Harmonic Analysis of Waves and Eigenfunctions

波与本征函数的调和分析本征函数的研究摘要本项目将研究低正则度规下双曲方程的解，以及这类规下本征函数的行为。我们的主要工作将是获得解和特征函数的Lp范数界，并量化这些界对介质的Holder平滑性的依赖。一个重要的问题是控制粗糙双曲方程解的能量传播。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，在粗糙的环境中，能量的某些定位是可能的。在这个项目中，我们将尝试在可能的能量集中程度和色散率上获得最佳结果。要使用的关键工具是波包分解，它提供了解的近似表示。频率相关的尺度参数将被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有边界流形上的波和特征函数的研究。这是通过在边界上反射度量来获得开集上的Lipschitz度量来实现的。所得到的测地线流的几何形状与短尺度边界相结合，建立了具有边界的流形上特征函数的最优Lp边界。本研究项目将研究粗糙介质中的静止振动模式和行波；粗糙介质是指控制波的速度的物理特性在点与点之间突然变化的介质。这将通过将行波表示为相干脉冲的总和来实现，每个脉冲都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。它们还能提供障碍物反射波的信息。我们将研究多重反射如何增加行波的浓度，并表明从凸障碍物反射的波与不反射的波以相似的速率分散。

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.