Propagation of Waves and Fronts in Complex Media

University Of Chicago, Chicago IL

Investigators

Abstract

DMS Award Abstract Award #: 0203537 PI: Ryzhik, Leonid Institution: University of Chicago Program: Applied Mathematics Program Manager: Catherine Mavriplis Title: Propagation of Waves and Fronts in Complex Media This 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