The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation

Hongxia Lin; Sabana; Qing Sun; Ruiqi You; Xiaochuan Guo; Hongxia Lin; Sabana; Qing Sun; Ruiqi You; Xiaochuan Guo

doi:10.3934/cam.2025005

Communications in Analysis and Mechanics

2025, Volume 17, Issue 1: 100-127. doi: 10.3934/cam.2025005

Previous Article Next Article

Research article

The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation

1.
School of Mathematical Sciences, Chengdu University of Technology, Chengdu 610059, P.R. China
2.
Geomathematics Key Laboratory of Sichuan Province, Chengdu University of Technology, Chengdu 610059, P.R. China

Received: 13 January 2024 Revised: 27 November 2024 Accepted: 07 January 2025 Published: 10 February 2025
35A01, 35Q35, 76D03

This paper studies a special 2D anisotropic incompressible Boussinesq equation in $ {\mathbb{T}}^2 $ with $ \mathbb{T} = [-\frac{1}{2}, \frac{1}{2}] $ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in $ {\mathbb{R}}^2 $ has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain $ \mathbb{T} \times \mathbb{R} $ has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the $ u_2 $ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity $ f $ into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain $ H^2 $-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in $ H^1 $ are also established.
- 2D anisotropic Boussinesq equations,
- partial dissipation,
- stability,
- vertical thermal diffusion
Citation: Hongxia Lin, Sabana, Qing Sun, Ruiqi You, Xiaochuan Guo. The stability and decay of 2D incompressible Boussinesq equation with partial vertical dissipation[J]. Communications in Analysis and Mechanics, 2025, 17(1): 100-127. doi: 10.3934/cam.2025005

Related Papers:

Abstract

This paper studies a special 2D anisotropic incompressible Boussinesq equation in $ {\mathbb{T}}^2 $ with $ \mathbb{T} = [-\frac{1}{2}, \frac{1}{2}] $ being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in $ {\mathbb{R}}^2 $ has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain $ \mathbb{T} \times \mathbb{R} $ has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the $ u_2 $ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity $ f $ into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain $ H^2 $-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in $ H^1 $ are also established.

