# A New Prodi–Serrin Type Regularity Criterion in Velocity Directions

• Benjamin Pineau
• Xinwei Yu
Article

## Abstract

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}
(1)
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}
(2)
with $$\frac{2}{p} + \frac{3}{q} \leqslant \frac{1}{2}, q > 6$$ and $${1 \leqslant r < \infty }$$.

## Notes

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
20. 20.
Lemarié-Rieusset, P.G.: Recent Developments in the Navier–Stokes Problem. Chapman & Hall/CRC, London (2002)
21. 21.
Lemarie-Rieusset, P.G.: The Navier–Stokes Problem in the 21st Century. Chapman & Hall/CRC, London (2016)
22. 22.
Leray, J.: On the motion of a viscous liquid filling space. Acta Math. 63, 193–248 (1934)
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)
25. 25.
O’Neil, R.: Convolution operators and l(p, q) spaces. Duke Math. J. 30(1), 129–142 (1963)
26. 26.
Prodi, G.: Un teorema di unicità per le equazioni di Navier–Stokes. Ann. Mat. Pura Appl. 4(48), 173–182 (1959)
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)
29. 29.
Sohr, H.: Zur regularitätstheorie der instationären gleichungen von Navier–Stokes. Math. Z. 184, 359–375 (1983)
30. 30.
Sohr, H.: A regularity class for the Navier–Stokes equations in Lorentz spaces. J. Evol. Equ. 1, 441–467 (2001)
31. 31.
Sohr, H.: The Navier-Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser, Basel (2001)
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)
33. 33.
Tran, C.V., Yu, X.: Regularity of Navier–Stokes flows with bounds for the pressure. Appl. Math. Lett. 67, 21–27 (2017)
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)