Skip to main content
SpringerLink
Log in

Regularity Criterion for the 3D Micropolar Fluid Equations in Terms of Pressure

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, we consider the regularity of weak solutions to the 3D incompressible micropolar fluid equations. It is proved that if the one directional derivative of the pressure, say \(\partial _{3}P\), satisfies

$$\begin{aligned} \partial _{3}P \in L^{\beta }(0,T;L^{\alpha }(\mathbb {R}^{3})) \quad \text { with } \frac{2}{\beta }+\frac{3}{\alpha }\le 2, \frac{3}{2}\le \alpha <\infty , \end{aligned}$$

then the corresponding weak solution \((u,\omega )\) is regular on [0, T].

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. Adams, R., Fournier, J.: Soblev Spaces, 2nd edn. Academic, New York (2003)

    Google Scholar 

  2. Berselli, L., Galdi, G.: Regularity criteria involving the pressure for the weak solutions of the Navier–Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cao, C., Titi, E.: Regularity criteria for the three-dimensional Navier–Stokes equaitons. India Univ. Math. J. 57, 2643–2661 (2008)

    Article  MATH  Google Scholar 

  4. Cao, C., Wu, J.: Two new regularity criteria for the 3D MHD equations. J. Differ. Equ. 248, 2263–2274 (2010)

    Article  MATH  Google Scholar 

  5. Chemin, J.: Perfect Incompressible Fluids. Oxford University Press, New York (1998)

    MATH  Google Scholar 

  6. da Veiga, H.B.: A new regularity class for the Navier–Stokes equations in \({\mathbb{R}} ^{n}\). Chin. Ann. Math. Ser. B 16, 407–412 (1995)

    MATH  Google Scholar 

  7. da Veiga, H.B.: A sufficient condition on the pressure for the regularity of weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 2, 99–106 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dong, B., Chen, Z.: Regularity criteria of weak solutions to the three-dimensional micropolar flows. J. Math. Phys. 50, 103525 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dong, B., Jia, Y., Chen, Z.: Pressure regularity criteria of the three-dimensional micropolar fluid flows. Math. Methods Appl. Sci. 34, 595–606 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dong, B., Li, J., Wu, J.: Global well-posedness and large-time decay for the 2D micropolar equations. J. Differ. Equ. 262, 3488–3523 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dong, B., Zhang, W.: On the regularity criterion for the three-dimensional micropolar fluid flows in Besov spaces. Nonlinear Anal. 73, 2334–2341 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dong, B., Zhang, Z.: Global regularity of the 2D micropolar fluid flows with zero angular viscosity. J. Differ. Equ. 249, 200–213 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dong, B., Zhang, Z.: On the weak-strong uniqueness of Koch–Tataru’s solution for the Navier–Stokes equations. J. Differ. Equ. 256, 2406–2422 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eringen, A.: Theory of micropolar fluids. J. Math. Mech. 16, 1–18 (1966)

    MathSciNet  Google Scholar 

  15. Escauriaza, L., Seregin, G., S̆verák, V.: \(L^{3,\infty }-\)solutions of Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58, 211–250 (2003)

    Article  MATH  Google Scholar 

  16. 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  MATH  Google Scholar 

  17. Gala, S.: On the regularity criteria for the three-dimensional micropolar fluid equations in the critical Morrey–Campanato space. Nonlinear Anal. Real World Appl. 12, 2142–2150 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gala, S.: A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of pressure. Math. Methods Appl. Sci. 34, 1945–1953 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Galdi, G., Rionero, S.: A note on the existence and uniqueness of solutions of micropolar fluid equations. Int. J. Eng. Sci. 14, 105–108 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jia, X., Zhou, Y.: A new regularity criterion for the 3D incompressible MHD equations in terms of one compenent of the gredient of pressure. J. Math. Anal. Appl. 396, 345–350 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jia, Y., Zhang, W., Dong, B.: Remarks on the regularity of the 3D micropolar fluid flows in terms of the pressure. Appl. Math. Lett. 24, 199–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. Chapman and Hall/CRC, London (2002)

    Book  MATH  Google Scholar 

  23. Lukaszewicz, G.: Micropolar Fluids: Theory and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Boston (1999)

    Book  MATH  Google Scholar 

  24. Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. 48, 173–182 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rojas-Medar, M.: Magnato-microplar fluid motion: existence and uniqueness of strong solution. Math. Nachr. 188, 301–319 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–195 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sohr, H.: A generalization of Serrin’s regularity criterion for the Navier–Stokes equations. Quaderni Di Math. 10, 321–347 (2002)

    MathSciNet  MATH  Google Scholar 

  28. Struwe, M.: On a Serrin-type regularity criterion for the Navier–Stokes equaitons in terms of the pressure. J. Math. Fluid Mech. 9, 235–242 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Xiang, Z., Yang, H.: On the regularity criteria for the 3D magneto-micropolar fluids in terms of one directional derivative. Bound. Value Probl. 2012, 139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yamaguchi, N.: Existence of global strong solution to the micropolar fluid system in a bounded domain. Math. Methods in Appl. Sci. 28, 1507–1526 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, Z., Li, P., Yu, G.: Regularity criteria for the 3D MHD equations via one directional derivative of the pressure. J. Math. Anal. Appl. 401, 66–71 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zhang, Z.: A regularity criterion for the three-dimensional micropolar fluid system in homogeneous Besov spaces. Electron. J. Differ. Equ. 104, 1–6 (2016)

    Google Scholar 

  33. Zhou, Y.: Regularity criteria in terms of pressure for the 3-D Navier–Stokes equations in a generic domain. Math. Ann. 328, 173–192 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhou, Y.: On the regularity criteria in terms of pressure for the Navier–Stokes equations in \(\mathbb{R}^{3}\). Proc. Am. Math. Soc. 134, 149–156 (2006)

    Article  MATH  Google Scholar 

  35. Zhou, Y.: On a regularity criterion in terms of the gradient of pressure for the Navier–Stokes equations in \(\mathbb{R}^{3}\). Z. Angew. Math. Phys. 57, 384–392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhou, Y.: Regularity criteria for the 3D MHD equations in terms of the pressure. Int. J. Non Linear Mech. 41, 1174–1180 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to acknowledge his thanks to the referees and the managing editor for valuable comments and suggestions to improve the presentation of this manuscript.

Author information

Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP) (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Qiao Liu

Authors
  1. Qiao Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiao Liu.

Additional information

Communicated by Syakila Ahmad.

This work is partially supported by the National Natural Science Foundation of China (11401202).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Q. Regularity Criterion for the 3D Micropolar Fluid Equations in Terms of Pressure. Bull. Malays. Math. Sci. Soc. 42, 1305–1317 (2019). https://doi.org/10.1007/s40840-017-0545-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0545-1

Keywords

Mathematics Subject Classification

Search

Navigation