FRG: Collaborative Research: Analysis of the Einstein Constraint Equations

FRG：合作研究：爱因斯坦约束方程的分析

• 批准号: 1262982
• 负责人: Michael Holst
• 金额: $ 25.15万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1262982&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Analysis; Einstein

AbstractAward: DMS 1265187, 1262982, 1263431, 1263544PI: Rafe R Mazzeo, Stanford University PI: Michael Holst, University of California - San Diego PI: Jim Isenberg, University of Oregon PI: David Maxell, University of AlaskaThe goal of this project is to understand the extent to which one can parametrize and construct initial data sets for the Einstein evolution equations. We plan to capitalize on the recent progress using the conformal method to obtain new existence results in the nonconstant mean curvature (non-CMC) setting, to understand the limits of these methods, and then to develop alternate techniques toward these same goals, including degree theory, a priori estiimates, and gluing methods. The Lichnerowicz equation, central to the conformal method, is a semilinear elliptic equation. Due to the mixed sign of its nonlinear exponents, it is of a type not yet fully understood. The full Lichnerowicz-Choquet-Bruhat-York set of equations is a more difficult coupled system which incorporates features presenting new analytic subtleties. The ultimate aim is to provide a complete parametrization of initial data sets, particularly in the non-CMC setting, not only on compact backgrounds but also for manifolds with asymptotically Euclidean, hyperbolic or cylindrical ends, all of which are highly relevant for physical applications. In exploring new methods, we plan to use new and advanced analytical tools, as well as increasingly accurate and flexible numerical simulation techniques. Technical advances made in the course of this project should have a substantial application to many other equations of this general type which play important roles in other parts of pure and applied mathematics and mathematical physics. Einstein's gravitational field theory is a remarkably accurate mathematical model of gravitational physics, which does an excellent job of predicting and modeling gravitational phenomena at both the astrophysical and cosmological scales. It is consistent with every known gravitational observation and experiment. From the point of view of underlying mathematics, Einstein's theory involves two very distinct types of equations. The study of dynamics of gravitational fields involves the analysis of the Einstein equations as a nonlinear system of time-dependent evolution partial differential equations (PDE), while the study of initial data sets representing gravitational states involves Riemannian geometry and the study of the Einstein constraint equations as a nonlinear system of time-independent PDE.The last decade has witnessed remarkable progress in under- standing both equations. This project focuses on developing a more complete understanding of the constraint equations. The study of the Einstein equations presents a very important point of contact between mathematics and physics, one which has motivated many advances in differential geometry and PDE on the one side, and which also has provided a compelling and accurate model of the physical world, both on the astrophysical and on the cosmological scales. This project has the potential for settling significant open questions in this area.

摘要奖项：DMS 1265187, 1262982, 1263431, 1263544 PI：Rafe R Mazzeo，斯坦福大学 PI：Michael Holst，加州大学圣地亚哥分校 PI：Jim Isenberg，俄勒冈大学 PI：David Maxell，阿拉斯加大学 该项目的目标是了解人们可以在多大程度上参数化和构建爱因斯坦演化的初始数据集方程。我们计划利用共形方法的最新进展，在非恒定平均曲率（非 CMC）设置中获得新的存在结果，了解这些方法的局限性，然后开发替代技术来实现这些相同的目标，包括度数理论、先验估计和粘合方法。 Lichnerowicz 方程是共形方法的核心，是一个半线性椭圆方程。由于其非线性指数的混合符号，它是一种尚未完全理解的类型。完整的 Lichnerowicz-Choquet-Bruhat-York 方程组是一个更困难的耦合系统，它包含了呈现新的分析细节的特征。最终目标是提供初始数据集的完整参数化，特别是在非 CMC 设置中，不仅适用于紧凑背景，而且适用于具有渐近欧几里德、双曲线或圆柱端的流形，所有这些都与物理应用高度相关。 在探索新方法时，我们计划使用新的、先进的分析工具，以及日益准确和灵活的数值模拟技术。该项目过程中取得的技术进步应该对这种一般类型的许多其他方程有实质性的应用，这些方程在纯数学和应用数学以及数学物理的其他部分中发挥着重要作用。爱因斯坦的引力场理论是一个非常精确的引力物理数学模型，它在天体物理和宇宙学尺度上的引力现象预测和建模方面做得非常出色。它与所有已知的引力观测和实验都是一致的。从基础数学的角度来看，爱因斯坦的理论涉及两种截然不同的方程类型。引力场动力学的研究涉及将爱因斯坦方程作为随时间演化偏微分方程（PDE）的非线性系统进行分析，而表示引力状态的初始数据集的研究则涉及黎曼几何和爱因斯坦约束的研究方程作为与时间无关的偏微分方程的非线性系统。过去十年在理解这两个方程方面取得了显着的进展。该项目的重点是更全面地理解约束方程。爱因斯坦方程的研究提出了数学和物理学之间非常重要的联系点，一方面推动了微分几何和偏微分方程的许多进步，另一方面也提供了物理世界的令人信服和准确的模型，两者在天体物理学和宇宙学尺度上。该项目有可能解决该领域的重大悬而未决的问题。