Skip to main content
SpringerLink
Log in

Some regularity criteria for the 3D generalized Navier–Stokes equations

  • Published:
Zeitschrift für angewandte Mathematik und Physik Aims and scope Submit manuscript

Abstract

We show some regularity criteria (Prodi–Serrin type regularity) to weak solutions of the 3D generalized Navier–Stokes equations in viewpoint of the velocity vector u or the vorticity vector \(\omega :=\nabla \times u\) in Lorentz space. Moreover, we briefly mention some results for coupled equations with Navier–Stokes equation (see Remark 1.5 and 1.8).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bosia, S., Pata, V., Robinson, J.C.: A weak-\(L^p\) Prodi–Serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)

    Article  MathSciNet  Google Scholar 

  2. Chae, D.: On the regularity conditions for the Navier–Stokes and related equations. Rev. Mat. Iberoam. 23, 371–384 (2007)

    Article  MathSciNet  Google Scholar 

  3. Chae, D., Lee, J.: On the geometric regularity conditions for the 3D Navier–Stokes equations. Nonlinear Anal. 151, 265–273 (2017)

    Article  MathSciNet  Google Scholar 

  4. Códorba, A., Códorba, D.: A maximum principle applied to quasi-geostrophic equations. Commun. Math. Phys. 249, 511–528 (2004)

    Article  MathSciNet  Google Scholar 

  5. Colombo, M., De Lellis, C., De Rosa, L.: Ill-posedness of Leray solutions for the hypodissipative Navier–Stokes equations. Commun. Math. Phys. 362, 659–688 (2017)

    Article  MathSciNet  Google Scholar 

  6. Colombo, M., Haffter, S.: Global regularity for the hyperdissipative Navier–Stokes equation below the critical order. Journal of Differential Equations. 75, 815–836 (2021)

  7. Constantin, P.: Geometric statistics in turbulence. SIAM Rev. 36, 73 (1994)

    Article  MathSciNet  Google Scholar 

  8. Dascaliuc, R., Grujić, Z., Jolly, M.S.: Effect of vorticity coherence on energy-enstrophy bounds for the 3D Navier–Stokes equations. J. Math. Fluid Mech. 17, 393–410 (2015)

    Article  MathSciNet  Google Scholar 

  9. De Rosa, L.: Infinitely many Leray–Hopf solutions for the fractional Navier–Stokes equations. Commun. Partial Differ. Equ. 44, 335–365 (2019)

    Article  MathSciNet  Google Scholar 

  10. Fan, J., Ozawa, T.: On the regularity criteria for the generalized Navier–Stokes equations and Lagrangian averaged Euler equations. Differ. Integral Equ. 21, 443–457 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Fan, J., Jiang, S., Nakamura, G., Zhou, Y.: Logarithmically improved regularity criteria for the Navier–Stokes and MHD equations. J. Math. Fluid Mech. 13, 557–571 (2011)

    Article  MathSciNet  Google Scholar 

  12. Fan, J., Zhou, Y.: Uniform local well-posedness for the density-dependent magnetohydrodynamic equations. Appl. Math. Lett. 24, 1945–1949 (2011)

    Article  MathSciNet  Google Scholar 

  13. Fan, J., Fukumoto, Y., Zhou, Y.: Logarithmically improved regularity criteria for the generalized Navier–Stokes and related equations. Kinet. Relat. Models 6, 545–556 (2013)

    Article  MathSciNet  Google Scholar 

  14. Fan, J., Alsaedi, A., Hayat, T., Nakamura, G., Zhou, Y.: A regularity criterion for the 3D generalized MHD equations. Math. Phys. Anal. Geom. 17, 333–340 (2014)

    Article  MathSciNet  Google Scholar 

  15. Fan, J., Jia, X., Nakamura, G., Zhou, Y.: On well-posedness and blowup criteria for the magnetohydrodynamics with the Hall and ion-slip effects. Z. Angew. Math. Phys. 66, 1695–1706 (2015)

    Article  MathSciNet  Google Scholar 

  16. Fan, J., Zhang, Z., Zhou, Y.: Local well-posedness for the incompressible full magneto-micropolar system with vacuum. Z. Angew. Math. Phys. 71, 1–11 (2020)

    Article  MathSciNet  Google Scholar 

  17. Jiang, Z., Wang, Y., Zhou, Y.: On regularity criteria for the 2D generalized MHD system. J. Math. Fluid Mech. 18, 331–341 (2016)

    Article  MathSciNet  Google Scholar 

  18. Jiang, Z., Ma, C., Zhou, Y.: Commutator estimates with fractional derivatives and local existence for the generalized MHD equations. Z. Angew. Math. Phys. 111 (2021)

