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

这个项目要做的工作涉及偏微分方程解的时间演化，特别是非线性波动方程。在这些研究中出现的基本问题是解决方案崩溃或破裂的可能性。对于某些方程，解是整体存在的，而对于另一些方程，有限时间后解将不存在。工作的一个重点是研究低正则解，即具有低正则数据的解。其动机来自于物理学中能量守恒的方程。如果局部存在，则守恒能量将保证在能量规范中无限期地存在。目前的结果要求大量衍生品必须是小额的，这让人看不出会出什么问题。除了研究解的存在周期外，还将努力描述解在爆炸开始时的行为，这可能是当今这一领域研究人员面临的最具挑战性的问题之一。物理学中的许多重要方程都可以写成非线性波动方程组。其中包括固定规范中的经典场论、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力学方程。对这些系统的解如何随时间发展的数学研究集中在数学分析中正在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.