Long-time asymptotics of $3$-D axisymmetric Navier-Stokes equations in critical spaces

TL;DR

This paper proves that global solutions to 3-D axisymmetric Navier-Stokes equations in critical spaces asymptotically become small, with specific decay of vorticity and velocity components, refining decay estimates for such solutions.

Contribution

It demonstrates that solutions in critical spaces tend to zero over time without smallness assumptions, providing new decay estimates for axisymmetric Navier-Stokes solutions.

Findings

Vorticity-to-radius ratio tends to zero as time approaches infinity.

Azimuthal velocity component normalized by sqrt(r) tends to zero over time.

Refined decay estimates for axisymmetric Navier-Stokes solutions.

Abstract

We show that any unique global solution (here we do not require any smallness condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some scaling invariant spaces must eventually become a small solution. In particular, we show that the limits of $\|\omega^\theta(t)/r\|_{L^1}$ and $\|u^\theta(t)/\sqrt r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can refine some decay estimates for the axisymmetric solutions.