Global well-posedness of 3-D nonhomogeneous incompressible MHD equations with bounded nonnegative density

In this paper, we consider a nonhomogeneous incompressible MHD equations in the torus T3 or a C2 simply connected bounded domain of R3. When the initial velocity and magnetic are in H1 and initial density is only bounded nonnegative, we first construct the existence of local in time solution for arbitrary large data and global in time for the model with some smallness conditions. Moreover, the uniqueness of the solution is also proved under some quite soft assumptions about its regularity through a Lagrangian approach. In particular, the initial vacuum is allowed.