Weissenbacher Max
Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ UK.
Arch Ration Mech Anal. 2024;248(6):118. doi: 10.1007/s00205-024-02060-1. Epub 2024 Nov 20.
We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner-Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis' results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.
我们研究了次极端和极端雷斯纳 - 诺德斯特龙时空外部的无质量弗拉索夫方程。我们证明，在次极端情况下矩以指数速率衰减，而在极端情况下以多项式速率衰减。沿着事件视界，这种多项式速率被证明是精确的。在极端情况下，我们表明，如果解及其一阶时间导数最初在事件视界的一个邻域内有支撑，那么能量动量张量某些分量的横向导数沿事件视界不会衰减。将极端情况下横向导数的不衰减与阿雷塔基斯关于波动方程不稳定性的工作进行了比较。与阿雷塔基斯关于波动方程的结果不同，后者利用了守恒律的层次结构，我们的证明完全基于测地线流的定量分析，并且守恒律在本工作中没有出现。
