Uniqueness and Stability for an Ideal Fluid

理想流体的独特性和稳定性

• 批准号: 0301531
• 负责人: Mikhail Vishik
• 金额: $ 16万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301531&HistoricalAwards=false
• 关键词: Uniqueness; Stability; Ideal; Fluid

This project addresses mathematical problems related to the motion of an ideal fluid. The question of uniqueness will be studied for the system of equations describing fluid motion with special emphasis on "rough" flows. In this situation, the trajectories of fluid particles may not be uniquely defined at some points. The vanishing viscosity limit is deeply connected with uniqueness and will be investigated for such classes of flows. Qualitative properties of a possible blow-up set will also be considered. Another line of research focuses on the linear stability of an ideal fluid. The main question to be addressed in this area is how the growth rate of a small perturbation is related to the location of the spectrum of small oscillations. The interrelation between linear instability and nonlinear instability for flows of an ideal incompressible fluid will be studied.Mathematical study of the basic models describing fluid motion forms a foundation for a number of applications, such as meteorology, geophysics, astrophysics, engineering, and the theory of turbulence. This research explores one of the two commonly used models describing the motion of fluids, and investigates whether the initial distribution of velocity in a fluid flow determines the future velocity distribution. Stability of general classes of fluid flows will be studied, and in particular, the question of whether stability can be determined by characteristics of small oscillations. Potential applications of this research include modeling of large and small scale structures in turbulence, as well as stability of vortices in the atmosphere and ocean.

这个项目解决了与理想流体运动有关的数学问题。我们将研究描述流体运动的方程组的唯一性问题，特别强调“粗糙”流动。在这种情况下，流体粒子的轨迹在某些点上可能不是唯一定义的。消失粘度极限与唯一性密切相关，将对这类流动进行研究。还将考虑可能爆破集的定性性质。另一个研究方向是关注理想流体的线性稳定性。在这个领域中要解决的主要问题是小扰动的增长率如何与小振荡的频谱位置有关。研究了理想不可压缩流体流动的线性不稳定性和非线性不稳定性之间的相互关系。描述流体运动的基本模型的数学研究形成了许多应用的基础，如气象学、地球物理学、天体物理学、工程学和湍流理论。本研究探讨了描述流体运动的两种常用模型之一，并探讨了流体流动中速度的初始分布是否决定了未来的速度分布。一般流体流动的稳定性将被研究，特别是稳定性是否可以由小振荡的特征决定的问题。本研究的潜在应用包括湍流中大尺度和小尺度结构的建模，以及大气和海洋中漩涡的稳定性。