Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier-Stokes system

摘要

In this paper, we investigate the global regularity to 3-D inhomogeneous incompressible Navier-Stokes system with axisymmetric initial data which does not have swirl component for the initial velocity. We first prove that the norm to the quotient of the inhomogeneity by r, namely controls the regularity of the solutions. Then we prove the global regularity of such solutions provided that the norm of is sufficiently small. Finally, with additional assumption that the initial velocity belongs to for some we prove that the velocity field decays to zero with exactly the same rate as the classical Navier-Stokes system.