We establish the global existence of weak solutions to isentropic compressible magnetohydrodynamic equations with ripped density in the whole plane provided the bulk viscosity coefficient is properly large. Moreover, we show that such solutions converge globally in time to a weak solution of the inhomogeneous incompressible magnetohydrodynamic equations when the bulk viscosity coefficient tends to infinity. In particular, the initial data can be arbitrarily large and vacuum states are allowed in interior regions. Our method relies on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality. To the best of our knowledge, this is the first result concerning incompressible limit of isentropic compressible magnetohydrodynamic equations for the large bulk viscosity.
Wang et al. (Wed,) studied this question.