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

摘要

In this paper, we investigate the global regularity to 3-D inhomogeneousincompressible Navier-Stokes system with axisymmetric initial data which doesnot 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 thesolutions. Then we prove the global regularity of such solutions provided thatthe $L^\infty$ norm of $a_0/r$ is sufficiently small. Finally, with additionalassumption that the initial velocity belongs to $L^p$ for some $p\in [1,2),$ weprove that the velocity field decays to zero with exactly the same rate as theclassical Navier-Stokes system.