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February  2022, 27(2): 863-882. doi: 10.3934/dcdsb.2021068

## Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

 Department of Applied Mathematics, Chengdu University of Technology, Chengdu 610059, P.R. China

* Corresponding author: mllpzh@126.com (Liangliang Ma)

Received  September 2020 Revised  November 2020 Published  February 2022 Early access  March 2021

Fund Project: The research of L. Ma was supported by the National Natural Science Foundation of China (No. 11571243, 11971331), China Scholarship Council (No. 202008515084), Opening Fund of Geomathematics Key Laboratory of Sichuan Province (No. scsxdz2020zd02), and Teacher's development Scientific Research Staring Foundation of Chengdu University of Technology (No. 10912-KYQD2019_07717)

Stability problem on perturbations near the hydrostatic balance is one of the important issues for Boussinesq equations. This paper focuses on the asymptotic stability and large-time behavior problem of perturbations of the 2D fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity. Since the linear portion of the Boussinesq equations plays a crucial role in the stability properties, we firstly study the linearized fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity and complete the following work: 1) assessing the stability and obtaining the precise large-time asymptotic behavior for solutions to the linearized system satisfied the perturbation; 2) understanding the spectral property of the linearization; 3) showing the $H^2$-stability for the linearized system, and prove that the $L^2$-norm of $\nabla{u}$ and $\Delta{u}$ (or $\nabla\theta$ and $\Delta\theta$), the $L^\varrho$-norm $(2<\varrho<\infty)$ of $u$ and $\nabla{u}$ (or $\theta$ and $\nabla\theta$) are all approaching to zero as $t\rightarrow\infty$ when $\alpha = 1$ and $\eta = 0$ (or $\nu = 0$ and $\beta = 1$). Secondly, we obtain the $H^1$-stability for the full nonlinear system and prove the $L^\varrho$-norm $(2<\varrho<\infty)$ of $\theta$ and the $L^2$-norm of $\nabla\theta$ approaching to zero as $t\rightarrow\infty$.

Citation: Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 863-882. doi: 10.3934/dcdsb.2021068
##### References:
 [1] A. Castro, D. Córdoba and D. Lear, On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term, Math. Models Methods Appl. Sci., 29 (2019), 1227–1277. doi: 10.1142/S0218202519500210. [2] P. Constantin and C. R. Doering, Infinite prandtl number convection, J. Stat. Phys., 94 (1999), 159-172.  doi: 10.1023/A:1004511312885. [3] Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics equations, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4. [4] C. R. Doering, J. Wu, K. Zhao and X. Zheng, Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, 376 (2018), 144-159.  doi: 10.1016/j.physd.2017.12.013. [5] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982. [6] L. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of alfvén waves, Ann. PDE, 4 (2018), 5-105.  doi: 10.1007/s40818-017-0041-9. [7] R. Ji, D. Li, Y. Wei and J. Wu, Stability of hydrostatic equilibrium to the 2D Boussinesq systems with partial dissipation, Appl. Math. Lett., 98 (2019), 392-3974.  doi: 10.1016/j.aml.2019.06.019. [8] R. Ji, H. Lin, J. Wu and L. Yan, Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation, Appl. Math. Lett., 94 (2019), 244-249.  doi: 10.1016/j.aml.2019.03.013. [9] R. Ji and J. Wu, The resistive magnetohydrodynamic equation near an equilibrium, J. Differential Equations, 268 (2020), 1854-1871.  doi: 10.1016/j.jde.2019.09.027. [10] F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034. [11] H. Lin, R. Ji, J. Wu and L. Yan, Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation, J. Funct. Anal., 279 (2020), 108519, 39 pp. doi: 10.1016/j.jfa.2020.108519. [12] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [14] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [15] X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020. [16] O. B. Said, U. R. Pandey and J. Wu, The stabilizing effect of the temperature on buoyancy-driven fluids, (2020), arXiv: 2005.11661. [17] A. Stefanov and J. Wu, A global regularity result for the 2D Boussinesq equations with critical dissipation, J. Anal. Math., 137 (2019), 269-290.  doi: 10.1007/s11854-018-0073-4. [18] L. Tao, J. Wu, K. Zhao and X. Zheng, Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion, Arch. Ration. Mech. Anal., 237 (2020), 585-630.  doi: 10.1007/s00205-020-01515-5. [19] J. P. Whitehead and C. R. Doering, Internal heating driven convection at infinite Prandtl number, J. Math. Phys., 52 (2011), 093101, 11 pp. doi: 10.1063/1.3637032. [20] B. Wen, N. Dianati, E. Lunasin, G. P. Chini and C. R. Doering, New upper bounds and reduced dynamical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2191-2199.  doi: 10.1016/j.cnsns.2011.06.039. [21] J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445. [22] D. Wei and Z. Zhang, Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.  doi: 10.2140/apde.2017.10.1361. [23] T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, (2014), arXiv: 1404.5681.

