Nonlinear Partial Differential Equations in Relativity and Geometry

相对论和几何中的非线性偏微分方程

• 批准号: 9704760
• 负责人: Gilbert Weinstein
• 金额: $ 6.86万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-06-15 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704760&HistoricalAwards=false
• 关键词: Nonlinear; Partial; Differential; Equations; Relativity

This project will address a number of problems, mainly in the following two related areas: harmonic maps, and the Einstein equations. The investigator will study the asymptotic behavior of harmonic maps into negatively curved spaces near prescribed singularities. This study is motivated by, and has applications to, the problem of stationary black holes in general relativity. The PI also proposes to generalize blow-up results for wave maps. Wave maps which develop singularities from regular initial data will be constructed from harmonic maps defined on hyperbolic space. In the area of general relativity, the investigator will continue his work on equilibrium configurations of co-axially rotating charged black holes. A uniqueness theorem is sought independent of the number of black holes present. Also proposed is the study of the topology of the space of initial data for the Einstein equations, and questions related to equilibrium configurations of rotating and nonrotating perfect fluid bodies. Harmonic maps are mappings between Riemannian manifolds (geometrical spaces) which satisfy nonlinear elliptic partial differential equations. They were first introduced on two dimensional domains by C. B. Morrey Jr. in the 1950's in his study of Plateau's Problem in Riemannian manifolds: to find a soap film which spans a given contour. Harmonic maps have since found numerous applications in physics, geometry, and other fields. Harmonic maps with prescribed singularities will be used in this project to model equilibrium configurations of co-axially rotating charged black holes. Understanding these configurations is an important step in the future study of the evolution equations. Wave maps are the dynamical counterparts of harmonic maps. They satisfy nonlinear hyperbolic partial differential equations. These equations have been studied by many researchers as a first step in a program to understand more complex evolution equations of classical field theory such as the Einstein field equations of general rel ativity. The Einstein field equations model the large scale evolution of spacetime. One of the interesting features of this class of nonlinear evolution equations is that solutions starting from regular data may sometimes yield blow-up after a finite time laps, leading to singularities. The investigator will construct and study wave maps with such singularities. Another problem, with applications to astrophysics, to be addressed in this project will be to construct a model for relativistic dense rotating stars in equilibrium.

这个项目将解决一些问题，主要是在以下两个相关的领域：谐波映射和爱因斯坦方程。研究者将研究进入负弯曲空间的调和映射在指定奇点附近的渐近行为。这项研究的动机是，并有应用，在广义相对论中的静态黑洞的问题。PI还建议推广波映射的爆破结果。从规则的初始数据发展奇点的波映射将由定义在双曲空间上的调和映射构造。在广义相对论领域，研究人员将继续他对同轴旋转带电黑洞平衡构型的研究。唯一性定理寻求独立的黑洞的数量。还提出了爱因斯坦方程的初始数据空间的拓扑结构的研究，以及旋转和非旋转的完美流体体的平衡配置有关的问题。调和映射是满足非线性椭圆型偏微分方程的黎曼流形（几何空间）之间的映射。 它们是由C. B。小莫雷在20世纪50年代在他的研究高原的问题在黎曼流形：找到一个肥皂膜跨越一个给定的轮廓。调和映射在物理学、几何学和其他领域有着广泛的应用。在这个计画中，我们将使用具有指定奇异性的调和映射来模拟同轴旋转带电黑洞的平衡组态。了解这些组态是今后发展方程研究的重要一步。波映射是调和映射的动力学对应物。它们满足非线性双曲型偏微分方程。这些方程已经被许多研究人员研究，作为理解经典场论更复杂演化方程（如广义相对论的爱因斯坦场方程）的第一步。爱因斯坦场方程模拟了时空的大尺度演化。这类非线性发展方程的一个有趣的特征是，从规则数据开始的解有时会在有限时间圈后产生爆破，导致奇点。研究者将构造和研究具有这种奇点的波图。另一个应用于天体物理学的问题，将在这个项目中解决，将是建立一个相对论性的高密度旋转恒星的平衡模型。