The Lagrangian averaged Navier-Stokes equations with rough initial data

Turbulent fluid flow is governed by the Navier-Stokes equations, given in their incompressible formulation as 0.0.1 6tu+u&dot1 u-n&utriu=-1p, where the incompressibility condition requires div u = 0, nu is a constant greater than zero due to the viscosity of the fluid and u is the velocity field of the fluid.Because of the difficulty of working with the Navier-Stokes equations, several different approximations of the Navier-Stokes equations have been developed. One recently derived approxmation is the Lagrangian Averaged Navier-Stokes equations, which are given in their incompressible, isotropic form as 0.0.26 tu+u&dot1u+ divta u-n&utriu=-1-a 2&utri-11p.This thesis will focus on three main areas. First, we seek local solutions to the Lagrangian Averaged Navier-Stokes equations with initial data in Sobolev space Hr,p( Rn ) with the goal of minimizing r. We generate these results by following the program of [6] for the Navier-Stokes equations. Following results of [8], we are able to turn the local solution into a global solution for the n = 3, p = 2 case.Secondly, we seek solutions to the Lagrangian Averaged Navier-Stokes equations for initial data in Besov space Brp,q , again following the broad outline of [6]. Finally, we get a global result for Besov spaces in the p = 2 case and a qualitatively different local result for general p by modifying the results in [18] for the homogeneous generalized Navier-Stokes equations.