Some Classes of Function Spaces, Their Properties, and Their Applications 2014View this Special Issue
Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations
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 pressure in the homogeneous Besov space .
This paper is devoted to establish a blow-up criterion of weak solutions to the Cauchy problem for 3-dimensional Boussinesq equations: where is the velocity, is the pressure, and is the small temperature deviations which depends on the density. is the viscosity, is called the molecular diffusivity, and . The above systems describe the evolution of the velocity field for a 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.
When the initial density is identically zero (or constant) and , then 1–4 reduces to the classical incompressible Euler equation: From the investigation of 5, we cannot expect to have a better theory for the Boussinesq system than that of the Euler equation. For the Euler equation, a well-known criterion for the existence of global smooth solutions is the Beale-Kato-Majda criterion . It states that the control of the vorticity of the fluid in is sufficient to get the global well posedness.
The Boussinesq equations 1–4 are of relevance to study a number of models coming from atmospheric or oceanographic turbulence where rotation and stratification play an important role. The scalar function may for instance represent temperature variation in a gravity field and the buoyancy force. For the regularity criteria of the Navier-Stokes equations, we can refer to Zhou et al. [2–9], Fan and Ozawa , He , Zhang and Chen , and Escauriaza et al. .
From the mathematical point of view, the global well posedness for two-dimensional Boussinesq equations which has recently drawn much attention seems to be in a satisfactory state. More precisely, global well posedness has been shown in various function spaces and for different viscosities; we refer, for example, to [14–19]. In contrast, in the case when , the Boussinesq system exhibits vorticity intensification and the global well-posedness issue remains an unsolved challenging open problem (except if is a constant, of course) which may be formally compared to the similar problem for the three-dimensional axisymmetric Euler equations with swirl.
In the three-dimensional case, there are only few results (see [20–24]). Hmidi and Rousset  proved the global well-posedness for the three-dimensional Euler-Boussinesq equations with axisymmetric initial data without swirl. Danchin and Paicu  obtained a global existence and uniqueness result for small data in Lorentz space.
Our purpose of this paper is to obtain a blow-up criterion of weak solutions in terms of Besov space.
Now, we state our result as follows.
Theorem 1. Assume that with in . Assume that the pressure satisfies the condition then the solution can be extended smoothly only up to .
Throughout this paper, we use the following usual notations. denotes the Lebesgue space and denotes the standard Sobolev space. denotes the space of bounded mean oscillations. is the homogeneous Besov space, where , .
Lemma 2. There exists a uniform positive constant , such that hold for all vectors with .
Lemma 3. From 1, one has
Lemma 4. Assume that ; one has the commutator estimate due to Kato and Ponce : with , .
Lemma 5 (the Gagliardo-Nirenberg inequality). Consider
3. Proof of Theorem 1
Proof of Theorem 1. Multiplying 1 by , using 3, and integrating in , we derive
Multiplying 2 by , using 3, and integrating in , we obtain
Combining 14 and 15, using the Gronwall inequality, we deduce that
Multiplying 1 by , using 3 and 7, and integrating in , we derive
Multiplying 2 by , using 3, and integrating in , we arrive at
Combining 18 and 19, using 9 and 16, we derive that
Choosing and setting
where is a small enough constant, such that
Next, we want to estimate the -norm of and .
Multiplying 1 by , integrating in , and using 3 and 11, we derive that that is, Multiplying 2 by , integrating in , and using 3 and 11, we derive that that is, Combining 26 and 28, using 16, we deduce that is, which implies that Last, we will estimate the -norm and -norm of and and use the operator to derive our goal.
Applying to 1 and then multiplying 1 with , we deduce that is, Similarly, applying to 2 and multiplying 2 by , we derive that is, Combining 33 and 35, we have that is, Choosing small enough, using 16, 23, and 24, we conclude that We complete the proof.
Conflict of Interests
The authors declare that they have no competing interests.
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final paper.
All authors thank the reviewers for their suggestions; this paper was in part supported by the NNSF of China (nos. 11326154 and U1304101), the NSFC Projects of Education Department of Henan Province (no. 14B110029), and the Program for Science and Technology Innovation Talents in University of Henan Province (no. 13HASTIT040).
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