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This paper concerns the Cauchy problem of the two-dimensional (2D)nonhomogeneous incompressible Magnetohydrodynamic (MHD) equations with vacuumas far field density. We establish the global existence and uniqueness ofstrong solutions to the 2D Cauchy problem on the whole space $\mathbb{R}^2$,provided that the initial density and the initial magnetic decay not too slowat infinity. In particular, the initial data can be arbitrarily large and theinitial density can contain vacuum states and even have compact support.Furthermore, we also obtain the large time decay rates of the gradients ofvelocity, magnetic and pressure.

Global Existence and Large Time Asymptotic Behavior of Strong Solutions  to the Cauchy Problem of 2D Density-Dependent Magnetohydrodynamic Equations  with Vacuum