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

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  March 2021 Early access  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.

Citation: Xuemin Deng, Yuelong Xiao, Aibin Zang. Global well-posedness of the $ n $-dimensional hyper-dissipative Boussinesq system without thermal diffusivity. Communications on Pure and 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.

[2]

N. BoardmanR. H. JiH. 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.

[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.

[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.

[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.

[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.

[7]

L. JinJ. FanG. 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.

[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.

[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. 

[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.

[11]

M. LaiR. 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.

[12]

A. LariosE. 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.

[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.

[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.

[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.

[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.

[17]

J. WuX. XuL. 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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

X. ZhaiB. 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.

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.

[2]

N. BoardmanR. H. JiH. 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.

[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.

[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.

[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.

[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.

[7]

L. JinJ. FanG. 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.

[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.

[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. 

[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.

[11]

M. LaiR. 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.

[12]

A. LariosE. 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.

[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.

[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.

[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.

[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.

[17]

J. WuX. XuL. 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.

[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.

[19]

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

[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.

[21]

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

[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.

[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.

[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.

[25]

X. ZhaiB. 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.

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