Skip to main content
Log in

A logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

This paper proves a logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain with the Navier-type boundary condition.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. R. A. Adams, J. J. F. Fournier. Sobolev Spaces. Pure and Applied Mathematics 140, Academic Press, New York, 2003.

    Google Scholar 

  2. J. Azzam, J. Bedrossian: Bounded mean oscillation and the uniqueness of active scalar equations. Trans. Am. Math. Soc. 567 2015), 3095–3118.

    MathSciNet  MATH  Google Scholar 

  3. H. Beirao da Veiga, L. C. Berselli: Navier-Stokes equations: Green’s matrices, vorticity direction, and regularity up to the boundary. J. Differ. Equation. 37 2009), 597–628.

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Beirao da Veiga, F. Crispo: Sharp inviscid limit results under Navier type boundary conditions. An V theory. J. Math. Fluid Mech. 37 2010), 397–411.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. Bendali, J. M. Dominguez, S. Gallic: A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains. J. Math. Anal. Appl. 107 (1985), 537–560.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. C. Berselli: On a regularity criterion for the solutions to the 3D Navier-Stokes equations. Differ. Integral Equ. 37 2002), 1129–1137.

    MathSciNet  MATH  Google Scholar 

  7. V. Georgescu: Some boundary value problems for differential forms on compact Riemannian manifolds. Ann. Mat. Pura Appl. (4) 122. 1979), 159–198.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Giga: Solutions for semilinear parabolic equations in LP and regularity of weak solutions of the Navier-Stokes system. J. Differ. Equation. 37 1986), 186–212.

    Article  MATH  Google Scholar 

  9. F. He, C. Ma, Y. Wang: On regularity for the Boussinesq system in a bounded domain. Appl. Math. Comput. 37 2016), 148–151.

    MathSciNet  MATH  Google Scholar 

  10. E. Hopf: Über die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen. Math. Nachr. 37 1951), 213–231. (In German.)

    MATH  Google Scholar 

  11. T. Huang, C. Wang, H. Wen: Strong solutions of the compressible nematic liquid crystal flow. J. Differ. Equation. 37 2012), 2222–2265.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 37 1934), 193–248. (In French.)

    Article  MathSciNet  MATH  Google Scholar 

  14. P.-L. Lions: Mathematical Topics in Fluid Mechanics. Vol. 2: Compressible Models. Oxford Lecture Series in Mathematics and Its Applications 10, Clarendon Press, Oxford, 1998.

    Google Scholar 

  15. A. Lunardi: Interpolation Theory. Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) 9, Edizioni della Normale, Pisa, 2009.

    Google Scholar 

  16. K. Nakao, Y. Taniuchi: An alternative proof of logarithmically improved Beale-Kato-Majda type extension criteria for smooth solutions to the Navier-Stokes equations. Nonlinear Anal, Theory Methods Appl, Ser. A, Theory Method. 37 2018), 48–55.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. Nakao, Y. Taniuchi: Brezis-Gallouet-Wainger type inequalities and blow-up criteria for Navier-Stokes equations in unbounded domains. Commun. Math. Phys. 37 2018), 951–973.

    Article  MathSciNet  MATH  Google Scholar 

  18. L. Nirenberg: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 37 1959), 115–162.

    MathSciNet  MATH  Google Scholar 

  19. W. von Wahl: Estimating ∇u by div u and curl u. Math. Methods Appl. Sci. 37 1992), 123–143.

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted the two referees for helpful comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yong Zhou.

Additional information

This work is partially supported by the National Natural Science Foundation of China (Grants No. 11171154 and 11801585).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fan, J., Jia, X. & Zhou, Y. A logarithmic regularity criterion for 3D Navier-Stokes system in a bounded domain. Appl Math 64, 397–407 (2019). https://doi.org/10.21136/AM.2019.0246-18

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2019.0246-18

Keywords

MSC 2010

Navigation