# On the global well-posedness of 3D axisymmetric MHD system with pure swirl magnetic field

## TL;DR

This paper proves the global well-posedness of the 3D axisymmetric MHD system with pure swirl magnetic field under small initial data conditions in certain scaling-invariant norms.

## Contribution

It establishes the global existence and uniqueness of solutions for the axisymmetric MHD system with specific initial data structures, extending understanding of well-posedness in critical regimes.

## Key findings

- Global well-posedness under small initial data norms
- Solution existence holds for nearly critical initial conditions
- Provides conditions for stability of the axisymmetric MHD system

## Abstract

In this paper, we consider the axisymmetric MHD system with nearly critical initial data having the special structure: $u_0=u_0^r e_r+\ut_0 e_\theta+u_0^z e_z, ~b_0=b_0^\theta e_\theta.$ We prove that, this system is global well-posed provided the scaling-invariant norms $\|r\ut_0\|_{L^\infty},~\|r^{-1} b^\theta_0\|_{L^{\frac32}}$ are sufficiently small.