| Title:
|
Some new regularity criteria for the Navier-Stokes equations containing gradient of the velocity (English) |
| Author:
|
Penel, Patrick |
| Author:
|
Pokorný, Milan |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
49 |
| Issue:
|
5 |
| Year:
|
2004 |
| Pages:
|
483-493 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Navier-Stokes equations |
| Keyword:
|
regularity of systems of PDE’s |
| MSC:
|
35B65 |
| MSC:
|
35Q30 |
| MSC:
|
35Q35 |
| MSC:
|
76D03 |
| MSC:
|
76D05 |
| idZBL:
|
Zbl 1099.35101 |
| idMR:
|
MR2086090 |
| DOI:
|
10.1023/B:APOM.0000048124.64244.7e |
| . |
| Date available:
|
2009-09-22T18:19:29Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134580 |
| . |
| Reference:
|
[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. MR 0673830, 10.1002/cpa.3160350604 |
| Reference:
|
[2] D. Chae, H. J. Choe: Regularity of solutions to the Navier-Stokes equation.Electron. J. Differential Equations 5 (1999), 1–7. MR 1673067 |
| Reference:
|
[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 |
| Reference:
|
[4] L. Escauriaza, G. Seregin, V. Šverák: On backward uniqueness for parabolic equations.Zap. Nauch. Seminarov POMI 288 (2002), 100–103. MR 1923546 |
| Reference:
|
[5] E. Hopf: Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen.Math. Nachrichten 4 (1951), 213–231. MR 0050423 |
| Reference:
|
[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 |
| Reference:
|
[7] J. Leray: Sur le mouvement d’un liquide visqueux emplissant l’espace.Acta Math. 63 (1934), 193–248. MR 1555394, 10.1007/BF02547354 |
| Reference:
|
[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. MR 1932862, 10.1007/s00021-002-8544-9 |
| Reference:
|
[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). |
| Reference:
|
[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 |
| Reference:
|
[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, MR 0109940 |
| Reference:
|
[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, MR 1958046 |
| Reference:
|
[13] V. Scheffer: Hausdorff measure and the Navier-Stokes equations.Comm. Math. Phys. 55 (1977), 97–112. Zbl 0357.35071, MR 0510154, 10.1007/BF01626512 |
| Reference:
|
[14] G. Seregin, V. Šverák: Navier-Stokes with lower bounds on the pressure.Arch. Ration. Mech. Anal. 163 (2002), 65–86. MR 1905137, 10.1007/s002050200199 |
| Reference:
|
[15] G. Seregin, V. Šverák: Navier-Stokes and backward uniqueness for the heat equation.IMA Preprint No. 1852 (2002). MR 1972005 |
| Reference:
|
[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. |
| Reference:
|
[17] E. M. Stein: Singular Integrals and Differentiability Properties of Functions.Princeton University Press, Princeton, 1970. Zbl 0207.13501, MR 0290095 |
| Reference:
|
[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). |
| . |
