Regularity Properties of Nonlinear Evolution Equations

本文的主要目标是发展新的分析方法来处理非线性波动方程的全局正则性的基本问题，如波动映射、杨-米尔斯方程和爱因斯坦真空方程。我们提出了两个具体的问题进行调查。第一个是关于波映射的临界整体适定性，第二个是建立爱因斯坦真空方程在H^2 $中的局部适定性。非线性波方程是我们的一些基本物理理论的核心，如广义相对论，电动力学，弹性力学等。尽管在上个世纪取得了很大的进展，我们对非线性波的认识仍然是初步的。该提案概述了我们预期取得重大进展的一些方向。主要的是爱因斯坦场方程。众所周知，这些方程可以产生黑洞和奇点。这一事实使得它必须发展一个理论，粗糙的解决方案，这些方程。 波映射方程可以看作是爱因斯坦方程的简化模型问题。我们希望更好地理解这些方程的粗糙解将有助于我们在与前方程的联系方面取得进展。

ABSTRACTThe main goal of the proposal is to develop new analytic methods to deal with the fundamental issue of global regularity for Nonlinear Wave Equations such as Wave Maps, Yang-Mills and the Einstein Vacuum Equations. We propose two specific problems to investigate. The first concerns the critical global well-posedness for Wave Maps and the second is to establish local well posedness in $H^2$ for the Einstein Vacuum equations.Nonlinear Wave equations are at the heart of some of our basic physical theory such as General Relativity, Electrodynamics, Elasticity etc. Despite a lot of progress made throughout last century our knowledge of nonlinear waves remains rudimentary. The proposal outlines some directions in which we expect to make considerable progress. The main one concerns the Einstein field equations. It is well known that these equations can develop black holes and singularities. This fact makes it imperative to develop a theory for rough solutions for these equations. The Wave Maps equations can be viewed as a simplified model problem for the Einstein equations. We hope that a better understanding of rough solutions of these equations will help us make progress in connection to the former equations also.