Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 219–225 | Cite as

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component



We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of u3 and ω3, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M.Pokorný (2004).


regularity criterion Navier-Stokes equation 

MSC 2010

35B65 35Q30 76D03 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. Beir˜ao da Veiga: A new regularity class for the Navier-Stokes equations in Rn. Chin. Ann. Math. Ser. B 16 (1995), 407–412.Google Scholar
  2. [2]
    C. Cao, E. S. Titi: Regularity criteria for the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57 (2008), 2643–2661.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    C. Cao, E. S. Titi: Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202 (2011), 919–932.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    L. Escauriaza, G. A. Serëgin, V. Shverak: L 3,∞-solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58 (2003), 211–250. (In English. Russian original.); translation from Usp. Mat. Nauk 58 (2003), 3–44.CrossRefMATHGoogle Scholar
  5. [5]
    E. Hopf: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231. (In German.)MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    I. Kukavica, M. Ziane: One component regularity for the Navier-Stokes equations. Nonlinearity 19 (2006), 453–469.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    I. Kukavica, M. Ziane: Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48 (2007), 065203, 10 pages.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934), 193–248. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    T. Ohyama: Interior regularity of weak solutions of the time-dependent Navier-Stokes equation. Proc. Japan Acad. 36 (1960), 273–277.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    P. Penel, M. Pokorný: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math., Praha 49 (2004), 483–493.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    P. Penel, M. Pokorný: On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations. J. Math. Fluid Mech. 13 (2011), 341–353.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    G. Prodi: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., IV. Ser. 48 (1959), 173–182. (In Italian.)CrossRefMATHGoogle Scholar
  13. [13]
    J. Serrin: The initial value problem for the Navier-Stokes equations. Nonlinear Problems. Proc. Symp., Madison, 1962, University of Wisconsin Press, Madison, Wisconsin, 1963, pp. 69–98.Google Scholar
  14. [14]
    Z. Skalák: A note on the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component. J. Math. Phys. 55 (2014), 121506, 6 pages.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton, 1970.Google Scholar
  16. [16]
    R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. American Mathematical Society, Chelsea Publishing, Providence, 2001.Google Scholar
  17. [17]
    Z. Zhang: An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component. Z. Angew. Math. Phys. 66 (2015), 1707–1715.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Z. Zhang, Z.-A. Yao, M. Lu, L. Ni: Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations. J. Math. Phys. 52 (2011), 053103, 7 pages.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Y. Zhou, M. Pokorný: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 50 (2009), 123514, 11 pages.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Y. Zhou, M. Pokorný: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23 (2010), 1097–1107.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.College of Mathematics and Computer SciencesGannan Normal UniversityZhanggong, JiangxiP.R. China

Personalised recommendations