  19. Lee, J.: Notes on the geometric regularity criterion of 3D Navier–Stokes system. J. Math. Phys. 53, 073103 (2012)

    Article  MathSciNet  Google Scholar 

  20. Loayza, M., Rojas-Medar, M.A.: A weak-Lp Prodi–Serrin type regularity criterion for the micropolar fluid equations. J. Math. Phys. 57, 021512 (2016)

    Article  MathSciNet  Google Scholar 

  21. Luo, Y.: On the regularity of generalized MHD equations. J. Math. Anal. Appl. 365, 806–808 (2010)

    Article  MathSciNet  Google Scholar 

  22. O’Neil, R.: Convolution operators and \(L(p, q)\) spaces. Duke Math. J. 30, 129–142 (1963)

    Article  MathSciNet  Google Scholar 

  23. Nakai, K.: Direction of vorticity and a refined regularity criterion for the Navier–Stokes equations with fractional Laplacian. J. Math. Fluid Mech. 21, 1–8 (2019)

    Article  MathSciNet  Google Scholar 

  24. Ni, L., Guo, Z., Zhou, Y.: Some new regularity criteria for the 3D MHD equations. J. Math. Anal. Appl. 396, 108–118 (2012)

    Article  MathSciNet  Google Scholar 

  25. Pan, N., Zhu, M.: A new regularity criterion for the 3D generalized Hall-MHD system with \(\beta \in (\frac{1}{2},1]\). J. Math. Anal. Appl. 445, 604–611 (2017)

    Article  MathSciNet  Google Scholar 

  26. Pineau, B., Yu, X.: On Prodi–Serrin type conditions for the 3D Navier–Stokes equations. Nonlinear Anal. 190, 111612 (2020)

    Article  MathSciNet  Google Scholar 

  27. Roulstone, I., Banos, B., Gibbon, J.D., Roubtsov, V.N.A.: Geometric interpretation of coherent structures in Navier–Stokes flows. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 2015–2021 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

    Google Scholar 

  29. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)

    Book  Google Scholar 

  30. Yamazaki, K.: Remarks on the regularity criteria of generalized MHD and Navier–Stokes systems. J. Math. Phys. 54, 011502 (2013)

    Article  MathSciNet  Google Scholar 

  31. Wan, R., Zhou, Y.: Global well-posedness for the 3D incompressible Hall-magnetohydrodynamic equations with Fujita–Kato type initial data. J. Math. Fluid Mech. 21, 1–16 (2019)

    Article  MathSciNet  Google Scholar 

  32. Wan, R., Zhou, Y.: Global well-posedness, BKM blow-up criteria and zero \(h\) limit for the 3D incompressible Hall-MHD equations. J. Differ. Equ. 267, 3724–3747 (2019)

    Article  MathSciNet  Google Scholar 

  33. Wang, J.: Balance of the vorticity direction and the vorticity magnitude in 3D fractional Navier–Stokes equations. arXiv:2001.04792

  34. Wu, J.: Lower bounds for an integral involving fractional Laplacians and the generalized Navier–Stokes equations in Besov spaces. Commun. Math. Phys. 263, 803–831 (2006)

    Article  MathSciNet  Google Scholar 

  35. Wu, J.: Generalized MHD equations. J. Differ. Equ. 195, 284–312 (2003)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Z., Wang, W., Yong, Z.: Global regularity criterion for the Navier–Stokes equations based on the direction of vorticity. Math. Methods Appl. Sci. 42, 7126–7134 (2019)

    Article  MathSciNet  Google Scholar 

  37. Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincare Anal. Non Lineaire 24, 491–505 (2007)

    Article  MathSciNet  Google Scholar 

  38. Zhou, Y., Fan, J.: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. 24, 691–708 (2012)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the very knowledgeable referee very much for his/her valuable comments and helpful suggestions. Jae-Myoung Kim was supported by Basic Science Research Program through the National Research Foundation(NRF) of Korea funded by the Ministry of Education (No. NRF-2020R1C1C1A01006521).

Author information

Authors and Affiliations

  1. Department of Mathematics Education, Andong National University, Andong, 36729, Republic of Korea

    Jae-Myoung Kim

Authors
  1. Jae-Myoung Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jae-Myoung Kim.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kim, JM. Some regularity criteria for the 3D generalized Navier–Stokes equations. Z. Angew. Math. Phys. 72, 118 (2021). https://doi.org/10.1007/s00033-021-01549-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00033-021-01549-z

Keywords

Mathematics Subject Classification

Search

Navigation