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

Abstract: 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 ${L^{2}(s,T;L^{\infty}(\Omega))}$ bound for the velocity field, where the axisymmetry of the solutions plays an important role.