论文标题
空间无穷大的度量行为对渐近平稳环境中尖的波衰减的影响
The effect of metric behavior at spatial infinity on pointwise wave decay in the asymptotically flat stationary setting
论文作者
论文摘要
当前的工作考虑了（1+3）维度的渐近平坦，固定，洛伦兹的空间上的波动方程的解决方案。我们研究了几何形状倾向于平坦的速率与溶液的点衰减速率之间的关系。在\ cite {tat2013}中研究了时空以$ | x |^{ - 1} $趋向平坦的情况，其中建立了$ t^{ - 3} $点衰减率。在这里，我们将结果扩展到以$ | x |^{ - κ} $的速度趋于平坦的几何形状，并以$κ\ in \ mathbb {n} $使用$κ\ ge 2 $建立了$ t^{ - κ-2} $的$ t^{ - κ-2} $。我们假设局部能量衰减估计值较弱，这限制了基础几何形状上允许的大地捕获。我们使用解决方案将解决方案的时间变换连接到Cauchy数据。最终，波动衰减的速率取决于分解的低频行为，这对背景几何趋于平坦的速率很敏感。
The current work considers solutions to the wave equation on asymptotically flat, stationary, Lorentzian spacetimes in (1+3) dimensions. We investigate the relationship between the rate at which the geometry tends to flat and the pointwise decay rate of solutions. The case where the spacetime tends toward flat at a rate of $|x|^{-1}$ was studied in \cite{tat2013}, where a $t^{-3}$ pointwise decay rate was established. Here we extend the result to geometries tending toward flat at a rate of $|x|^{-κ}$ and establish a pointwise decay rate of $t^{-κ-2}$ for $κ\in \mathbb{N}$ with $κ\ge 2$. We assume a weak local energy decay estimate holds, which restricts the geodesic trapping allowed on the underlying geometry. We use the resolvent to connect the time Fourier Transform of a solution to the Cauchy data. Ultimately the rate of pointwise wave decay depends on the low frequency behavior of the resolvent, which is sensitive to the rate at which the background geometry tends to flat.
