Harmonic Analysis of Waves and Eigenfunctions

波的谐波分析与本征函数--拟研究摘要。本计画将研究双曲型方程在低正则度规下的解，以及这种度规的本征函数的行为。我们的主要努力将是获得Lp范数的解决方案和本征函数的界限，并量化这些界限的保持器光滑的媒体上的依赖。一个重要的问题是控制粗糙双曲型方程解的能量传播。对于粗糙介质，测地线流是不适定的，几何光学方法失效。先前的研究表明，在粗糙环境中，能量的某些局部化是可能的。在本项目中，我们将尝试获得可能的能量集中程度和分散率的最佳结果。要使用的关键工具是波包分解，提供近似的解决方案。 频率相关的标度参数将被调用来建立粗糙介质中的波表现出经典色散的短时间尺度。研究者还将把上述方法应用于研究有边界流形上的波和本征函数。这是通过在边界上反射度量以获得开集上的Lipschitz度量来完成的。由此产生的测地线流的几何形状结合短尺度边界，建立最佳Lp边界的特征函数的流形上的boundary.This研究项目将研究固定的振动模式和粗糙介质中的行波;一个粗糙的介质是一个物理支配波的速度突然改变从一点到另一点。这将使用行波表示为相干脉冲的总和来完成，每个脉冲以非常简单的方式移动。研究人员能够量化振动模式和波可以集中的程度，这取决于底层介质的粗糙度。这些结果对于研究非线性波的相互作用以及振动和波的外部可观测性问题具有重要意义。它们还提供了有关障碍物反射波的信息。我们将研究多次反射如何增加行波的集中度，并表明反射离开凸障碍物的波以与没有反射的波相似的速率传播。

Harmonic Analysis of Waves and EigenfunctionsAbstract of Proposed ResearchHart F. SmithThis project will study solutions of hyperbolic equations in the setting of metrics of low regularity, and the behavior of eigenfunctions for such metrics. Our primary efforts will be to obtain Lp norm bounds on solutions and eigenfunctions, and quantifying the dependence of these bounds on the Holder smoothness of the media. An important issue is controlling the propagation of energy for solutions of rough hyperbolic equations. For rough media the geodesic flow is ill-posed, and geometric optics methods break down. Prior research has shown that some localization of energy is possible in rough settings. In this project we shall attempt to obtain optimal results on the possible degree of energy concentration and the rate of dispersion. The key tool to be used is a wave packet decomposition that provides approximate representations of the solution. Frequency dependent scaling arguments will be invoked to establish the short time scales on which waves in rough media exhibit classical dispersion. The investigator will also adapt the above methods to the study of waves and eigenfunctions on manifolds with boundary. This is done by reflecting the metric across the boundary to obtain a Lipschitz metric on an open set. The geometry of the resulting geodesic flow combines with the short scale bounds to establish optimal Lp bounds for eigenfunctions on manifolds with boundary.This research project will investigate stationary vibrational modes and travelling waves in rough media; a rough medium being one where the physics which governs the speed of waves changes abruptly from point to point. This will be done using a representation of travelling waves as a sum of coherent pulses, each of which moves in a very simple fashion. The investigator is able to quantify the degree to which vibrational modes and waves can concentrate, depending on the roughness of the underlying media. Such results are important for the study of nonlinear wave interactions and questions of the outside observability of vibrations and waves. They also provide information on the reflection of waves off obstacles. We will study how multiple reflections increase the concentration of travelling waves, and also show that waves reflecting off convex obstacles disperse at similar rates to waves travelling without reflection.