Harmonic Analysis and Hyperbolic Partial Differential Equations

摘要：--------------------------------------------------------研究者主要研究低正则性指标下双曲型方程解的行为。关键工具是通过波包技术构建具有最小正则度量的线性波动方程的近似解，然后用于建立精确解的Strichartz和相关估计。本工作的一个应用是具有低索波正则性初始数据的拟线性双曲方程的适定性。在联合工作中，研究者建立了一般的准线性方程在非齐次性下的二次增长的最佳可能结果；提议的工作是研究在特殊情况下放松正则性假设，比如爱因斯坦真空方程，在这种情况下，零条件表明更好的结果应该成立。波包技术也被用于建立黎曼流形上具有有限可微度量的特征函数的范数估计。提出的研究包括建立具有Lp缩紧曲率的紧流形的最佳可能界。本文还应用上述方法建立了具有Dirichlet条件的凹凸障碍混合型波动方程的范数估计解。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与无障碍情况下相同的解范数估计。拟议的研究涉及波在粗糙介质中的传播；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。通过研究一类特定的局部孤立波的性质，研究者能够回答关于在这种介质中传播的一般波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，可以认为波与自身相互作用的情况。一个这样的例子是从爱因斯坦的广义相对论中产生的引力场方程，其中空间的几何形状本身就是这个方程的对象。当考虑到该理论可能导致什么样的奇点时，必然会产生粗糙的解，从而产生粗糙的介质。该研究对粗糙介质的基本振动模式的研究也具有启示意义。众所周知，在这种介质中的折射可以导致能量的高度集中，这可以通过检查这些模式来检测。正在做的工作是把可能的能量集中程度与底层介质的粗糙度联系起来。这项研究在研究波对障碍物的反射方面也有很好的应用。研究粗糙介质的技术被用来证明反射凸障碍物的波的扩散程度必须与不反射的波的传播程度相同。

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.