Harmonic Analysis of Waves and Eigenfunctions

波的调和分析和特征函数建议的研究摘要Hart F.Smith这个项目将研究双曲方程在低正则性指标设置下的解，以及这些指标的特征函数的行为。我们的主要工作将是获得解和特征函数的Lp范数界，并量化这些范数界对介质的Holder光滑性的依赖。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。先前的研究已经表明，在粗糙的环境中，能量的某些局部化是可能的。在这个项目中，我们将尝试获得关于可能的能量集中度和分散率的最佳结果。要使用的关键工具是提供解的近似表示的波包分解。频率相关的标度自变量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带边界流形上的波和本征函数的研究。这是通过将度量反射到边界以获得开集上的Lipschitz度量来完成的。由此产生的测地线流的几何形状与短尺度界相结合，建立了带边界流形上特征函数的最佳Lp界。本研究项目将研究粗糙介质中的定常振动模式和行波；粗糙介质是指控制波速的物理从点到点突然变化的介质。这将使用行波表示为相干脉冲的和来实现，每个相干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的集中程度，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。它们还提供有关海浪从障碍物上反射的信息。我们将研究多次反射如何增加行波的集中度，并证明从凸起障碍物反射的波的色散速度与没有反射的波的传播速度相似。

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.