Navier–Stokes equations: Green's matrices, vorticity direction, and regularity up to the boundary

We consider the initial–boundary value problem for the 3D Navier–Stokes equations. The physical domain is a bounded open set with a smooth boundary on which we assume a condition of free-boundary type. We show that if a suitable hypothesis on the vorticity direction is assumed, then weak solutions are regular. The main tool we use in the proof is an explicit representation of the velocity in terms of the vorticity, by means of Green's matrices.