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

这个项目要做的工作涉及时间演变 偏微分方程的解，特别是 非线性波动方程 其中出现的基本问题是 这些研究是崩溃或爆炸的可能性， 解决方案 对于某些方程，有全局存在性： 解决方案，而对于其他人来说，解决方案将不存在 在有限的时间之后。 这项工作的一个重点是研究 低正则性解，即具有低正则性的解 数据 动机来自于物理学中的方程， 能量守恒。 如果可以建立本地存在， 守恒能量将保证存在于能量范数中， 无限期 目前的结果要求大量的 衍生品的规模应该很小，不知道什么地方会出问题。在 除了研究解的存在期外， 在描述溶液的行为时， 在爆炸的开始，也许是最具挑战性的一个 目前，这一领域的研究人员面临着许多问题。 物理学中许多重要的方程可以写成 非线性波动方程组 其中包括古典 固定规范场论，连续介质力学和 爱因斯坦、杨米尔斯的经典方程和 非线性弹性 数学研究如何 这些系统的解决方案随着时间的推移而发展， 数学研究中最重要的问题 分析.

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.