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We consider the initial boundary value problem for the 3D Navier-Stokes equations under a slip type boundary condition. Roughly speaking, we are concerned with regularity results, up to the boundary, under suitable assumptions on the directions of velocity and vorticity. Our starting point is a recent, interesting, result obtained by Berselli and Córdoba concerning the ``near orthogonal case''. We also consider a ``near parallel case''.
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