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Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. (English) Zbl 1212.35354
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 \(\operatorname {div}(u/| u| )\) 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 P. Constantin and C. 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.

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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[1] J.T. Beale, T. Kato, A. Majda: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94 (1984), 61–66. · Zbl 0573.76029
[2] H. Beirão da Veiga: A new regularity class for the Navier-Stokes equations in \(\mathbb{R}\)n. Chin. Ann. Math., Ser. B 16 (1995), 407–412. · Zbl 0837.35111
[3] Dongho Chae, Hi-Jun Choe: Regularity of solutions to the Navier-Stokes equation. Electron. J. Differ. Equ. No. 05 (1999). · Zbl 0923.35117
[4] P. Constantin, C. Fefferman: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42 (1993), 775–789. · Zbl 0837.35113
[5] E.B. Fabes, B. F. Jones, N.M. Rivière: The initial value problem for the Navier-Stokes equations with data in L p . Arch. Ration. Mech. Anal. 45 (1972), 222–240. · Zbl 0254.35097
[6] C. He: Regularity for solutions to the Navier-Stokes equations with one velocity component regular. Electron. J. Differ. Equ. No. 29 (2002). · Zbl 0993.35072
[7] E. Hopf: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231. (In German.) · Zbl 0042.10604
[8] L. Iskauriaza, G.A. Serëgin, V. Shverak: L 3,solutions of Navier-Stokes equations and backward uniqueness. Usp. Mat. Nauk 58 (2003), 3–44. (In Russian.)
[9] H. Kozono, Y. Taniuchi: Bilinear estimates in BMO and the Navier-Stokes equations. Math. Z. 235 (2000), 173–194. · Zbl 0970.35099
[10] J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta. Math. 63 (1934), 193–248. (In French.) · JFM 60.0726.05
[11] P. Penel, M. Pokorný: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math. 49 (2004), 483–493. · Zbl 1099.35101
[12] J. Serrin: The initial value problem for the Navier-Stokes equations. Nonlinear Probl., Proc. Sympos. Madison 1962 (R. Langer, ed.). Univ. Wisconsin Press, Madison, 1963, pp. 69–98.
[13] M. Struwe: On partial regularity results for the Navier-Stokes equations. Commun. Pure Appl. Math. 41 (1988), 437–458. · Zbl 0632.76034
[14] Y. Zhou: 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. · Zbl 1166.35359
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