Instability of Fluid Flows

流体运动的不稳定性是这项提议的主要主题。虽然它在早期是流体力学的一个经典和相当实验的分支，但现在它是现代分析流体力学的一个活跃的发展领域，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受欧拉方程支配。由于方程中存在的非线性的特殊结构，即使是最基本的问题也似乎很难用严格的数学方法回答。其中一个问题是证明定常流体流动的Lyapunov线性化方法的合理性。特别是，如果线性化的欧拉方程有不稳定谱，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下是不稳定的？研究者研究了这个问题和其他与不稳定性和谱有关的问题。这些问题包括：寻找一般流体流动的不稳定短波扰动问题；欧拉方程的频谱分析以及在零粘性范围内与弱粘性流动频谱的关系。研究人员使用欧拉方程的现代WKB型渐近分析、伪微积分以及协循环理论和流体动力学之间的新发展联系来研究这些问题。这一特定项目中涉及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。大液团的不稳定性是这些过程的本质所固有的，而大尺度的不稳定性往往是从小尺度发展而来的。研究人员提出了这种小尺度不稳定性的机理，并给出了精确的数学描述。了解流体中的不稳定性对于大气科学和地球物理中的一系列问题都很重要。

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.