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Decay for the 3D anisotropic MHD-Boussinesq equations near the hydrostatic equilibrium

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Abstract

This paper focuses on the 3D anisotropic MHD-Boussinesq equations with horizontal dissipation for perturbations near the hydrostatic equilibrium. The main purpose of this paper is to establish the global stability and the decay rates of the system emanating from a small data.

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References

  1. Bian, D.: Initial boundary value problem for two-dimensional viscous Boussinesq equations for MHD convection. Discrete Contin. Dyn. Syst. 9, 1591–1611 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bian, D., Dai, S., Mao, J.: Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field with vertical dissipation. Appl. Math. Lett. 121, 107415 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bian, D., Liu, J.: Initial-boundary value problem to 2D Boussinesq equations for MHD convection with stratification effects. J. Differ. Equ. 263, 8074–8101 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bian, D., Mao, J., Pu, X.: Stability of hydrostatic equilibrium for the 2D BMHD system with partial dissipation. Commun. Pure Appl. Anal. 21, 3441–3462 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brandolese, L.: Characterization of solutions to dissipative systems with sharp algebraic decay. SIAM J. Math. Anal. 48(3), 1616–1633 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Guterres, R., Niche, C., Perusato, C., Zingano, P.: Upper and lower \(\dot{H}^m\) estimates for solutions to parabolic equations. J. Differ. Equ. 356, 407–431 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ji, R., Wu, J., Yang, W.: Stability and optimal decay for the 3D Navier–Stokes equations with horizontal dissipation. J. Differ. Equ. 290, 57–77 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ji, R., Yan, L., Wu, J.: Optimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balance. Calc. Var. Partial Differ. Equ. 61(4), 136 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, Y.: Global weak solutions to the three-dimensional inviscid Boussinesq system in the presence of magnetic field. Z. Angew. Math. Phys. 70(6), 172 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, H., Bian, D., Pu, X.: Global well-posedness of the 3D Boussinesq-MHD system without heat diffusion. Z. Angew. Math. Phys. 70(3), 81 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  12. Larios, A., Pei, Y.: On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion. J. Differ. Equ. 263, 1419–1450 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, X., Tan, Z., Xu, S.: Global existence and decay estimates of solutions to the MHD-Boussinesq system with stratification effects. Nonlinearity 35(12), 6067–6097 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, X., Zhang, Q.: Time decay and stability of solutions for the 3D density-dependent incompressible Boussinesq system. Discrete Contin. Dyn. Syst. Ser. B 28(4), 2694–2717 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  15. Majda, A., Bertozzi, A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  16. Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids. Appl. Math. Sci., vol. 96. Springer, New York (1994)

    MATH  Google Scholar 

  17. Melo, W., Perusato, C., Guterres, R., Nunes, J.: Large time decay for the magnetohydrodynamics system in \(H^s(\mathbb{R}_n)\). Acta Appl. Math. 168, 1–16 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mulone, G., Rionero, S.: Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem. Arch. Ration. Mech. Anal. 166(3), 197–218 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Trève, Y., Manley, O.: Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Bénard convection. Physica D 4(3), 319–342 (1982)

    Article  MATH  Google Scholar 

  20. Wiegner, M.: Decay results for weak solutions of the Navier–Stokes equations on \(\mathbb{R}^n\). J. Lond. Math. Soc. 35(3), 303–313 (1987)

    Article  MATH  Google Scholar 

  21. Wu, J., Zhu, Y.: Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium. Adv. Math. 377, 107466 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ye, Z.: Global regularity of 2D temperature-dependent MHD-Boussinesq equations with zero thermal diffusivity. J. Differ. Equ. 293, 447–481 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yu, Y., Fei, M.: Global well-posedness for the 2D MHD-Boussinesq system with temperature-dependent diffusion. Appl. Math. Lett. 106, 106399 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhai, X., Chen, Z.: Global well-posedness for the MHD-Boussinesq system with the tempera-ture-dependent viscosity. Nonlinear Anal. Real World Appl. 44, 260–282 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors want to express their sincere thanks to the editors and the referees for their invaluable comments and suggestions which helped improve the paper greatly. This work was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 12001004) and the NSF of Anhui Province (No. 2108085QA12).

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  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, People’s Republic of China

    Wenjuan Wang & Yan Jia

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  1. Wenjuan Wang
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  2. Yan Jia
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Correspondence to Yan Jia.

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Wang, W., Jia, Y. Decay for the 3D anisotropic MHD-Boussinesq equations near the hydrostatic equilibrium. Z. Angew. Math. Phys. 74, 127 (2023). https://doi.org/10.1007/s00033-023-02011-y

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