### Electronic Research Archive

2020, Issue 4: 1375-1393. doi: 10.3934/era.2020073

# The regularized Boussinesq equations with partial dissipations in dimension two

• Received: 01 January 2020 Published: 31 July 2020
• Primary: 35Q35; Secondary: 76D03

• 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[J]. Electronic Research Archive, 2020, 28(4): 1375-1393. doi: 10.3934/era.2020073

### Related Papers:

• 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.

 [1] Global regularity results for the 2D Boussinesq equations with partial dissipation. J. Differential Equations (2016) 260: 1893-1917. [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 [3] Nonlinear Schrödinger evolution equations. Nonlinear Anal. (1980) 4: 677-681. [4] A note on limiting cases of Sobolev embedding and convolution inequalities. Comm. Partial Differential Equations (1980) 5: 773-789. [5] The initial problem for the Boussinesq equations with data in $L^{p}$. Lect. Notes Math. (1980) 771: 129-144. [6] Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. (2011) 226: 1803-1822. [7] Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. (2013) 208: 985-1004. [8] Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. (2006) 203: 497-513. [9] Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. (2012) 233: 1618-1645. [10] (1998) Perfect Incompressible Fluids.Oxford Lecture Ser. Math. Appl., vol. 14, The Clarendon Press/Oxford Univ. Press. [11] Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. (1998) 81: 5338-5341. [12] The three dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory. J. Dyn. Differ. Equ. (2002) 14: 1-35. [13] Global well-posedness for Euler-Boussinesq system with critical dissipation. Comm. Partial Differential Equations (2011) 36: 420-445. [14] Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. (2005) 12: 1-12. [15] The 2D incompressible Boussinesq equations with general critical dissipation. SIAM J. Math. Anal. (2014) 46: 3426-3454. [16] Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure Appl. Math. (1988) 41: 891-907. [17] BKM's criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin. Dyn. Syst. (2009) 25: 575-583. [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 [19] (2002) Vorticity and Incompressible Flow.Cambridge University Press. [20] Global well-posedness for the LANS-$\alpha$ equations on bounded domains. R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. (2001) 359: 1449-1468. [21] The anisotropic Lagrangian averaged Euler and Navier-Stokes equations. Arch. Ration. Mech. Anal. (2003) 166: 27-46. [22] J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. [23] Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation. J. Differential Equations (2016) 260: 6716-6744. [24] On the Cauchy problems for certain Boussinesq-$\alpha$ equations. Proc. Roy. Soc. Edinburgh Sect. A (2010) 140: 319-327.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8