Schwarzschild背景下无质量Dirac场的精确衰减估计
Sharp decay estimates for massless Dirac fields on a Schwarzschild background
摘要：
本文研究了Schwarzschild背景下无质量Dirac场的显式渐近分布。首先，我们证明了狄拉克场的自旋$s=\pm \frac{1}{2}$分量的正定能量的一致界和一个积分的局部能量衰减估计从对称双曲波系统。基于这些估计，我们进一步表明，这些组件有全球逐点衰减$fv^{-3/2-s}τ^{-5/2+s}$作为一个上限和下限外的黑洞，与功能$f$有限的，明确表示的初始数据和坐标。这就建立了史瓦西黑洞外无质量狄拉克场的受限普莱斯定律的有效性。
英文摘要：
We consider the explicit asymptotic profile of massless Dirac fields on a Schwarzschild background. First, we prove for the spin $s=\pm \frac{1}{2}$ components of the Dirac field a uniform bound of a positive definite energy and an integrated local energy decay estimate from a symmetric hyperbolic wave system. Based on these estimates, we further show that these components have globally pointwise decay $fv^{-3/2-s}τ^{-5/2+s}$ as both an upper and a lower bound outside the black hole, with function $f$ finite and explicitly expressed in terms of the initial data and the coordinates. This establishes the validity of the conjectured Price's law for massless Dirac fields outside a Schwarzschild black hole.
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