Global strong solutions to the incompressible Navier–Stokes equations with density-dependent viscosity

This paper is concerned with the 3D incompressible Navier–Stokes equations with density-dependent viscosity in a smooth bounded domain. The global well-posedness of strong solutions is established for the case when the bound of density is suitably small, or when the total mass is small with large oscillations. The vacuum is allowed in both cases.