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Keywords:
regularity criterion; Navier-Stokes equation
regularity criterion; Navier-Stokes equation
Summary:
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] Veiga, H. Beirão da: 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] Cao, C., Titi, E. S.: 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] 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 (2011), 919-932. DOI 10.1007/s00205-011-0439-6 | MR 2854673 | Zbl 1256.35051
[4] Escauriaza, L., Serëgin, G. A., Shverak, V.: $L_{3,\infty}$-solutions of Navier-Stokes equations and backward uniqueness. Russ. Math. Surv. 58 (2003), 211-250. English. Russian original translation from Usp. Mat. Nauk 58 2003 3-44. DOI 10.1070/RM2003v058n02ABEH000609 | MR 1992563 | Zbl 1064.35134
[5] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213-231 German. DOI 10.1002/mana.3210040121 | MR 0050423 | Zbl 10.1002/mana.3210040121
[6] Kukavica, I., Ziane, M.: 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] Kukavica, I., Ziane, M.: 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] Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), 193-248 French \99999JFM99999 60.0726.05. DOI 10.1007/BF02547354 | MR 1555394
[9] Ohyama, T.: 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] Penel, P., Pokorný, M.: 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] Penel, P., Pokorný, M.: 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] Prodi, G.: Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl., IV. Ser. 48 (1959), 173-182 Italian. DOI 10.1007/BF02410664 | MR 0126088 | Zbl 0148.0802
[13] Serrin, J.: 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] Skalák, Z.: 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] Stein, E. M.: 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] Temam, R.: 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] Zhang, Z.: 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] Zhang, Z., Yao, Z.-A., Lu, M., Ni, L.: 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] Zhou, Y., Pokorný, M.: 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] Zhou, Y., Pokorný, M.: 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
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