On the global well-posedness of 3-D axi-symmetric Navier-Stokes system with small swirl component

TL;DR

This paper establishes the conditions under which the 3-D axi-symmetric Navier-Stokes equations are well-posed globally, especially focusing on small initial swirl components in critical spaces.

Contribution

It proves global well-posedness for the 3-D axi-symmetric Navier-Stokes system with small initial data and small swirl component in almost critical spaces.

Findings

Global well-posedness with small initial data

Well-posedness with small swirl component in almost critical spaces

Local well-posedness in critical Lebesgue spaces

Abstract

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier-Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.