Existence and non-uniqueness of classical solutions to the axially symmetric stationary Navier-Stokes equations in an exterior cylinder

Abstract: In this paper, we show existence and non-uniqueness on the axially symmetric stationary Navier-Stokes equations in an exterior periodic cylinder. On the boundary of the cylinder, the horizontally swirl velocity is subject to the perturbation of a rotation, the horizontally radial velocity is subject to the perturbation of an interior sink, while the vertical velocity is the perturbation of zero. At infinity, the flow stays at rest. We construct a solution to such problem, whose principal part admits a critical decay for the horizontal components and a supercritical decay for the vertical component of the velocity. This existence result is related to the 2D Stokes paradox and an open problem raised by V. I. Yudovich in [Eleven great problems of mathematical hydrodynamics, Mosc. Math. J. 3 (2003), no. 2, 711--737], where Problem 2 states that: Show (spatially) global existence theorems for stationary and periodic flows. Moreover, if the horizontally radial-sink velocity is relatively large ($\nu<-2$ in our setting), then the solution to this problem is non-unique.