2006/12/06 by Blue, P., Soffer, A.
#35Q75 #58J45 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0612168
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 (H3) initial data \citeBSterbenz and slower decay for rough (H1+ε) initial data \citeBlueSoffer3. 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 ∫I |ϕ|4 d4vol < C for H1+ε initial data with additional decay towards infinite and the bifurcation sphere. Here ϕ 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∈\Reals, ω∈ S2, and d4vol 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^∞ estimates cannot hold at this level of regularity.