

This manuscript is a lightly reformatted version of my 2017 PhD thesis. I am posting it on arXiv at the request of my advisor, Sergiu Klainerman, who noted that it has been useful to some students. The content largely reflects the thesis in its original form. This thesis details a method for proving global boundedness and decay results for nonlinear wave equations on black hole spacetimes. The method is applied to five example problems of increasing difficulty. The first problem, which addresses the semilinear wave equation on Minkowski space, is quite simple and should be accessible to a reader who is still new to the field of partial differential equations. The final problem, which was posed by Ionescu and Klainerman in [IK14], constitutes a step toward proving stability for slowly rotating Kerr black holes. The remaining intermediate problems are: a semilinear wave equation on the Schwarzschild spacetime, a semilinear wave equation on any subextremal Kerr spacetime with the additional assumption of axisymmetry, and a restriction of the final problem to the Schwarzschild case. The method used in this thesis is based on a few particular developments that may be useful for other related problems. These include: a new method for constructing Morawetz-type estimates that is fairly robust (insofar as it may be successfully applied to all five problems), a strategy based on a decay hierarchy for energy estimates on uniformly spacelike hypersurfaces using, in particular, a notion of weak decay, and a technique for handling certain terms with factors that are singular on an axis of symmetry.
Paper Structure (251 sections, 308 theorems, 2303 equations, 3 figures)

This paper contains 251 sections, 308 theorems, 2303 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Foundational, prior, and related works
  3. The general method of this thesis
  4. The main idea
  5. The steps involved in the general method
  6. A description of each problem in this thesis
  7. Problem 1: Waves in the Minkowski spacetime
  8. Problem 2: Waves in the Schwarzschild spacetime
  9. Problem 3: Axisymmetric waves in subextremal Kerr spacetimes
  10. The model problem of ionescu2014global
  11. Problem 4: axisymmetric perturbations of the nontrivial wave map in the Schwarzschild spacetime
  12. Problem 5: axisymmetric perturbations of the nontrivial wave map in slowly rotating Kerr spacetimes
  13. An outline of this thesis
  14. Preliminaries
  15. Geometric calculations in coordinates
  16. ...and 236 more sections

Key Result

Proposition 2.1.5

Equation (divergence_thm_eqn) is consistent with equation (divergence_eqn).

Figures (3)

  • Figure 1: The modified time foliation $\Sigma_t$ enters the event horizon and coincides with the Boyer-Lindquist foliation for $r>r_H+\delta_H$.
  • Figure 2: Plots of $w$ and $\tilde{K}^{rr}=\frac{1}{2}r^3\partial_r\left(\frac{u}{r^2}\right)$ for $r\ge 2M$. In the region $r>r_*$, the function $w$ is chosen so that $\tilde{K}^{rr}$ is constant. In the region $r<r_*$, $w$ is chosen to be constant (this is required by condition (\ref{['s:u_cond_3_eqn']})). The value $r_*=4M$ is chosen so that $w$ remains $C^1$.
  • Figure 3: Plots of $w$ and $\tilde{K}^{rr}=\frac{1}{2}r^3\partial_r\left(\frac{u}{r^2+a^2}\right)$ for $r\ge r_H$. In the region $r>r_*$, the function $w$ is chosen so that $\tilde{K}^{rr}$ is constant. In the region $r<r_*$, $w$ is chosen to be constant (this is required by condition (\ref{['k:u_cond_3_eqn']})). The value $r_*$ is chosen so that $w$ remains $C^1$.

Theorems & Definitions (683)

  • Example 2.1.1
  • Example 2.1.2
  • Example 2.1.3
  • Example 2.1.4
  • Proposition 2.1.5
  • proof
  • Example 2.1.6
  • Proposition 2.2.1
  • Definition 2.2.2
  • Proposition 2.2.3
  • ...and 673 more