Some Regularity Criteria for the 3D Boussinesq Equations in the Class

We consider the three-dimensional Boussinesq equations and obtain some regularity criteria via the velocity gradient (or the vorticity, or the deformation tensor) and the temperature gradient. 1. Introduction Consider the following three-dimensional (3D) Boussinesq equations: Here, is the fluid velocity, is a scalar pressure, and is the temperature, while and are the prescribed initial velocity and temperature, respectively. When , (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are challenging open problems; see [1–3]. Initialed by [4, 5], there have been a lot of literatures devoted to finding sufficient conditions to ensure the smoothness of the solutions; see [6–18] and so forth. Since the convective terms are similar in Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1); see [19–23] and so forth. Motivated by [7], we will consider the regularity criteria for (1) and extend the result of [7] to the case of Boussinesq equations. Before stating the precise result, let us recall the weak formulation of (1). Definition 1. Let , , and . A measurable pair is said to be a weak solution of (1) in , provided that (1) , , ; ;(2)(1)1,2,3 are satisfied in the sense of distributions;(3)the energy inequality ？for all . Now, our main result reads the following. Theorem 2. Let with in the sense of distributions, . Suppose that is a weak solution of (1) in , and then the solution . Due to the divergence-free condition of the fluid velocity , we have Thus, Here, is the Riesz transformation. Using (5), we can deduce easily from Theorem 2 the following. Corollary 3. Let with in the sense of distributions, . Suppose that is a weak solution of (1) in , and or then the solution . Here, is the vorticity and + is the deformation tensor (the symmetric tensor of the rate of strain). The rest of this paper is organized as follows. In Section 2, we recall the definition of Besov spaces and the related interpolation inequalities. Section 3 is devoted to proving Theorem 2. 2. Preliminaries We first introduce the Littlewood-Paley decomposition. Let be the Schwartz class of rapidly decreasing functions. For , its Fourier transform is defined as 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 have the following Littlewood-Paley