On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case

This note concerns the sufficient condition for regularity of solutions to the evolution Navier-Stokes equations known in the literature as Prodi-Serrin's condition. H.-O. Bae and H.J. Choe proved in a 1997 paper that, in the whole space R3, it is merely sufficient that two components of the velocity satisfy the above condition. Below, we extend the result to the half-space case R+n under slip boundary conditions. We show that it is sufficient that the velocity component parallel to the boundary enjoys the above condition. Flat boundary geometry is not essential, as shown in a forthcoming paper in cylindrical domains, prepared in collaboration with J. Bemelmans and J. Brand.