Advertisement

The global solutions of axisymmetric Navier–Stokes equations with anisotropic initial data

  • Hui ChenEmail author
  • Daoyuan Fang
  • Ting Zhang
Article
  • 55 Downloads

Abstract

Consider the Cauchy problem of incompressible Navier–Stokes equations in three dimension with axisymmetric initial data. If the swirl component satisfies some certain regularity criteria in weighted spaces, the existence of the time-global solution has been proved in our previous work. However, it is more or less evident that the solution is anisotropic in vertical direction (i.e., the direction parallel to the axis of symmetry) and the horizontal directions (in the plane perpendicular to the axis of symmetry). In this paper, we are interested in constructing regularity criteria and time-global solution in anisotropic Lebesgue spaces.

Keywords

Axisymmetric Navier–Stokes equations Regularity criteria Global solution Anisotropic 

Mathematics Subject Classification

Primary 35Q30 Secondary 76D03 76D05 76N10 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his (her) careful reading and kind suggestions on the manuscript.

References

  1. 1.
    Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction. Grundlehren der Mathematischen Wissenschaften, No. 223. Springer, Berlin (1976)zbMATHGoogle Scholar
  2. 2.
    Carlen, E.A., Loss, M.: Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the \(2\)-D Navier–Stokes equation. Duke Math. J. 81(1), 135–157 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chae, D., Lee, J.: On the regularity of the axisymmetric solutions of the Navier–Stokes equations. Math. Z. 239(4), 645–671 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, C., Strain, R.M., Yau, H., Tsai, T.: Lower bound on the blow-up rate of the axisymmetric Navier–Stokes equations. Int. Math. Res. Not. 2008(9), 31 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Chen, C., Strain, R.M., Yau, H., Tsai, T.: Lower bounds on the blow-up rate of the axisymmetric Navier–Stokes equations II. Comm. Partial Differ. Equ. 34(1–3), 203–232 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, H., Fang, D., Zhang, T.: Global axisymmetric solutions of three dimensional inhomogeneous incompressible Navier–Stokes system with nonzero swirl. Arch. Ration. Mech. Anal. 223(2), 817–843 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, H., Fang, D., Zhang, T.: Regularity of 3D axisymmetric Navier–Stokes equations. Discrete Contin. Dyn. Syst. 37(4), 1923–1939 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen, Q., Zhang, Z.: Regularity criterion of axisymmetric weak solutions to the 3D Navier–Stokes equations. J. Math. Anal. Appl. 331(2), 1384–1395 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Escauriaza, L., Seregin, G., Šverák, V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169(2), 147–157 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fabes, E.B., Jones, B.F., Rivieère, N.M.: The initial value problem for the Navier–Stokes equations with data in \(L^{p}\). Arch. Ration. Mech. Anal. 45, 222–240 (1972)CrossRefGoogle Scholar
  11. 11.
    Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)CrossRefGoogle Scholar
  12. 12.
    Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hou, Thomas Y., Li, C.: Dynamic stability of the three-dimensional axisymmetric Navier–Stokes equations with swirl. Comm. Pure Appl. Math. 61(5), 661–697 (2008)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Koch, G., Nadirashvili, N., Seregin, G.A., S̆verák, V.: Liouville theorems for the Navier–Stokes equations and applications. Acta Math. 203(1), 83–105 (2009)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kreml, O., Pokorný, M.: A regularity criterion for the angular velocity component in axisymmetric Navier–Stokes equations. Electron. J. Differ. Equ. 2017(08), 10 (2017)zbMATHGoogle Scholar
  16. 16.
    Kubica, A., Pokorný, M., Zajaczkowski, W.: Remarks on regularity criteria for axially symmetric weak solutions to the Navier–Stokes equations. Math. Methods Appl. Sci. 35(3), 360–371 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ladyženskaja, O.A.: Unique global solvability of the three-dimensional Cauchy problem for the Navier–Stokes equations in the presence of axial symmetry. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. 7, 155–177 (1968)MathSciNetGoogle Scholar
  18. 18.
    Lei, Z., Zhang, Qi S.: A Liouville theorem for the axially-symmetric Navier–Stokes equations. J. Funct. Anal. 261(8), 2323–2345 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lei, Z., Zhang, Qi S.: Criticality of the axially symmetric Navier–Stokes equations. Pac. J. Math. 289(1), 169–187 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Leonardi, S., Málek, J., Nečas, J., Pokorný, M.: On axially symmetric flows in \(\mathbb{R}^{3}\). Z. Anal. Anwendungen 18(3), 639–649 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC Research Notes in Mathematics, Vol. 431. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  22. 22.
    Leray, J.: Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’Hydrodynamique. Journal de Mathématiques Pures et Appliquées 12, 1–82 (1933)zbMATHGoogle Scholar
  23. 23.
    Liu, J., Wang, W.: Characterization and regularity for axisymmetric solenoidal vector fields with application to Navier–Stokes equation. SIAM J. Math. Anal. 41(5), 1825–1850 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Neustupa, J., Pokorný, M.: Axisymmetric flow of Navier–Stokes fluid in the whole space with non-zero angular velocity component. Math. Bohem. 126(2), 469–481 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Prodi, G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Math. Pura Appl. 48, 173–182 (1959)CrossRefGoogle Scholar
  26. 26.
    Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Struwe, M.: On partial regularity results for the Navier–Stokes equations. Comm. Pure Appl. Math. 41(4), 437–458 (1988)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Takahashi, S.: On interior regularity criteria for weak solutions of the Navier–Stokes equations. Manuscr. Math. 69(3), 237–254 (1990)MathSciNetCrossRefGoogle Scholar
  29. 29.
    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)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wei, D.: Regularity criterion to the axially symmetric Navier–Stokes equations. J. Math. Anal. Appl. 435(1), 402–413 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhang, P., Zhang, T.: Global axisymmetric solutions to three-dimensional Navier–Stokes system. Int. Math. Res. Not. 2014(3), 610–642 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of ScienceZhejiang University of Science and TechnologyHangzhouPeople’s Republic of China
  2. 2.School of Mathematical SciencesZhejiang UniversityHangzhouPeople’s Republic of China

Personalised recommendations