The Navier-Stokes- equations belong to the family of LES (Large Eddy Simulation)
models whose fundamental idea is to capture the influence of the small scales on the large ones
without computing all the whole range present in the flow. The constant is a regime flow parameter
that has the dimension of the smallest scale being resolvable by the model. Hence, when = 0, one
recovers the classical Navier-Stokes equations for a flow of viscous, incompressible, Newtonian fluids.
Furthermore, the Navier-Stokes- equations can also be interpreted as a regularization of the Navier-
Stokes equations, where stands for the regularization parameter.
In this paper we first present the Navier-Stokes- equations on bounded domains with no-slip
boundary conditions by means of the Leray regularization using the Helmholtz operator. Then we
study the problem of relating the behavior of the Galerkin approximations for the Navier-Stokes-
equations to that of the solutions of the Navier-Stokes equations on bounded domains with no-slip
boundary conditions. The Galerkin method is undertaken by using the eigenfunctions associated
with the Stokes operator. We will derive local- and global-in-time error estimates measured in terms
of the regime parameter and the eigenvalues. In particular, in order to obtain global-in-time error
estimates, we will work with the concept of stability for solutions of the Navier-Stokes equations in
terms of the L2 norm.Ministerio de Educación y Ciencia JC2011-041
