Global Well-Posedness and exponential decay of 3D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum

Abstract We establish the global well-posedness of strong solutions for the inhomogeneous magnetohydrodynamic equations with density-dependent viscosity in three-dimensional bounded domains, where the initial density is permitted to vanish. By skillfully employing various inequalities and scaling techniques, we demonstrate the existence and uniqueness of a global strong solution provided that ||σ( ρ 0 )|| L q is suitably small. Furthermore, we derive exponential decay rates for the solution. Importantly, no compatibility conditions need to be imposed on the initial data, even when vacuum states are present.