Blow-up criterion for 3-D compressible magnetohydrodynamics with vacuum and zero resistivity

We study strong solutions of the equations of compressible magnetohydrodynamics with zero resistivity in a domain Ω ⊂ R 3 . We establish a criterion for possible breakdown of such solutions at a finite time in terms of both ‖ ∇ u ‖ L 1 ( 0 , T ; L ∞ ) and ‖ θ ‖ L ∞ ( 0 , T ; L ∞ ) . More precisely, if a solution of 3D compressible magnetohydrodynamics with zero resistivity is initially regular and loses its regularity at some later time, then the loss of regularity implies growth without bound of both ‖ ∇ u ‖ L 1 ( 0 , T ; L ∞ ) and ‖ θ ‖ L ∞ ( 0 , T ; L ∞ ) as the critical time approaches. In addition, initial vacuum states are allowed in our cases.