Analytical and Numerical Study of Certain Models of Turbulence

In my thesis I study two different models of turbulence. The first part of my research concerns the, so-called, shell models of turbulence. Shell models had attracted a lot of interest as useful phenomenological models that mimics certain features of the Navier-Stokes equations (NSE). The shell models capture different aspects of the real turbulent flows, such as the energy cascade, an anomalous scaling, etc. At the same time they are an infinite set of ordinary differential equations with a simple quadratic non-linearity, which can be effectively studied numerically. In my thesis I initiate the analytical study of shell models. I obtain some analytic results concerning the existence of unique regular solutions of the, so called, viscous Sabra shell model of turbulence and the finite dimensionality of its underlying long-term dynamics. Furthermore, I continue this line