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⊂R3Ω⊂R3. We establish a criterion for possible breakdown of such solutions at a finite time in terms of both ‖∇u‖L1(0,T;L∞)‖∇u‖L1(0,T;L∞) and ‖θ‖L∞(0,T;L∞)‖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‖L1(0,T;L∞)‖∇u‖L1(0,T;L∞) and ‖θ‖L∞(0,T;L∞)‖θ‖L∞(0,T;L∞) as the critical time approaches. In addition, initial vacuum states are allowed in our cases.