Global solutions to 3D compressible MHD equations with partial magnetic diffusion

TL;DR

This paper proves the global existence of strong solutions for 3D compressible viscous MHD equations with partial magnetic diffusion, addressing a significant open problem in mathematical fluid dynamics.

Contribution

It establishes the first global well-posedness results for these equations with horizontal magnetic diffusion and small initial data.

Findings

Global solutions exist under small initial data

Use of anisotropic Sobolev inequalities and sharp estimates

Addresses open problem in MHD equations with partial diffusion

Abstract

The global existence of strong solutions to the compressible viscous magnetohydrodynamic (MHD) equations in $\mathbb{R}^3$ remains a significant open problem. When there is no magnetic diffusion, even small data global well-posedness is unknown. This study investigates the Cauchy problem in $\mathbb{R}^3$ for the compressible viscous MHD equations with horizontal magnetic diffusion. Using various anisotropic Sobolev inequalities and sharp estimates, we establish the existence of global solutions under small initial data within the Sobolev space framework.