A blow-up criterion for the 3D compressible magnetohydrodynamics in terms of density

TL;DR

This paper establishes a criterion based on density and magnetic field bounds that determines whether solutions to the 3D compressible MHD equations can be extended globally, aiding understanding of solution blow-up conditions.

Contribution

It provides a new blow-up criterion for 3D compressible MHD equations based on density and magnetic field bounds, advancing the analysis of solution longevity.

Findings

Bounded density away from vacuum prevents blow-up.

Bounded magnetic field in L-infinity norm ensures solution continuation.

The criterion links physical quantities to mathematical solution behavior.

Abstract

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in $\R^3$. We establish a blow-up criterion for the local strong solutions in terms of the density and magnetic field. Namely, if the density is away from vacuum ($\rho= 0$) and the concentration of mass ($\rho=\infty$) and if the magnetic field is bounded above in terms of $L^\infty$-norm, then a local strong solution can be continued globally in time.