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: 1263431
• 负责人: James Isenberg
• 金额: $ 20.22万
• 依托单位: University of Oregon Eugene
• 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 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1263431&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.

Abstractaward：DMS 1265187，1262982，1263431，12635444PI：Rafe R Mazzeo，Stanford University PI：Michael Holst，加利福尼亚州 - 圣地亚哥PI：Jim Isenberg：Jim Isenberg，俄勒冈大学PI：俄勒冈大学的目标：Alaskathe of Alaskathe的范围，以了解该项目的范围，以了解范围的范围。爱因斯坦进化方程。我们计划利用保形方法利用最近的进度来获得新的存在，从而在非稳定均值曲率（非CMC）设置中，了解这些方法的限制，然后开发针对这些相同目标的替代技术，包括学位理论，先验性估计性和粘合方法。 Lichnerowicz方程是共形方法的核心，是半线性椭圆方程。由于其非线性指数的混合迹象，它的类型尚未完全理解。 Lichnerowicz-Choquet-Bruhat-york方程组的完整耦合系统结合了呈现新的分析微妙的功能。最终的目的是提供初始数据集的完整参数化，尤其是在非CMC设置中，不仅在紧凑的背景下，而且在渐近的欧几里得，双曲线或圆柱形末端的歧管上，所有这些都与物理应用高度相关。 在探索新方法时，我们计划使用新的和高级的分析工具，以及越来越准确且灵活的数值模拟技术。在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式有实质性的应用，这些方程在纯数学和应用数学和数学物理学的其他部分中都起着重要作用。爱因斯坦的重力场理论是一个非常准确的重力物理学数学模型，在天体物理和宇宙学量表上预测和建模引力现象方面表现出色。它与每个已知的重力观察和实验一致。从基本数学的角度来看，爱因斯坦的理论涉及两种非常不同的方程式。对引力场动力学的研究涉及对爱因斯坦方程的分析，该方程是一个非线性的时间依赖性进化局部微分方程（PDE），而对代表引力状态的初始数据集的研究涉及Riemannian的几何形状，涉及Riemannian的几何形状，并且对EINSTEIN的约束方程式的研究都在持续依赖于时态依赖于PED。方程式。该项目着重于对约束方程的更完整理解。对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接触点，这在一侧促进了差异几何和PDE的许多进步，并且还提供了物理世界的引人入胜且准确的模型，包括在天体物理学和宇宙学量表上。该项目有可能在该领域解决重大的开放问题。