Mathematical Sciences: Nonlinear Waves and Chaotic Interfaces

本研究项目的主要目的是了解非线性流体波在多个空间维度上的相互作用。冲击波、湍流混合区、火焰前锋、凝固前锋以及流体和材料界面提供了这里所讨论的波的类型的例子。研究人员首先将少量波在一点相遇并以自相似方式传播的常规情况分析为具有一定速度和一定相对角的行波。这个问题与一维黎曼问题有关，但比一维黎曼问题更困难，研究人员和同事对一维黎曼问题有先验经验。其次，他们考虑了混沌高维波相互作用的情况，其特征是界面不稳定、混合区和对初始条件的敏感依赖。在这种混乱的情况下，他们识别出基本模式，这些模式相互作用，形成一个有效的统计集合。混沌情况的研究依赖于重整化群，以及减少有效自由度数的方法。研究人员和同事发现了重整化群的固定点，以表征混沌混合制度，这为研究提供了方向。波传播研究中最困难的方面之一是非线性波相互作用的计算。这个项目是研究这种相互作用的数学结构，并利用这种分析来改进对波浪相互作用很重要的流动进行建模的计算机程序。这类流动在工业、军事和科学应用中广泛存在，包括超音速飞行、板材成形和模压、天气预报、流体湍流、弹丸侵彻装甲、冲击波和天体物理。所有这些领域的特点都是对波的非线性相互作用产生的细节结构的敏感依赖，因此有许多共同的数学方面。在研究波相互作用的抽象性质中获得的知识可以应用于所有这些领域，并将导致改进的计算机模拟和更好地理解在强波相互作用过程中发生的过程。

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.