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This paper concerns the 3-dimensional Lagrangian Navier-Stokes $\alpha$ modeland the limiting Navier-Stokes system on smooth bounded domains with a class ofvorticity-slip boundary conditions and the Navier-slip boundary conditions. Itestablishes the spectrum properties and regularity estimates of the associatedStokes operators, the local well-posedness of the strong solution and globalexistence of weak solutions for initial boundary value problems for suchsystems. Furthermore, the vanishing $\alpha$ limit to a weak solution of thecorresponding initial-boundary value problem of the Navier-Stokes system isproved and a rate of convergence is shown for the strong solution.

On 3D Lagrangian Navier-Stokes $\alpha$ model with a Class of Vorticity-Slip Boundary conditions