Instability of Fluid Flows

流体流动的不稳定性

• 批准号: 0604050
• 负责人: Roman Shvydkoy
• 金额: $ 9.49万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604050&HistoricalAwards=false
• 关键词: Instability; Fluid; Flows

Shvydkoy0604050 Instability of fluid motion is the main subject of thisproposal. Although it was a classical and rather experimentalbranch of the hydrodynamics in the early days, it now stands asan actively developing area of modern analytical fluid dynamics,and it provides many challenging problems to mathematicians. Themotion of an ideal incompressible fluid is governed by the Eulerequation. Because of the particular structure of the nonlinearitypresent in the equation it appears to be notoriously difficult toanswer even the most basic questions in a rigorous mathematicalway. One of these questions is to justify the Lyapunovlinearization method for stationary fluid flows. In particular,if the Euler equation linearized about a given equilibrium hasunstable spectrum, does this imply that the equilibrium isunstable in the nonlinear sense, say, in the basic energy norm?The investigator studies this and other questions related toinstability and spectra. These include: the question of findingunstable shortwave perturbations for general fluid flows;analysis of the spectrum for the Euler equations and therelationship with the spectrum of weakly viscous flows in thelimit of vanishing viscosity. The investigator examines theseproblems using modern WKB-type asymptotic analysis for the Eulerequations, pseudo-differential calculus, as well as newlydeveloped connections between the cocycle theory and fluiddynamics. The questions addressed in this particular project areclosely related to the fundamental problems of environmentalscience such as weather prediction, climate change, and oceanicmotion. Instability of large fluid masses is inherent in thenature of those processes, while large scale instabilities oftengrow out of small scales. The investigator presents mechanismsfor such small scale instabilities and gives them a precisemathematical description. Understanding instabilities in fluidflows is important for a broad range of questions in atmosphericscience and geophysics.

Shvydkoy0604050 流体运动的不稳定性是本提案的主要主题。尽管它在早期是流体力学的一个经典的、相当实验性的分支，但现在它已成为现代分析流体动力学的一个活跃发展的领域，它为数学家提供了许多具有挑战性的问题。理想不可压缩流体的运动由欧拉方程控制。由于方程中存在的非线性的特殊结构，以严格的数学方式回答最基本的问题似乎是出了名的困难。这些问题之一是证明静态流体流动的李亚普诺夫线性化方法的合理性。特别是，如果关于给定平衡线性化的欧拉方程具有不稳定谱，这是否意味着平衡在非线性意义上（例如基本能量范数）不稳定？研究者研究这个问题以及与不稳定性和谱相关的其他问题。其中包括：寻找一般流体流动的不稳定短波扰动问题；欧拉方程谱的分析以及在消失粘度极限下与弱粘性流谱的关系。研究人员使用现代 WKB 型欧拉方程渐近分析、伪微分演算以及新开发的余循环理论和流体动力学之间的联系来研究这些问题。 这个特定项目解决的问题与天气预报、气候变化和海洋运动等环境科学的基本问题密切相关。大流体质量的不稳定性是这些过程的本质所固有的，而大规模的不稳定性往往是从小规模中产生的。研究人员提出了这种小规模不稳定性的机制，并给出了精确的数学描述。了解流体流动的不稳定性对于大气科学和地球物理学中的广泛问题非常重要。