Local existence and blow-up criterion for the generalized Boussinesq equations in Besov spaces

2012 ◽  
Vol 36 (1) ◽  
pp. 86-98 ◽  
Author(s):  
Hua Qiu ◽  
Yi Du ◽  
Zheng'an Yao
Keyword(s):  
Besov Spaces ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Blow Up Criterion

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

2010 ◽  
Vol 73 (3) ◽  
pp. 806-815 ◽  
Author(s):  
Hua Qiu ◽  
Yi Du ◽  
Zheng’an Yao
Keyword(s):  
Besov Spaces ◽  
Blow Up ◽  
Boussinesq Equations ◽  
Blow Up Criterion

scholarly journals Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

1999 ◽  
Vol 155 ◽  
pp. 55-80 ◽  
Author(s):  
Dongho Chae ◽  
Sung-Ki Kim ◽  
Hee-Seok Nam
Keyword(s):  
Existence And Uniqueness ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Passive Scalar ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Local Existence And Uniqueness ◽  
Local Solutions ◽  
Blow Up Criterion

AbstractIn this paper we prove the local existence and uniqueness of C1+γ solutions of the Boussinesq equations with initial data υ0, θ0 ∈ C1+γ, ω0, ∇θ0 ∈ Lq for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C1+γ solutions of the Boussinesq equations.

scholarly journals Remarks on Pressure Blow-Up Criterion of the 3D Zero-Diffusion Boussinesq Equations in Margin Besov Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Min Fu ◽  
Chao Cai
Keyword(s):  
Besov Spaces ◽  
Smooth Solution ◽  
Blow Up ◽  
Three Dimensional ◽  
Boussinesq Equations ◽  
Energy Methods ◽  
Blow Up Criterion

This study is focused on the pressure blow-up criterion for a smooth solution of three-dimensional zero-diffusion Boussinesq equations. With the aid of Littlewood-Paley decomposition together with the energy methods, it is proved that if the pressure satisfies the following condition on margin Besov spaces,π(x,t)∈L2/(2+r)(0,T;B˙∞,∞r)forr=±1,then the smooth solution can be continually extended to the interval(0,T⁎)for someT⁎>T. The findings extend largely the previous results.

Local existence and blow-up criterion for the Boussinesq equations

1997 ◽  
Vol 127 (5) ◽  
pp. 935-946 ◽  
Author(s):  
Dongho Chae ◽  
Hee-Seok Nam
Keyword(s):  
Global Existence ◽  
External Force ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Local Existence ◽  
Boussinesq Equations ◽  
Passive Scalar ◽  
Smooth Solutions ◽  
Local Existence And Uniqueness ◽  
Blow Up Criterion

SynopsisIn this paper, we prove local existence and uniqueness of smooth solutions of the Boussinesq equations. We also obtain a blow-up criterion for these smooth solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

scholarly journals Blow-up criterion for the density dependent inviscid Boussinesq equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Li Li ◽  
Yanping Zhou
Keyword(s):  
Velocity Field ◽  
Energy Method ◽  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Smooth Solutions ◽  
Density Dependent ◽  
Priori Estimates ◽  
Blow Up Criterion

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.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
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.

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

2020 ◽  
Vol 17 (03) ◽  
pp. 501-557
Author(s):  
Hao Li ◽  
Yachun Li
Keyword(s):  
Blow Up ◽  
Initial Density ◽  
Local Existence ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Hydrodynamic Equations ◽  
Open Set ◽  
The Cauchy Problem ◽  
Local Strong Solutions ◽  
Blow Up Criterion

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

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