论文标题
Schwarzschild空间上无质量弗拉索夫方程的衰减估计值
Decay estimates for the massless Vlasov equation on Schwarzschild spacetimes
论文作者
论文摘要
我们考虑了施瓦尔柴尔德黑洞外部通信领域的无质量弗拉索夫方程的解决方案。通过对Dafermos和Rodnianski的R^P加权能量方法进行了广泛的使用来研究波动方程，我们证明了通过精心挑选的Foliation的vlasov场F的非分类能量通量是任意衰减。这种方法的一个重要步骤是证明非分类综合局部能量衰减。为此，我们特别利用了事件视野附近的红移效果。但是，光子球处的陷阱需要在速度变量中失去可积分性的Epsilon。然后，通过功能不平等获得了F的速度平均值的衰减估计值，该功能不平等得到了研究，适合于弗拉索夫田地的研究，这使我们能够处理径向衍生物缺乏保护法。
We consider solutions to the massless Vlasov equation on the domain of outer communications of the Schwarschild black hole. By adapting the r^p-weighted energy method of Dafermos and Rodnianski, used extensively in order to study wave equations, we prove arbitrary decay for a non-degenerate energy flux of the Vlasov field f through a well-chosen foliation. An essential step of this methodology consists in proving a non-degenerate integrated local energy decay. For this, we take in particular advantage of the red-shift effect near the event horizon. The trapping at the photon sphere requires however to lose an epsilon of integrability in the velocity variable. Pointwise decay estimates on the velocity average of f are then obtained by functional inequalities, adapted to the study of Vlasov fields, which allow us to deal with the lack of conservation law for the radial derivative.
