Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes

Léo Bigorgne

Published 2020-06-05Version 1

We consider solutions to the massless Vlasov equation on the domain of outer communications of the Schwarschild black hole. By adapting the r^p-weighted energy method of Dafermos and Rodnianski, used extensively in order to study wave equations, we prove arbitrary decay for a non-degenerate energy flux of the Vlasov field f through a well-chosen foliation. An essential step of this methodology consists in proving a non-degenerate integrated local energy decay. For this, we take in particular advantage of the red-shift effect near the event horizon. The trapping at the photon sphere requires however to lose an epsilon of integrability in the velocity variable. Pointwise decay estimates on the velocity average of f are then obtained by functional inequalities, adapted to the study of Vlasov fields, which allow us to deal with the lack of conservation law for the radial derivative.