A New Prodi–Serrin Type Regularity Criterion in Velocity Directions

  • Benjamin Pineau
  • Xinwei Yu


In this article we generalize (Vasseur in Appl Math 54(1):47–52, 2009) to Lorentz spaces. More specifically, we prove the following. Let u be a Leray–Hopf solution to the Navier–Stokes equation with viscosity \(\nu \) and initial value \(u_0 \in L^2 ({\mathbb {R}}^3)\). Then there is \(c_0 > 0\) such that u is smooth beyond \(T > 0\) if
$$\begin{aligned} \left\| {{\mathrm{div}}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, \infty } (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < c_0 \nu ^{1 - \frac{1}{p}} \Vert u_0 \Vert _{L^2}^{- 1} \end{aligned}$$
with \(\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}\), \(q > 6\). We also show that u remains smooth beyond \(T > 0\) if
$$\begin{aligned} \left\| {\mathrm{div}} \left( \frac{u}{| u |} \right) \right\| _{L^{p, r} (0, T ; L^{q, \infty } ({\mathbb {R}}^3))} < \infty \end{aligned}$$
with \(\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}, q > 6\) and \({1 \leqslant r < \infty }\).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bosia, S., Conti, M., Pata, V.: A regularity criterion for the Navier–Stokes equations in terms of the pressure gradient. Open Math. 12(7), 1015–1025 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berselli, L.C.: Some criteria concerning the vorticity and the problem of global regularity for the 3D Navier-Stokes equations. Ann. Univ. Ferrara 55, 209 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berselli, L.C., Galdi, G.P.: Regularity criteria involving the pressure for the weak solutions to the Navier–Stokes equations. Proc. Am. Math. Soc. 130, 3585–3595 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bosia, S., Pata, V., Robinson, J.C.: A weak-\(L^p\) prodi-serrin type regularity criterion for the Navier–Stokes equations. J. Math. Fluid Mech. 16, 721–725 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bjorland, C., Vasseur, A.: Weak in space, log in time improvement of the Ladyzhenskaja–Prodi–Serrin criteria. J. Math. Fluid Mech 13(2), 259–269 (2011)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chan, C.H.: Smoothness criterion for Navier-Stokes equations in terms of regularity along the streamlines. Methods Appl. Anal. 17(1), 81–104 (2010)MathSciNetMATHGoogle Scholar
  7. 7.
    Chae, D., Lee, J.: Regularity criterion in terms of pressure for the Navier–Stokes equations. Nonlinear Anal. Theory Methods Appl. 46(5), 727–735 (2001)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chamorro, D., Lemarié-Rieusset, P.-G.: Real interpolation method, Lorentz spaces and refined Sobolev inequalities. J. Funct. Anal. 265(12), 3219–3232 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cao, C., Titi, E.S.: Regularity criteria for the three-dimensional Navier–Stokes equations. Indiana Univ. Math. J. 57(6), 2643–2661 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cao, C., Titi, E.S.: Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202, 919–932 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Chan, C.H., Vasseur, A.: Log improvement of the Prodi–Serrin criteria for Navier–Stokes equations. Methods Appl. Anal. 14(2), 197–212 (2007)MathSciNetMATHGoogle Scholar
  12. 12.
    Chen, Q., Zhang, Z.: Regularity criterion via the pressure on weak solutions to the 3D Navier–Stokes equations. Proc. Am. Math. Soc. 135, 1829–1837 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    da Veiga, H.B.: A new regularity class for the Navier–Stokes equations in \({\mathbb{R}}^n\). Chin. Ann. Math. 16B(4), 407–412 (1995)MATHGoogle Scholar
  14. 14.
    Escauriaza, L., Seregin, G., Sverák, V.: \(L_{3, \infty }\)-solutions of Navier–Stokes equations and backward uniqueness. Russ. Math. Surv. 58(2), 211–250 (2003)CrossRefMATHGoogle Scholar
  15. 15.
    Fan, J., Ozawa, T.: Regularity criterion for weak solutions to the Navier–Stokes equations in terms of the gradient of the pressure. J. Inequal. Appl. 2008, 412678 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Giga, Y.: Solutions for semilinear parabolic equations in \(L^p\) and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)ADSCrossRefMATHGoogle Scholar
  17. 17.
    Grafakos, L.: Classical fourier analysis. In: Graduate Texts in Mathematics, vol. 249, 3rd edn. Springer (2014)Google Scholar
  18. 18.
    Han, B., Lei, Z., Li, D., Zhao, N.: Sharp one component regularity for Navier–Stokes. arXiv:1708.04119 (August 2017)
  19. 19.
    Kozono, H., Taniuchi, Y.: Bilinear estimates in BMO and the Navier–Stokes equations. Math. Z. 235(1), 173–194 (2000)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC, London (2002)CrossRefMATHGoogle Scholar
  21. 21.
    Lemarie-Rieusset, P.G.: The Navier–Stokes Problem in the 21st Century. Chapman & Hall/CRC, London (2016)CrossRefMATHGoogle Scholar
  22. 22.
    Leray, J.: On the motion of a viscous liquid filling space. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Neustupa, J., Penel, P.: Regularity of a suitable weak solution to the Navier–Stokes equations as a consequence of regularity of one velocity component. In: Sequeira, A. (ed.) Applied Nonlinear Analysis, pp. 391–402. Kluwer Academic/Plenum Publishers, New York (1999)Google Scholar
  24. 24.
    Núñez, M.: Regularity criteria for the Navier–Stokes equations involving the ratio pressure-gradient of velocity. Math. Methods Appl. Sci. 33(3), 323–331 (2009)MathSciNetMATHGoogle Scholar
  25. 25.
    O’Neil, R.: Convolution operators and l(p, q) spaces. Duke Math. J. 30(1), 129–142 (1963)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Prodi, G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 4(48), 173–182 (1959)CrossRefMATHGoogle Scholar
  27. 27.
    Robinson, J.C., Rodrigo, J.L., Sadowski, W.: The three-dimensional Navier–Stokes equations: Classical theory. In: Cambridge Studies in Advanced Mathematics, vol. 157. Cambridge University Press (2016)Google Scholar
  28. 28.
    Serrin, J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Ration. Mech. Anal. 9, 187–191 (1962)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Sohr, H.: Zur regularitätstheorie der instationären gleichungen von Navier–Stokes. Math. Z. 184, 359–375 (1983)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Sohr, H.: A regularity class for the Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Sohr, H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)CrossRefMATHGoogle Scholar
  32. 32.
    Tran, C.V., Yu, X.: Note on Prodi–Serrin–Ladyzhenskaya type regularity criteria for the Navier–Stokes equations. J. Math. Phys. 58(1), 11501 (2017)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Tran, C.V., Yu, X.: Regularity of Navier–Stokes flows with bounds for the pressure. Appl. Math. Lett. 67, 21–27 (2017)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Vasseur, A.: Regularity criterion for 3D Navier–Stokes equations in terms of the direction of the velocity. Appl. Math. 54(1), 47–52 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations