Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains

The nonhomogeneous incompressible Magnetohydrodynamic Equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with slip boundary conditions. The key is the constraint of an additional initial value condition $B_0\in L^p(1⩽p<12\mathrm{/}7)$, which increase decay-in-time rates of the solutions, thus we obtain the global existence of strong solutions provided the gradient of the initial velocity and initial magnetic field is suitably small. In particular, the initial density is allowed to contain vacuum states and large oscillations. Moreover, the large-time behavior of the solution is also shown.