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arXiv · 1409.2953

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

Hammadi Abidi·Ping Zhang
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Abstract

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.

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Hammadi Abidi, Ping Zhang. 2014-09-10. Global smooth axisymmetric solutions of 3-D Inhomogenenous incompressible Navier-Stokes system. https://arxiv.org/abs/1409.2953

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