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We revisit the 2D Cauchy problem of nonhomogeneous heat conductingmagnetohydrodynamic (MHD) equations in $\mathbb{R}^2$. For the initial densityallowing vacuum at infinity, we derive the global existence and uniqueness ofstrong solutions provided that the initial density and the initial magneticdecay not too slowly at infinity. In particular, the initial data can bearbitrarily large. This improves our previous work where the initial densityhas non-vacuum states at infinity. The result could also be viewed as anextension of the study in L{\"u}-Xu-Zhong for the inhomogeneous case to thefull inhomogeneous situation. The method is based on delicate spatial weightedestimates and the structural characteristic of the system under consideration.As a byproduct, we get the global existence of strong solutions to the 2DCauchy problem for nonhomogeneous heat conducting Navier-Stokes equations withvacuum at infinity.

journal of mathematical fluid mechanics

Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat  conducting Navier-Stokes and magnetohydrodynamic equations with vacuum at  infinity