Global Existence of Strong Solutions to Incompressible MHD

We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for incompressible MHD equations in a bounded smooth domain of three spatial dimensions with initial density being allowed to have vacuum, in particular, the initial density can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\mathrm{\mid }{\sqrt{\rho }}_0{u}_0{\mathrm{\mid }}_{L^2(\mathrm{\Omega })}^2+\mathrm{\mid }{H}_0{\mathrm{\mid }}_{L^2(\mathrm{\Omega })}^2$ and $\mathrm{\mid }\mathrm{\nabla }{u}_0{\mathrm{\mid }}_{L^2(\mathrm{\Omega })}^2+\mathrm{\mid }\mathrm{\nabla }{H}_0{\mathrm{\mid }}_{L^2(\mathrm{\Omega })}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.