Global strong solutions for 3D viscous incompressible heat conducting magnetohydrodynamic flows with non-negative density

We study an initial boundary value problem for the nonhomogeneous heat conducting magnetohydrodynamic fluids with non-negative density. Firstly, it is shown that for the initial density allowing vacuum, the strong solution to the problem exists globally if the gradients of velocity and magnetic field satisfy ‖∇u‖L4(0,T;L2)+‖∇b‖L4(0,T;L2)<∞‖∇u‖L4(0,T;L2)+‖∇b‖L4(0,T;L2)<∞. Then, under some smallness condition, we prove that there is a unique global strong solution to the 3D viscous incompressible heat conducting magnetohydrodynamic flows. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equations.