Global well-posedness of 3-D density-dependent incompressible MHD equations with variable resistivity

In this paper, we investigate the global existence of weak solutions to 3-D inhomogeneous incompressible MHD equations with variable viscosity and resistivity, which is sufficiently close to $1$ in $L^{\mathrm{\infty }}({\mathbb{R}}^3),$ provided that the initial density is bounded from above and below by positive constants, and both the initial velocity and magnetic field are small enough in the critical space ${\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)\mathrm{.}$ Furthermore, if we assume in addition that the kinematic viscosity equals $1,$ and both the initial velocity and magnetic field belong to ${\dot{B}}_{2,1}^{\frac{1}{2}}({\mathbb{R}}^3),$ we can also prove the uniqueness of such solution.