Axisymmetric flows in the exterior of a cylinder

TL;DR

This paper establishes the existence of unique global solutions to the 3D Navier-Stokes equations for axisymmetric flows outside a cylinder with slip boundary conditions, under specific initial data and decay conditions.

Contribution

It provides the first proof of global solutions for axisymmetric Navier-Stokes flows in an exterior cylindrical domain with slip boundary conditions.

Findings

Existence of unique global solutions for specified initial data.

Solutions are axisymmetric and satisfy decay conditions.

Results extend understanding of Navier-Stokes in exterior domains.

Abstract

We study an initial-boundary value problem of the three-dimensional Navier-Stokes equations in the exterior of a cylinder $\Pi=\{x=(x_{h}, x_3)\ |\ |x_{h} |>1\}$, subject to the slip boundary condition. We construct unique global solutions for axisymmetric initial data $u_0\in L^{3}\cap L^{2}(\Pi)$ satisfying the decay condition of the swirl component $ru^{\theta}_{0}\in L^{\infty}(\Pi)$.