Global Regular Axially Symmetric Solutions to the Navier–Stokes Equations: Part 2

The axially symmetric solutions to the Navier–Stokes equations are considered in a bounded cylinder $\Omega \subset {\mathbb{R}}^3$ with the axis of symmetry. $S_1$ is the boundary of the cylinder parallel to the axis of symmetry, and $S_2$ is perpendicular to it. We have two parts of $S_2$. On $S_1$ and $S_2$, we impose vanishing of the normal component of velocity and the angular component of vorticity. Moreover, we assume that the angular component of velocity vanishes on $S_1$ and the normal derivative of the angular component of velocity vanishes on $S_2$. We prove the existence of global regular solutions. To prove this, the coordinate of velocity along the axis of symmetry must vanish on it. We have to emphasize that the technique of weighted spaces applied to the stream function plays a crucial role in the proof of global regular axially symmetric solutions. The paper is a generalization of Part 1, where the periodic boundary conditions are prescribed on $S_2$. The transformation is not trivial because it needs to examine many additional boundary terms and derive new estimates.