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

Abstract

We study an initial boundary value problem for the 3D magnetohydrodynamics (MHD) equations of compressible fluids in ${\mathbb{R}}^3$. 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.

15 pages