Harmonic Analysis and Hyperbolic Partial Differential Equations

PI：Hart Smith，华盛顿大学DMS-0140499Abstract：--------------------------------------------------------The研究员，主要研究双曲型方程在低正则度规设置下的解的行为。关键工具是通过波包技术构造具有最小正则度量的线性波动方程的近似解，然后用这些近似解建立Strichartz和相关的精确解估计。这项工作的一个应用是研究初值为低Sobolev正则性的拟线性双曲型方程的适定性。在联合工作中，研究者建立了一般拟线性方程在非齐次性中具有二次增长的最佳可能结果；拟议的工作研究了在特殊情况下放松正则性假设，例如爱因斯坦真空方程，其中零条件表明更好的结果应该成立。波包技术也被用来建立具有有限可微性度量的黎曼流形上特征函数的范数估计。所提出的研究包括为具有Lp Pinch曲率的紧致流形建立最佳可能的界。研究人员还利用上述方法建立了凸障碍上具有Dirichlet条件的混合型波动方程解的范数估计。这是通过跨越边界反射度量来实现的，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的范数估计。拟议的研究涉及到波在粗糙介质中传播的研究；粗糙介质是指控制波速度的物理从点到点突然变化的介质。通过研究一种特殊的局域孤立波族的性质，研究者能够回答在这种介质中传播的一般波可能出现的能量集中的问题。这项工作在非线性波动方程的研究中有重要的应用，也就是波可以被认为是与自己相互作用的情况。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方程的对象。当考虑到理论可能导致什么样的奇点时，必然会出现粗糙的解，因此也就是粗糙的媒介。该研究对研究粗糙介质中的基本振动模也有一定的指导意义。众所周知，这种介质中的折射会导致能量的高度集中，可以通过检查这些模式来检测到。人们正在研究如何将可能的能量集中程度与底层介质的粗糙度联系起来。这项研究在研究波浪对障碍物的反射方面也得到了应用。用来研究粗糙介质的技术被用来证明，在凸形障碍物上反射的波必须像没有反射的波一样扩散到相同的程度。

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.