Fine large-time asymptotics for the axisymmetric Navier--Stokes equations

We examine the large-time behavior of axisymmetric solutions without swirl of the Navier--Stokes equation in $\mathbb{R}^3$. We construct higher-order asymptotic expansions for the corresponding vorticity. The appeal of this work lies in the simplicity of the applied techniques: Our approach is completely based on standard $L^2$-based entropy methods.