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

TL;DR

This paper investigates the long-range behavior of axisymmetric steady Navier-Stokes flows outside an infinite cylinder with periodic boundary conditions, establishing decay estimates and Liouville-type theorems under finite Dirichlet integral conditions.

Contribution

It provides new decay estimates and Liouville-type theorems for axisymmetric solutions without swirl in an exterior cylindrical domain, extending understanding of steady Navier-Stokes solutions.

Findings

Derived pointwise decay estimates for vorticity at infinity.

Proved Liouville-type theorems under finite Dirichlet integral conditions.

Linked the problem to two-dimensional exterior flow problems.

Abstract

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