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March  2021, 20(3): 1229-1240. doi: 10.3934/cpaa.2021018

Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity

Xuemin Deng 1, , Yuelong Xiao 1, and  Aibin Zang 2,,

1. 

Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan, 411100, China
2. 

The Center of Applied Mathematics, Yichun University, Yichun, Jiangxi, 336000, China

* Corresponding author

Received  August 2020 Revised  December 2020 Published  February 2021

Fund Project: The work of Aibin Zang is partially supported by School of Mathematics and Computational Science in Xiangtan University as he visited the second author and supported in part by National Natural Science Foundation of China (Grant no. 11771382). The research of Yuelong Xiao is partially supported by National Natural Science Foundation of China (Grant no. 11871412, 11771300). The study of Xuemin Deng Supported by Hunan Provincial Innovation Foundation For Postgraduate (Grant no. XDCX2021B096)

In this paper, we start to investigate the global existence and uniqueness of weak solutions of the $ n $-dimensional ($ n\geq3 $) hyper-dissipative Boussinesq system without thermal diffusivity in the periodic domain $ \mathbb{T}^n $ with the initial data $ u_0\in L^2(\mathbb{T}^n) $ and $ \theta_0 \in L^2(\mathbb{T}^n)\cap L^{\frac{4n}{n+2}}(\mathbb{T}^n) $. Then we focus on the vanishing thermal diffusion limit and obtain the convergent result in the sense of $ L^2 $-norm. Ultimately, we also prove the global regularity of this system in the case of 3-dimension.

Keywords: Boussinesq equations, global well-posedness, Fourier series, spectral decomposition, vanishing thermal diffusion limit.
Mathematics Subject Classification: Primary: 35P05, 35Q35.
Citation: Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1229-1240. doi: 10.3934/cpaa.2021018
References:
[1]

H. Abidi and P. Zhang, On the global well-posedness of 2-D Boussinesq system with variable viscosity, Adv. Math, 305 (2017), 1202-1249.  doi: 10.1016/j.aim.2016.09.036.  Google Scholar

[2]

N. Boardman, R. H. Ji, H. Qiu and J. Wu, Uniqueness of weak solutions to the Boussinesq equations without thermal diffusion, Commun. Math. Sci, 17 (2019), 1595-1624.  doi: 10.4310/CMS.2019.v17.n6.a5.  Google Scholar

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C. Cao and J. Wu, Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation, Archive for Rational Mechanics & Analysis, 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar

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L. He, Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains with applications, J. Funct. Anal., 262 (2012), 3430-3464.  doi: 10.1016/j.jfa.2012.01.017.  Google Scholar

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L. Jin, J. Fan, G. Nakamura and Y. Zhou, Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition, Bound. Value Probl., 2012 (2012), 20-24.  doi: 10.1186/1687-2770-2012-20.  Google Scholar

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Q. Jiu and H. Yu, Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations, Acta Mathematicae Applicatae Sinica, English Series, 32 (2016), 1-16.  doi: 10.1007/s10255-016-0539-z.  Google Scholar

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I. Kukavica and W. Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity, J. Dyn. Differ. Equ., 32 (2020), 2061-2077.  doi: 10.1007/s10884-019-09802-w.  Google Scholar

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M. Lai, R. Pan and K. Zhao, Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.  Google Scholar

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A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 9 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[13]

C. Li and T. Hou, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. S., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

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K. Tosio and P. Gustavo, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure. Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

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J. Wu, X. Xu, L. Xue and Z. Ye, Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation, Commun. Math. Sci., 14 (2016), 1963-1997.  doi: 10.4310/CMS.2016.v14.n7.a9.  Google Scholar

[18]

J. Wu, X. Xu and Z. Ye, The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusion, Journal De Math$\acute{e}$matiques Pures Et Appliqu$\acute{e}$s, 115 (2018), 187–217. doi: 10.1016/j.matpur.2018.01.006.  Google Scholar

[19]

Z. Xiang and W. Yan, Global regularity of solutions to the Boussinesq equations with fractional diffusion, Adv. Differ. Equ., 18 (2013), 1105-1128.   Google Scholar

[20]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system, Appl. Math., 60 (2015), 109-133.  doi: 10.1007/s10492-015-0087-5.  Google Scholar

[21]

V. Yudovich, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz, 3 (1963), 1032–1066.  Google Scholar

[22]

Z. Ye, A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci., 35 (2015), 112-120.  doi: 10.1016/S0252-9602(14)60144-2.  Google Scholar

[23]

Z. Ye, Remarks on the improved regularity criterion for the 2D Euler-Boussinesq equations with supercritical dissipation, Zeitschrift F$\ddot{u}$r Angewandte Mathematik Und Physik, 67 (2016), 149–156. doi: 10.1007/s00033-016-0742-z.  Google Scholar

[24]

Z. Ye, On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation, Appl. Math. Lett., 90 (2019), 1-7.  doi: 10.1016/j.aml.2018.10.009.  Google Scholar

[25]

