Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
Uniform energy bound and asymptotics for the Maxwell field on a slowly rotating Kerr black hole exterior
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DOI：
10.1142/s0219891615500204
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发表时间：
2015-12-01
影响因子：
0.7
通讯作者：
Blue, Pieter
中科院分区：
文献类型：
--
作者：
Andersson, Lars;Blue, Pieter
我们考虑旋转非常缓慢的克尔黑洞外部的麦克斯韦方程。对于这个系统，我们证明了常数 t 的每个超曲面上正定能量的有界性。我们还证明了每个解对于平稳库仑解的收敛性。我们将通解分为带电部分、库仑部分和不带电部分。库仑解的收敛源于以下事实：不带电部分满足 Morawetz 估计，即空间局部能量密度在时间上可积。对于不变的部分，我们研究了一个组件的完整麦克斯韦方程和 Fackerell-Ipser 方程。为了处理 Fackerell-Ipser 方程，我们对 t 使用傅里叶变换。对于 Fackerell-Ipser 方程，我们证明了一种精炼的 Morawetz 估计，该估计控制 3/2 导数，且在轨道零测地线附近没有损失。
We consider the Maxwell equation in the exterior of a very slowly rotating Kerr black hole. For this system, we prove the boundedness of a positive definite energy on each hypersurface of constant t. We also prove the convergence of each solution to a stationary Coulomb solution. We separate a general solution into the charged, Coulomb part and the uncharged part. Convergence to the Coulomb solutions follows from the fact that the uncharged part satisfies a Morawetz estimate, i.e. that a spatially localized energy density is integrable in time. For the unchanged part, we study both the full Maxwell equation and the Fackerell-Ipser equation for one component. To treat the Fackerell-Ipser equation, we use a Fourier transform in t. For the Fackerell-Ipser equation, we prove a refined Morawetz estimate that controls 3/2 derivatives with no loss near the orbiting null geodesics.