Global well-posedness of the two-dimensional incompressible magnetohydrodynamics system with variable density and electrical conductivity

Studied in this paper is the Cauchy problem of the two-dimensional magnetohydrodynamics system with inhomogeneous density and electrical conductivity. It is shown that the 2-D incompressible inhomogeneous magnetohydrodynamics system with a constant viscosity is globally well-posed for a generic family of the variations of the initial data and an inhomogeneous electrical conductivity. Moreover, it is established that the system is globally well-posed in the critical spaces if the electrical conductivity is homogeneous.