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On regularity criteria for the 3D Navier-Stokes equations involving the ratio of the vorticity and the velocity. (English) Zbl 1372.35220
Summary: This note concerns regularity criteria for the Navier-Stokes equations. It is proved that if the solution satisfies $$\int_0^T\frac{\|\omega(\tau)\|_{L^s}^{\frac{2s}{2s-3}}}{\| \mathbf u(\tau)\|_{L^3}^{f(s)}}\mathrm d\tau<\infty$$ for $$\frac{3}{2}<s<\infty$$ and suitable function $$f(s)$$, then the solution is regular on $$(0,T]$$.

##### MSC:
 35Q30 Navier-Stokes equations 35B65 Smoothness and regularity of solutions to PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
##### Keywords:
regularity criteria; Navier-Stokes equations; vorticity
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##### References:
 [1] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05 [2] Hopf, E., Über die anfangwertaufgaben für die hydromischen grundgleichungen, Math. Nachr., 4, 213-321, (1951) [3] Eskauriaza, L.; Serëgin, G. A.; Šverák, V., $$L_{3, \infty}$$-solutions of Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58, 211-250, (2003) · Zbl 1064.35134 [4] Prodi, G., Un teorema di unicitá per le equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48, 173-182, (1959) · Zbl 0148.08202 [5] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9, 187-195, (1962) · Zbl 0106.18302 [6] Beirão da Veiga, H., A new regularity class for the Navier-Stokes equations in $$\mathbb{R}^n$$, Chinese Ann. Math. Ser. B, 16, 407-412, (1995) · Zbl 0837.35111 [7] Tran, C. V.; Yu, X. W., Depletion of nonlinearity in the pressure force driving Navier-Stokes flows, Nonlinearity, 28, 1295-1306, (2015) · Zbl 1312.76013 [8] C.V. Tran, Pressure moderation and effective pressure in Navier-Stokes flows, (submitted for publication). · Zbl 1349.76046 [9] Kato, T., Strong $$L^p$$-solutions of the Navier-Stokes equation in $$\mathbb{R}^m$$, with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073 [10] Zhou, Y., On regularity criteria in terms of pressure for the Navier-Stokes equations in $$\mathbb{R}^3$$, Proc. Amer. Math. Soc., 134, 149-156, (2006) · Zbl 1075.35044
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