Nonlinear Wave Progatation

提案9801558非线性波传播项目摘要Markus Keel许多有趣的非线性演化方程的共同困难 就是所谓的L-p不适定性 也就是说，相应的线性时间演化不能被有效地 在每个固定时间与初始数据进行比较 使用单个L-p空间（指数为p的勒贝格空间） 当p不等于2时。 (When p = 2时，通常有一个守恒量， 能量提供了这样的估计。） 这个障碍存在于以下方程中： 经典场论的高维类似物 用Glimm方法处理一维双曲型方程组。 我们提出了两种分析方法来处理非线性波动方程的L-p不适定性， 非线性薛定谔方程 第一个是相应线性问题的时空估计。 我们建议合作学习 和T.中薛定谔方程和波动方程的Tao端点Schehartz估计 空间维度分别为2和3。 这将是一个延伸， 陶和作者以前的工作。 我们建议第二个研究广义Sobolev估计和 保角紧化方法 我们将在 与C合作。Sogge和H.史密斯，在 在时间上全局存在的特殊问题 小初值拟线性波动方程组 在障碍物的外部。 最后，我们想把这些技术的问题， 如何选择规范的影响规律性的解决方案， 双曲杨-米尔斯方程的上下文。 目前许多有趣的系统背后的物理学 被调查（例如，流体流动或 LIGO引力波探测器）导致科学家在非线性数学中遇到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的非线性偏微分方程时， 自然的问题是“在长时间内解决方案有多大”，甚至 “是否存在一种解决办法”。我们只有非常不完整的 这些问题的答案。 本提案旨在进一步了解有前途的技术 在非线性双曲型PDE中，最终可能导致 物理系统。 例如，上面的问题 关于端点，Eschhartz估计可以揭示波的衰减特性。 提到的障碍问题 上面是史瓦西空间稳定性的玩具模型 在广义相对论中。

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.