Global strong solutions to the inhomogeneous incompressible Navier-Stokes system in the exterior of a cylinder

In this paper, the global strong axisymmetric solutions for the inhomogeneous incompressible Navier-Stokes system are established in the exterior of a cylinder subject to the Dirichlet boundary conditions. Moreover, the vacuum is allowed in these solutions. One of the key ingredients of the analysis is to obtain the $L^2(s,T;L^{\mathrm{\infty }}(\mathrm{\Omega }))$ bound for the velocity field, where the axisymmetry of the solutions plays an important role.