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A local energy estimate for wave equations on metrics asymptotically close to Kerr

Hans Lindblad, Mihai Tohaneanu

TL;DR

<3-5 sentence high-level summary>

Abstract

In this article we prove a local energy estimate for the linear wave equation on metrics with slow decay to a Kerr metric with small angular momentum. As an application, we study the quasilinear wave equation $\Box_{g(u, t, x)} u = 0$ where the metric $g(u, t, x)$ is close (and asymptotically equal)to a Kerr metric with small angular momentum $g(0,t,x)$. Under suitable assumptions on the metric coefficients, and assuming that the initial data for $u$ is small enough, we prove global existence and decay of the solution $u$.

A local energy estimate for wave equations on metrics asymptotically close to Kerr

TL;DR

<3-5 sentence high-level summary>

Abstract

In this article we prove a local energy estimate for the linear wave equation on metrics with slow decay to a Kerr metric with small angular momentum. As an application, we study the quasilinear wave equation where the metric is close (and asymptotically equal)to a Kerr metric with small angular momentum . Under suitable assumptions on the metric coefficients, and assuming that the initial data for is small enough, we prove global existence and decay of the solution .

Paper Structure

This paper contains 46 sections, 27 theorems, 463 equations.

Table of Contents

  1. Introduction
  2. Statement of the results and history
  3. Assumptions on the metric
  4. The Local energy decay estimate
  5. The history
  6. Applications
  7. The heuristics
  8. The Schwarzschild case
  9. Perturbations of Schwarschild
  10. Trapped geodesics and a pseudo differential operator vanishing on the trapped set
  11. Positive commutator estimates
  12. Kerr with small angular momentum $a$
  13. Perturbations of Kerr
  14. The proof
  15. The Schwarzschild multiplier written as a pseudo differential operators
  16. ...and 31 more sections

Key Result

Theorem 1

Let $u$ solve the inhomogeneous linear wave equation $\Box_g u = F$ in $\mathcal{M}$, where $g$ is a Lorentzian metric satisfying the conditions rdecayintro, cpt1intro, cpt2intro and holderintro or alternatively the conditions in section 4. Then for any $0\leq\tt_0 < \tt_1$ where $\kappa^2=\kappa_0+\kappa_1^2$ and the implicit constant is independent of $\tt_0$, $\tt_1$, $\epsilon$. Here $B(F,u)$

Theorems & Definitions (50)

  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 1
  • Remark 2
  • Lemma 3
  • ...and 40 more