[Estimations de décroissance intégrées, relativement non dégénérées, pour les champs de Vlasov sans masse sur les espaces-temps de Schwarzschild]
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 implies that $V_+f$ remains bounded, 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 the approach of [Mav24, DHRT22] used to study quasi-linear wave equations on black hole spacetimes.
Dans cet article, nous utilisons une fonction poids reflétant le phénomène de concentration des géodésiques causales piégées, dont la trajectoire est instable. Il s’avère qu’une projection $V_+$, sur l’espace tangent de la variété isotrope, du gradient symplectique associé à cette fonction poids possède de bonnes propriétés de commutation avec l’opérateur de Vlasov pour les particules de masse nulle. Cela implique que $V_+f$ reste bornée pour tout solution lisse de l’équation de Vlasov sans masse.
Une modification judicieusement choisie de $V_+$ nous permet ensuite de construire une norme $W^{1,1}_{x,p}$ à poids pour laquelle toute solution lisse de l’équation de Vlasov sans masse satisfait une estimation de décroissance intégrée relativement non dégénérée. En combinant cela avec la méthode d’énergie à poids $r^p$ de Dafermos-Rodnianski, nous démontrons que la norme d’énergie décroît au cours du temps. Cette norme permet notamment de contrôler le tenseur énergie-impulsion $\mathrm{T}[f]$ ainsi que toutes ses dérivées d’ordre $1$.
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Keywords: Schwarzschild spacetime, massless Vlasov fields, decay estimates, relativistic kinetic theory
Mots-clés : Espace-temps de Schwarzschild, champs de Vlasov sans masse, estimations de décroissance, théorie cinétique relativiste
Léo Bigorgne 1 ; Renato Velozo Ruiz 2
CC-BY 4.0
@article{JEP_2026__13__437_0,
author = {L\'eo Bigorgne and Renato Velozo Ruiz},
title = {Relatively non-degenerate integrated decay estimates for massless {Vlasov} fields on {Schwarzschild} spacetimes},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {437--518},
year = {2026},
publisher = {\'Ecole polytechnique},
volume = {13},
doi = {10.5802/jep.331},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.331/}
}
TY - JOUR AU - Léo Bigorgne AU - Renato Velozo Ruiz TI - Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes JO - Journal de l’École polytechnique — Mathématiques PY - 2026 SP - 437 EP - 518 VL - 13 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.331/ DO - 10.5802/jep.331 LA - en ID - JEP_2026__13__437_0 ER -
%0 Journal Article %A Léo Bigorgne %A Renato Velozo Ruiz %T Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes %J Journal de l’École polytechnique — Mathématiques %D 2026 %P 437-518 %V 13 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.331/ %R 10.5802/jep.331 %G en %F JEP_2026__13__437_0
Léo Bigorgne; Renato Velozo Ruiz. Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 437-518. doi: 10.5802/jep.331
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