Mathematical Sciences: Global Properties of Nonlinear Hyperbolic Equations Arising in Mathematical Physics

这个项目继续了首席研究员在数学物理中出现的与非线性偏微分方程式有关的问题上的工作。这些方程通常是双曲型的。这项工作的目的包括发展证明经典场方程解的整体存在性、稳定性和渐近性态结果的方法。需要考虑的问题包括爱因斯坦真空方程的Minkowski解、SchwartzChild解和Kerr解的全局非线性稳定性。此外，还将研究一般初始条件下杨-米尔斯方程的渐近行为。我们还将在小初始条件下建立Maxwell-Dirac方程的整体存在性和渐近结果。二线研究包括分析三维空间中拟线性波动方程的奇点的形成。众所周知，某些类型的方程的解在有限时间内会出现奇点。在爆炸开始之前，人们对这些奇点的性质知之甚少，而三维奇点是所有奇点中了解最少的。要克服的困难之一是设想一种可视化解决方案的方法，其奇点只有在指数级长的时间段过去后才会出现。我们将继续研究半线性波动方程整体光滑解的新方法。在波动方程中，非线性项是未知函数的幂，如果幂小于5，则解是无限光滑的。对于更高的国家来说，几乎什么都不知道，尽管人们相信一些版本的单一地方能源估计将提供解决这个问题所需的信息。除了在数学物理中的自然应用之外，这项工作还将提供适用于分析大类微分方程的方法。

This project continues work of the principal investigator on problems related to nonlinear partial differential equations arising in mathematical physics. The equations are typically of hyperbolic type. Objectives of this work include the development of methods for proving global existence, stability and asymptotic behavior results for classical field equations. Among the problems to be considered are the global nonlinear stability of Minkowski, Schwartzchild and Kerr solutions of the Einstein vacuum equations. Also the asymptotic behavior of the Yang-Mills equations for general initial conditions will be studied. Work will also be done on establishing global existence and asymptotic results for the Maxwell-Dirac equations subject to small initial conditions. A second line investigation involves analysis of the formation of singularities for quasilinear wave equations in three space dimensions. It has been known for some time that solutions to certain classes of equations have developed singularities in finite time . Very little is known about the nature of these singularities near the onset of blow up and the three dimensional case is the least understood of all. One of the difficulties to be overcome is conceiving a means for visualizing solutions whose singularities only occur after exponentially long periods of time have elapsed. Work will continue toward developing new methods for the study of global smooth solutions for semilinear wave equations. In wave equations where the nonlinear term is a power of the unknown function, it is known that solutions remain smooth indefinitely if the power is less than five. For higher powers almost nothing is known, although it is believed that some version of singular local energy estimates will provide the information necessary for the resolution of this problem. In addition to natural applications to mathematical physics, this work will provide methods which will be applicable to the analysis of large classes of differential equations.