Blow-up criteria for 3D Boussinesq equations in the multiplier space

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in $\mathbb{R}^{N}\ (N\geq 2)$ R N ( N ≥ 2 ) . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

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A blow-up criterion for 3D Boussinesq equations in Besov spaces

Nonlinear Analysis ◽

10.1016/j.na.2010.04.021 ◽

2010 ◽

Vol 73 (3) ◽

pp. 806-815 ◽

Cited By ~ 16

Author(s):

Hua Qiu ◽

Yi Du ◽

Zheng’an Yao

Keyword(s):

Besov Spaces ◽

Blow Up ◽

Boussinesq Equations ◽

Blow Up Criterion

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

Journal of Function Spaces ◽

10.1155/2015/303025 ◽

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.

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On the Well-Posedness of the Boussinesq Equation in the Triebel-Lizorkin-Lorentz Spaces

Abstract and Applied Analysis ◽

10.1155/2012/573087 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Zhaoyin Xiang ◽

Wei Yan

Keyword(s):

Boussinesq Equation ◽

Blow Up ◽

Boussinesq Equations ◽

Lorentz Spaces ◽

Well Posedness ◽

Smooth Solutions ◽

Blow Up Criterion

We establish the local well-posedness and obtain a blow-up criterion of smooth solutions for the Boussinesq equations in the framework of Triebel-Lizorkin-Lorentz spaces. The main ingredients of our proofs are Littlewood-Paley decomposition and the paradifferential calculus.

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