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This paper concerns the Cauchy problem of the nonhomogeneous incompressiblemagnetohydrodynamic (MHD) equations on the whole two-dimensional (2D) spacewith vacuum as far field density. In particular, the initial density can havecompact support. We prove that the 2D Cauchy problem of the nonhomogeneousincompressible MHD equations admits a unique local strong solution provided theinitial density and the initial magnetic decay not too slow at infinity.

On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum