Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity


In this paper, the authors first consider the global well-posedness of 3-D Boussinesq system, which has variable kinematic viscosity yet without thermal conductivity and buoyancy force, provided that the viscosity coefficient is sufficiently close to some positive constant in L and the initial velocity is small enough in \(\dot{B}_{3,1}^0(\mathbb{R}^3)\). With some thermal conductivity in the temperature equation and with linear buoyancy force θe3 on the velocity equation in the Boussinesq system, the authors also prove the global well-posedness of such system with initial temperature and initial velocity being sufficiently small in L1(ℝ3) and \(\dot{B}_{3,1}^0(\mathbb{R}^3)\) respectively.

This is a preview of subscription content, log in to check access.


  1. [1]

    Abidi, H., Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Rev. Mat. Iberoam., 23, 2007, 537–586.

  2. [2]

    Abidi, H., Sur l’unicité pour le système de Boussinesq avec diffusion non linéaire, J. Math. Pures Appl., 91, 2009, 80–99.

  3. [3]

    Abidi, H. and Hmidi, T., On the global well-posedness for Boussinesq system, J. Differential Equations, 233, 2007, 199–220.

  4. [4]

    Abidi, H. and Zhang, P., On the global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity, Sci. China Math., 58, 2015, 1129–1150.

  5. [5]

    Abidi, H. and Zhang, P., On the well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity, J. Differential Equations, 259, 2015, 3755–3802.

  6. [6]

    Abidi, H. and Zhang, P., On the global well-posedness of 2-D boussinesq system with variable viscosity, Adv. Math., 305, 2017, 1202–1249.

  7. [7]

    Bahouri, H., Chemin, J. Y. and Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Heidelberg, 2011.

  8. [8]

    Bergh, J. and Löfstrom, J., Interpolation Spaces, An Introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer-Verlag, Berlin-New York, 1976.

  9. [9]

    Bony, J. M., Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. Ecole Norm. Sup., 14, 1981, 209–246.

  10. [10]

    Cao, C. and Wu, J., Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208, 2013, 985–1004.

  11. [11]

    Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 2006, 497–513.

  12. [12]

    Córdoba, A. and Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Coram. Math. Phys., 249, 2004, 511–528.

  13. [13]

    Cwikel, M., On (L p0(A 0), L p1 (A 1))θ,q, Proc. Amer. Math. Soc., 44, 1974, 286–292.

  14. [14]

    Danchin, R. and Paicu, M., Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 2008, 261–309.

  15. [15]

    Danchin, R. and Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 2011, 421–457.

  16. [16]

    Desjardins, B., Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Ration. Mech. Anal, 137, 1997, 135–158.

  17. [17]

    Díaz, J. I. and Galiano, G., Existence and uniqueness of solutions of the Boussinesq system with nonlinear thermal diffusion, Topol. Methods Nonlinear Anal., 11, 1998, 59–82.

  18. [18]

    Grafakos, L., Classical Fourier Analysis, 2nd ed., Graduate Texts in Mathematics, 249, Springer-Verlag, New York, 2008.

  19. [19]

    Hmidi, T., On a maximum principle and its application to logarithmically critical Boussinesq system, Anal. PDE, 4, 2011, 247–284.

  20. [20]

    Hmidi, T. and Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12, 2007, 461–480.

  21. [21]

    Hmidi, T., Keraani, S. and Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differential Equations, 249, 2010, 2147–2174.

  22. [22]

    Hmidi, T., Keraani, S. and Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Coram. Partial Differential Equations, 36, 2011, 420–445.

  23. [23]

    Hou, T. and Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 2005, 1–12.

  24. [24]

    Huang, J. and Paicu, M., Decay estimates of global solutions to 2D incompressible inhomogeneous Navier-Stokes equations with variable viscosity, Discrete Contin. Dyn. Syst., 34(11), 2011, 4647–4669.

  25. [25]

    Lemarié-Rieusset, P. G., Recent Developments in the Navier-Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002.

  26. [26]

    Meyer, Y., Wavelets, Paraproducts and Navier-Stokes, Current Developments in Mathematics, 1996, International Press, Cambridge, 1999.

  27. [27]

    Meyer, Y., Ondelettes et Opérateurs, Tome, 3, Hermann, Paris, 1991.

  28. [28]

    Pedloski, J., Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.

  29. [29]

    Planchon, F., An extension of the Beale-Kato-Majda criterion for the Euler equations, Coram. Math. Phys., 232, 2003, 319–326.

  30. [30]

    Rodrigues, J. F., Weak solutions for thermoconvective flows of Boussinesq-Stefan type, Mathematical Topics in Fluid Mechanics, Pitman Res. Notes Math. Ser., 274, Longman Sci. Tech., Harlow, 1992, 93–116.

  31. [31]

    Schonbek, M. E., Large time behaviour of solutions to the Navier-Stokes equations, Coram. Partial Differential Equations, 11, 1986, 733–763.

  32. [32]

    Wang, C. and Zhang, Z., Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228, 2011, 43–62.

  33. [33]

    Wu, J., Lower bounds for an integral involving fractional Laplacians and the generalized Navier-Stokes equations in Besov spaces, Coram. Math. Phys., 263, 2006, 803–831.

Download references


Part of this work was done when we were visiting Morningside Center of Mathematics, CAS, in the summer of 2013. We appreciate the hospitality and the financial support from the Center.

Author information

Correspondence to Hammadi Abidi or Ping Zhang.

Additional information

Dedicated to Professor Andrew J. Majda for the 70th birthday

This work was supported by the National Natural Science Foundation of China (Nos. 11731007, 11688101) and Innovation Grant from National Center for Mathematics and Interdisciplinary Sciences.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abidi, H., Zhang, P. On the Global Well-Posedness of 3-D Boussinesq System with Variable Viscosity. Chin. Ann. Math. Ser. B 40, 643–688 (2019).

Download citation


  • Boussinesq systems
  • Littlewood-Paley theory
  • Variable viscosity
  • Maximal regularity of heat equation

2000 MR Subject Classification

  • 35Q30
  • 76D03