Schwarzschild-de Sitter时空膨胀区域上的线性波：前向渐近性与无穷远散射
Linear waves on the expanding region of Schwarzschild-de Sitter spacetimes: forward asymptotics and scattering from infinity
AI总结：
本研究针对Schwarzschild-de Sitter时空膨胀区域的线性波动方程，通过新型加权能量估计证明了解的渐近展开特性及散射解的存在唯一性，结果可推广至Kerr de Sitter族等膨胀时空。
摘要：
我们研究Schwarzschild-de Sitter时空宇宙学区域上线性波动方程的解。我们证明，所有足够正则的有限能量线性方程解在未来边界附近都具有特定的有限阶渐近展开式。具体而言，我们证明该渐近展开式中的若干项恒等于零，这一结果通过捕捉宇宙学区域全局膨胀的新型加权高阶能量估计得以实现。此外，我们证明了膨胀区域上线性波动方程散射解的存在性与唯一性。给定无穷远处的两组散射数据，我们构造出与前向解具有相同渐近性的解。证明过程涉及构造波动方程的渐近解，以及适用于后向问题的新型加权能量估计。该散射结果可推广至一大类膨胀时空，包括Kerr de Sitter族。
英文摘要：
We study solutions to the linear wave equation on the cosmological region of Schwarzschild-de Sitter spacetimes. We show that all sufficiently regular finite-energy solutions to the linear equation possess a particular finite-order asymptotic expansion near the future boundary. Specifically, we prove that several terms in this asymptotic expansion are identically zero. This is accomplished with new weighted higher-order energy estimates that capture the global expansion of the cosmological region. Furthermore we prove existence and uniqueness of scattering solutions to the linear wave equation on the expanding region. Given two pieces of scattering data at infinity, we construct solutions that have the same asymptotics as forward solutions. The proof involves constructing asymptotic solutions to the wave equation, as well as a new weighted energy estimate that is suitable for the backward problem. This scattering result extends to a large class of expanding spacetimes, including the Kerr de Sitter family.
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