A blow-up criterion of strong solutions to the 3D compressible MHD equations with vacuum

In this paper, we prove a blow-up criterion of strong solutions to 3D viscous isentropic compressible magnetohydrodynamic equations. It is shown that if ρ and H satisfy ‖ ρ ‖ L ∞ ( 0 , T ; L ∞ ) + ‖ H ‖ L ∞ ( 0 , T ; L r ) < ∞ , for any 24 / 5 ≤ r ≤ ∞ and 3 μ > λ , then the strong solutions to the Cauchy problem of the compressible magnetohydrodynamic equations can exist globally on [ 0 , T ] . In addition, initial vacuum is allowed.