Decay estimates for massless Vlasov fields on Schwarzschild spacetimes

We will present a commutation vector field approach for the solutions to the massless Vlasov equation on the exterior of a Schwarzschild spacetime. The approach satisfies two properties. First, it provides decay estimates for the energy flux induced by the energy-momentum tensor of Vlasov fields and its derivatives. Secondly, it is compatible with the methods used to study wave equations on black hole spacetimes.

For this, we make use of a weight function that captures the concentration properties of the geodesic flow. By using a well-chosen modification of its symplectic gradient, we construct a norm for which any smooth solution to the massless Vlasov equation satisfies an integrated energy decay estimate without relative degeneration, which in turn implies exponential decay of the energy flux. This is joint work with Renato Velozo.