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 $W_{x,p}^{1,1}$ 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 $r^p$-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 $\mathrm{T}[f]$ 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.

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Cited by in corpus (1)

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