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 p in the homogeneous Besov space ?̇?0 ∞,∞ .


Introduction
This paper is devoted to establish a blow-up criterion of weak solutions to the Cauchy problem for 3-dimensional Boussinesq equations: ∇ ⋅  = 0, = 0 :  =  0 () , = 0 () , where  is the velocity,  is the pressure, and  is the small temperature deviations which depends on the density. ≥ 0 is the viscosity, ] ≥ 0 is called the molecular diffusivity, and  3 = (0, 0, 1)  .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  0 is identically zero (or constant) and  = 0, then (1)-( 4) reduces to the classical incompressible Euler equation:   +  ⋅ ∇ + ∇ = 0, ∇ ⋅  = 0,  (, )| =0 =  0 () . (5) 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 [1].It states that the control of the vorticity of the fluid  = curl  in  1 (0, ;  ∞ ) 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  3 the buoyancy force.For the regularity criteria of the Navier-Stokes equations, we can refer to Zhou et al. [2][3][4][5][6][7][8][9], Fan and Ozawa [10], He [11], Zhang and Chen [12], and Escauriaza et al. [13].
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][15][16][17][18][19].In contrast, in the case when  = ] = 0, the Boussinesq system exhibits vorticity intensification and the global well-posedness issue remains an unsolved challenging open problem (except if  0 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][21][22][23][24]).Hmidi and Rousset [23] proved the global wellposedness for the three-dimensional Euler-Boussinesq equations with axisymmetric initial data without swirl.Danchin and Paicu [20] 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.
The paper is organized as follows.We first state some important inequalities in Section 2. We will prove Theorem 1 in Section 3.