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

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 $\Vert {\omega }^{\theta }(t)/r{\Vert }_{L^1}$ and $\Vert u^{\theta }(t)/\sqrt{r}{\Vert }_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can refine some decay estimates for the axisymmetric solutions.

References in corpus (1)

• On the Cauchy problem for axi-symmetric vortex rings