X. Zhai, B. Dong and Z. Chen, Global well-posedness for 2D Boussinesq system with the temperature-dependent viscosity and supercritical dissipation, J. Differ. Equ., 267 (2019), 364-387.  doi: 10.1016/j.jde.2019.01.011.  Google Scholar

show all references

References:
[1]

H. Abidi and P. Zhang, On the global well-posedness of 2-D Boussinesq system with variable viscosity, Adv. Math, 305 (2017), 1202-1249.  doi: 10.1016/j.aim.2016.09.036.  Google Scholar

[2]

N. Boardman, R. H. Ji, H. Qiu and J. Wu, Uniqueness of weak solutions to the Boussinesq equations without thermal diffusion, Commun. Math. Sci, 17 (2019), 1595-1624.  doi: 10.4310/CMS.2019.v17.n6.a5.  Google Scholar

[3]

C. Cao and J. Wu, Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation, Archive for Rational Mechanics & Analysis, 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar

[4]

D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-513.  doi: 10.1016/j.aim.2005.05.001.  Google Scholar

[5]

L. He, Smoothing estimates of 2D incompressible Navier-Stokes equations in bounded domains with applications, J. Funct. Anal., 262 (2012), 3430-3464.  doi: 10.1016/j.jfa.2012.01.017.  Google Scholar

[6]

T. Hmidi and S. Keraani, On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J, 58 (2009), 1591-1618.  doi: 10.1512/iumj.2009.58.3590.  Google Scholar

[7]

L. Jin, J. Fan, G. Nakamura and Y. Zhou, Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition, Bound. Value Probl., 2012 (2012), 20-24.  doi: 10.1186/1687-2770-2012-20.  Google Scholar

[8]

Q. Jiu and H. Yu, Global well-posedness for 3D generalized Navier-Stokes-Boussinesq equations, Acta Mathematicae Applicatae Sinica, English Series, 32 (2016), 1-16.  doi: 10.1007/s10255-016-0539-z.  Google Scholar

[9]

I. Kukavica and W. Wang, Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity, Pure Appl. Funct. Anal., 5 (2020), 27-45.   Google Scholar

[10]

I. Kukavica and W. Wang, Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity, J. Dyn. Differ. Equ., 32 (2020), 2061-2077.  doi: 10.1007/s10884-019-09802-w.  Google Scholar

[11]

M. Lai, R. Pan and K. Zhao, Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations, Arch. Ration. Mech. Anal., 199 (2011), 739-760.  doi: 10.1007/s00205-010-0357-z.  Google Scholar

[12]

A. Larios, E. Lunasin and E. S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 9 (2013), 2636-2654.  doi: 10.1016/j.jde.2013.07.011.  Google Scholar

[13]

C. Li and T. Hou, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. S., 12 (2005), 1-12.  doi: 10.3934/dcds.2005.12.1.  Google Scholar

[14]

J. Robinson, J. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations: Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781139095143.  Google Scholar

[15]

K. Tosio and P. Gustavo, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure. Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[16]

C. Wang and Z. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Adv. Math., 228 (2011), 43-62.  doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[17]

J. Wu, X. Xu, L. Xue and Z. Ye, Regularity results for the 2D Boussinesq equations with critical or supercritical dissipation, Commun. Math. Sci., 14 (2016), 1963-1997.  doi: 10.4310/CMS.2016.v14.n7.a9.  Google Scholar

[18]

J. Wu, X. Xu and Z. Ye, The 2D Boussinesq equations with fractional horizontal dissipation and thermal diffusion, Journal De Math$\acute{e}$matiques Pures Et Appliqu$\acute{e}$s, 115 (2018), 187–217. doi: 10.1016/j.matpur.2018.01.006.  Google Scholar

[19]

Z. Xiang and W. Yan, Global regularity of solutions to the Boussinesq equations with fractional diffusion, Adv. Differ. Equ., 18 (2013), 1105-1128.   Google Scholar

[20]

K. Yamazaki, On the global regularity of N-dimensional generalized Boussinesq system, Appl. Math., 60 (2015), 109-133.  doi: 10.1007/s10492-015-0087-5.  Google Scholar

[21]

V. Yudovich, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz, 3 (1963), 1032–1066.  Google Scholar

[22]

Z. Ye, A note on global well-posedness of solutions to Boussinesq equations with fractional dissipation, Acta Math. Sci., 35 (2015), 112-120.  doi: 10.1016/S0252-9602(14)60144-2.  Google Scholar

[23]

Z. Ye, Remarks on the improved regularity criterion for the 2D Euler-Boussinesq equations with supercritical dissipation, Zeitschrift F$\ddot{u}$r Angewandte Mathematik Und Physik, 67 (2016), 149–156. doi: 10.1007/s00033-016-0742-z.  Google Scholar

[24]

Z. Ye, On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation, Appl. Math. Lett., 90 (2019), 1-7.  doi: 10.1016/j.aml.2018.10.009.  Google Scholar

[25]

X. Zhai, B. Dong and Z. Chen, Global well-posedness for 2D Boussinesq system with the temperature-dependent viscosity and supercritical dissipation, J. Differ. Equ., 267 (2019), 364-387.  doi: 10.1016/j.jde.2019.01.011.  Google Scholar

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