  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
$L^\infty$ norm to the quotient of the inhomogeneity by $r,$ namely
$a/r\eqdefa\bigl(1/\r-1\bigr)\bigl/r,$ controls the regularity of the
solutions. Then we prove the global regularity of such solutions provided that
the $L^\infty$ norm of $a_0/r$ is sufficiently small. Finally, with additional
assumption that the initial velocity belongs to $L^p$ for some $p\in [1,2),$ we
prove that the velocity field decays to zero with exactly the same rate as the
classical Navier-Stokes system.

Analysis of PDEs