Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl

We study the long-time behaviour of axisymmetric solutions without swirl for the threedimensional Navier-Stokes equations in the whole space. Assuming that the initial vorticity is sufficiently localised, we compute explicitly the leading terms in the asymptotic expansion of the solution, both for the vorticity and the velocity field. In particular, we identify optimal temporal decay rates depending on the spatial localisation of the initial data. Our approach relies on accurate $L^p$-$L^q$ estimates for the linearised evolution equation and its Taylor expansion in self-similar variables.