On stochastic shell models of turbulence

In this thesis we study shell models of turbulence, and in particular, their well-posedness and statistical properties.;Shell Models are simplified phenomenological models that capture some properties of turbulent fluid flows. They approximate the Navier-Stokes equations in its Fourier form by limiting the interactions between different modes to a finite number of neighbors in the nonlinear term. They preserve certain properties of the original equations such as the energy spectrum. Shell models are written as infinite system of coupled nonlinear scalar valued ordinary differential equations. Our work is related to some of the popular shell models such as the GOY, Sabra and dyadic model.;In the study of 3D turbulence, among the quantities of major interest are the asymptotic exponents zetap of the p-order structure functions. There is a general agreement on zeta p for p large and zeta3 = 1. The value of zeta2 is less clear as there is no analytical proof while certain numerical simulations indicate a value larger than 2/3 opposed to zetap =2/3 claimed by Kolmogorov theory. Our aim in this thesis is to investigate such relationships using simpler models than the original Navier-Stokes equations-the shell models of turbulence.;We consider a viscous dyadic model driven by additive white noise in time. We first prove existence and uniqueness of solutions using a path-wise approach. We derive some estimates related to the expectation of the solution and study its regularity. Moreover, we show the existence of invariant measures and establish a simple balance relation among the mean rates of energy injection, dissipation and flux. On the basis of the balance relation and assuming that only the first component of the noise term to be nonzero in our stochastic shell model, we prove some results related to K41 theory including a monotonicity property and a boundedness property. Finally, we give sufficient and necessary conditions for on zeta2 <2/3 and zeta2=2/3. The latter result is based on an assumption on the ratio related to structure functions and flux asymptotic exponent that is not derived from the balance relation.