Czechoslovak Mathematical Journal, Vol. 68, No. 1, pp. 219-225, 2018
Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component
Zujin Zhang
Received August 7, 2016. First published October 10, 2017.
Abstract: We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of $u_3$ and $\omega_3$, which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M. Pokorný (2004).
References: [1] 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. MR 1380578 | Zbl 0837.35111 [2] C. Cao, E. S. Titi: Regularity criteria for the three-dimensional Navier-Stokes equations. Indiana Univ. Math. J. 57 (2008), 2643-2661. DOI 10.1512/iumj.2008.57.3719 | MR 2482994 | Zbl 1159.35053 [3] C. Cao, E. S. Titi: Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor. Arch. Ration. Mech. Anal. 202 (2011), 919-932. DOI 10.1007/s00205-011-0439-6 | MR 2854673 | Zbl 1256.35051 [4] L. Escauriaza, G. A. Serëgin, V. Shverak: $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58 (2003), 211-250. (In English. Russian original.); translation from Usp. Mat. Nauk 58 (2003), 3-44. DOI 10.1070/RM2003v058n02ABEH000609 | MR 1992563 | Zbl 1064.35134 [5] E. Hopf: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213-231. (In German.) DOI 10.1002/mana.3210040121 | MR 0050423 | Zbl 10.1002/mana.3210040121 [6] I. Kukavica, M. Ziane: One component regularity for the Navier-Stokes equations. Nonlinearity 19 (2006), 453-469. DOI 10.1088/0951-7715/19/2/012 | MR 2199398 | Zbl 1149.35069 [7] I. Kukavica, M. Ziane: Navier-Stokes equations with regularity in one direction. J. Math. Phys. 48 (2007), 065203, 10 pages. DOI 10.1063/1.2395919 | MR 2337002 | Zbl 1144.81373 [8] J. Leray: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193-248. (In French.) DOI 10.1007/BF02547354 | MR 1555394 | JFM 60.0726.05 [9] T. Ohyama: Interior regularity of weak solutions of the time-dependent Navier-Stokes equation. Proc. Japan Acad. 36 (1960), 273-277. DOI 10.3792/pja/1195524029 | MR 0139856 | Zbl 0100.22404 [10] P. Penel, M. Pokorný: Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. Appl. Math., Praha 49 (2004), 483-493. DOI 10.1023/B:APOM.0000048124.64244.7e | MR 2086090 | Zbl 1099.35101 [11] P. Penel, M. Pokorný: On anisotropic regularity criteria for the solutions to 3D Navier-Stokes equations. J. Math. Fluid Mech. 13 (2011), 341-353. DOI 10.1007/s00021-010-0038-6 | MR 2824487 | Zbl 1270.35354 [12] G. Prodi: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., IV. Ser. 48 (1959), 173-182. (In Italian.) DOI 10.1007/BF02410664 | MR 0126088 | Zbl 0148.0802 [13] J. Serrin: The initial value problem for the Navier-Stokes equations. Nonlinear Problems Proc. Symp., Madison, 1962, University of Wisconsin Press, Madison, Wisconsin (1963), 69-98. MR 0150444 | Zbl 0115.08502 [14] Z. Skalák: A note on the regularity of the solutions to the Navier-Stokes equations via the gradient of one velocity component. J. Math. Phys. 55 (2014), 121506, 6 pages. DOI 10.1063/1.4904836 | MR 3390527 | Zbl 1308.35177 [15] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton (1970). DOI 10.1515/9781400883882 | MR 0290095 | Zbl 0207.13501 [16] R. Temam: Navier-Stokes Equations. Theory and Numerical Analysis. American Mathematical Society, Chelsea Publishing, Providence (2001). DOI 10.1090/chel/343 | MR 1846644 | Zbl 0981.35001 [17] Z. Zhang: An almost Serrin-type regularity criterion for the Navier-Stokes equations involving the gradient of one velocity component. Z. Angew. Math. Phys. 66 (2015), 1707-1715. DOI 10.1007/s00033-015-0500-7 | MR 3377710 | Zbl 1325.35147 [18] Z. Zhang, Z.-A. Yao, M. Lu, L. Ni: Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations. J. Math. Phys. 52 (2011), 053103, 7 pages. DOI 10.1063/1.3589966 | MR 2839081 | Zbl 1317.35180 [19] Y. Zhou, M. Pokorný: On a regularity criterion for the Navier-Stokes equations involving gradient of one velocity component. J. Math. Phys. 50 (2009), 123514, 11 pages. DOI 10.1063/1.3268589 | MR 2582610 | Zbl 05772327 [20] Y. Zhou, M. Pokorný: On the regularity of the solutions of the Navier-Stokes equations via one velocity component. Nonlinearity 23 (2010), 1097-1107. DOI 10.1088/0951-7715/23/5/004 | MR 2630092 | Zbl 1190.35179
Affiliations: Zujin Zhang, College of Mathematics and Computer Sciences, Gannan Normal University, Shangxue Avenue, Ganzhou 341000, Zhanggong, Jiangxi, P. R. China, e-mail: zhangzujin361@163.com