References

[1]	P. Constantin, C. R. Doering, Heat transfer in convective turbulence, Nonlinearity, 9 (1996), 1049. https://doi.org/10.1088/0951-7715/9/4/013 doi: 10.1088/0951-7715/9/4/013
[2]	A. Majda, A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2001. https://doi.org/10.1017/CBO9780511613203
[3]	A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, American Mathematical Society, 2003. http://dx.doi.org/10.1090/cln/009 doi: 10.1090/cln/009
[4]	J. Pedlosky, Geophysical Fluid Dynamics, 2 Eds., New York: Springer-Verlag, 1987. https://doi.org/10.1007/978-1-4612-4650-3
[5]	D. Adhikari, C. Cao, J. Wu, X. Xu, Small global solutions to the damped two-dimensional Boussinesq equations, J. Differ. Equ., 256 (2014), 3594–3613. https://doi.org/10.1016/j.jde.2014.02.012 doi: 10.1016/j.jde.2014.02.012
[6]	D. Adhikari, C. Cao, H. Shang, J. Wu, X. Xu, Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 260 (2016), 1893–1917. https://doi.org/10.1016/j.jde.2015.09.049 doi: 10.1016/j.jde.2015.09.049
[7]	L. Brandolese, M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Am. Math. Soc., 364 (2012), 5057–5090. https://doi.org/10.1090/S0002-9947-2012-05432-8 doi: 10.1090/S0002-9947-2012-05432-8
[8]	B. Dong, Z. Ye, X. Zhai, Global regularity for the 2D Boussinesq equations with temperature-dependent viscosity, J. Math. Fluid Mech., 22 (2020). https://doi.org/10.1007/s00021-019-0463-0 doi: 10.1007/s00021-019-0463-0
[9]	T. Hmidi, S. Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations, 12 (2007), 461–480. https://doi.org/10.48550/arXiv.0711.3198 doi: 10.48550/arXiv.0711.3198
[10]	N. Ju, Global regularity and long-time behavior of the solutions to the 2D Boussinesq equations without diffusivity in a bounded domain, J. Math. Fluid Mech., 19 (2017), 105–121. https://doi.org/10.1007/s00021-016-0277-2 doi: 10.1007/s00021-016-0277-2
[11]	C. Khor, X. Xu, Temperature patches for a generalised 2D Boussinesq system with singular velocity, J. Nonlinear. Sci., 33 (2023). https://doi.org/10.1007/s00332-022-09886-7 doi: 10.1007/s00332-022-09886-7
[12]	J. Wu, X. Xu, L. Xue, Z. Ye, Regularity results for the 2d Boussinesq equations with critical and supercritical dissipation, Comm. Math. Sci., 14 (2016), 1963–1997. https://doi.org/10.4310/CMS.2016.v14.n7.a9 doi: 10.4310/CMS.2016.v14.n7.a9
[13]	Z. Ye, Global regularity for a 3D Boussinesq model without thermal diffusion, Z. Angew. Math. Phys., 68 (2017). https://doi.org/10.1007/s00033-017-0832-6 doi: 10.1007/s00033-017-0832-6
[14]	C. R. Doering, J. Wu, K. Zhao, X. Zheng, Long time behavior of the two dimensional Boussinesq equations without buoyancy diﬀusion, Phys. D., 376 (2018), 144–159. https://doi.org/10.1016/j.physd.2017.12.013 doi: 10.1016/j.physd.2017.12.013
[15]	L. Tao, J. Wu, K. Zhao, X. Zheng, Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diﬀusion, Arch. Ration. Mech. Anal., 237 (2020), 585–630. https://doi.org/10.1007/s00205-020-01515-5 doi: 10.1007/s00205-020-01515-5
[16]	O. Ben Said, U. Pandey, J. Wu, The stabilizing eﬀect of the temperature on buoyancy-driven ﬂuids, Indiana University Math. J., 71 (2020), 2605–2645. https://doi.org/10.1512/iumj.2022.71.9070 doi: 10.1512/iumj.2022.71.9070
[17]	D. Adhikari, O. Ben Said, U. R. Pandey, J. Wu, Stability and large-time behavior for the 2D Boussineq system with horizontal dissipation and vertical thermal diﬀusion, Nonlinear Differ. Equ. Appl., 29 (2022). https://doi.org/10.1007/s00030-022-00773-4 doi: 10.1007/s00030-022-00773-4
[18]	D. Chen, Q. Liu, Stability and large Time behavior of the 2D boussinesq equations with mixed partial dissipation near hydrostatic equilibrium, Acta Appl. Math., 181 (2022). https://doi.org/10.1007/s10440-022-00525-7 doi: 10.1007/s10440-022-00525-7
[19]	K. Kang, J. Lee, D. Nguyen, Global well-posedness and stability of the 2D Boussinesq equations with partial dissipation near a hydrostatic equilibrium, J. Differ. Equ., 393 (2024), 1–57. https://doi.org/10.1016/j.jde.2024.02.016 doi: 10.1016/j.jde.2024.02.016
[20]	S. Lai, J. Wu, Y. Zhong, Stability and large-time behavior of the 2D Boussinesq equations with partial dissipation, J. Differ. Equ., 271 (2021), 764–796. https://doi.org/10.1016/j.jde.2020.09.022 doi: 10.1016/j.jde.2020.09.022
