Analytic aspects of long-time dynamics in general relativity

广义相对论的爱因斯坦方程是一个几何的、非线性的偏微分方程，具有基本的双曲性质。在过去的几十年里，特别是最近几年，人们已经看到了大量的活动，应用数学分析的方法来理解这个方程解的存在性、唯一性、连续依赖性和长期行为。这包括证明空的空间和一些宇宙学模型的稳定性，而黑洞和奇点的稳定性仍然是主要的开放问题。关键方法包括通过足够强大的方法来证明线性方程的衰变扩展到非线性问题和理解非线性的结构。本计画将应用这些技术来研究广义相对论中的长期动力学问题。

The Einstein equation of general relativity is a geometric, nonlinear PDE with a fundamentally hyperboliccharacter. The last few decades, and particularly recent years, have seen a lot of activity in applying themethods of mathematical analysis to understanding the existence, uniqueness, continuous dependence, andlong-term behaviour of solutions to this equation. This has included proof of the stability of empty space andsome cosmological models, while the stability of black holes and singularities remain major open problems.Key methods include proving decay for linear equations via methods that are sufficiently robust to extend tononlinear problems and understanding the structure of nonlinearities. This project will apply these techniques toinvestigate problems in long-term dynamics in general relativity using these techniques.