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 case = 0, (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][2][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][7][8][9][10][11][12][13][14][15][16] and so forth. Since the convective terms (u⋅∇)u 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.
The Scientific World Journal then the solution (u, ) can be extended after time = .

Remark 2. The Osgood type condition (4) is weaker than (3).
Notice that, for ∈ [2, ∞), 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.

Preliminaries
Let S(R 3 ) be the Schwartz class of rapidly decreasing functions. For ∈ S(R 3 ), its Fourier transform F =̂is defined bŷ( Let us choose a nonnegative radial function ∈ S(R 3 ) such that and let For ∈ Z, the Littlewood-Paley projection operators anḋ Δ are, respectively, defined by = * ,Δ = * .
Observe thatΔ = − −1 . Also, it is easy to check that if in the 2 sense. By telescoping the series, we thus have the following Littlewood-Paley decomposition: for all ∈ 2 (R 3 ), where the summation is the 2 sense. Notice thatΔ then from Young's inequality, it readily follows thaṫ where 1 ⩽ ⩽ ⩽ ∞ and is an absolute constant independent of and .

Proof of Theorem 1
This section is devoted to proving Theorem 1. From standard continuity arguments, we need to only provide the uniform 1 bounds of the solution (u, ).
Taking the inner products of (1) 1 with −Δu, (1) 2 with −Δ , 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 The Scientific World Journal 3 where is positive integral to be determined later on. Plugging (20) into , we see that For 1 , we dominate as For 2 , we have Finally, for 3 , we estimate as (by (13) and Gagliardo-Nireberg inequality) (17)) .