Global regular axially symmetric solutions to the Navier–Stokes equations. Part 3

The series of papers denoted by Part 1, Part 2, and Part 3 is devoted to motions of axially symmetric solutions to the Navier–Stokes equations in a cylindrical domain. Part 1 and Part 2 deal with motions such that $\psi_1=\psi/r$, where $\psi$ is the stream function and $r$ is the radius, vanishes on the axis of symmetry. The boundary of the cylinder equals $S_1\cup S_2$, where $S_1$ is parallel to the axis of symmetry and $S_2$ is perpendicular to it. It is clear that $S_2$ has two parts. In Part 1 it was assumed that the normal component of velocity, angular components of velocity and vorticity vanish on $S_1$ and the periodic boundary conditions were assumed on $S_2$. In Part 2 the same boundary conditions were imposed on $S_1$ as in Part 1 and it was assumed that the normal component of velocity, angular component of vorticity and the normal derivative of the angular component of velocity vanish on $S_2$. In this paper, Part 3, the same boundary conditions as in Part 2 are imposed but the restriction $\psi_1=0$ on the axis of symmetry is dropped. This is done by looking for solutions to the problem under study in the form $v=v'+\stackrel1{v}+\stackrel2{v}$, $p=p'+\stackrel1{p}+\stackrel2{p}$, where $(\stackrel1{v},\stackrel1{p})$ is the solution from Part 2 and $(\stackrel2{v},\stackrel2{p})$ is the solution without swirl proved by O. A. Ladyzhenskaya. Finally, $(v',p')$ is added because otherwise $(v,p)$ cannot satisfy the Navier–Stokes equations.