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

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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.