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doi: 10.3934/dcdsb.2021068
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Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

Liangliang Ma ,

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 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 $.

Keywords: Fractional Boussinesq equations, hydrostatic, stability, large-time behavior.
Mathematics Subject Classification: Primary: 35Q35, 35Q86; Secondary: 76D03, 76D50.
Citation: Liangliang Ma. Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar

[13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
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J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

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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.  Google Scholar

[16]

O. B. Said, U. R. Pandey and J. Wu, The stabilizing effect of the temperature on buoyancy-driven fluids, (2020), arXiv: 2005.11661. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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.  Google Scholar

[2]

P. Constantin and C. R. Doering, Infinite prandtl number convection, J. Stat. Phys., 94 (1999), 159-172.  doi: 10.1023/A:1004511312885.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/cln/009.  Google Scholar

[13] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
[14]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar

[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.  Google Scholar

[16]

O. B. Said, U. R. Pandey and J. Wu, The stabilizing effect of the temperature on buoyancy-driven fluids, (2020), arXiv: 2005.11661. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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