Mathematical Sciences: Nonlinear Waves and Chaotic Interfaces

点击翻译按钮获取中文摘要

The main purpose of this research project is to understand the interactions in multiple space dimensions of nonlinear fluid waves. Shock waves, turbulent mixing zones, flame fronts, solidification fronts, and fluid and material interfaces provide examples of the types of waves considered here. The investigators analyze first the regular case, in which a small number of waves meet at a point and propagate in a self-similar fashion, as a traveling wave with definite velocity and with definite relativeangles. This problem is related to, but more difficult than, the one-dimensional Riemann problem, with which the investigators and coworkers have prior experience. Secondly, they consider the case of chaotic higher dimensional wave interactions, characterized by unstable interfaces, mixing regions, and sensitive dependence on initial conditions. In this chaotic case, they identify elementary modes, which interact to form an effective statistical ensemble. The study of the chaotic case depends upon the renormalization group, and methods to reduce the number of effective degrees of freedom. Fixed points of the renormalization group found by the investigators and coworkers to characterize chaotic mixing regimes suggest directions for research. One of the most difficult aspects of the study of wave propagation is the computation of nonlinear wave interactions. This project is to study the mathematical structure of such interactions and to use this analysis to improve computer programs that model flows where wave interactions are important. Flows of this type occur in a wide variety of industrial, military, and scientific applications, including supersonic flight, sheet metal forming and die pressing, weather prediction, fluid turbulence, projectile penetration of armor, blast waves, and astrophysics. All of these areas are characterized by sensitive dependence on the detailed structures produced by the nonlinear interaction of waves, and thus share many common mathematical aspects. Knowledge gained in studying the abstract properties of wave interactions can be applied to all of these areas, and will lead to improved computer simulations and a better understanding of the processes occurring during strong wave interactions.