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On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions

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Abstract

In this paper, we first prove the local existence of strong solutions to the 3D Boussinesq equations in a bounded domain with Navier boundary conditions. Then we show the global stability of strong large solutions under a suitable integral condition.

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References

  1. Adhikari, D., Cao, C., Wu, J.: Global regularity results for the 2D Boussinesq equations with partial dissipation. J. Differ. Equ. 260(2), 1893–1917 (2016)

    Article  MathSciNet  Google Scholar 

  2. Benvenutti, M., Ferreira, L.: Global stability of large solutions for the Navier–Stokes equations with Navier boundary conditions. Nonlinear Anal. Real World Appl. 43(10), 308–322 (2018)

    Article  MathSciNet  Google Scholar 

  3. Berselli, L.C., Spirito, S.: On the Boussinesq system: regularity criteria and singular limits. Methods Appl. Anal. 18(4), 391–416 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Brandolese, L., Schonbek, M.E.: Large time decay and growth for solutions of a viscous Boussinesq system. Trans. Amer. Math. Soc. 364(10), 5057–5090 (2012)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Chae, D., Nam, H.-S.: Local existence and blow-up criterion for the Boussinesq equations. Proc. Roy. Soc. Edinburgh Sect. A 127(5), 935–946 (1997)

    Article  MathSciNet  Google Scholar 

  7. Cao, C., Wu, J.: Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 208(3), 985–1004 (2013)

    Article  MathSciNet  Google Scholar 

  8. Doering, C.R., Gibbon, J.D.: Applied Analysis of the Navier-Stokes Equations. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (1995)

  9. Danchin, R., Paicu, M.: Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136(2), 261–309 (2008)

    Article  MathSciNet  Google Scholar 

  10. Ferreira, L.C.F., Villamizar-Roa, E.J.: On the stability problem for the Boussinesq equations in weak-\(L^p\) spaces. Comm. Pure Appl. Anal. 9(3), 667–684 (2010)

  11. Guo, L., Li, Y., Hou C.: On the Boussinesq system: local well-posedness of the strong solution and inviscid limits. Bound. Value Probl., Paper No. 177, 11 pp. (2019)

  12. Hmidi, T., Keraani, S.: On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. Adv. Differ. Equ. 12(4), 461–480 (2007)

    MathSciNet  MATH  Google Scholar 

  13. Hou, T.Y., Li, C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)

    Article  MathSciNet  Google Scholar 

  14. Hu, W., Wang, Y., Wu, J., Xiao, B., Yuan, J.: Partially dissipative 2D Boussinesq equations with Navier type boundary conditions. Phys. D 376/377, 39–48 (2018)

  15. Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys. 54(8), 081507, 10 (2013)

  16. Kukavica, I., Wang, W.: Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity. Pure Appl. Funct. Anal. 5(1), 27–45 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Kukavica, I., Wang, W.: Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity. J. Dynam. Differential Equations 32(4), 2061–2077 (2020)

  18. Lu, J.: Local existence for Boussinesq equations with slip boundary condition in a bounded domain. J. Appl. Math. Phys. 5, 1951–1963 (2017)

    Article  Google Scholar 

  19. Li, X., Cai, X.: The global \(L^2\) stability of solutions to three dimensional MHD equations. Acta Math. Sci. 33(1), 247–267 (2013)

    Article  MathSciNet  Google Scholar 

  20. Larios, A., Lunasin, E., Titi, E.S.: Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion. J. Differ. Equ. 255(9), 2636–2654 (2013)

    Article  MathSciNet  Google Scholar 

  21. Li, Z., Wang, W.: Norm inflation for the Boussinesq system. arXiv:1912.06114 (2019)

  22. Liu, X., Li, Y.: On the stability of global solutions to the 3D Boussinesq system. Nonlinear Anal. 17(95), 580–591 (2014)

    Article  MathSciNet  Google Scholar 

  23. Ponce, G., Racke, R., Sideris, T., Titi, E.: Global stability of large solutions to the 3D Navier–Stokes equations. Comm. Math. Phys. 159(2), 329–341 (1994)

  24. Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Reprint of the 1984 edition. AMS Chelsea Publishing, Providence, RI (2001)

  25. Wang, W.: On the global regularity for a 3D Boussinesq model without thermal diffusion. Z. Angew. Math. Phys. 70, 174 (2019)

    Article  MathSciNet  Google Scholar 

  26. Wang, W., Yue, H.: Almost sure existence of global weak solutions for the Boussinesq equations. Dyn. Part. Differ. Equ. 17(2), 165–183 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank his advisor Igor Kukavica for useful discussions. The author is also very thankful to the anonymous referee for many insightful suggestions and comments which have greatly improved this manuscript. This work was supported in part by the NSF grant DMS-1907992.

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85712, USA

    Weinan Wang

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Wang, W. On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions. Arch. Math. 116, 445–456 (2021). https://doi.org/10.1007/s00013-020-01557-x

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