Multiscale Analysis of Hyperbolic Partial Differential Equations

该奖项支持的研究涉及线性和非线性双曲型偏微分方程解的定性行为。这些方程描述了信号以有限速度传播的情况下的波传播，例如声学、电磁学、可压缩流体……等。第一组问题涉及非线性内层。这样的层代表波，其波前在时空中占据光滑表面附近的一个非常薄的区域。波前将解的值相差很大的区域分开。传播火焰锋面和反应锋面的例子说明了这一点。数学结果将导致算法的近似，其误差趋于零，随着宽度的减少。提出的问题是平面层在相对于宽度倒数较大的时刻的行为。人们期望看到衍射效应，实际上对于这种类型的层没有已知的数学结果。对于弱耗散边界层，很可能会有类似的结果。第二个问题与贝伦杰有关。S完美匹配层算法在无界域中的波传播计算。仔细分析表明，这两层并非完全匹配。鉴于这些方法的广泛使用，对这一现象及其后果的详细分析是重要的。第三类问题涉及到借助伪微分算子推导出稳定性估计的情况。这样的操作人员散布支撑，使得他们似乎不适合研究支撑。本课题旨在研究可对称双曲系统类空间超曲面上柯西问题的尖锐有限速度和唯一性。最后，我们将研究当周期和波长相当时短波长的波在扰动周期介质中的传播。我们感兴趣的是预期会发生衍射效应的长距离传播。该研究项目解决了波传播现象的数学模型中出现的问题，即信号可以以有限速度传播并相互作用以及与所穿越的介质相互作用的物理情况。这些信号可以采用波阵面（如照相机发出的球形闪光、拍手发出的声波或内燃机的火焰阵）或射线（如光学）等形式。这类现象有一类常见的数学模型，它们在化学工程、雷达设备和超声波收发器的设计和使用、计算机图形学和光纤等各种应用中都得到了非常成功的应用。这个项目有四个部分。第一个将增加对相互作用的波前模型的理解。第二部分将研究一种非常广泛使用的计算方法，用于无界空间区域（例如雷达信号从平面反弹）中的波现象，并试图解释一些使该方法不像通常假设的那样可靠的异常情况。在第三部分，提议者将建立一个广泛的类别的模型确实表现出有限的信号速度在所有可能的情况下。最后，第四个项目旨在更好地理解通过光子材料长距离传播的光信号。PI将继续通过系列讲座、出版笔记、咨询、指导论文学生以及与工程学校和工业实验室的密切联系来开展教育工作。

The research supported by this award concerns the qualitative behavior of solutions of linear and nonlinear hyperbolic partial differential equations. Such equations describe wave propagation in situations when signals travel at finite speed, e.g. acoustics, electromagnetism, compressible fluids, ... etc. The first set of problems concerns nonlinear internal layers. Such layers represent waves which have wave fronts occupying a very thin region about a smooth surface in space time. The wave front separates regions where the values of the solution are quite different. The examples of propagating flame fronts and reaction fronts give the idea. The mathematical results will lead to algorithms for approximations whose error tends to zero as the width decreases. The problem proposed is the behavior of planar layers at times which are large relative to the reciprocal of the width. One expects to see diffractive effects and there are virtually no known mathematical results for this type for layers. It is likely that there will be analogous results for weakly dissipative boundary layers. The second problems concern Berenger?s perfectly matched layer algorithms for the computation of wave propagation in unbounded domains. Careful analysis indicates that the layers are not perfectly matched. The detailed analysis of this phenomenon and its consequences is important given the wide use of these methods. The third family of problems concern situations where stability estimates are derived with the aid of pseudodifferential operators. Such operators spread supports which makes them seemingly inappropriate for the study of supports. The project proposes the study of sharp finite speed and uniqueness of the Cauchy problem at space like hypersurfaces for symmetrizable hyperbolic systems. Finally the propagation of short wavelength waves through perturbed periodic media will be studied when the period and wavelength are of comparable size. The interest is in propagation for long distances on which diffractive effects are expected to take place.This research project addresses questions that arise in mathematical models for wave propagation phenomena, i.e. physical situations in which signals can travel at finite speed and interact with one another and with the medium that they traverse. Such signals can take the form of wave fronts (such as the spherical light flash from a camera, sound waves made by clapping hands, or a flame front in a combustion engine) or of rays (as in optics), among others. There is a common class of mathematical models for such phenomena that has been employed very successfully in applications as diverse as chemical engineering, the design and use of radar equipment and ultrasound transceivers, computer graphics, and fiber optics. There are four parts to this project. The first will add to the understanding of models for wave fronts that interact. The second part will examine a very widely used computational method for wave phenomena in unbounded spatial regions (e.g. radar signals bouncing off a plane) and try to explain some anomalies that make this method less reliable than is generally assumed. In the third part, the proposer will establish that a wide class of models does indeed exhibit the finite signal speed in all possible situations. Finally, the fourth project aims at a better understanding of light signals that travel through photonic materials over long distances. The PI will continue his educational efforts through lecture series, published notes, counseling, mentoring of thesis students, and close contact with laboratories in engineering schools and industry.