GEOMETRY AND PDE ANALYSIS IN GENERAL RELATIVITY.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本研究项目的目标是研究几何、非线性偏微分方程（PDE）分析和广义相对论（GR）交叉领域的一些主题。这些主题有一个共同的主题：研究爱因斯坦方程在物理有趣情况下的全局柯西问题的解，并描述这些解的渐近性。要做到这一点，必须解决洛伦兹几何和黎曼几何中的问题，并且需要研究曲率和几何量，特别是沿零超曲面的曲率和几何量。最近由首席研究员（PI）得到的结果，推广了D. Christodoulou和S. Klainerman的工作，证明了爱因斯坦真空（EV）方程在渐近平坦设置下解的整体存在性和唯一性，在更一般的条件下导致边界估计。后者表明，就初始数据在无穷远处衰减的假设而言，PI的主要定理中的条件是尖锐的。此外，新的结果描述了这些更一般情况下的渐近行为。这些新结果与证明中使用和发展的几何和解析理论一起，为进一步研究渐近平坦初始数据下EV方程的物理有趣解提供了所需的工具。因此，PI计划计算允许非各向同性质量的时空渐近性。这将以直接的方式从PI的工作中获得的结果。这将使以后解决另一个主要问题成为可能，即在更一般的情况下研究零无穷处的角动量。针对后一个目标，本项目将在三种具体情况下研究EV方程的柯西问题，将新的工具和结果应用于这些未探索的情况，以描述EV方程相应解的渐近性。广义相对论是在牛顿万有引力理论和牛顿运动方程的基础上，对空间、时间和引力的统一理论。广义相对论的定律是爱因斯坦方程，将时空的曲率与其物质含量联系起来。探索这些方程将有助于我们更好地理解宇宙。揭示几何、分析和物理之间复杂的相互作用是GR中最丰富的挑战之一。本项目旨在研究这种相互作用。在此过程中，开发的数学工具有望在分析许多结构相似的非线性偏微分方程方面取得成果。本文研究渐近平坦系统。孤立的引力系统，如双星、星团和星系，可以用爱因斯坦真空方程的渐近平坦解在广义相对论中描述，因为它们可以被认为在物质的支持之外有一个渐近平坦的区域。考虑到引力辐射，理解这类系统的渐近性是非常重要的。而著名的引力波探测实验就是基于对这种渐近平坦系统的分析。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The goal of this research project is to investigate a number of topics in the intersection of geometry, the analysis of nonlinear partial differential equations (PDE's) and general relativity (GR). These topics share a common theme: to investigate solutions of the global Cauchy problem for the Einstein equations for physically interesting situations and to describe the asymptotics of these solutions. To do this, problems in Lorentzian and Riemannian geometry have to be solved, and the study of the curvature and geometric quantities especially along null hypersurfaces is required. Recent results obtained by the principal investigator (PI), extending work by D. Christodoulou and S. Klainerman, prove global existence and uniqueness of solutions of the Einstein vacuum (EV) equations in an asymptotically flat setting under more general conditions leading to borderline estimates. The latter indicate that the conditions in the PI's main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Moreover, the new results describe the asymptotic behaviour for these more general situations. These new results together with the geometric and analytic theories used and developed in the proof provide the tools required to further investigate physically interesting solutions of the EV equations for asymptotically flat initial data. Thus, the PI plans to compute the asymptotics of spacetimes allowing non-isotropic mass. This will follow in a straightforward manner from the results obtained in the PI's work. It will make it possible to later on attack another main problem, namely the investigation of angular momentum at null infinity for more general settings. Aiming at this latter goal, this project will study the Cauchy problem for the EV equations in three specific cases, applying the new tools and results to these unexplored situations in order to describe the asymptotics of the corresponding solutions of the EV equations. General relativity, as a unified theory of space, time and gravitation, is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. The laws of GR are the Einstein equations, linking the curvature of spacetime to its matter content. Exploring these equations will lead to a better understanding of the universe. To unravel the complex interplay between geometry, analysis and physics is one of the richest challenges in GR. This project aims to investigate this interaction. Along the way, the mathematical tools developed promise to bear fruit in the analysis of the many structurally similar nonlinear PDE's. The proposed research studies asymptotically flat systems. Isolated gravitating systems such as binary stars, clusters of stars and galaxies can be described in GR by asymptotically flat solutions of the Einstein vacuum equations, as they can be thought of as having an asymptotically flat region outside the support of the matter. It is very important to understand the asymptotics of such systems also in view of gravitational radiation. And the well-known experiments to detect gravitational waves are based on the analysis of such asymptotically flat systems.