Instabilities in fluid motion

[9970977]弗里德兰德——对于控制无粘流体运动的欧拉方程的稳定谱的详细结构，我们所知甚少。即使这是一个经典问题，也是理解湍流的基础问题，情况也是如此。弗里德兰德将继续利用“流体李亚普诺夫指数”的概念，该指数为检测频谱基本部分的高频不稳定性提供了有效手段。在离散谱中也可能出现不稳定性，在这种情况下，目前还不可能有一般的方法。本文将对具有“猫眼”型结构的特定二维流动的特征值问题进行分析，该结构似乎是一类包含振荡区和双曲驻点的有趣流动的典型。Friedlander还提出扩展Friedlander等人证明在一定条件下PDE的线性不稳定性意味着非线性不稳定性的结果的适用范围。Friedlander等人最近证明，在二维空间中，任何与非简并不稳定流足够“接近”的稳定流也是不稳定的。现在将研究三维流动的不稳定鲁棒性这一较难的问题。本文将利用自由边界问题的方法研究一种非零通量环面上的新型流动。液体或气体的稳定性问题是一个可以通过复杂的数学技术来解决的物理问题的重要例子。答案有直接的物理解释：在环境中不可避免的干扰下，稳定的流动是强大的，而不稳定的流动会迅速破裂。流体流动的稳定性或不稳定性问题是研究包括海洋和大气在内的物理现象的基础。在许多科学家看来，波浪和不稳定性是我们尝试使用数学模型进行长期天气预报和更好地了解全球气候变化的核心。然而，与构成所有流体行为模型基础的基本方程有关的重要未决问题。本建议寻求与流体不稳定性有关的一些问题的答案。

9970977FriedlanderRemarkably little is known about the detailed structure of the stability spectrum of the Euler equations that govern the motion of an inviscid fluid. This is the case even though it is a classical problem and one fundamental to an understanding of turbulent flow. Friedlander will continue to exploit the concept of a "fluid Lyapunov exponent" which provides an effective means for detecting high frequency instabilities in the essential part of the spectrum. Instabilities may also arise in the discrete spectrum and in this situation no general approach is, at present, possible. Analysis of the eigenvalue problem will be given for a specific 2-dimensional flow with a "cat's eye" type structure which appears to typify an interesting class of flows containing both oscillatory regions and hyperbolic stagnation points. Friedlander also proposes to extend the range of applicability of a result of Friedlander et al that proves under certain conditions that linear instability in a PDE implies nonlinear instability. Friedlander et al recently proved that in two dimensions any steady flow that is sufficiently "close" to a nondegenerate unstable flow is also unstable. The harder problem of robustness of instability for 3-dimensional flows will now be investigated. A novel type of flow on a torus with nonzero flux will be investigated using techniques from free boundary problems.The issue of stability of liquids or gases presents an important example of a physical question that may be addressed through sophisticated mathematical techniques. The answers have direct physical interpretations: stable flows are robust under inevitable disturbances in the environment while unstable flows break up rapidly. The question of stability or instability of a fluid flow is fundamental to studies of physical phenomena including those in the oceans and the atmosphere. In the view of many scientists, waves and instabilities lie at the heart of our attempts to use mathematical models for long term weather prediction and to better understand global climate change. However there are important open questions connected with the fundamental equations that underlie all models for fluid behavior. This proposal seeks the answers to some of these questions related to fluid instabilities.