  In this paper we present analytical studies of three-dimensional viscous and
inviscid simplified Bardina turbulence models with periodic boundary
conditions. The global existence and uniqueness of weak solutions to the
viscous model has already been established by Layton and Lewandowski. However,
we prove here the global well-posedness of this model for weaker initial
conditions. We also establish an upper bound to the dimension of its global
attractor and identify this dimension with the number of degrees of freedom for
this model. We show that the number of degrees of freedom of the long-time
dynamics of the solution is of the order of $(L/l_d)^{12/5}$, where $L$ is the
size of the periodic box and $l_d$ is the dissipation length scale-- believed
and defined to be the smallest length scale actively participating in the
dynamics of the flow. This upper bound estimate is smaller than those
established for Navier-Stokes-$\alpha$, Clark-$\alpha$ and
Modified-Leray-$\alpha$ turbulence models which are of the order $(L/l_d)^{3}$.
Finally, we establish the global existence and uniqueness of weak solutions to
the inviscid model. This result has an important application in computational
fluid dynamics when the inviscid simplified Bardina model is considered as a
regularizing model of the three-dimensional Euler equations.
