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An Osgood Type Regularity Criterion for the 3D Boussinesq Equations
We consider the three-dimensional Boussinesq equations, and obtain an Osgood type regularity criterion in terms of the velocity gradient.
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.  obtained Serrin type regularity condition:
The extension to the multiplier spaces was established by the same authors in . For the Besov-type regularity criterion, Fan and Zhou  and Ishimura and Morimoto  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  for generalized models.
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 .
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  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.
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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