Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity. (English) Zbl 1099.35101

Summary: We study the nonstationary Navier-Stokes equations in the entire three-dimensional space and give some criteria on certain components of gradient of the velocity which ensure its global-in-time smoothness.


35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
Full Text: DOI EuDML


[1] L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), 771-831. · Zbl 0509.35067
[2] D. Chae, H. J. Choe: Regularity of solutions to the Navier-Stokes equation. Electron. J. Differential Equations 5 (1999), 1-7. · Zbl 0923.35117
[3] C. L. Berselli, G. P. Galdi: Regularity criterion involving the pressure for weak solutions to the Navier-Stokes equations. Dipartimento di Matematica Applicata, Università di Pisa, Preprint No. 2001/10. · Zbl 1075.35031
[4] L. Escauriaza, G. Seregin, V. Šverák: On backward uniqueness for parabolic equations. Zap. Nauch. Seminarov POMI 288 (2002), 100-103. · Zbl 1068.35090
[5] E. Hopf: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen. Math. Nachrichten 4 (1951), 213-231. · Zbl 0042.10604
[6] K. K. Kiselev, O. A. Ladyzhenskaya: On existence and uniqueness of solutions of the solutions to the Navier-Stokes equations. Izv. Akad. Nauk SSSR 21 (1957), 655-680.
[7] J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934), 193-248. · JFM 60.0726.05
[8] J. Neustupa, J. Nečas: New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations. J. Math. Fluid Mech. 4 (2002), 237-256. · Zbl 1010.35081
[9] J. Neustupa, A. Novotný, P. Penel: A remark to interior regularity of a suitable weak solution to the Navier-Stokes equations. CIM Preprint No. 25 (1999).
[10] J. Neustupa, P. Penel: Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes Equations. Mathematical Fluid Mechanics (Recent Results and Open Problems), J. Neustupa, P. Penel (eds.), Birkhäuser-Verlag, Basel, 2001, pp. 237-268. · Zbl 1027.35094
[11] L. Nirenberg: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 123 13 (1959), 115-162. · Zbl 0088.07601
[12] M. Pokorný: On the result of He concerning the smoothness of solutions to the Navier-Stokes equations. Electron. J. Differential Equations (2003), 1-8. · Zbl 1014.35073
[13] V. Scheffer: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55 (1977), 97-112. · Zbl 0357.35071
[14] G. Seregin, V. Šverák: Navier-Stokes with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163 (2002), 65-86. · Zbl 1002.35094
[15] G. Seregin, V. Šverák: Navier-Stokes and backward uniqueness for the heat equation. IMA Preprint No. 1852 (2002). · Zbl 1024.76011
[16] J. Serrin: The initial boundary value problem for the Navier-Stokes equations. Nonlinear Problems, R. E. Langer (ed.), University of Wisconsin Press, 1963. · Zbl 0115.08502
[17] E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970. · Zbl 0207.13501
[18] Y. Zhou: A new regularity result for the Navier-Stokes equations in terms of the gradient of one velocity component. Methods and Applications in Analysis
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.