Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 539278, 9 pages

doi:10.1155/2012/539278

Research Article

Blow-Up Criteria for Three-Dimensional Boussinesq Equations in Triebel-Lizorkin Spaces

Minglei Zang

School of Mathematics and Information Science, Yantai University, Yantai 264005, China

Received 28 September 2012; Accepted 30 October 2012

Academic Editor: Hua Su

Copyright © 2012 Minglei Zang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We establish a new blow-up criteria for solution of the three-dimensional Boussinesq equations in Triebel-Lizorkin spaces by using Littlewood-Paley decomposition.

1. Introduction and Main Results

In this paper, we consider the regularity of the following three-dimensional incompressible Boussinesq equations:

where denotes the fluid velocity vector field, is the scalar pressure, is the scalar temperature, is the constant kinematic viscosity, is the thermal diffusivity, and , while and are the given initial velocity and initial temperature, respectively, with . Boussinesq systems are widely used to model the dynamics of the ocean or the atmosphere. They arise from the density-dependent fluid equations by using the so-called Boussinesq approximation which consists in neglecting the density dependence in all the terms but the one involving the gravity. This approximation can be justified from compressible fluid equations by a simultaneous low Mach number/Froude number limit; we refer to [1] for a rigorous justification. It is well known that the question of global existence or finite-time blow-up of smooth solutions for the 3D incompressible Boussinesq equations. This challenging problem has attracted significant attention. Therefore, it is interesting to study the blow-up criterion of the solutions for system (1.1).

Recently, Fan and Zhou [2] and Ishimura and Morimoto [3] proved the following blow-up criterion, respectively:

Subsequently, Qiu et al. [4] obtained Serrin-type regularity condition for the three-dimensional Boussinesq equations under the incompressibility condition. Furthermore, Xu et al. [5] obtained the similar regularity criteria of smooth solution for the 3D Boussinesq equations in the Morrey-Campanato space.

Our purpose in this paper is to establish a blow-up criteria of smooth solution for the three-dimensional Boussinesq equations under the incompressibility condition in Triebel-Lizorkin spaces.

Now we state our main results as follows.

Theorem 1.1. Let , be the smooth solution to the problem (1.1) with the initial data for . If the solution satisfies the following condition

then the solution can be extended smoothly beyond .

Corollary 1.2. Let , be the smooth solution to the problem (1.1) with the initial data for . If the solution satisfies the following condition

then the solution can be extended smoothly beyond .

Remark 1.3. By Corollary 1.2, we can see that our main result is an improvement of (1.2).

2. Preliminaries and Lemmas

The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition. So, we first introduce the Littlewood-Paley decomposition and Triebel-Lizorkin spaces.

Let be the Schwartz class of rapidly decreasing function. Given , its Fourier transform is definedby

Let be a couple of smooth functions valued in such that is supported in the ball , is supported in the shell , and

Denoting , , and , we define the dyadic blocks as

Definition 2.1. Let be the space of temperate distribution such that

The formal equality

holds in and is called the homogeneous Littlewood-Paley decomposition. It has nice properties of quasi-orthogonality

Let us now define the homogeneous Besov spaces and Triebel-Lizorkin spaces; we refer to [6, 7] for more detailed properties.

Definition 2.2. Letting , the homogeneous Besov space is defined by

Here

and denotes the dual space of .

Definition 2.3. Let ,  ,  and  , and for ,  ,  and  , the homogeneous Triebel-Lizorkin space is defined by

Here

for and , the space is defined by means of Carleson measures which is not treated in this paper. Notice that by Minkowski’s inequality, we have the following inclusions:

Also it is well known that

Throughout the proof of Theorem 1.1 in Section 3, we will use the following interpolation inequality frequently:

Lemma 2.4. Let . Then there exists a constant independent of such that for

Remark 2.5. From the above Beinstein estimate, we easily know that

3. Proofs of the Main Results

In this section, we prove Theorem 1.1. For simplicity, without loss of generality, we assume .

Proof of Theorem 1.1. Differentiating the first equation and the second equation of (1.1) with respect to , and multiplying the resulting equations by and , respectively, then by integrating by parts over we get

Noting the incompressibility condition , since

then the above equations (3.1) can be rewritten as

Adding up (3.3), then we have

Firstly, for the third term , by Hölder’s inequality and Young’s inequality, we get

The other terms are bounded similarly. For simplicity, we detail the term . Using the Littlewood-Paley decomposition (2.5), we decompose as follows:

Here is a positive integer to be chosen later. Plugging (3.6) into produces that

For , using the Hölder inequality, (2.12), and (2.15), we obtain that

For , from the Hölder inequality and (2.15), it follows that

Here denotes the conjugate exponent of . Since by the Gagliardo-Nirenberg inequality and the Young inequality, we have

For , from the Hölder and Young inequalities, (2.12), (2.15), and Gagliardo-Nirenberg inequality, we have

Plugging (3.8), (3.10), and (3.11) into (3.7) yields

Similarly, we also obtain the estimate

Putting (3.5), (3.12), and (3.13) into (3.4) yields

Now we take in (3.14) such that

that is,

Then (3.14) implies that

Applying the Gronwall inequality twice, we have

for all . This completes the proof of Theorem 1.1.

Proof of Corollary 1.2. In Theorem 1.1, taking , and combining (2.12) with the classical Riesz transformation is bounded in , we can prove it.

References

E. Feireisl and A. Novotný, “The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system,” Journal of Mathematical Fluid Mechanics, vol. 11, no. 2, pp. 274–302, 2009.

J. Fan and Y. Zhou, “A note on regularity criterion for the 3D Boussinesq system with partial viscosity,” Applied Mathematics Letters, vol. 22, no. 5, pp. 802–805, 2009.

N. Ishimura and H. Morimoto, “Remarks on the blow-up criterion for the 3-D Boussinesq equations,” Mathematical Models & Methods in Applied Sciences, vol. 9, no. 9, pp. 1323–1332, 1999.

H. Qiu, Y. Du, and Z. Yao, “Serrin-type blow-up criteria for 3D Boussinesq equations,” Applicable Analysis, vol. 89, no. 10, pp. 1603–1613, 2010.

F. Xu, Q. Zhang, and X. Zheng, “Regularity Criteria of the 3D Boussinesq Equations in the Morrey-Campanato Space,” Acta Applicandae Mathematicae, vol. 121, pp. 231–240, 2012.

H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983.

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, New York, NY, USA, 1976.

J. Y. Chemin, Perfect Incompressible Fluids, vol. 14 of Oxford Lectures Series in Mathematics and its Applications, Oxford Science Publications, Oxford, UK, 1998.