Navier Stokes Equations Essay
In the attempt of describing and understanding natural phenomena, mathematicians and physicists develop various models. Among these models, the Navier-Stokes equations are often used to describe the motion of fluids. Various forms of the Navier-Stokes equations have wide range of applications, such as, modeling weather, ocean currents, vortex, water flow in a pipe, blood flow in the body and air flow around a wing. Turbulence is where a fluid flow exhibits a chaotic behavior with time. It is widely believed that the Navier-Stokes equations are of great use when describing turbulence. The nonlinear term of the Navier-
Stokes equations is the main contributor to the turbulence that the equations model and also the term that creates mathematical difficulties and challenges.
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These are obtained by the Itô formula and stochastic analysis. Our aim in this thesis is to investigate such balance laws in one of the simplest settings related to fluid dynamics, shell models of turbulence.
Shell models are simplified Fourier systems with respect to the Navier-Stokes equations, where the interactions among different modes are preserved only between neighbors. In most of these shell models the interactions among the neighbors are limited to either three or five neighbors. Thus, shell models are drastic simplifications of the Navier-Stokes equa- tions. In particular, the GOY and Sabra models are some of the most interesting and most popular examples of simplified phenomenological models of turbulence. Although departing from reality, they capture some essential statistical properties and features of turbulent flows, like the energy spectrum, the enstrophy cascade and the power law decay of the turbulent flows in some range of wave numbers-the inertial range, see [?].
We now describe how the content of this thesis has been organized.
This thesis is divided into 3 parts. The first part, Chapter 2 is devoted to some preliminaries.
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