show all references

##### References:
 [1] A. Castro, D. Córdoba and D. Lear, On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term, Math. Models Methods Appl. Sci., 29 (2019), 1227–1277. doi: 10.1142/S0218202519500210. [2] P. Constantin and C. R. Doering, Infinite prandtl number convection, J. Stat. Phys., 94 (1999), 159-172.  doi: 10.1023/A:1004511312885. [3] Y. Cai and Z. Lei, Global well-posedness of the incompressible magnetohydrodynamics equations, Arch. Ration. Mech. Anal., 228 (2018), 969-993.  doi: 10.1007/s00205-017-1210-4. [4] C. R. Doering, J. Wu, K. Zhao and X. Zheng, Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, 376 (2018), 144-159.  doi: 10.1016/j.physd.2017.12.013. [5] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982. [6] L. He, L. Xu and P. Yu, On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of alfvén waves, Ann. PDE, 4 (2018), 5-105.  doi: 10.1007/s40818-017-0041-9. [7] R. Ji, D. Li, Y. Wei and J. Wu, Stability of hydrostatic equilibrium to the 2D Boussinesq systems with partial dissipation, Appl. Math. Lett., 98 (2019), 392-3974.  doi: 10.1016/j.aml.2019.06.019. [8] R. Ji, H. Lin, J. Wu and L. Yan, Stability for a system of the 2D magnetohydrodynamic equations with partial dissipation, Appl. Math. Lett., 94 (2019), 244-249.  doi: 10.1016/j.aml.2019.03.013. [9] R. Ji and J. Wu, The resistive magnetohydrodynamic equation near an equilibrium, J. Differential Equations, 268 (2020), 1854-1871.  doi: 10.1016/j.jde.2019.09.027. [10] F. Lin, L. Xu and P. Zhang, Global small solutions of 2-D incompressible MHD system, J. Differential Equations, 259 (2015), 5440-5485.  doi: 10.1016/j.jde.2015.06.034. [11] H. Lin, R. Ji, J. Wu and L. Yan, Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation, J. Funct. Anal., 279 (2020), 108519, 39 pp. doi: 10.1016/j.jfa.2020.108519. [12] A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009. [13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002. [14] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [15] X. Ren, J. Wu, Z. Xiang and Z. Zhang, Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion, J. Funct. Anal., 267 (2014), 503-541.  doi: 10.1016/j.jfa.2014.04.020. [16] O. B. Said, U. R. Pandey and J. Wu, The stabilizing effect of the temperature on buoyancy-driven fluids, (2020), arXiv: 2005.11661. [17] A. Stefanov and J. Wu, A global regularity result for the 2D Boussinesq equations with critical dissipation, J. Anal. Math., 137 (2019), 269-290.  doi: 10.1007/s11854-018-0073-4. [18] L. Tao, J. Wu, K. Zhao and X. Zheng, Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion, Arch. Ration. Mech. Anal., 237 (2020), 585-630.  doi: 10.1007/s00205-020-01515-5. [19] J. P. Whitehead and C. R. Doering, Internal heating driven convection at infinite Prandtl number, J. Math. Phys., 52 (2011), 093101, 11 pp. doi: 10.1063/1.3637032. [20] B. Wen, N. Dianati, E. Lunasin, G. P. Chini and C. R. Doering, New upper bounds and reduced dynamical modeling for Rayleigh-Bénard convection in a fluid saturated porous layer, Commun. Nonlinear Sci. Numer. Simul., 17 (2012), 2191-2199.  doi: 10.1016/j.cnsns.2011.06.039. [21] J. Wu, Y. Wu and X. Xu, Global small solution to the 2D MHD system with a velocity damping term, SIAM J. Math. Anal., 47 (2015), 2630-2656.  doi: 10.1137/140985445. [22] D. Wei and Z. Zhang, Global well-posedness of the MHD equations in a homogeneous magnetic field, Anal. PDE, 10 (2017), 1361-1406.  doi: 10.2140/apde.2017.10.1361. [23] T. Zhang, An elementary proof of the global existence and uniqueness theorem to 2-D incompressible non-resistive MHD system, (2014), arXiv: 1404.5681.
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