Improved decay rates with small regularity loss for the wave equation about a Schwarzschild black hole
P. Blue, A. Soffer
TL;DR
This work analyzes the decoupled wave equation outside a Schwarzschild black hole, where photon-sphere trapping complicates decay for low-regularity data. By combining Morawetz-type local decay, angular modulation, and phase-space observables within a conformal-charge framework, the authors establish uniform bounds on the conformal charge and derive global space-time estimates, including an $L^4$ space-time bound for the solution. They extend the approach to warped-product manifolds with closed geodesics and show small-data nonlinear waves can be controlled and globally bounded, highlighting the photon sphere's influence analogous to closed geodesics in Riemannian geometry. The results rely on a sequence of localized density and commutator techniques (Heisenberg relations, phase-space observables, and temporal refinements) to overcome regularity losses and to obtain robust decay and integrability in time.
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 ($H^{1+ε}$) 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 $H^{1+ε}$ initial data with additional decay towards infinite and the bifurcation sphere. Here $\tildeφ$ refers to a solution of the wave equation. $I$ denotes the exterior region of the Schwarzschild solution, which can be expressed in coordinates as $r>2M$, $t\in\Reals, ω\in S^2$, 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.