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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
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[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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