Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes

In this article, we make use of a weight function capturing the concentration phenomenon of unstable future-trapped causal geodesics. A projection <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> , 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 implies that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> </mml:msub> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> remains bounded, for any smooth solution <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> to the massless Vlasov equation. By identifying a well-chosen modification of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> , we are able to construct a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>W</mml:mi> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>r</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">T</mml:mi> <mml:mo>[</mml:mo> <mml:mi>f</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> as well as all its first order derivatives. The method developed in this paper is in particular compatible with the approach of [Mav24, DHRT22] used to study quasi-linear wave equations on black hole spacetimes.