Blow-up criterion for the compressible magnetohydrodynamic equations with vacuum

Abstract

In this paper, the 3-D compressible MHD equations with initial vacuum or infinity electric conductivity is considered. We prove that the $L^\infty$ norms of the deformation tensor $D(u)$ and the absolute temperature $\theta$ control the possible blow-up (see [5][18][20]) for strong solutions, which means that if a solution of the compressible MHD equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of $D(u)$ and $\theta$ as the critical time approaches. The viscosity coefficients are only restricted by the physical conditions. Our criterion (see (\ref{eq:2.911})) is similar to [17] for $3$-D incompressible Euler equations, [10] for $3$-D compressible isentropic Navier-stokes equations and [22]for $3$-D compressible isentropic MHD equations.