Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Bigorgne, Léo Ruiz, Renato Velozo |
| author_facet | Bigorgne, Léo Ruiz, Renato Velozo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes Bigorgne, Léo Ruiz, Renato Velozo Analysis of PDEs 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. |
| title | Relatively non-degenerate integrated decay estimates for massless Vlasov fields on Schwarzschild spacetimes |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2501.09730 |