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

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

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References

  1. Abidi, H.: Résultats de régularité de solutions axisymétriques pour le système de Navier-Stokes. Bull. Sci. Math. 132, 592–624 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abidi, H., Hmidi, T., Keraani, S.: On the global well-posedness for the axisymmetric Euler equations. Math. Ann. 347, 15–41 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Abidi, H., Hmidi, T., Keraani, S.: On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete Contin. Dyn. Syst. 29, 737–756 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  4. Abidi, H., Gui, G., Zhang, P.: Stability to the global large solutions of the 3-D inhomogeneous Navier–Stokes equations. Commun. Pure. Appl. Math. 64, 832–881 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Abidi, H., Gui, G., Zhang, P.: Well-posedness of 3-D inhomogeneous Navier–Stokes equations with highly oscillatory initial velocity field. J. Math. Pures Appl. (9) 100, 166–203 (2013)

  6. Badiale, M., Tarantello, G.: A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239, 645–671 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Danchin, R.: Axisymmetric incompressible flows with bounded vorticity. Russ. Math. Survey 62, 73–94 (2007)

    Article  MathSciNet  Google Scholar 

  9. Danchin, R., Mucha, P.B.: A Lagrangian approach for the incompressible Navier–Stokes equations with variable density. Commun. Pure. Appl. Math. 65, 1458–1480 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Danchin, R., Zhang, P.: Inhomogeneous Navier–Stokes equations in the half-space, with only bounded density. J. Funct. Anal. 267, 2371–2436 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education Inc, Upper Saddle River (2004)

    MATH  Google Scholar 

  12. He, C., Miyakawa, T.: On two-dimensional Navier–Stokes flows with rotational symmetries. Funkcial. Ekvac. 49, 163–192 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hmidi, T., Rousset, F.: Global well-posedness for the Euler–Boussinesq system with axisymmetric data. J. Funct. Anal. 260, 745–796 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kim, H.: A blow-up criterion for the nonhomogeneous incompressible Navier–Stokes equations. SIAM J. Math. Anal. 37, 1417–1434 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ladyz̆enskaja, O.A.: Unique global solvability of the three-dimensional Cauchy problem for the Navier-Stokes equations in the presence of axial symmetry (Russian). Zap. Nauc̆n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 7, 155–177 (1968)

  16. Ladyzhenskaya, O., Solonnikov, V.: The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. J. Sov. Math. 9, 697–749 (1978)

    Article  MATH  Google Scholar 

  17. Leonardi, S., Málek, J., Nec̆as, J., Pokorny, M.: On axially symmetric flows in \(\mathbb{R}^3\). Z. Anal. Anwendungen 18, 639–649 (1999)

  18. Lions, P.L.: Mathematical topics in fluid mechanics, vol. 1. Incompressible models, Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1996)

  19. Miao, C., Zheng, X.: On the global well-posedness for the Boussinesq system with horizontal dissipation. Commun. Math. Phys. 321, 33–67 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  20. Paicu, M., Zhang, P., Zhang, Z.: Global well-posedness of inhomogeneous Navier–Stokes equations with bounded density. Commun. Partial Differ. Equ. 38, 1208–1234 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  21. Schonbek, M.: Large time behavior of solutions to Navier–Stokes equations. Commun. Partial Differ. Equ. 11, 733–763 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  22. Simon, J.: Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21, 1093–1117 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ukhovskii, M.R., Iudovich, V.I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech. 32, 52–61 (1968)

    Article  MathSciNet  Google Scholar 

  24. Zhang, P., Zhang, T.: Global axisymmetric solutions to three-dimensional Navier–Stokes system. Int. Math. Res. Not., IMRN2014, no. 3, pp. 610–642

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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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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences de Tunis Campus Universitaire, 2092, Tunis, Tunisia

    Hammadi Abidi

  2. Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China

    Ping Zhang

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  1. Hammadi Abidi
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  2. Ping Zhang
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Correspondence to Ping Zhang.

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Communicated by F. H. Lin.

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Abidi, H., Zhang, P. Global smooth axisymmetric solutions of 3-D inhomogeneous incompressible Navier–Stokes system. Calc. Var. 54, 3251–3276 (2015). https://doi.org/10.1007/s00526-015-0902-6

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  • DOI: https://doi.org/10.1007/s00526-015-0902-6

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