Instability of Fluid Flows

流体运动的不稳定性是本提案的主要主题。虽然它在早期是流体力学的一个经典的和相当实验性的分支，但它现在是现代分析流体力学的一个积极发展的领域，它给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。由于方程中非线性的特殊结构，用严格的数学方法来回答最基本的问题似乎是出了名的困难。其中一个问题是如何证明稳态流体流动的李雅普非线性化方法。特别是，如果欧拉方程线性化了一个给定的平衡有不稳定的谱，这是否意味着平衡在非线性意义上是不稳定的，比如说，在基本能量范数上？研究者研究这个问题和其他与不稳定性和光谱有关的问题。这包括：寻找一般流体流动的不稳定短波扰动的问题；分析了欧拉方程的谱及其与弱粘性流动在消失黏度极限下的谱的关系。研究者使用现代wkb型渐近分析欧拉方程，伪微分学，以及新发展的循环理论和流体动力学之间的联系来检查这些问题。在这个特殊的项目中解决的问题与环境科学的基本问题密切相关，如天气预报、气候变化和海洋运动。大流体团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由小尺度产生的。研究者提出了这种小尺度不稳定性的机制，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的广泛问题是重要的。

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.