On the Global Regularity Properties of Nonlinear Wave Equations

非线性演化偏微分方程理论的中心问题之一是解的规律性或分解问题。在流体力学中，这类最著名的问题涉及三维欧拉方程和纳维-斯托克斯方程的规律性。另一个众所周知的问题是找到可压缩欧拉方程的全局广义解的正确概念，包括激波不连续。在广义相对论中也出现了类似的问题。规律性问题由彭罗斯提出，其代号为“宇宙审查”。它粗略地指出，爱因斯坦场方程的稳定解应该排除“裸奇点”，即没有被黑洞隐藏的奇点。今天，这些问题中的大多数都是遥不可及的。我的目标是通过将我的注意力集中在杨-米尔斯场方程和波图场方程的简化情况上，帮助开发解决这些基本问题的工具。众所周知，杨-米尔斯方程是我们最成功的粒子物理理论，即所谓的标准模型的核心。当我们寻找爱因斯坦方程的特殊解时，波动映射方程自然出现。控制流体和气体行为的欧拉方程和纳维-斯托克斯方程以及广义相对论的爱因斯坦场方程的解的分解问题具有根本意义。在前者的情况下，这个问题与流体的高度不稳定行为密切相关，并与控制湍流和预测天气等实际问题有关。就广义相对论而言，这个问题对我们理解宇宙有着深远的意义，因为黑洞和奇点是广义相对论最重要、也是最不为人所知的预言。虽然从物理角度来看，这些理论似乎非常不同，但在数学上它们有很多相似之处。由于这些问题的非线性特征，它们在整个世纪都导致了新的定性技术的发展，数学家在其中发挥了主导作用。其中一些技术现在在实践中经常使用。最终的目标还很遥远，如果这个建议能帮助解决这些难题，研究者会认为它是一个巨大的成功。

9706754 Klainerman One of the central issues in the theory of nonlinear Partial Differential Equations of evolution is that of regularity or break-down of solutions. The best known problems of this type, in fluid mechanics, concern the regularity of the three dimensional Euler and Navier-Stokes equations. Another well known problem is that of finding a correct concept of global generalized solutions, including shock discontinuities, to compressible Euler equations. Similar questions arise also in General Relativity. The problem of regularity has been formulated, by Penrose, under the code name of "Cosmic Censorship". It states, roughly, that the stable solutions of the Einstein Field Equations should exclude "naked singularities", i.e. singularities not hidden by black holes. Most of these problems are today out of reach. My objective is to help develop tools for attacking these fundamental problems by concentrating my attention on the simplified situation of the Yang-Mills and Wave Maps field equations. As is known, the Yang-Mills equations are at the heart of our most successful particle physics theory, the so called standard model. The Wave Maps equations appear naturally when we look for special solutions of the Einstein equations. The question of break-down of solutions to the Euler and Navier-Stokes equations governing the behavior of fluids and gases and of the Einstein field equations of General Relativity is of fundamental significance. In the case of the former, this problem is intimately tied to the highly unstable behavior of fluids and has bearing on the practical problems of taming turbulence and predicting the weather. In the case of General Relativity, the problem has deep significance to our understanding of the Universe, since black holes and singularities are the most important, and least understood, predictions of General Relativity. While from a physical point of view these theories seem very different, mathematically they bear a lot of similarities. Due to the non-linear character of these problems, they have led, throughout this century, to the development of new qualitative techniques in which mathematicians have played a leading role. Some of these techniques are now routinely used in practice. The final goals remain distant, and the investigator would consider this proposal a ringing success if it can help make a dint on these difficult problems.