The fluid equations, shell models and the limit of vanishing viscosity

虽然所有的“真实的”流体至少是非常弱的粘性，但在许多物理情况下，粘性消失的微妙极限是相关的状态。特别是，开创性的想法Kolmogorov和Onsager关于统计理论的湍流是基于预测的行为的流体范围内的粘度消失。 欧拉方程和Navier-Stokes方程由于其特殊的非线性性质而在数学上非常具有挑战性。本项目解决了流体方程中出现的一些非线性现象，特别是与消失粘度极限有关的现象。所谓的“壳模型”进行了研究：这些都是由一个无限系统的非线性耦合常微分方程，虽然不那么复杂的欧拉和Navier-Stokes方程保留其非线性的某些重要特征。它已被证明在以前的NSF支持的工作，无粘模型的行为，精确的Onsager湍流流体的湍流。在该奖项将支持的项目中，建议证明粘性模型的Kolmogorov“定律”，即粘性系统的平均耗散率保持正值，并在粘性极限消失时收敛于平均湍流耗散率。另一个研究方向是流体运动的不稳定性，以及粘性极限消失时线性和非线性不稳定性之间的相互关系。模拟流体运动的微分方程的数学研究是许多应用的重要基础，如海洋学、气象学、天体物理学和工程学。一个特别重要的问题是湍流流体的行为。根据该奖项，弗里德兰德将使用数学技术来研究在发达湍流中发生的能量级联到越来越小的结构。她还将通过研究在大多数流体环境中发生的不可避免的不稳定性。这种不稳定的流动会破裂，并可能形成发展湍流的触发器。

Although all "real" fluids are at least very weakly viscous, in many physical situations the subtle limit of vanishing viscosity is the relevant regime. In particular, the seminal ideas of Kolmogorov and Onsager concerning the statistical theories of turbulence are based on predictions about the behaviour of fluids in the range of vanishing viscosity. The Euler and Navier-Stokes equations are very challenging mathematically because the particular nature of their nonlinearity.This project addresses some of the nonlinear phenomena that arise in the fluid equations, particularly those connected with the limit of vanishing viscosity. So called "shell models" are studied: these are composed of an infinite system of nonlinearly coupled ordinary differential equations which, although less complex than the Euler and Navier-Stokes equations retain certain important features of their nonlinearity. It has been shown in previous NSF-supported work that the inviscid model behaves precisely as Onsager conjectured for a turbulent fluid. In the project that will be supported by this award, it is proposed to prove Kolmogorov's "law" for the viscous model, namely that the average dissipation rate of the viscous system stays positive and converges to the average turbulent dissipation rate in the vanishing viscosity limit. Another line of research concerns instabilities in fluid motion, and the interrelation between linear and nonlinear instability in the vanishing viscosity limit.The mathematical study of the differential equations that model fluid motion forms an essential foundation for many applications, such as oceanography, meteorology, astrophysics and engineering. An issue of particular importance is the behaviour of turbulent fluids. Under this award, Friedlander will be using mathematical techniques to study the cascade of energy to smaller and smaller structures that occurs in developed turbulence. She will also by studying the inevitable instabilities that occur in most fluid environments. Such unstable flows break up and may form the trigger for developed turbulence.