A note on optimal decay rates for the axisymmetric D-solutions to the steady Navier-Stokes equations

TL;DR

This paper establishes the optimal decay rates for axisymmetric solutions to the steady Navier-Stokes equations in three dimensions, revealing a decay rate of 1/r for velocity and a dichotomy in the swirl component's decay behavior.

Contribution

It provides the first optimal decay estimates for axisymmetric D-solutions without swirl and characterizes the decay dichotomy of the swirl component.

Findings

Velocity decays at rate 1/|x| in space.

Swirl component exhibits a decay dichotomy with two possible behaviors.

Other velocity components decay optimally at rate 1/|x|.

Abstract

In this paper, we investigate the decay properties of an axisymmetric D-solutions to stationary incompressible Navier-Stokes systems in $\mathbb{R}^3$. We obtain the optimal decay rate $|{\bf u}(x)|\leq \frac{C}{|x|+1}$ for axisymmetric flows without swirl. Furthermore, we find a dichotomy for the decay rates of the swirl component $u_{\theta}$, that is, either $O(\frac{1}{r+1})\leq |u_{\theta}(r,z)|\leq \frac{C\log(r+1)}{(r+1)^{1/2}}$ or $|u_{\theta}(r,z)|\leq \frac{C r}{(\rho+1)^3}$, where $\rho=\sqrt{r^2+z^2}$. In the latter case, we can further deduce that the other two components of the velocity field also attain the optimal decay rates: $|u_r(r,z)|+ |u_{z}(r,z)|\leq \frac{C}{\rho+1}$. We do not require any small assumptions on the forcing term.