On the global well-posedness of the 3D axisymmetric resistive MHD equations

TL;DR

This paper establishes the global well-posedness of 3D axisymmetric resistive MHD equations with zero viscosity for specific Sobolev and Besov initial data regularities, advancing understanding of these complex fluid models.

Contribution

It proves global well-posedness for resistive MHD equations with axisymmetric initial data in both Sobolev and Besov critical regularities, addressing cases where classical criteria are not established.

Findings

Global well-posedness in Sobolev spaces for s>5/2.

Global well-posedness in Besov spaces for p between 2 and infinity.

Analysis of the problem where Beale-Kato-Majda criterion does not apply.

Abstract

In this paper, we prove the global well-posedness for the three-dimensional magnetohydrodynamics (MHD) equations with zero viscosity and axisymmetric initial data. First, we analyze the problem corresponding to the Sobolev regularities $ H^s\times H^{s-2}$, with $ s>5/2$. Second, we address the same problem but for the Besov critical regularities $ B_{p,1}^{3/p+1}\times B_{1,p}^{3/p-1}$, $2\leq p\leq \infty$. This case turns out to be more subtle as the Beale-Kato-Majda criterion is not known to be valid for rough regularities.