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A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment
Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations
Department of Applied Mathematics, Chengdu University of Technology, Chengdu 610059, P.R. China |
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 $.
References:
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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. |
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P. Constantin and C. R. Doering,
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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. |
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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. |
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A. E. Gill, Atmosphere-Ocean Dynamics, Academic Press, London, 1982.
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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. |
[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. |
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A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.
![]() ![]() |
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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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