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Local energy decay for scalar fields on time dependent non-trapping backgrounds

Jason Metcalfe, Jacob Sterbenz, Daniel Tataru

TL;DR

The paper addresses how local energy decay bounds for scalar waves persist on time-dependent, nontrapping, asymptotically flat backgrounds, providing a full spectral picture in the stationary regime and a robust, time-dependent trichotomy in the almost stationary, almost symmetric setting. It develops exterior reductions, high/medium/low frequency analyses via Carleman and related techniques, and connects LE bounds to resolvent and spectral properties, with stable behavior under AF perturbations. The results establish conditions under which local energy decay holds, identify obstructions (eigenvalues and resonances), and deliver a framework for exponential growth/decay dichotomies/trichotomies that underpin decay in broad nonstationary backgrounds, with implications for Strichartz and related dispersive estimates. This work lays the groundwork toward handling trapping scenarios (e.g., black holes) in future studies by first mastering the nontrapping, time-dependent case.

Abstract

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a "two point" local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.

Local energy decay for scalar fields on time dependent non-trapping backgrounds

TL;DR

The paper addresses how local energy decay bounds for scalar waves persist on time-dependent, nontrapping, asymptotically flat backgrounds, providing a full spectral picture in the stationary regime and a robust, time-dependent trichotomy in the almost stationary, almost symmetric setting. It develops exterior reductions, high/medium/low frequency analyses via Carleman and related techniques, and connects LE bounds to resolvent and spectral properties, with stable behavior under AF perturbations. The results establish conditions under which local energy decay holds, identify obstructions (eigenvalues and resonances), and deliver a framework for exponential growth/decay dichotomies/trichotomies that underpin decay in broad nonstationary backgrounds, with implications for Strichartz and related dispersive estimates. This work lays the groundwork toward handling trapping scenarios (e.g., black holes) in future studies by first mastering the nontrapping, time-dependent case.

Abstract

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a "two point" local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.

Paper Structure

This paper contains 29 sections, 36 theorems, 298 equations.

Table of Contents

  1. Introduction
  2. The geometric setup
  3. Local energy norms
  4. Energy estimates
  5. Main results
  6. The stationary case.
  7. The nonstationary case
  8. An overview of the paper
  9. Exterior estimates for asymptotically flat perturbations of Minkowski
  10. High frequency analysis
  11. Carleman estimates and the medium frequency analysis
  12. Carleman estimates for wave equations
  13. The medium frequency estimate
  14. Low frequency analysis
  15. Two point local energy decay and energy growth
  16. ...and 14 more sections

Key Result

Lemma 1.5

Let $g$ be an $\epsilon$-slowly varying asymptotically flat metric which satisfies the condition nt2 in Definition NT_def. If $\epsilon$ is sufficiently small, then the vector field $\partial_t$ is uniformly timelike.

Theorems & Definitions (70)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Definition 1.6
  • Theorem 1.7: MT
  • Definition 1.8
  • Definition 1.9
  • ...and 60 more