Propagation of Waves and Fronts in Complex Media

获奖摘要获奖编号：0203537PI: Ryzhik， leonid机构：芝加哥大学项目：应用数学项目经理：Catherine mavriplis标题：复杂介质中的波和锋的传播本工作涉及波和锋在异质介质中的传播的数学研究。Ryzhik计划集中研究两个领域：随机介质中时间反转脉冲的重新聚焦和流动运动对燃烧和化学反应过程的影响。声波时间反转是一项最新的实验技术，它可以将来自局部震源的信号重新聚焦到其原始位置，即使它已经在距离震源很远的地方被记录下来。这种方法已经成功地应用于许多领域，从医学（用于肾结石的破坏）到海洋中声波的长距离传播、细胞通信和嘈杂环境中的成像。值得注意的是，与均匀域相比，时间反转和重传播信号的重聚焦非常鲁棒，并且在高度非均匀的随机介质中得到了极大的改善。Ryzhik打算在他之前关于随机和其他复杂介质中的波传播的工作的基础上，推进对这一重要现象的数学理解。本研究的第二个主要组成部分是对流体中燃烧过程的数学分析，以及流体运动对燃烧的影响。众所周知，流体的搅拌既可以加速化学反应，也可以阻止化学反应，这取决于特定问题的性质。物理学家、工程师和数学家都对这些现象进行了深入的研究。然而，在这一领域很少有定量严谨的数学结果。Ryzhik打算继续他在这一领域的工作，专注于模型的数学研究，这些模型既考虑到流动通过改进混合对化学反应的影响，也考虑到燃烧通过改变流体的温度和密度对流体的影响。这项研究的潜在影响是发展我们对各种环境下湍流燃烧过程的理解。解决上述问题所必需的数学工具包括均匀化、随机系数方程的平均、极大值原理和偏微分方程解的估计。日期：2002年4月25日

DMS Award AbstractAward #: 0203537PI: Ryzhik, LeonidInstitution: University of ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Propagation of Waves and Fronts in Complex MediaThis work is concerned with the mathematical study of propagation of waves and fronts in heterogeneous media. Ryzhik plans to concentrate on two areas of research: refocusing of time-reversed pulses in random media and the effect of flow motion on combustion and chemical reaction processes. Time reversal of acoustic waves is a recent experimental technique that refocuses a signal that arrived from a localized source at its original location even if it has been recorded at a very long distance from the source. This method has been successfully applied in a number of applications, that range from medicine, where it was used for destruction of kidney stones, to long distance propagation of acoustic waves in the ocean, cellular communications, and imaging in noisy environments. Remarkably, refocusing of the time-reversed and re-propagated signal is very robust, and is greatly improved in a highly heterogeneous random medium as compared to a uniform domain. Ryzhik intends to build on his previous work on wave propagation in random and other complex media to advance the mathematical understanding of this important phenomenon.The second major component of the proposed research is the mathematical analysis of the combustion processes in fluids, and the effect of fluid motion on combustion. It is well known that stirring by a fluid flow may both speed up chemical reactions and stop them, depending on the nature of a particular problem. These phenomena have been intensively studied by physicists, engineers and mathematicians alike. Nevertheless, there are few quantitative rigorous mathematical results in the area. Ryzhik intends to continue his work in this field, concentrating on mathematical studies of models that take into account both the affect of the flow on the chemical reaction via improved mixing, and the effect of combustion on the fluid, through changes in temperature and density of the fluid. The potential impact of this research is development of our understanding of combustion processes in turbulent flows in various settings. The mathematical tools that are necessary to address the above problems include homogenization, averaging of equations with random coefficients, maximum principles and estimates on solutions of partial differential equations.Date: April 25, 2002