Schwarzschild背景下wave equation的能流和衰变
A note on energy currents and decay for the wave equation on a Schwarzschild background
摘要：
在最近的工作中，我们已经证明了统一衰减边界的波动方程$\Box_gϕ=0$的解决方案上的史瓦西外部，特别是，统一的逐点估计$|ϕ|\le Cv_+^{-1}$，它持有整个域的外部通信，其中$v$是一个先进的Eddington-Finkelstein坐标，$v_+=\max\{v,1\}$，和$C$是一个常数依赖于一个Sobolev规范的初始数据。证明中的一个关键估计需要分解成球谐函数。我们在这里给这个估计的另一种证明，不需要这样的分解。
英文摘要：
In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_gϕ=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|ϕ|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\max\{v,1\}$, and $C$ is a constant depending on a Sobolev norm of initial data. A crucial estimate in the proof required a decomposition into spherical harmonics. We here give an alternative proof of this estimate not requiring such a decomposition.
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