Uniqueness and Stability for an Ideal Fluid

这个项目解决了与理想流体运动有关的数学问题。 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，流体粒子的轨迹可能在某些点处不是唯一定义的。 消失的粘性极限与唯一性密切相关，并将对此类流动进行研究。 一个可能的爆破集的定性性质也将被考虑。 另一个研究方向集中在理想流体的线性稳定性上。 在这方面要解决的主要问题是如何增长率的一个小扰动是有关的位置的频谱的小振荡。 本课程将研究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构成了许多应用的基础，如气象学、物理学、天体物理学、工程学和湍流理论。 本研究探讨了两种常用的描述流体运动的模型之一，并研究了流体流动中的初始速度分布是否决定了未来的速度分布。 将研究流体流动的一般类别的稳定性，特别是稳定性是否可以由小振荡的特征来确定的问题。 这项研究的潜在应用包括湍流中大尺度和小尺度结构的建模，以及大气和海洋中涡旋的稳定性。

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.