Mathematical Sciences: Existence and Blow-up of Solutions of Nonlinear Wave Equations

本课题的工作涉及到偏微分方程，特别是非线性波动方程解的时间演化。在这些研究中出现的基本问题是解决方案破裂或爆炸的可能性。有些方程具有解的整体存在性，而另一些方程在有限时间后解将不存在。工作的一个重点是研究低正则性解，即具有低正则性数据的解。其动机来自于能量守恒的物理方程。如果能建立局部存在，则守恒能量将保证无限地存在于能量范数中。目前的结果要求大量的衍生品必须是小的，这样就没有可能出错的感觉。除了研究解的存在期外，还将尝试描述解在爆炸开始时的行为，这可能是当今该领域研究人员面临的最具挑战性的问题之一。物理学中许多重要的方程都可以写成非线性波动方程的系统。这些包括经典的定规场论、连续介质力学和爱因斯坦、杨·米尔斯的经典方程以及非线性弹性方程。这些系统的解如何随时间变化的数学研究，集中在数学分析中研究的一些最重要的问题上。

Work to be done on this project concerns the time evolution of solutions to partial differential equations, especially nonlinear wave equations. The basic question which arises in these studies is the possibility of breakdown or blow-up of solutions. For some equations one has global existence of solutions whereas for others the solution will fail to exist after some finite time. One focus of the work is in the study of low regularity solutions, that is, solutions with low regularity data. The motivation is from equations in physics in which energy is conserved. If local existence can be established, the conserved energy will guarantee existence in the energy norm indefinitely. Present results require that a large number of derivatives be small, leaving no sense of what can go wrong. In addition to studying the period of existence of solutions, work will also be done in trying to describe the behavior of solutions at the onset of blow-up, perhaps one of the most challenging problems facing researchers in this area today. Many important equations in physics can be written as systems of nonlinear wave equations. These include classical field theory in a fixed gauge, continuum mechanics and the classical equations of Einstein, Yang Mills and equations of nonlinear elasticity. The mathematical study of how the solutions to these systems progress with time centers on some of the most important questions under investigation in mathematical analysis.