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

  • Hammadi Abidi
  • Ping ZhangEmail author
Article

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 \(a/r\buildrel \hbox {def}\over =(1/\rho -1)/r,\) 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.

Mathematics Subject Classification

35Q30 76D03 

Notes

Acknowledgments

We would like to thank Rapha\(\ddot{e}\)l Danchin and Guilong Gui for profitable discussions on this topic. Part of this work was done when we were visiting Morningside Center of Mathematics, CAS, in the summer of 2014. We appreciate the hospitality and the financial support from the Center. P. Zhang is partially supported by NSF of China under Grant 11371037, the fellowship from Chinese Academy of Sciences and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Département de MathématiquesFaculté des Sciences de Tunis Campus UniversitaireTunisTunisia
  2. 2.Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina

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