  This paper concerns the 3-dimensional Lagrangian Navier-Stokes $\alpha$ model
and the limiting Navier-Stokes system on smooth bounded domains with a class of
vorticity-slip boundary conditions and the Navier-slip boundary conditions. It
establishes the spectrum properties and regularity estimates of the associated
Stokes operators, the local well-posedness of the strong solution and global
existence of weak solutions for initial boundary value problems for such
systems. Furthermore, the vanishing $\alpha$ limit to a weak solution of the
corresponding initial-boundary value problem of the Navier-Stokes system is
proved and a rate of convergence is shown for the strong solution.
