论文标题
在固定空间时的阻尼波方程的局部能量衰减
Integrated Local Energy Decay for the Damped Wave Equation on Stationary Space-Times
论文作者
论文摘要
我们证明了在（1 + 3）维度中的固定，渐近平坦的空间时间上阻尼波动方程的局部能量衰减。局部能量衰减构成了这种几何背景的分散偏微分方程的强大工具。通过利用几何控制条件来处理被捕获的轨迹，我们能够恢复高频估计而不会造成任何损失。然后，我们可以在中和低频方案中使用Metcalfe，Sterbenz和Tataru的工作中的已知估计值，以建立局部能量衰减。这概括了Bouclet和Royer从渐近的欧几里得流形到完整的Lorentzian案例中建立的集成版本的结果。
We prove integrated local energy decay for the damped wave equation on stationary, asymptotically flat space-times in (1 + 3) dimensions. Local energy decay constitutes a powerful tool in the study of dispersive partial differential equations on such geometric backgrounds. By utilizing the geometric control condition to handle trapped trajectories, we are able to recover high frequency estimates without any loss. We may then apply known estimates from the work of Metcalfe, Sterbenz, and Tataru in the medium and low frequency regimes in order to establish local energy decay. This generalizes the integrated version of results established by Bouclet and Royer from the setting of asymptotically Euclidean manifolds to the full Lorentzian case.
