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

尽管所有“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制度。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消失粘度范围内的行为的预测。欧拉方程和纳维-斯托克斯方程在数学上非常具有挑战性，因为它们的非线性特性。本项目解决了流体方程中出现的一些非线性现象，特别是那些与消失粘度极限有关的非线性现象。我们研究了所谓的“壳模型”：壳模型是由一个无限的非线性耦合常微分方程系统组成的，它虽然没有欧拉方程和纳维-斯托克斯方程复杂，但仍保留了其非线性的某些重要特征。在以前nsf支持的工作中已经证明，无粘模型的行为与Onsager对湍流的推测完全一致。在本奖项拟资助的项目中，拟证明黏性模型的Kolmogorov“定律”，即黏性系统的平均耗散率在消失黏性极限下保持正并收敛于平均湍流耗散率。另一个研究方向涉及流体运动的不稳定性，以及消失粘度极限下线性和非线性不稳定性之间的相互关系。流体运动微分方程的数学研究是许多应用的基础，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动环境中不可避免的不稳定性。这种不稳定的气流破裂，可能会引发发达的湍流。

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.