On 3D Lagrangian Navier-Stokes $\alpha$ model with a Class of   Vorticity-Slip Boundary conditions
  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. 
Yuelong Xiao and Zhouping Xin
math.AP