[Paper Review] Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric

Jean‐François Bony, Dietrich Häfner|ArXiv.org|Jun 4, 2007

Black Holes and Theoretical Physics22 references6 citations

TL;DR

This paper establishes a resonance expansion for solutions to the wave equation in the De Sitter–Schwarzschild spacetime, showing that the local energy decays polynomially with logarithmic angular derivative loss or exponentially with an ε-derivative loss. The dominant contribution arises from a zero resonance, and the error term's decay rate is tightly controlled via spectral theory and meromorphic continuation of the resolvent.

ABSTRACT

We describe an expansion of the solution of the wave equation in the De Sitter - Schwarzschild metric in terms of resonances. The main term in the expansion is due to a zero resonance. The error term decays polynomially if we permit a logarithmic derivative loss in the angular directions and exponentially if we permit an small derivative loss in the angular directions.

Motivation & Objective

• To analyze local energy decay for the wave equation in the De Sitter–Schwarzschild metric using resonance theory.
• To determine the precise decay rates of the local energy by expanding the solution in terms of resonances.
• To quantify the trade-off between decay rate and derivative loss in angular directions.
• To establish sharpness of decay estimates by testing the threshold for logarithmic derivative losses.
• To contrast the De Sitter–Schwarzschild case with the Schwarzschild case, where resonance accumulation prevents direct application of the method.

Proposed method

• Utilizes the theory of resonances to decompose the solution of the wave equation into contributions from poles of the meromorphically continued resolvent.
• Applies the Sá Barreto–Zworski result on resonance localization in the De Sitter–Schwarzschild metric to identify the zero resonance as the main term.
• Employs contour integration in the complex plane to express the propagator in terms of spectral projections and error terms.
• Uses weighted estimates in angular frequency spaces via the operator $\langle -\Delta_\omega \rangle^M$ to control derivative losses.
• Applies the radial decomposition $u = \sum_\ell u_\ell$ to reduce the problem to angular momentum modes and control $\ell$-dependent norms.
• Establishes bounds on the error term via estimates on the resolvent $\widehat{R}_\chi^\ell(\lambda)$ in the complex plane, leveraging polynomial and logarithmic growth bounds.

Experimental results

Research questions

• RQ1 What is the rate of local energy decay for the wave equation in the De Sitter–Schwarzschild spacetime?
• RQ2 How does the presence of a zero resonance influence the asymptotic behavior of the solution?
• RQ3 What is the optimal trade-off between decay rate and derivative loss in angular directions?
• RQ4 Can the resonance method be applied to the Schwarzschild case, and if not, why?
• RQ5 What is the sharp threshold for derivative loss in angular directions beyond which decay estimates fail?

Key findings

• The local energy decays polynomially with a logarithmic derivative loss in angular directions: $\|E_1(t)\| \lesssim e^{-\mu t} \|\langle -\Delta_\omega \rangle^M \chi_0 u\|_{{\mathcal{E}}^{\rm mod}}$ with $M = C\mu/2$.
• The local energy decays exponentially with an $\varepsilon$-derivative loss: $\|E_1(t)\| \lesssim e^{-\mu t} \|\chi_0 u\|_{{\mathcal{E}}^{\rm mod}}$ under $\varepsilon$-loss in angular derivatives.
• For initial data in the complement of a one-dimensional space, the local energy is integrable with a $\langle \ln \langle -\Delta_\omega \rangle \rangle^\alpha$ derivative loss for $\alpha > 1$, and this is nearly optimal since the estimate fails for $\alpha < 1/2$.
• The main contribution to the solution expansion comes from a zero resonance, which dominates the long-time behavior.
• The method does not extend directly to the Schwarzschild case due to the possible accumulation of resonances at the origin.
• The error term in the resonance expansion satisfies $\|I_1\| \lesssim e^{-\mu t} \ell^{C\mu} \|u\|_{{\mathcal{E}}^{\rm mod}}$ and $\|I_2\| \lesssim e^{-\mu t} \|u\|_{{\mathcal{E}}^{\rm mod}}$, leading to the stated decay estimates.