Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes
Abstract
In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection V+, on the tangent space of the null-shell, of the associated symplectic gradient turns out to enjoy good commutation properties with the massless Vlasov operator. This reflects that V+f decays exponentially locally near the photon sphere, for any smooth solution f to the massless Vlasov equation. By identifying a well-chosen modification of V+, we are able to construct a Wx,p1,1 weighted norm for which any smooth solution to the massless Vlasov equation verifies an integrated local energy decay estimate without relative degeneration. Together with the rp-weighted energy method of Dafermos--Rodnianski, we establish time decay for the energy norm. This norm allows for the control of the energy-momentum tensor T[f] as well as all its first order derivatives. The method developed in this paper is in particular compatible with approaches recently developed for the study of quasi-linear wave equations on black hole spacetimes.