Nonlinear Wave Progatation

建议9801558非线性波传播计划概述马库斯·基尔许多有趣的非线性发展方程共有的一个困难是所谓的L-p不适定性。也就是说，当p不等于2时，不能使用单个L-p空间(指数为p的勒贝格空间)在每个固定时间有效地将相应的线性时间演化与初始数据进行比较。(当p=2时，通常存在能量守恒定律，它正好提供了这样的估计。)这个障碍存在于从经典场论到用Glimm方法在一维空间中处理的双曲组的高维类似物的各种方程中。我们建议在非线性波动方程和非线性薛定谔方程的背景下研究两种处理L-p不适定性的分析方法。第一个是相应线性问题的时空估计。我们建议与陶渊明合作，分别在2维和3维空间研究薛定谔方程和波动方程的端点Strichartz估计。这将是陶渊明和作者之前工作的延伸。其次，我们提出研究广义Soblev估计和共形紧化方法。我们将与C.Sogge和H.Smith合作，研究在障碍物外部具有小初始数据的拟线性波动方程组的全局时间存在性的具体问题。最后，我们想把这些技巧应用到双曲型Yang-Mills方程中规范的选择如何影响解的正则性的问题。目前正在研究的许多有趣系统(例如流体流动或LIGO引力波探测器)背后的物理原理让科学家们在非线性数学中遇到了许多非常困难的问题。当我们考虑流体力学和广义相对论中出现的非线性偏微分方程组时，一些自然的问题是“一个解在很长时间内有多大”，甚至“一个解是否永远存在”。对于这些问题，我们只有非常不完整的答案。本提案旨在进一步了解可能最终应用于物理系统的非线性双曲偏微分方程组中有前途的技术。例如，上面关于终点Strichartz估计的问题可能有助于揭示波的衰减特性。上面提到的障碍问题是广义相对论中Schwarzschild空间稳定性的玩具模型。

proposal 9801558 NONLINEAR WAVE PROPAGATION PROJECT SUMMARY Markus Keel A difficulty common to many interesting nonlinear evolution equations is so-called L-p ill-posedness. That is, the corresponding linear time evolutions cannot be effectively compared at each fixed time to the initial data using a single L-p space (Lebesgue space with exponent p) when p is not equal to 2. (When p = 2, there is often a conservation of energy which provides just such an estimate.) This obstacle is present in equations ranging from classical field theory to the higher dimensional analogues of hyperbolic systems dealt with in 1 space dimension by Glimm's method. We propose to study two analytical methods developed to deal with L-p ill-posedness in the context of nonlinear wave equations and nonlinear Schrodinger equations. The first is space-time estimates for the corresponding linear problems. We propose to study in collaboration with T. Tao the endpoint Strichartz estimates for the Schrodinger and wave equation in space dimensions 2 and 3, respectively. This would be an extension of previous work by Tao and the author. We propose second to study generalized Sobolev estimates and the method of conformal compactification. We will be looking, in collaboration with C. Sogge and H. Smith, at the specific problem of global in time existence for a system of quasilinear wave equations with small initial data in the exterior of an obstacle. Finally, we would like to bring these techniques to the question of how the choice of gauge affects regularity of the solution in the context of the hyperbolic Yang-Mills equations. The physics behind many interesting systems currently being investigated (for example, fluid flow or the LIGO gravitational wave detectors) leads scientists to a number of very difficult questions in nonlinear mathematics. When we consider the nonlinear partial differential equations arising in fluid mechanics and general relativity, some natural questions are "how big is a solution at long times" or even "does a solution exist for all times". We have only very incomplete answers to these questions. The present proposal aims to further understand promising techniques in nonlinear hyperbolic pde which might eventually be brought to physical systems. For example, the question above regarding endpoint Strichartz estimates may shed light on the decay properties of waves. The obstacle problem mentioned above is a toy model for the stability of the Schwarzschild space in general relativity.