Advertisement

Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient

  • Zhengguang GuoEmail author
  • Yafei Li
  • Zdenĕk Skalák
Article

Abstract

In this paper we establish regularity conditions for the three dimensional incompressible Navier-Stokes equations in terms of one entry of the velocity gradient tensor, say for example, \( \partial _{3}u_{3}\). We show that if \(\partial _{3}u_{3}\) satisfies certain integrable conditions with respect to time and space variables in anisotropic Lebesgue spaces, then a Leray-Hopf weak solution is actually regular. The anisotropic Lebesgue space helps us to almost reach the Prodi-Serrin level 2 in certain special case. Moreover, regularity conditions on non-diagonal element of gradient tensor \(\partial _1 u_3\) are also established, which covers some previous literature.

Keywords

Navier-Stokes equations Regularity criteria One entry of velocity gradient 

Mathematics Subject Classification

35Q35 35B65 76D05 

Notes

Acknowledgements

The authors would like to thank the anonymous referee for his/her very helpful comments on the initial version of this manuscript which make this paper more readable. The first author was partially supported by National Natural Science Foundation of China, under Grant No. 11301394, and China Postdoctoral Science Foundation, under Grant Nos. 2017M620149 and 2018T110387. The third author was supported by the Grant Agency of the Czech republic through Grant 18-09628S and by the Czech Academy of Sciences through RVO: 67985874.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Beirao da Veiga, H.: A new regularity class for the Navier-Stokes equations in \(\mathbb{R}^{n}\). Chin. Ann. Math. Ser. B 16, 407–412 (1995)zbMATHGoogle Scholar
  2. 2.
    Cao, C.: Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete Contin. Dyn. Syst. 26, 1141–1151 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cao, C., Titi, E.: Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Rational Mech. Anal. 202, 919–932 (2011)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, C., Titi, E.: Regularity criteria for the three dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57, 2643–2661 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chemin, J., Zhang, P.: On the critical one component regularity for 3-D Navier-Stokes system. Ann. Sci. Éc. Norm. Supér. 49, 131–167 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chemin, J., Zhang, P., Zhang, Z.: On the critical one component regularity for 3-D Navier-Stokes system: general case. Arch. Rational Mech. Anal. 224, 871–905 (2017)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Chae, D.: On the regularity conditions of suitable weak solutions of the 3D Navier-Stokes equations. J. Math. Fluid Mech. 12, 171–180 (2010)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Chae, D., Kang, K., Lee, J.: On the interior regularity of suitable weak solutions to the Navier-Stokes equations. Commun. Part. Differ. Equ. 32, 1189–1207 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chae, D., Lee, J.: On the geometric regularity conditions for the 3D Navier-Stokes equations. Nonlinear Anal. 151, 265–273 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Escauriaza, L., Seregin, G., Sverak, V.: Backward uniqueness for parabolic equations. Arch. Rational Mech. Anal. 169, 147–157 (2003)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Guo, Z., Caggio, M., Skalak, Z.: Regularity criteria for the Navier-Stokes equations based on one component of velocity. Nonlinear Anal. Real World Appl. 35, 379–396 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kukavica, I., Ziane, M.: Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48, 10 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kukavica, I., Ziane, M.: One component regularity for the Navier-Stokes equations. Nonlinearity 19, 453–469 (2006)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Leray, J.: Sur le mouvement d’un liquide visquenx emplissant l’escape. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Neustupa, J., Novotny, A., Penel, P.: An interior regularity of a weak solution to the Navier-Stokes equations in dependence on one component of velocity, Topics in mathematical fluid mechanics. Quad. Mat. 10, 163–183 (2002)zbMATHGoogle Scholar
  16. 16.
    Prodi, G.: Un teorema di unicita per el equazioni di Navier-Stokes. Ann. Mat. Pura. Appl. 48, 173–182 (1959)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pokorny, M.: On the result of He concerning the smoothness of solutions to the Navier-Stokes equations. Electron. J. Differ. Equ. 11, 1–8 (2003)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Penel, P., Pokorny, M.: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. 49, 483–493 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Qian, C.: A generalized regularity criterion for 3D Navier-Stokes equations in terms of one velocity component. J. Differ. Equ. 260, 3477–3494 (2016)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Serrin, J.: The initial value problems for the Navier-Stokes equations. In: Langer, R.E. (ed.) Nonlinear Problems. University of Wisconsin Press, Wisconsin (1963)zbMATHGoogle Scholar
  21. 21.
    Sohr, H.: The Navier-Stokes Equations, An Elementary Functional Analytic Approach. Birkhanser Verlag, Boston (2001)zbMATHGoogle Scholar
  22. 22.
    Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 8, 3–24 (1969)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Wolf, J.: A regularity criterion of Serrin-type for the Navier-Stokes equations involving the gradient of one velocity component. Analysis (Berlin) 5, 259–292 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Zhou, Y.: A new regularity criterion for weak solutions to the Navier-Stokes equations. J. Math. Pures Appl. 84, 1496–1514 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zhang, Z., Yao, Z., Li, P., Guo, C., Lu, M.: Two new regularity criteria for the 3D Navier-Stokes equations via two entries of the velocity gradient tensor. Acta Appl. Math. 123, 43–52 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zhang, Z., Zhong, D., Huang, X.: A refined regularity criterion for the Navier-Stokes equations involving one non-diagonal entry of the velocity gradient. J. Math. Anal. Appl. 453, 1145–1150 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zheng, X.: A regularity criterion for the tridimensional Navier-Stokes equations in term of one velocity component. J. Differ. Equ. 256, 283–309 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhou, Y., Pokorny, M.: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phy. 50, 11 (2009)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhou, Y., Pokorny, M.: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinerarity 23, 1097–1107 (2010)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhang, Z.: An improved regularity criterion for the Navier-Stokes equations in terms of one directional derivative of the velocity field. Bull. Math. Sci. 8, 33–47 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Natural SciencesShanghai Jiao Tong UniversityMinhang District, ShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsWenzhou UniversityWenzhouPeople’s Republic of China
  3. 3.Czech Academy of SciencesInstitute of HydrodynamicsPrague 6Czech Republic

Personalised recommendations