Global Strong solution with vacuum to the 2D nonhomogeneous incompressible MHD system

In this paper, we first prove the unique global strong solution with vacuum to the two dimensional nonhomogeneous incompressible MHD system, as long as the initial data satisfies some compatibility condition. As a corollary, the global existence of strong solution with vacuum to the 2D nonhomogeneous incompressible Navier-Stokes equations is also established. Our main result improves all the previous results where the initial density need to be strictly positive. The key idea is to use some critical Sobolev inequality of logarithmic type, which is originally due to Brezis-Wainger.