Blow-up criterion for the compressible magnetohydrodynamic equations with vacuum

In this paper, the 3-D compressible magnetohydrodynamic (MHD) equations with initial vacuum or infinite electric conductivity is considered. We prove that the L∞L∞ norms of the deformation tensor D(u)D(u) and the absolute temperature θ control the possible blow-up (see [18] and [23]) of strong solutions, especially for the non-resistive MHD system when the magnetic diffusion vanishes. This conclusion 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)D(u) or θ as the critical time approaches. The viscosity coefficients are only restricted by the physical conditions. Our criterion (see (1.17)) is similar to [17] for 3-D incompressible Euler equations and to [12] for 3-D compressible isentropic Navier–Stokes equations.