Price's Law on Nonstationary Space-Times
Price's Law on Nonstationary Space-Times
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DOI：
10.1016/j.aim.2012.03.010
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发表时间：
2011-04
影响因子：
1.7
通讯作者：
Jason Metcalfe;D. Tataru;M. Tohaneanu
中科院分区：
文献类型：
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作者：
Jason Metcalfe;D. Tataru;M. Tohaneanu
本文研究了三维空间中一类非平稳渐近平坦背景上波动方程解的逐点衰减性质。在假定均匀能量界和局部能量衰减的弱形式随时间向前推进的假设下，我们建立了线性波的t−3局部均匀衰减率(普赖斯定律，Price(1972)[54])。作为推论，我们还证明了Kerr度规的某些小扰动的普赖斯定律。这一结果以前是由第二作者在(Tataru[65])中关于静止背景建立的。这项工作的动机是真空爱因斯坦方程Kerr/Schwarzschild解的非线性稳定性问题，这似乎需要一种更稳健的方法来证明线性衰减估计。
In this article, we study the pointwise decay properties of solutions to the wave equation on a class of nonstationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t−3local uniform decay rate (Price’s law, Price (1972) [54]) for linear waves. As a corollary, we also prove Price’s law for certain small perturbations of the Kerr metric. This result was previously established by the second author in (Tataru [65]) on stationary backgrounds. The present work was motivated by the problem of nonlinear stability of the Kerr/Schwarzschild solutions for the vacuum Einstein equations, which seems to require a more robust approach to proving linear decay estimates.