Regularity of Axially Symmetric Flows in a Half-Space in Three Dimensions

We consider a vector field in a half-space which vanishes at . If a vector field is invariant under rotation around axis, we say that it is axially symmetric, in other words, for every rotation about the axis. If, moreover, an axially symmetric vector field is invariant under reflection by every plane containing axis, we say it is axially symmetric with no swirl, that is to say, is axially symmetric and for every reflection as above. In this paper we study the regularity of axially symmetric solutions with no swirl of the Navier-Stokes equations in . When is replaced by , it is known that such solutions are regular (see [3], [7] and [15]). Our main result is that suitable weak solutions of the Navier-Stokes equations are locally Hölder continuous up to the boundary