[21]	S. Lai, J. Wu, X. Xu, J. Zhang, Y. Zhong, Optimal decay estimates for 2D Boussinesq equations with partial dissipation, J. Nonlinear. Sci., 31 (2021). https://doi.org/10.1007/s00332-020-09672-3 doi: 10.1007/s00332-020-09672-3
[22]	A. Larios, E. Lunasin, E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 255 (2013), 2636–2654. https://doi.org/10.1016/j.jde.2013.07.011 doi: 10.1016/j.jde.2013.07.011
[23]	A. Castro, D. Córdoba, 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. https://doi.org/10.1142/S0218202519500210 doi: 10.1142/S0218202519500210
[24]	I. Kukavica, W. Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity, J. Dyn. Differ. Equ., 32 (2020), 2061–2077. https://doi.org/10.1007/s10884-019-09802-w doi: 10.1007/s10884-019-09802-w
[25]	C. Miao, L. Xue, On the global well-posedness of a class of Boussinesq-Navier-Stokes systems, Nonlinear Differ. Equ. Appl., 18 (2011), 707–735. https://doi.org/10.1007/s00030-011-0114-5 doi: 10.1007/s00030-011-0114-5
[26]	P. Dreyfuss, H. Houamed, Uniqueness result for the 3-D Navier-Stokes-Boussinesq equations with horizontal dissipation, J. Math. Fluid Mech., 23 (2021). https://doi.org/10.1007/s00021-020-00547-x doi: 10.1007/s00021-020-00547-x
[27]	D. Fang, W. Le, T. Zhang, Global solutions of 3D axisymmetric Boussinesq equations with nonzero swirl, Nonlinear Anal., 166 (2018), 48–86. https://doi.org/10.1016/j.na.2017.10.008 doi: 10.1016/j.na.2017.10.008
[28]	F. Jian, D. Chen, X. Chen, Stability and decay rate of smooth solutions for the 3D Boussinesq equation with partial horizontal dissipation and thermal damping, Nonlinear Anal., 234 (2023), 113317. https://doi.org/10.1016/j.na.2023.113317 doi: 10.1016/j.na.2023.113317
[29]	R. Ji, L. Yan, J. Wu, Optimal decay for the 3D anisotropic Boussinesq equations near the hydrostatic balance, Calc. Var., 61 (2022). https://doi.org/10.1007/s00526-022-02242-3 doi: 10.1007/s00526-022-02242-3
[30]	H. Liu, H. Gao, Global well-posedness and long time decay of the 3D Boussinesq equations, J. Differ. Equ., 263 (2017), 8649–8665. https://doi.org/10.1016/j.jde.2017.08.049 doi: 10.1016/j.jde.2017.08.049
[31]	H. Shang, L. Xu, Stability near hydrostatic equilibrium to the three-dimensional Boussinesq equations with partial dissipation, Z. Angew. Math. Phys., 72 (2021). https://doi.org/10.1007/s00033-021-01495-w doi: 10.1007/s00033-021-01495-w
[32]	J. Wu, Q. Zhang, Stability and optimal decay for a system of 3D anisotropic Boussinesq equations, Nonlinearity, 34 (2021), 5456. https://doi.org/10.1088/1361-6544/ac08e9 doi: 10.1088/1361-6544/ac08e9
[33]	C. Cao, J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985–1004. https://doi.org/10.1007/S00205-013-0610-3 doi: 10.1007/S00205-013-0610-3
[34]	B. Dong, J. Wu, X. Xu, N. Zhu, Stability and exponential decay for the 2D anisotropic Boussinesq equations with horizontal dissipation. Calc. Var. Partial Differential Equations, 60 (2021). https://doi.org/10.1007/s00526-021-01976-w doi: 10.1007/s00526-021-01976-w
[35]	D. Bian, S. Dai, J. Mao, Stability of Couette flow for 2D Boussinesq system in a uniform magnetic field with vertical dissipation, Applied Mathematics Letters, 121 (2021), 107515. https://doi.org/10.1016/j.aml.2021.107415 doi: 10.1016/j.aml.2021.107415
[36]	H. Lin, Q. Sun, S. Liu, H. Zhang, The Stability and Decay for the 2D Incompressible Euler-Like Equations, J. Math. Fluid Mech., 25 (2023). https://doi.org/10.1007/s00021-023-00824-5 doi: 10.1007/s00021-023-00824-5
[37]	M. Schonbek, T. Schonbek, E. Süli, Large-time behaviour of solutions to the magneto-hydrodynamics equations, Math. Ann., 304 (1996), 717–756. https://doi.org/10.1007/BF01446316 doi: 10.1007/BF01446316
[38]	M. Schonbek, M. Wiegner, On the decay of higher-order norms of the solutions of Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 677–685. https://doi.org/10.1017/S0308210500022976 doi: 10.1017/S0308210500022976
[39]	C. Cao, J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diﬀusion, Adv. in Math., 226 (2011), 1803–1822. https://doi.org/10.1016/j.aim.2010.08.017 doi: 10.1016/j.aim.2010.08.017
[40]	T. Tao, Nonlinear dispersive equations: local and global analysis, American Mathematical Society, 2006. http://dx.doi.org/10.1090/cbms/106 doi: 10.1090/cbms/106

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)