A BKM type blow-up criterion of strong solutions to the 3D inhomogeneous incompressible magnetohydrodynamic equations with initial vacuum

In this paper, we consider the regularity of 3D inhomogeneous incompressible magnetohydrodynamic (MHD) equations with initial vacuum, and establish a new regularity criterion in terms of only infinite modulus of gradient of velocity, which is known for the incompressible/compressible MHD equations. As far as we know, the result obtained in this paper is the first BKM type criterion for 3D inhomogeneous incompressible MHD equations with initial vacuum. In addition, we establish the blow-up criterion for the inhomogeneous incompressible Navier-Stokes equations.