Czechoslovak Mathematical Journal, Vol. 69, No. 4, pp. 1165-1175, 2019


On ratio improvement of Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes system

Zujin Zhang, Chupeng Wu, Yong Zhou

Received March 8, 2018.   Published online August 7, 2019.

Abstract:  This paper concerns improving Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes system, in the sense of multiplying certain negative powers of scaling invariant norms.
Keywords:  regularity criteria; Navier-Stokes equations
Classification MSC:  35B65, 35Q30, 76D03


References:
[1] H. Beirão da Veiga: A new regularity class for the Navier-Stokes equations in $\bbR^n$. Chin. Ann. Math., Ser. B 16 (1995), 407-412. MR 1380578 | Zbl 0837.35111
[2] P. Constantin, C. Fefferman: Direction of vorticity and the problem of global regularity for the Navier-Stokes equations. Indiana Univ. Math. J. 42 (1993), 775-789. DOI 10.1512/iumj.1993.42.42034 | MR 1254117 | Zbl 0837.35113
[3] 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; translation from Usp. Mat. Nauk 58 (2003), 3-44. DOI 10.1070/rm2003v058n02abeh000609 | MR 1992563 | Zbl 1064.35134
[4] H. Fujita, T. Kato: On the Navier-Stokes initial value problem. I. Arch. Ration. Mech. Anal. 16 (1964), 269-315. DOI 10.1007/bf00276188 | MR 0166499 | Zbl 0126.42301
[5] E. Hopf: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213-321. (In German.) DOI 10.1002/mana.3210040121 | MR 0050423 | Zbl 0042.10604
[6] T. Kato: Strong $L^p$-solutions of the Navier-Stokes equation in $\bbR^m$, with applications to weak solutions. Math. Z. 187 (1984), 471-480. DOI 10.1007/bf01174182 | MR 0760047 | Zbl 0545.35073
[7] 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
[8] 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 0148.08202
[9] J. C. Robinson, J. L. Rodrigo, W. Sadowski: The Three-Dimensional Navier-Stokes Equations. Classical Theory. Cambridge Studies in Advanced Mathematics 157, Cambridge University Press, Cambridge (2016). DOI 10.1017/cbo9781139095143 | MR 3616490 | Zbl 1358.35002
[10] R. Schoen, S.-T. Yau: Lectures on Differential Geometry. Conference Proceedings and Lecture Notes in Geometry and Topology 1, International Press, Cambridge (1994). MR 1333601 | Zbl 0830.53001
[11] J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9 (1962), 187-195. DOI 10.1007/bf00253344 | MR 0136885 | Zbl 0106.18302
[12] H. Sohr, W. von Wahl: On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations. Manuscr. Math. 49 (1984), 27-59. DOI 10.1007/bf01174870 | MR 0762786 | Zbl 0567.35069
[13] C. V. Tran, X. Yu: Depletion of nonlinearity in the pressure force driving Navier-Stokes flows. Nonlinearity 28 (2015), 1295-1306. DOI 10.1088/0951-7715/28/5/1295 | MR 3346162 | Zbl 1312.76013
[14] C. V. Tran, X. Yu: Pressure moderation and effective pressure in Navier-Stokes flows. Nonlinearity 29 (2016), 2290-3005. DOI 10.1088/0951-7715/29/10/2990 | MR 3551051 | Zbl 1349.76046
[15] C. V. Tran, X. Yu: Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations. J. Math. Phys. 58 (2017), 011501, 10 pages. DOI 10.1063/1.4974020 | MR 3597368 | Zbl 1355.76017
[16] A. Vasseur: Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity. Appl. Math., Praha 54 (2009), 47-52. DOI 10.1007/s10492-009-0003-y | MR 2476020 | Zbl 1212.35354
[17] Z. Zhang, X. Yang: Navier-Stokes equations with vorticity in Besov spaces of negative regular indices. J. Math. Anal. Appl. 440 (2016), 415-419. DOI 10.1016/j.jmaa.2016.03.037 | MR 3479607 | Zbl 1339.35221
[18] Z. Zhang, Y. Zhou: On regularity criteria for the 3D Navier-Stokes equations involving the ratio of the vorticity and the velocity. Comput. Math. Appl. 72 (2016), 2311-2314. DOI 10.1016/j.camwa.2016.08.031 | MR 3564413 | Zbl 1372.35220
[19] Y. Zhou: Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. Math. Ann. 328 (2004), 173-192. DOI 10.1007/s00208-003-0478-x | MR 2030374 | Zbl 1054.35062
[20] Y. Zhou: A new regularity criterion for the Navier-Stokes equations in terms of the direction of vorticity. Monatsh. Math. 144 (2005), 251-257. DOI 10.1007/s00605-004-0266-z | MR 2130277 | Zbl 1072.35148
[21] Y. Zhou: Direction of vorticity and a new regularity criterion for the Navier-Stokes equations. ANZIAM J. 46 (2005), 309-316. DOI 10.1017/s1446181100008270 | MR 2124925 | Zbl 1072.35565

Affiliations:   Zujin Zhang (corresponding author), Chupeng Wu, School of Mathematics and Computer Science, Gannan Normal University, Shangxue Avenue, Ganzhou 341000, Zhanggong, Jiangxi, P. R. China, e-mail: zhangzujin361@163.com, 229429387@qq.com; Yong Zhou, School of Mathematics, Sun Yat-sen University, Zhuhai, P. R. China, e-mail: zhouyong3@mail.sysu.edu.cn


 
PDF available at: