The Scientific World Journal

Volume 2014 (2014), Article ID 563084, 4 pages

http://dx.doi.org/10.1155/2014/563084

Research Article

An Osgood Type Regularity Criterion for the 3D Boussinesq Equations

Qiang Wu,1,2 Lin Hu,² and Guili Liu³

¹School of Resources and Safety Engineering, Central South University, Changsha, Hunan 410075, China

²Jiangxi University of Science and Technology, Ganzhou, Jiangxi 341000, China

³Department of Basic Teaching, Harbin Finance University, Harbin 150030, Heilongjiang, China

Received 2 November 2013; Accepted 2 February 2014; Published 11 March 2014

Academic Editors: D. Baleanu and H. Jafari

Copyright © 2014 Qiang Wu et al. 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 consider the three-dimensional Boussinesq equations, and obtain an Osgood type regularity criterion in terms of the velocity gradient.

1. Introduction

In this paper, we consider the following three-dimensional (3D) Boussinesq equations with the incompressibility condition:

where is the fluid velocity, is a scalar pressure, and is the scalar temperature, while and are the prescribed initial velocity and temperature, respectively, with .

In case , (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are important open problems; see [1–3]. Starting with [4, 5], there have been a lot of literatures devoted to finding sufficient conditions (which now are called regularity criteria) to ensure the smoothness of the solutions; see [6–16] and so forth. Since the convective terms are the same in the Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1). In particular, Qiu et al. [17] obtained Serrin type regularity condition:

The extension to the multiplier spaces was established by the same authors in [18]. For the Besov-type regularity criterion, Fan and Zhou [19] and Ishimura and Morimoto [20] showed the following regularity conditions:

Zhang [21, 22] then considers the regularity criterion in terms of the pressure or its gradient. The readers are also referred to [23] for generalized models.

Motivated by [24–26], we will improve (3) as in the following.

Theorem 1. Let . Assume that is the smooth solution to (1) with the initial data for . If

then the solution can be extended after time . Here, denotes the Fourier localization operator and .

Remark 2. The Osgood type condition (4) is weaker than (3). Notice that, for , we have

The rest of this paper is organized as follows. In Section 2, we recall the definition of Besov spaces and some interpolation inequalities. Section 3 is devoted to proving Theorem 1.

2. Preliminaries

Let be the Schwartz class of rapidly decreasing functions. For , its Fourier transform is defined by

Let us choose a nonnegative radial function such that

and let

For , the Littlewood-Paley projection operators and are, respectively, defined by

Observe that . Also, it is easy to check that if , then

in the sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition:

for all , where the summation is the sense. Notice that

then from Young's inequality, it readily follows that

where and is an absolute constant independent of and .

Let , , ; the homogeneous Besov space is defined by the full-dyadic decomposition such that

where

and is the dual space of

Also, it is well known that

We refer to [27] for more detailed properties.

3. Proof of Theorem 1

This section is devoted to proving Theorem 1. From standard continuity arguments, we need to only provide the uniform bounds of the solution .

Taking the inner products of (1) with , (1) with , we obtain by adding together that

For , we use Hölder’s inequality to get

For , applying the Littilewood-Paley decomposition as in (11), we get

where is positive integral to be determined later on. Plugging (20) into , we see that

For , we dominate as

For , we have

Finally, for , we estimate as

Gathering (22), (23), and (24) together and plugging them into (21), we deduce

Substituting (19) and (25) into (18), we find

Taking

where is the largest integer smaller than and , then (26) implies that

Applying Gronwall inequality three times, we deduce

Recalling (4), we see the solution is uniformly bounded in . This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the NSF of China (no. 11326238, no. 11326138, and no. 11101101), the Science Foundation of Jiangxi Provincial Department of Education (no. GJJ13374 and no. GJJ13658), and the Youth Natural Science Foundation of Jiangxi Province (20132BAB211007).

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