Well-posedness of the three-dimensional Lagrangian averaged Navier-Stokes equations

In this dissertation, we study the well-posedness of the three-dimensional Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. There are two types of LANS-$\alpha$ equations: the anisotropic version in which the fluctuation tensor is a dynamic variable that is coupled with the evolution equations for the mean velocity, and the isotropic version in which the covariance tensor is assumed to be a constant multiple of the identity matrix. We prove the global-in-time existence and uniqueness of weak solutions to the isotropic LANS-$\alpha$ equations for the case of no-slip boundary conditions. When the anisotropic LANS-$\alpha$ equations are written with the Stokes projector viscosity term, we show the local-in-time well-posedness of the anisotropic equations in the periodic box. We numerically compute strong solutions to the anisotropic equations in the laminar channel and pipe by considering steady fluid flow solutions to the Navier-Stokes equations.