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

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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∞ norm to the quotient of the inhomogeneity by r, namely a/r(1/-1)/r, controls the regularity of the solutions. Then we prove the global regularity of such solutions provided that the L∞ norm of a₀/r is sufficiently small. Finally, with additional assumption that the initial velocity belongs to Lᵖ for some p∈ [1,2), we prove that the velocity field decays to zero with exactly the same rate as the classical Navier-Stokes system.