Vorticity and regularity for flows under the Navier boundary condition

Abstract

In reference [13], by Constantin and Fefferman, a quite simple geometrical assumption on the direction of the vorticity is shown to be sufficient to guarantee the regularity of the weak solutions to the evolution Navier--Stokes equations in the whole of $\mathbf R^3$. Essentially, the solution is regular if the direction of the vorticity is Lipschitz continuous with respect to the space variables. In reference [8], among other side results, the authors prove that $1/2$-Hölder continuity is sufficient.<br> A main open problem remains of the possibility of extending the same kind of results to boundary value problems. Here, we succeed in making this extension to the well known Navier (or slip) boundary condition in the half-space $\mathbf R^3$. It is worth noting that the extension to the non-slip boundary condition remains open. See [7].