On the Continuous Periodic Weak Solutions of Boussinesq Equations

2018 ◽  
Vol 50 (1) ◽  
pp. 1120-1162 ◽  
Author(s):  
Tao Tao ◽  
Liqun Zhang
Keyword(s):  
Weak Solutions ◽  
Boussinesq Equations ◽  
Periodic Weak Solutions

The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

2020 ◽  
Vol 18 (04) ◽  
pp. 639-682 ◽  
Author(s):  
Yichen Dai ◽  
Weiwei Hu ◽  
Jiahong Wu ◽  
Bei Xiao
Keyword(s):  
Thermal Diffusion ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Boussinesq Equations ◽  
Convergence Theory ◽  
Diffusion Limit ◽  
Periodic Functions ◽  
Whole Space ◽  
Complete Set ◽  
Periodic Weak Solutions

The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate.

scholarly journals Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Zhaohui Dai ◽  
Xiaosong Wang ◽  
Lingrui Zhang ◽  
Wei Hou
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Interpolation Inequality ◽  
Embedding Theorem ◽  
The Earth ◽  
The Cauchy Problem ◽  
Homogeneous Besov Space ◽  
Blow Up Criterion

The Boussinesq equations describe the three-dimensional incompressible fluid moving under the gravity and the earth rotation which come from atmospheric or oceanographic turbulence where rotation and stratification play an important role. In this paper, we investigate the Cauchy problem of the three-dimensional incompressible Boussinesq equations. By commutator estimate, some interpolation inequality, and embedding theorem, we establish a blow-up criterion of weak solutions in terms of the pressurepin the homogeneous Besov spaceḂ∞,∞0.

scholarly journals Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature

2020 ◽  
Vol 40 (6) ◽  
pp. 3737-3765
Author(s):  
Tianwen Luo ◽  
◽  
Tao Tao ◽  
Liqun Zhang ◽  
◽  
...  
Keyword(s):  
Weak Solutions ◽  
Boussinesq Equations ◽  
Finite Energy ◽  
2D Boussinesq Equations
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