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We study the asymptotic behavior of solutions to the steady Navier-Stokesequations outside of an infinite cylinder in $\mathbb{R}^3$. We assume that theflow is periodic in $x_3$-direction and has no swirl. This problem is closelyrelated with two-dimensional exterior problem. Under a condition on thegeneralized finite Dirichlet integral, we give a pointwise decay estimate ofthe vorticity at the spatial infinity. Moreover, we prove a Liouville-typetheorem only from the condition of the generalized finite Dirichlet integral.

Asymptotic behavior and Liouville-type theorems for axisymmetric  stationary Navier-Stokes equations outside of an infinite cylinder with a  periodic boundary condition