Global regularity for the 3D compressible magnetohydrodynamics with general pressure

TL;DR

This paper proves that under certain conditions on density, local solutions to 3D compressible MHD equations can be extended globally, even with relaxed pressure assumptions and no magnetic field in the criterion.

Contribution

It establishes a new blow-up criterion for 3D compressible MHD equations that depends solely on density, relaxing pressure assumptions and removing magnetic field constraints.

Findings

Global regularity is guaranteed if density stays away from vacuum and infinity.

New a priori estimates are developed for the 3D compressible MHD system.

The criterion applies without magnetic field influence in the blow-up condition.

Abstract

We address the compressible magnetohydrodynamics (MHD) equations in $\mathbb{R}^3$ and establish a blow-up criterion for the local strong solutions in terms of the density only. Namely, if the density is away from vacuum ($\rho= 0$) and the concentration of mass ($\rho=\infty$), then a local strong solution can be continued globally in time. The results generalise and strengthen the previous ones in the sense that there is no magnetic field present in the criterion and the assumption on the pressure is significantly relaxed. The proof is based on some new a priori estimates for three-dimensional compressible MHD equations.