Harmonic Analysis and Hyperbolic Partial Differential Equations

调和分析和双曲偏微分方程

• 批准号: 0140499
• 负责人: Hart Smith
• 金额: $ 22.19万
• 依托单位: University of Washington
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0140499&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Hyperbolic; Partial; Differential

PI: Hart Smith, University of WashingtonDMS - 0140499Abstract:--------------------------------------------------------The investigator's research focuses on the behavior of solutions tohyperbolic equations in the setting of metrics of low regularity.The key tool is the construction, through wave packet techniques, ofapproximate solutions for linear wave equations with minimally regularmetrics, which are then used to establish Strichartz and related estimatesfor exact solutions. One application of this work is to well-posednessfor quasi-linear hyperbolic equations with initial data of low Sobolevregularity. In joint work the investigator has established a best possibleresult for general quasi-linear equations with quadratic growth in theinhomogeneity; proposed work investigates relaxing the regularity assumptionin special cases such as the Einstein vacuum equation, where a null conditionindicates that better results should hold. Wave packet techniques arealso being used to establish norm estimates for eigenfunctions onRiemannian manifolds with metrics of limited differentiability.The proposed research includes establishing best possible boundsfor compact manifolds with Lp pinched curvature. The investigatoris also adapting the above methods to establish norm estimateson solutions to mixed-type wave equations with Dirichlet conditions on aconvex obstacle. This is carried out by reflecting the metric across theboundary to obtain a Lipschitz metric on an open set. The geometry ofthe resulting geodesic flow suggests that wave packet techniques can beused to establish the same norm estimates on solutions as hold in thenon-obstacle case.The proposed research involves the study of waves traveling in rough media;a rough medium being one where the physics which governs the speed of waveschanges abruptly from point to point. By studying the properties of aspecial family of localized solitary waves, the investigator is able toanswer questions about the possible concentration of energy that can occurfor general waves traveling in such media. This work has importantapplications in the study of nonlinear wave equations; that is, situationswhere the wave can be considered to interact with itself. One such exampleis the gravitational field equation arising from Einstein's general theoryof relativity, where the geometry of space itself is the object of theequation. Rough solutions, and thus a rough media, necessarily arise whenconsidering what kind of singularities the theory can lead to. The researchalso has implications for investigating the fundamental vibrational modesin rough media. It is known that refraction in such media can lead to highconcentrations of energy that can be detected by examining these modes.Work is being done to relate the possible degree of concentration of energyto the roughness of the underlying media. The research finds applicationsas well in studying the reflection of waves off obstacles. The techniquesdeveloped to study rough media are being used to show that waves reflectingoff of convex obstacles must diffuse to the same degree as do wavestraveling without reflection.

主要研究者：哈特史密斯，华盛顿大学DMS-0140499摘要：--这一工作的一个应用是拟线性双曲型方程的适定性与低Sobolevregularity的初始数据。在联合工作中，研究人员已经建立了一个最好的结果，一般准线性方程的二次增长的不均匀性;拟议的工作调查放松的正则性approption在特殊情况下，如爱因斯坦真空方程，其中一个空条件表明，更好的结果应该持有。波包技术也被用来建立黎曼流形上具有有限可微度量的本征函数的范数估计，所提出的研究包括建立具有Lp压缩曲率的紧致流形的最佳可能界。本文还将上述方法应用于凸障碍物上Dirichlet条件下混合型波动方程解的模估计。这是通过在边界上反射度量来实现的，以获得开集上的Lipschitz度量。由此产生的测地线流的几何形状表明，波包技术可以用来建立相同的范数估计的解决方案，在then-obstacle的情况下举行。拟议的研究涉及研究波在粗糙介质中行进;一个粗糙的介质是一个物理支配的速度waveschanges突然从点到点。通过研究一类特殊的局域孤立波的性质，研究者能够回答关于一般波在这种介质中传播时可能发生的能量集中的问题。这项工作在非线性波动方程的研究中有重要的应用，也就是说，波可以被认为是与自身相互作用的情况。一个这样的例子是引力场方程产生于爱因斯坦的广义相对论，其中空间几何本身是方程的对象。粗糙的解决方案，从而粗糙的媒体，必然会出现当考虑什么样的奇点理论可以导致。该研究对研究粗糙介质中的基本振动模式也有一定的意义。众所周知，这种介质中的折射可以导致能量的高度集中，通过检查这些模式可以检测到能量的高度集中。人们正在做的工作是将能量集中的可能程度与底层介质的粗糙度联系起来。该研究也可用于研究波浪在障碍物上的反射。为研究粗糙介质而开发的技术正在被用来表明，凸面障碍物反射的波必须与没有反射的波传播达到相同的程度。