Previous |  Up |  Next


Navier-Stokes equations; regularity of systems of PDE’s
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.
[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. DOI 10.1002/cpa.3160350604 | MR 0673830
[2] D.  Chae, H. J. Choe: Regularity of solutions to the Navier-Stokes equation. Electron. J.  Differential Equations 5 (1999), 1–7. MR 1673067
[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. MR 1920038
[4] L.  Escauriaza, G. Seregin, V. Šverák: On backward uniqueness for parabolic equations. Zap. Nauch. Seminarov POMI 288 (2002), 100–103. MR 1923546
[5] E. Hopf: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen. Math. Nachrichten 4 (1951), 213–231. MR 0050423
[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. (Russian) MR 0100448
[7] J.  Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934), 193–248. DOI 10.1007/BF02547354 | MR 1555394
[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. DOI 10.1007/s00021-002-8544-9 | MR 1932862
[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. In: Mathematical Fluid Mechanics (Recent Results and Open Problems), J. Neustupa, P. Penel (eds.), Birkhäuser-Verlag, Basel, 2001, pp. 237–268. MR 1865056
[11] L.  Nirenberg: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 123 13 (1959), 115–162. MR 0109940 | 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. MR 1958046 | Zbl 1014.35073
[13] V.  Scheffer: Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55 (1977), 97–112. DOI 10.1007/BF01626512 | MR 0510154 | Zbl 0357.35071
[14] G.  Seregin, V. Šverák: Navier-Stokes with lower bounds on the pressure. Arch. Ration. Mech. Anal. 163 (2002), 65–86. DOI 10.1007/s002050200199 | MR 1905137
[15] G.  Seregin, V. Šverák: Navier-Stokes and backward uniqueness for the heat equation. IMA Preprint No.  1852 (2002).
[16] J.  Serrin: The initial boundary value problem for the Navier-Stokes equations. In: Nonlinear Problems, R. E. Langer (ed.), University of Wisconsin Press, 1963.
[17] E. M.  Stein: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970. MR 0290095 | 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 (to appear).
Partner of
EuDML logo