Global smooth axisymmetric solutions of 3-D inhomogeneous 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 \(L^\infty \) norm to the quotient of the inhomogeneity by r, namely \(a/r\buildrel \hbox {def}\over =(1/\rho -1)/r,\) 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.