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.
Veiled singularities for the spherically symmetric massless Einstein-Vlasov system
本文继续研究在[RV]中引发的无碰撞物质坍缩中裸奇点的形成。证明了Einstein-Vlasov系统的某些类非光滑解的存在性。这些解是自相似的，因此不是渐近平坦的。为了得到与物理更相关的解，尝试以适当的方式截断这些解，使它们渐近平坦是有意义的。这项任务，这是原来是技术上具有挑战性的，将在本文中进行。[RV]A. D. Rendall和J.J.L. Velázquez，无质量Einstein-Vlasov系统的一类尘埃自相似解。Annales Henri Poincare 12，919-964，（2011）.
Future stability of expanding spatially homogeneous FLRW solutions of the spherically symmetric Einstein--massless Vlasov system with spatial topology $\mathbb{R}^3$
A proof of the instability of AdS for the Einstein--massless Vlasov system
Dafermos-Holzegel于2006年提出的AdS不稳定性猜想近年来吸引了大量的数值和启发式研究。继Bizon-Rostworowski的开创性工作之后，研究工作主要集中在球对称爱因斯坦-标量场系统的研究上。AdS在最简单的球对称环境中，即在爱因斯坦零尘埃系统中不稳定性的第一个严格证明在[G. Moschalek，A proof of the instability of AdS for the Einstein-null dust system with an inner mirror，2017，arXiv：1704.08681]。在这项工作中，演化被限制在放置在$r=r_{0}>0$的内部镜子的外部，以避免发生在$r=0$的爱因斯坦零尘埃系统的轻微破裂;在这种设置中，额外的考虑需要$r_{0}$缩小到初始扰动大小的$0$。本文建立了球对称下Einstein-Vlasov无质量系统的AdS不稳定性。这将是第一次证明AdS不稳定性猜想的爱因斯坦-物质系统，这是良好的设置为正常的初始数据在标准意义上，没有增加一个内部的镜子。新的困难与爱因斯坦-无质量的弗拉索夫系统（如需要控制的路径上的非径向测地线在一个大曲率制度）迫使我们开发一种策略的证明，这是根本不同的情况下所采用的爱因斯坦-零尘埃系统，虽然仍然是基于相互作用的物质束在物理空间。
Coulomb Sectors and Scattering for Maxwell-Higgs Fields on Schwarzschild and Slowly Rotating Kerr Backgrounds
Theoretical Physics Laboratory, Theoretical High Energy Physics Research Division, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung（万隆理工学院数学与自然科学学院理论物理实验室、理论高能物理研究部）
