# Article

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Keywords:
Navier-Stokes; fluid mechanics; regularity; PRodi-Serrin criteria
Summary:
In this short note we give a link between the regularity of the solution $u$ to the 3D Navier-Stokes equation and the behavior of the direction of the velocity $u/|u|$. It is shown that the control of ${\rm Div}(u/|u|)$ in a suitable $L_t^p(L_x^q)$ norm is enough to ensure global regularity. The result is reminiscent of the criterion in terms of the direction of the vorticity, introduced first by Constantin and Fefferman. However, in this case the condition is not on the vorticity but on the velocity itself. The proof, based on very standard methods, relies on a straightforward relation between the divergence of the direction of the velocity and the growth of energy along streamlines.
References:
[1] Beale, J. T., Kato, T., Majda, A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94 (1984), 61-66. DOI 10.1007/BF01212349 | MR 0763762 | Zbl 0573.76029
[2] H. Beirão da Veiga: A new regularity class for the Navier-Stokes equations in $\Bbb R^n$. Chin. Ann. Math., Ser. B 16 (1995), 407-412. MR 1380578
[3] Dongho Chae, Hi-Jun Choe: Regularity of solutions to the Navier-Stokes equation. Electron. J. Differ. Equ. No. 05 (1999). MR 1673067
[4] Constantin, P., Fefferman, C.: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42 (1993), 775-789. DOI 10.1512/iumj.1993.42.42034 | MR 1254117 | Zbl 0837.35113
[5] Fabes, E. B., Jones, B. F., Rivière, N. M.: The initial value problem for the Navier-Stokes equations with data in $L^p$. Arch. Ration. Mech. Anal. 45 (1972), 222-240. DOI 10.1007/BF00281533 | MR 0316915
[6] He, C.: Regularity for solutions to the Navier-Stokes equations with one velocity component regular. Electron. J. Differ. Equ. No. 29 (2002). MR 1907705 | Zbl 0993.35072
[7] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213-231 German. DOI 10.1002/mana.3210040121 | MR 0050423
[8] Iskauriaza, L., Serëgin, G. A., Shverak, V.: $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58 (2003), 3-44 Russian. MR 1992563
[9] Kozono, H., Taniuchi, Y.: Bilinear estimates in ${BMO}$ and the Navier-Stokes equations. Math. Z. 235 (2000), 173-194. DOI 10.1007/s002090000130 | MR 1785078 | Zbl 0970.35099
[10] Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta. Math. 63 (1934), 193-248 French. DOI 10.1007/BF02547354 | MR 1555394
[11] Penel, P., Pokorný, M.: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. 49 (2004), 483-493. DOI 10.1023/B:APOM.0000048124.64244.7e | MR 2086090 | Zbl 1099.35101
[12] Serrin, J.: The initial value problem for the Navier-Stokes equations. Nonlinear Probl., Proc. Sympos. Madison 1962 R. Langer Univ. Wisconsin Press Madison (1963), 69-98. MR 0150444 | Zbl 0115.08502
[13] Struwe, M.: On partial regularity results for the Navier-Stokes equations. Commun. Pure Appl. Math. 41 (1988), 437-458. DOI 10.1002/cpa.3160410404 | MR 0933230 | Zbl 0632.76034
[14] Zhou, Y.: A new regularity criterion for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods Appl. Anal. 9 (2002), 563-578. MR 2006605 | Zbl 1166.35359

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