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

尽管所有“真正的”流体至少具有非常弱的粘性，但在许多物理情况下，粘度消失的微妙极限是相关的区域。特别是，Kolmogorov和Onsager关于湍流统计理论的开创性思想是基于对流体在粘度消失范围内的行为的预测。Euler方程和Navier-Stokes方程由于其特殊的非线性性质，在数学上是非常具有挑战性的。这个项目解决了流体方程中出现的一些非线性现象，特别是那些与消失粘性极限有关的现象。研究了所谓的“壳模型”：这些模型是由一个无限的非线性耦合常微分方程组组成的，这些常微分方程组虽然没有欧拉和纳维斯托克斯方程那么复杂，但保留了它们的某些重要的非线性特征。NSF以前支持的工作已经表明，无粘模型的行为与Onsager对湍流的猜想完全相同。在该奖项支持的项目中，建议证明粘性模型的柯尔莫戈罗夫定律，即粘性系统的平均耗散率保持正值，并在消失的粘性极限内收敛于平均湍流耗散率。另一项研究涉及流体运动的不稳定性，以及在消失的粘性极限中线性和非线性不稳定性之间的相互关系。对模拟流体运动的微分方程的数学研究构成了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动流体的行为。根据这一奖项，弗里德兰德将使用数学技术来研究在发达的湍流中出现的能量级联到越来越小的结构中。她还将通过研究大多数不稳定环境中不可避免的不稳定性来实现这一目标。这种不稳定的流动会破裂，并可能形成形成湍流的触发因素。

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.