The regularized Boussinesq equations with partial dissipations in dimension two

Hua Qiu; Zheng-An Yao; Hua Qiu; Zheng-An Yao

doi:10.3934/era.2020073

Electronic Research Archive

2020, Volume 28, Issue 4: 1375-1393. doi: 10.3934/era.2020073

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The regularized Boussinesq equations with partial dissipations in dimension two

Hua Qiu ^{1
,
,},
Zheng-An Yao ^{2
,}

1.
Department of Mathematics, South China Agricultural University, Guangzhou 510642, China
2.
School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

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.
- Boussinesq-α equations,
- global existence,
- partial dissipation
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

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Abstract

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.

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