Global well-posedness for axisymmetric MHD system with only vertical viscosity

TL;DR

This paper proves the global well-posedness of a 3D axisymmetric MHD system with only vertical viscosity, leveraging flow structure and vertical diffusion effects to establish high regularity of solutions.

Contribution

It introduces a novel approach to prove global regularity for an MHD system with anisotropic viscosity using axisymmetric structure and vertical diffusion effects.

Findings

Established boundedness of vertical derivatives of velocity.

Proved high regularity and global existence of solutions.

Demonstrated the effectiveness of anisotropic diffusion in MHD systems.

Abstract

In this paper, we are concerned with the global well-posedness of a tri-dimensional MHD system with only vertical viscosity in velocity equation for the large axisymmetric initial data. By making good use of the axisymmetric structure of flow and the maximal smoothing effect of vertical diffusion, we show that $\displaystyle\sup_{2\leq p<\infty}\int_0^t\frac{\|\partial_{z}u(\tau)\|_{L^p}^{2}}{p^{3/4}}\,\mathrm{d}\tau<\infty$. With this regularity for the vertical first derivative of velocity vector field, we further establish losing estimates for the anisotropy tri-dimensional MHD system to get the high regularity of $(u,b)$, which guarantees that $\int_0^t\|\nabla u(\tau)\|_{L^\infty}\,\mathrm{d}\tau<\infty$. This together with the classical commutator estimate entails the global regularity of a smooth solution.