The Problem of Evolution in General Relativity

摘要：该提案的主要目标是研究广义相对论爱因斯坦场方程以及其他重要物理系统的规律性。本文着重讨论了“有界平方可积曲率猜想”，根据该猜想，爱因斯坦-真空方程的初值问题可以很好地求解，其初始数据集的曲率只有平方可积边界。PI相信，这是迈向爱因斯坦方程普遍规律性理论的重要基础一步。通过关注最小正则性假设的问题，方程可以唯一地连续，希望我们至少可以找到一些理解奇异行为开始所需的工具。数学物理学的基本任务之一是提供黑洞和广义相对论爱因斯坦方程解的奇点的完整、定性描述。首席研究员已经建立了一个程序，他希望该程序将为这种理论提供严格的数学基础。本建议的主要目标是建立最小规则假设，这些假设需要由初始条件验证，以便人们仍然可以理解爱因斯坦场方程的物理解。该提案还设想了与其他重要的数学物理方程相关的研究，例如那些控制电磁场中无碰撞粒子行为的方程。

PI: Sergiu Klainerman, Princeton UniversityDMS-0245368Abstract:The main goal of the proposal is to investigate the regularity properties of the Einstein field equations of General Relativity as well as those of other important physical systems. Most emphasis be given to the `` Bounded square integrable curvature conjecture'' according to which the initial value problem for the Einstein-vacuum equations is well posed with only square integrable bounds on the curvature of its initial data set. The PI believes that this is an essential foundational step towards a generalregularity theory of the Einstein equations. By concentrating on the issue of minimal regularity assumptions for which the equations can be uniquely continued it is hoped that we can find, at least, some of the tools needed in understanding the onset of singular behavior.One of the fundamental tasks of mathematical physics is to provide a full, qualitative, description of black holes and singularities of solutions to the Einstein equations in General Relativity. The principal investigator has established a programwhich he hopes that will provide the rigorous mathematical foundations of such a theory. The main goal of this present proposal is to establish the minimal regularity assumptions that needs to be verified by the initial conditions such that one can still make sense of a physical solution to the Einstein field equations. The proposal also envisages research in connection to other important equations of mathematical physics such as those governing the behavior of collisionless particles in an electromagnetic field.