A Blow-up Criterion for the Two-Dimensional Nonhomogeneous Incompressible Magnetohydrodynamic Equations with Vacuum

This paper establishes a blow-up criterion for strong solutions to the Cauchy problem of the two-dimensional nonhomogeneous incompressible magnetohydrodynamic(MHD) equations with vacuum as far field density. The criterion is only in terms of the spatial weighted density, and is independent of both the velocity and the magnetic field. In particular, the initial data is not imposed any additional compatibility conditions and the initial density can contain vacuum states and even have compact support.