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doi: 10.3934/era.2020073

The regularized Boussinesq equations with partial dissipations in dimension two

1. 

Department of Mathematics, South China Agricultural University, Guangzhou 510642, China

2. 

School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author: Hua Qiu

Received  January 2020 Published  July 2020

Fund Project: The first author is partially supported by National Natural Science Foundation of China (Grant No. 11126266), Natural Science Foundation of GuangDong Province (Grant No. 2016A030313390) and SCAU Fund for High-level University Building. The second author is partially supported by the National Natural Science Foundation of China (Grant Nos. 11971496 and 11431015). Part of this work was done when H. Qiu was visiting Department of Mathematics at Oklahoma State University in 2018. H. Qiu appreciates the hospitality of Department of Mathematics at OSU and would like to thank Professor Jiahong Wu for his helpful discussion and encouragement. H. Qiu's research is supported partly by China Scholarship Council. This work is also supported in part by the Visiting Program of Chern Institute of Mathematics (CIM) at Nankai University. We appreciate the hospitality of CIM

The incompressible Boussinesq system plays an important role in modelling geophysical fluids and studying the Raleigh-Bernard convection. We consider the regularized model (also named as Boussinesq-$ \alpha $ model) to the Boussinesq equations. We consider the Cauchy problem of a two-dimensional regularized Boussinesq model with vertical dissipation in the horizontal regularized velocity equation and horizontal dissipation in the vertical regularized velocity equation and prove that this system has a unique global classical solution. Next, we consider a two-dimensional Boussinesq-$ \alpha $ model with only vertical thermal diffusion and establish a Beale-Kato-Majda type regularity condition of smooth solution for this system.

Citation: Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, doi: 10.3934/era.2020073
References:
[1]

D. AdhikariC. CaoH. ShangJ. WuX. Xu and Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differential Equations, 260 (2016), 1893-1917.  doi: 10.1016/j.jde.2015.09.049.  Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

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H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

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H. Brézis and S. Wainger, A note on limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

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C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

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C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar

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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

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D. Chae and J. Wu, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 233 (2012), 1618-1645.   Google Scholar

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S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

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C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

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T. HmidiS. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

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

[15]

Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454.  doi: 10.1137/140958256.  Google Scholar

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T. Kato and G. Ponce, 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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Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[18]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003. doi: 10.1090/cln/009.  Google Scholar

[19] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
[20]

J. E. Marsden and S. Shkoller, Global well-posedness for the LANS-$\alpha$ equations on bounded domains, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[21]

J. E. Marsden and S. Shkoller, The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 166 (2003), 27-46.  doi: 10.1007/s00205-002-0207-8.  Google Scholar

[22]

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

[23]

Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differential Equations, 260 (2016), 6716-6744.  doi: 10.1016/j.jde.2016.01.014.  Google Scholar

[24]

Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-$\alpha$ equations, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 319-327.  doi: 10.1017/S0308210509000122.  Google Scholar

show all references

References:
[1]

D. AdhikariC. CaoH. ShangJ. WuX. Xu and Z. Ye, Global regularity results for the 2D Boussinesq equations with partial dissipation, J. Differential Equations, 260 (2016), 1893-1917.  doi: 10.1016/j.jde.2015.09.049.  Google Scholar

[2]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343. Springer, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[3]

H. Brézis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), 677-681.  doi: 10.1016/0362-546X(80)90068-1.  Google Scholar

[4]

H. Brézis and S. Wainger, A note on limiting cases of Sobolev embedding and convolution inequalities, Comm. Partial Differential Equations, 5 (1980), 773-789.  doi: 10.1080/03605308008820154.  Google Scholar

[5]

J. R. Cannon and E. DiBenedetto, The initial problem for the Boussinesq equations with data in $L^{p}$, Lect. Notes Math., 771 (1980), 129-144.   Google Scholar

[6]

C. Cao and J. Wu, Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226 (2011), 1803-1822.  doi: 10.1016/j.aim.2010.08.017.  Google Scholar

[7]

C. Cao and J. Wu, Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch. Ration. Mech. Anal., 208 (2013), 985-1004.  doi: 10.1007/s00205-013-0610-3.  Google Scholar

[8]

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

[9]

D. Chae and J. Wu, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 233 (2012), 1618-1645.   Google Scholar

[10] J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Ser. Math. Appl., vol. 14, The Clarendon Press/Oxford Univ. Press, New York, 1998.   Google Scholar
[11]

S. ChenC. FoiasD. D. HolmE. OlsonE. S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81 (1998), 5338-5341.  doi: 10.1103/PhysRevLett.81.5338.  Google Scholar

[12]

C. FoiasD. D. Holm and E. S. Titi, The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dyn. Differ. Equ., 14 (2002), 1-35.  doi: 10.1023/A:1012984210582.  Google Scholar

[13]

T. HmidiS. Keraani and F. Rousset, Global well-posedness for Euler-Boussinesq system with critical dissipation, Comm. Partial Differential Equations, 36 (2011), 420-445.  doi: 10.1080/03605302.2010.518657.  Google Scholar

[14]

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

[15]

Q. JiuC. MiaoJ. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J. Math. Anal., 46 (2014), 3426-3454.  doi: 10.1137/140958256.  Google Scholar

[16]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Commun. Pure Appl. Math., 41 (1988), 891-907.  doi: 10.1002/cpa.3160410704.  Google Scholar

[17]

Z. Lei and Y. Zhou, BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity, Discrete Contin. Dyn. Syst., 25 (2009), 575-583.  doi: 10.3934/dcds.2009.25.575.  Google Scholar

[18]

A. Majda, Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9, AMS/CIMS, 2003. doi: 10.1090/cln/009.  Google Scholar

[19] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2002.   Google Scholar
[20]

J. E. Marsden and S. Shkoller, Global well-posedness for the LANS-$\alpha$ equations on bounded domains, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 359 (2001), 1449-1468.  doi: 10.1098/rsta.2001.0852.  Google Scholar

[21]

J. E. Marsden and S. Shkoller, The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Ration. Mech. Anal., 166 (2003), 27-46.  doi: 10.1007/s00205-002-0207-8.  Google Scholar

[22]

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

[23]

Z. Ye and X. Xu, Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation, J. Differential Equations, 260 (2016), 6716-6744.  doi: 10.1016/j.jde.2016.01.014.  Google Scholar

[24]

Y. Zhou and J. Fan, On the Cauchy problems for certain Boussinesq-$\alpha$ equations, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 319-327.  doi: 10.1017/S0308210509000122.  Google Scholar

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