Time-Asymptotic Study of a Viscous Axisymmetric Fluid without Swirl

TL;DR

This paper analyzes the long-time behavior of axisymmetric, swirl-free solutions to the 3D Navier-Stokes equations, explicitly computing asymptotic expansions and decay rates based on initial data localization.

Contribution

It provides explicit asymptotic expansions and decay rates for solutions without swirl, using precise $L^p$-$L^q$ estimates and self-similar analysis.

Findings

Derived explicit asymptotic expansions for vorticity and velocity.

Identified optimal decay rates depending on initial data localization.

Established a framework for analyzing long-time behavior of axisymmetric flows.

Abstract

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.