Science China Mathematics

, Volume 62, Issue 6, pp 1175–1204 | Cite as

Global strong solutions to 3-D Navier-Stokes system with strong dissipation in one direction

  • Marius Paicu
  • Ping ZhangEmail author


We consider three-dimensional incompressible Navier-Stokes equations (NS) with different viscous coefficients in the vertical and horizontal variables. In particular, when one of these viscous coefficients is large enough compared with the initial data, we prove the global well-posedness of this system. In fact, we obtain the existence of a global strong solution to (NS) when the initial data verifies an anisotropic smallness condition which takes into account the different roles of the horizontal and vertical viscosity.


anisotropic Navier-Stokes equations Littlewood-Paley theory well-posedness 


35Q30 76D03 


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  1. 1.
    Avrin J D. Large-eigenvalue global existence and regularity results for the Navier-Stokes equations. J Differential Equations, 1996, 127: 365–390MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bahouri H, Chemin J Y, Danchin R. Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der Mathematischen Wissenschaften, vol. 343. Berlin-Heidelberg: Springer-Verlag, 2011Google Scholar
  3. 3.
    Bony J-M. Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann Sci Éc Norm Supér (4), 1981, 14: 209–246CrossRefzbMATHGoogle Scholar
  4. 4.
    Bourgain J, Pavlović N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D. J Funct Anal, 2008, 255: 2233–2247MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cannone M, Meyer Y, Planchon F. Solutions autosimilaires des équations de Navier-Stokes. Séminaire Équations aux Dérivées Partielles (Polytechnique) (1993–1994), Exposé VIII, 1994Google Scholar
  6. 6.
    Chemin J-Y. Remarques sur l’existence globale pour le systéme de Navier-Stokes incompressible. SIAM J Math Anal, 1992, 23: 20–28MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chemin J-Y, Desjardins B, Gallagher I, et al. Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations. Oxford Lecture Series in Mathematics and Its Applications, vol. 32. Oxford: The Clarendon Press/Oxford University Press, 2006Google Scholar
  8. 8.
    Chemin J-Y, Gallagher I. On the global wellposedness of the 3-D Navier-Stokes equations with large initial data. Ann Sci Éc Norm Supér (4), 2006, 39: 679–698MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chemin J-Y, Gallagher I. Large, global solutions to the Navier-Stokes equations, slowly varying in one direction. Trans Amer Math Soc, 2010, 362: 2859–2873MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chemin J-Y, Gallagher I, Zhang P. Sums of large global solutions to the incompressible Navier-Stokes equations. J Reine Angew Math, 2013, 681: 65–82MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chemin J-Y, Zhang P. Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable. Comm Partial Differential Equations, 2015, 40: 878–896MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fujita H, Kato T. On the Navier-Stokes initial value problem I. Arch Ration Mech Anal, 1964, 16: 269–315MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gallagher I. The tridimensional Navier-Stokes equations with almost bidimensional data: Stability, uniqueness, and life span. Int Math Res Not IMRN, 1997, 18: 919–935MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kato T. Strong L p-solutions of the Navier-Stokes equation in ℝm, with applications to weak solutions. Math Z, 1984, 187: 471–480MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Koch H, Tataru D. Well-posedness for the Navier-Stokes equations. Adv Math, 2001, 157: 22–35MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Leray J. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math, 1933, 63: 193–248MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu Y, Zhang P. Global well-posedness of 3-D anisotropic Navier-Stokes system with large vertical viscous coefficient. ArXiv:1708.04731, 2017Google Scholar
  18. 18.
    Paicu M. Équation anisotrope de Navier-Stokes dans des espaces critiques. Rev Mat Iberoam, 2005, 21: 179–235MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Paicu M, Zhang P. Global solutions to the 3-D incompressible anisotropic Navier-Stokes system in the critical spaces. Comm Math Phys, 2011, 307: 713–759MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Raugel G, Sell G R. Navier-Stokes equations on thin 3D domains, I: Global attractors and global regularity of solutions. J Amer Math Soc, 1993, 6: 503–568MathSciNetzbMATHGoogle Scholar
  21. 21.
    Zhang P, Zhang T. Regularity of the Koch-Tataru solutions to Navier-Stokes system. Sci China Math, 2012, 55: 453–464MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhang T. Erratum to: Global wellposed problem for the 3-D incompressible anisotropic Navier-Stokes equations in an anisotropic space. Comm Math Phys, 2010, 295: 877–884MathSciNetCrossRefzbMATHGoogle Scholar

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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de BordeauxUniversité Bordeaux 1BordeauxFrance
  2. 2.Academy of Mathematics and Systems Science and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina

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