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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 2(350), Pages 45–78 (Mi umn610)  

This article is cited in 41 scientific papers (total in 41 papers)

Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness

O. A. Ladyzhenskaya

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: This paper presents the main results concerning solubility of the basic initial-boundary value problem and the Cauchy problem for the three-dimensional non-stationary Navier–Stokes equations, together with a list of what to prove in order to solve the sixth problem of the “seven problems of the millennium” proposed on the Internet at the site http://claymath.org/.

DOI: https://doi.org/10.4213/rm610

Full text: PDF file (463 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2003, 58:2, 251–286

Bibliographic databases:

UDC: 517.9
MSC: Primary 35Q30, 76P05; Secondary 35A05, 35A07
Received: 15.02.2003

Citation: O. A. Ladyzhenskaya, “Sixth problem of the millennium: Navier–Stokes equations, existence and smoothness”, Uspekhi Mat. Nauk, 58:2(350) (2003), 45–78; Russian Math. Surveys, 58:2 (2003), 251–286

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    5. Shugaev F.V., Terentiev E.N., Shtemenko L.S., Nikolaeva O.A., Pavlova T.A., Dokukina O.I., “On the problem of beam focusing in the turbulent atmosphere - art. no. 67470K”, Optics in Atmospheric Propagation and Adaptive Systems X, Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), 6747, 2007, K7470–K7470  isi
    6. Cao Chongsheng, Titi E.S., “Regularity criteria for the three-dimensional Navier–Stokes equations”, Indiana Univ. Math. J., 57:6 (2008), 2643–2661  crossref  mathscinet  zmath  isi  elib
    7. Sheretov Yu.V., “O svoistvakh reshenii kvazigidrodinamicheskikh uravnenii v barotropnom priblizhenii”, Vestn. Tverskogo gos. un-ta. Ser.: Prikladnaya matematika, 2009, no. 14, 5–19  elib
    8. A. E. Mamontov, “Globalnaya razreshimost mnogomernykh uravnenii szhimaemoi nenyutonovskoi zhidkosti, transportnoe uravnenie i prostranstva Orlicha”, Sib. elektron. matem. izv., 6 (2009), 120–165  mathnet  mathscinet  elib
    9. Andrianov I.V., Awrejcewicz J., Weichert D., “Improved continuous models for discrete media”, Math. Probl. Eng., 2010 (2010), 986242, 35 pp.  crossref  zmath  isi  elib
    10. Cao Chongsheng, “Sufficient conditions for the regularity to the 3D Navier–Stokes equations”, Discrete Contin. Dyn. Syst., 26:4 (2010), 1141–1151  crossref  mathscinet  zmath  isi  elib
    11. Wang Shu, “On a New 3D Model for Incompressible Euler and Navier–Stokes Equations”, Acta Mathematica Scientia, 30:6 (2010), 2089–2102  crossref  mathscinet  zmath  isi
    12. Cao Ch., Titi E.S., “Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor”, Arch Ration Mech Anal, 202:3 (2011), 919–932  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Sheretov Yu.V., “Edinstvennost klassicheskogo resheniya osnovnoi nachalno-kraevoi zadachi dlya kvazigidrodinamicheskikh uravnenii”, Vestnik Tverskogo gosudarstvennogo universiteta. Seriya: Prikladnaya matematika, 2011, no. 20, 7–20  mathnet  elib
    14. Sheretov Yu.V., “Edinstvennost resheniya kvazigidrodinamicheskikh uravnenii v priblizhenii melkoi vody”, Vestnik Tverskogo gosudarstvennogo universiteta. Seriya: Prikladnaya matematika, 2011, no. 22, 7–28  mathnet  elib
    15. Sivaguru S. Sritharan, Kumarasamy Sakthivel, “Martingale solutions for stochastic Navier–Stokes equations driven by Lévy noise”, EECT, 1:2 (2012), 355  crossref  mathscinet  zmath  isi
    16. S. V. Zakharov, “Regular asymptotics of a generalized solution of the stationary Navier–Stokes system”, Proc. Steklov Inst. Math. (Suppl.), 281, suppl. 1 (2013), 146–151  mathnet  crossref  isi  elib
    17. Gishlarkaev V.I., “Uravneniya nave-stoksa: nekotorye metody i ikh prilozheniya k problemam predstavleniya reshenii”, Vestnik akademii nauk chechenskoi respubliki, 2012, no. 2, 13–31  elib
    18. Fang D., Qian Ch., “The Regularity Criterion for 3D Navier–Stokes Equations Involving One Velocity Gradient Component”, Nonlinear Anal.-Theory Methods Appl., 78 (2013), 86–103  crossref  mathscinet  zmath  isi
    19. Tobias Grafke, Rainer Grauer, Thomas C. Sideris, “Turbulence properties and global regularity of a modified Navier–Stokes equation”, Physica D: Nonlinear Phenomena, 2013  crossref  mathscinet  isi
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    21. Biferale L., Titi E.S., “On the Global Regularity of a Helical-Decimated Version of the 3D Navier–Stokes Equations”, J. Stat. Phys., 151:6 (2013), 1089–1098  crossref  mathscinet  zmath  adsnasa  isi  elib
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    23. Akhmetov R.G., Kutluev R.R., “Vortex Structure Around the Cylinder At a Flow of Viscous Fluid”, Applied Non-Linear Dynamical Systems, Springer Proceedings in Mathematics & Statistics, 93, ed. Awrejcewicz J., Springer-Verlag Berlin, 2014, 151–159  crossref  mathscinet  isi
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    28. Akysh A.Sh., “The Simplest Maximum Principle For Navier–Stokes Equations”, Bull. Karaganda Univ-Math., 83:3 (2016), 8–12  isi
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