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Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes

Benjamin Dodson, Sam Looi

Abstract

We prove quantitative scattering for the three-dimensional defocusing energy-critical quintic wave equation on a class of asymptotically flat, possibly non-stationary perturbations of Minkowski space, by establishing the first explicit global $L^8_{t,x}$ bound in this variable-coefficient setting. Earlier work in this setting proved scattering only qualitatively. For \[ Pu=u^5,\qquad P=\partial_α(g^{αβ}\partial_β), \] we show that, under smallness, decay, and regularity assumptions on the metric, and assuming a priori $\dot H^5\times\dot H^4$ and $L^2\times\dot H^{-1}$ bounds on the solution, the critical spacetime norm $\|u\|_{L^8_{t,x}(\mathbb R\times\mathbb R^3)}$ satisfies an explicit exponential-type estimate in terms of the energy and the a priori bound. This upgrades the qualitative scattering theory in this setting to a quantitative one. The main difficulty is to control geometric error terms over long times. We handle this by splitting the Duhamel history into recent past and remote past. For the recent past, we prove a variable-coefficient interaction Morawetz estimate that yields, on every sufficiently long interval, a time at which the recent nonlinear contribution is small. For the remote past, we prove a dispersive estimate from integrated local energy decay together with a transfer of pointwise decay from large radius to large time. Combining these estimates gives the explicit global $L^8_{t,x}$ bound.

Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes

Abstract

We prove quantitative scattering for the three-dimensional defocusing energy-critical quintic wave equation on a class of asymptotically flat, possibly non-stationary perturbations of Minkowski space, by establishing the first explicit global bound in this variable-coefficient setting. Earlier work in this setting proved scattering only qualitatively. For we show that, under smallness, decay, and regularity assumptions on the metric, and assuming a priori and bounds on the solution, the critical spacetime norm satisfies an explicit exponential-type estimate in terms of the energy and the a priori bound. This upgrades the qualitative scattering theory in this setting to a quantitative one. The main difficulty is to control geometric error terms over long times. We handle this by splitting the Duhamel history into recent past and remote past. For the recent past, we prove a variable-coefficient interaction Morawetz estimate that yields, on every sufficiently long interval, a time at which the recent nonlinear contribution is small. For the remote past, we prove a dispersive estimate from integrated local energy decay together with a transfer of pointwise decay from large radius to large time. Combining these estimates gives the explicit global bound.

Paper Structure

This paper contains 10 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Preliminaries, and the remote past
  3. The remote past
  4. The recent past, and an interaction Morawetz estimate
  5. A dispersive estimate
  6. Proof of \ref{['3.4']}.
  7. The principle of $r$ to $t$ decay transfer
  8. Proof of \ref{['3.10']}
  9. Returning to the main proof
  10. Uniform energy bounds and integrated local energy decay for high orders

Key Result

Theorem 1.1

Let $(\mathbb{R}^{1+3}, g)$ be a spacetime where the metric $g^{\alpha\beta} = m^{\alpha\beta} + h^{\alpha\beta}$ is a perturbation of the Minkowski metric $m$. Let $P = \partial_{\alpha}(g^{\alpha\beta}\partial_{\beta})$ be the associated d'Alembertian. We assume the metric perturbation $h$ satisfi with initial data $(u_0, u_1) \in \dot{H}^1(\mathbb{R}^3) \times L^2(\mathbb{R}^3)$ satisfying the

Theorems & Definitions (22)

  • Theorem 1.1: Quantitative Scattering for the Energy-Critical Wave Equation on Asymptotically Flat Spacetimes
  • proof : Proof of \ref{['eq:L_eight']}
  • Proposition 2.1: Finite decomposition via Strichartz estimates
  • proof
  • Proposition 2.2: Negligibility/smallness of the remote-past contribution
  • proof
  • Proposition 3.1: Smallness of the recent-past contribution
  • Proposition 3.2: Existence of a "quiet" time
  • Lemma 4.1
  • proof
  • ...and 12 more