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

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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.

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References

  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)

    MATH  Google Scholar 

  2. Cao, C.: Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete Contin. Dyn. Syst. 26, 1141–1151 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. 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 

  10. Escauriaza, L., Seregin, G., Sverak, V.: Backward uniqueness for parabolic equations. Arch. Rational Mech. Anal. 169, 147–157 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Kukavica, I., Ziane, M.: Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48, 10 (2007)

    Article  MathSciNet  Google Scholar 

  13. Kukavica, I., Ziane, M.: One component regularity for the Navier-Stokes equations. Nonlinearity 19, 453–469 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  14. Leray, J.: Sur le mouvement d’un liquide visquenx emplissant l’escape. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    MATH  Google Scholar 

  21. Sohr, H.: The Navier-Stokes Equations, An Elementary Functional Analytic Approach. Birkhanser Verlag, Boston (2001)

    MATH  Google Scholar 

  22. Troisi, M.: Teoremi di inclusione per spazi di Sobolev non isotropi. Ricerche Mat. 8, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  24. Zhou, Y.: A new regularity criterion for weak solutions to the Navier-Stokes equations. J. Math. Pures Appl. 84, 1496–1514 (2005)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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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.

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Communicated by D. Chae.

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Guo, Z., Li, Y. & Skalák, Z. Regularity Criteria of The Incompressible Navier-Stokes Equations via Only One Entry of Velocity Gradient. J. Math. Fluid Mech. 21, 35 (2019). https://doi.org/10.1007/s00021-019-0441-6

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