Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole

Abstract

We continue our study of the decoupled wave equation in the exterior of a spherically symmetric, Schwarzschild, black hole. Because null geodesics on the photon sphere orbit the black hole, extra effort must be made to show that the high angular momentum components of a solution decay sufficiently fast, particularly for low regularity initial data. Previous results are rapid decay for regular ($H^3$) initial data \cite{BSterbenz} and slower decay for rough ($H^{1+\epsilon }$) initial data \cite{BlueSoffer3}. Here, we combine those methods to show boundedness of the conformal charge. From this, we conclude that there are bounds for global in time, space-time norms, in particular \int_I |\tildeϕ|^4 d^4vol < C for $H^{1+\epsilon }$ initial data with additional decay towards infinite and the bifurcation sphere. Here $\tilde{\varphi }$ refers to a solution of the wave equation. $I$ denotes the exterior region of the Schwarzschild solution, which can be expressed in coordinates as $r>2M$, $t\in \mathbb{R},\omega \in S^2$, and $d^4\text{vol}$ is the natural 4-dimensional volume induced by the Schwarzschild pseudo-metric. We also demonstrate that the photon sphere has the same influence on the wave equation as a closed geodesic has on the wave equation on a Riemannian manifold. We demonstrate this similarity by extending our techniques to the wave equation on a class of Riemannian manifolds. Under further assumptions, the space-time estimates are sufficient to prove global bounds for small data, nonlinear wave equations on a class of Riemannian manifolds with closed geodesics. We must use global, space-time integral estimates since $L^{\infty }$ estimates cannot hold at this level of regularity.

40 pages, 1 figure

References in corpus (7)

• Schwarzschild black hole lensing
• Global existence for the Einstein vacuum equations in wave coordinates
• Uniform Decay of Local Energy and the Semi-Linear Wave Equation on Schwarzchild Space
• Decay of Solutions of the Wave Equation in the Kerr Geometry
• Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates
• Phase Space Analysis on some Black Hole Manifolds
• The wave equation on the Schwarzschild metric II: Local decay for the spin 2 Regge Wheeler equation

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• Strichartz estimates on Schwarzschild black hole backgrounds
• Improved decay for solutions to the linear wave equation on a Schwarzschild black hole
• Phase Space Analysis on some Black Hole Manifolds
• The Wave Equation on Extreme Reissner-Nordström Black Hole Spacetimes: Stability and Instability Results
• Monotonic Local Decay Estimates
• Decay and non-decay of the local energy for the wave equation in the De Sitter - Schwarzschild metric
• Localized energy estimates for wave equations on high dimensional Schwarzschild space-times