×

zbMATH — the first resource for mathematics

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations. (English) Zbl 1355.76017
Summary: In this article, we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical \( L^{r}(0,T;L^{s})\) regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms \(\|u\|_{L^3}\) and \(\|u\|_{\dot{H}^{1 / 2}}\).{
©2017 American Institute of Physics}

MSC:
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI Link
References:
[1] Benbernou, S., A note on the regularity criterion in terms of pressure for the Navier-Stokes equations, Appl. Math. Lett., 22, 9, 1438-1443, (2009) · Zbl 1182.35179
[2] 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) · Zbl 1075.35031
[3] Bosia, S.; Pata, V.; Robinson, J. C., A weak-Lp Prodi-Serrin type regularity criterion for the Navier-Stokes equations, J. Math. Fluid Mech., 16, 4, 721-725, (2014) · Zbl 1307.35186
[4] Cao, C.; Titi, E. S., Regularity criteria for the three-dimensional Navier-Stokes equations, Indiana Univ. Math. J., 57, 6, 2643-2661, (2008) · Zbl 1159.35053
[5] Cao, C.; Qin, J.; Titi, E. S., Regularity criterion for solutions of three-dimensional turbulent channel flows, Commun. Partial Differ. Equations, 33, 419-428, (2008) · Zbl 1151.76012
[6] 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) · Zbl 1256.35051
[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) · Zbl 1007.35064
[8] 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) · Zbl 1126.35047
[9] Escauriaza, L.; Seregin, G.; Šverák, V., L_{3,∞}-solutions of Navier-Stokes equations and backward uniqueness, Usp. Mat. Nauk, 58, 2, 3-44, (2003), 10.4213/rm609; Escauriaza, L.; Seregin, G.; Šverák, V., L_{3,∞}-solutions of Navier-Stokes equations and backward uniqueness, Usp. Mat. Nauk, 58, 2, 3-44, (2003), 10.1070/rm2003v058n02abeh000609;
[10] Fabes, E. B.; Jones, B. F.; Riviere, N. M., The initial value problem for the Navier-Stokes equations with data in Lp, Arch. Ration. Mech. Anal., 45, 222-248, (1972) · Zbl 0254.35097
[11] Fan, J.; Jiang, S.; Nakamura, G.; Zhou, Y., Logarithmically improved regularity criteria for the Navier-Stokes and MHD equations, J. Math. Fluid Mech., 13, 4, 557-571, (2011) · Zbl 1270.35339
[12] Fan, J.; Jiang, S.; Ni, G., On regularity criteria for the n-dimensional Navier-Stokes equations in terms of the pressure, J. Differ. Equations, 244, 11, 2963-2979, (2008) · Zbl 1143.35081
[13] Gala, S., Remarks on regularity criterion for weak solutions to the Navier-Stokes equations in terms of the gradient of the pressure, Appl. Anal.: Int. J., 92, 1, 96-103, (2013) · Zbl 1284.35313
[14] Giga, Y., Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system, J. Differ. Equations, 62, 2, 186-212, (1986) · Zbl 0577.35058
[15] Guo, Z.; Gala, S., Remarks on logarithmical regularity criteria for the Navier-Stokes equations, J. Math. Phys., 52, 6, 63503, (2011) · Zbl 1317.35174
[16] Guo, Z.; Gala, S., A note on the regularity criteria for the Navier-Stokes equations, Appl. Math. Lett., 25, 3, 305-309, (2012) · Zbl 1239.35124
[17] Guo, Z.; Wittwer, P.; Wang, W., Regularity issue of the Navier-Stokes equations involving the combination of pressure and velocity field, Acta Appl. Math., 123, 1, 99-112, (2013) · Zbl 1280.35091
[18] He, X.; Gala, S., Regularity criterion for weak solutions to the Navier-Stokes equations in terms of the pressure in the class \(L^2(0, T; \dot{B}_{\infty, \infty}^{- 1}(R^3))\) · Zbl 1231.35146
[19] Chan, C. H.; Vasseur, A., Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations, Methods Appl. Anal., 14, 2, 197-212, (2007) · Zbl 1198.35175
[20] Kato, T., Strong Lp-solutions of the Navier-Stokes equation in Rm, with applications to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[21] Ladyzhenskaya, O. A., On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5, 169-185, (1967); Ladyzhenskaya, O. A., On uniqueness and smoothness of generalized solutions to the Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5, 169-185, (1967); · Zbl 0194.12805
[22] Leray, J., On the motion of a viscous liquid filling space, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05
[23] Ohyama, T., Interior regularity of weak solutions to the Navier-Stokes equation, Proc. Jpn. Acad., 36, 273-277, (1960) · Zbl 0100.22404
[24] Prodi, G., Un teorema di unicità per le equazioni di Navier-Stokes, Ann. Math. Pura Appl., 4, 48, 173-182, (1959) · Zbl 0148.08202
[25] Serrin, J., On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 9, 187-191, (1962) · Zbl 0106.18302
[26] Struwe, M., On partial regularity results for the Navier-Stokes equations, Commun. Pure Appl. Math., 41, 4, 437-458, (1988) · Zbl 0632.76034
[27] Struwe, M., On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure, J. Math. Fluid Mech., 9, 2, 235-242, (2007) · Zbl 1131.35060
[28] Suzuki, T., A remark on the regularity of weak solutions to the Navier-Stokes equations in terms of the pressure in Lorentz spaces, Nonlinear Anal.: Theory, Methods Appl., 75, 9, 3849-3853, (2012) · Zbl 1239.35115
[29] Tran, C. V.; Yu, X., Pressure moderation and effective pressure in Navier-Stokes flows, Nonlinearity, 29, 2990-3005, (2016) · Zbl 1349.76046
[30] Zhang, X.; Jia, Y.; Dong, B.-Q., On the pressure regularity criterion of the 3D Navier-Stokes equations, J. Math. Anal. Appl., 393, 2, 413-420, (2012) · Zbl 1248.35151
[31] Zhou, Y., Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain, Math. Ann., 328, 1, 173-192, (2004) · Zbl 1054.35062
[32] Zhou, Y., On regularity criteria in terms of pressure for the Navier-Stokes equations in R3, Proc. Am. Math. Soc., 134, 149-156, (2006) · Zbl 1075.35044
[33] Zhou, Y.; Gala, S., Logarithmically improved regularity criteria for the Navier-Stokes equations in multiplier spaces, J. Math. Anal. Appl., 356, 2, 498-501, (2009) · Zbl 1172.35063
[34] Zhou, Y.; Lei, Z., Logarithmically improved criteria for Euler and Navier-Stokes equations, Commun. Pure Appl. Anal., 12, 6, 2715-2719, (2013) · Zbl 1267.35173
[35] Zhu, X., A regularity criterion for the Navier-Stokes equations in the multiplier spaces, Abstract Appl. Anal., 2012, 1 · Zbl 1242.35188
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.