Blow-Up Criteria for Three-Dimensional Boussinesq Equations in Triebel-Lizorkin Spaces

We establish a new blow-up criteria for solution of the three-dimensional Boussinesq equations in Triebel-Lizorkin spaces by using Littlewood-Paley decomposition.


Introduction and Main Results
In this paper, we consider the regularity of the following three-dimensional incompressible Boussinesq equations: where u u 1 x, t , u 2 x, t , u 3 x, t denotes the fluid velocity vector field, P P x, t is the scalar pressure, θ x, t is the scalar temperature, μ > 0 is the constant kinematic viscosity, κ > 0 is the thermal diffusivity, and e 3 0, 0, 1 T , while u 0 and θ 0 are the given initial velocity and initial temperature, respectively, with ∇•u 0 0. Boussinesq systems are widely used to model the dynamics of the ocean or the atmosphere.They arise from the density-dependent fluid equations by using the so-called Boussinesq approximation which consists in neglecting the density dependence in all the terms but the one involving the gravity.This approximation can be justified from compressible fluid equations by a simultaneous low Mach number/Froude number limit; we refer to 1 for a rigorous justification.It is well known that the question of global existence or finite-time blow-up of smooth solutions for the 3D incompressible Boussinesq equations.This challenging problem has attracted significant attention.Therefore, it is interesting to study the blow-up criterion of the solutions for system 1.1 .
Recently, Fan and Zhou 2 and Ishimura and Morimoto 3 proved the following blow-up criterion, respectively: Subsequently, Qiu et al. 4 obtained Serrin-type regularity condition for the threedimensional Boussinesq equations under the incompressibility condition.Furthermore, Xu et al. 5 obtained the similar regularity criteria of smooth solution for the 3D Boussinesq equations in the Morrey-Campanato space.Our purpose in this paper is to establish a blow-up criteria of smooth solution for the three-dimensional Boussinesq equations under the incompressibility condition ∇ • u 0 0 in Triebel-Lizorkin spaces.Now we state our main results as follows.
Theorem 1.1.Let u 0 , θ 0 ∈ H 1 R 3 , u •, t , θ •, t be the smooth solution to the problem 1.1 with the initial data u 0 , θ 0 for 0 t < T. If the solution u satisfies the following condition ∇u ∈ L p 0, T; Ḟq, 2q/3 R 3 , 2 p then the solution u, θ can be extended smoothly beyond t T .
Corollary 1.2.Let u 0 , θ 0 ∈ H 1 R 3 , u •, t , θ •, t be the smooth solution to the problem 1.1 with the initial data u 0 , θ 0 for 0 t < T. If the solution u satisfies the following condition then the solution u, θ can be extended smoothly beyond t T .
Remark 1.3.By Corollary 1.2, we can see that our main result is an improvement of 1.2 .

Preliminaries and Lemmas
The proof of the results presented in this paper is based on a dyadic partition of unity in Fourier variables, the so-called homogeneous Littlewood-Paley decomposition.So, we first introduce the Littlewood-Paley decomposition and Triebel-Lizorkin spaces.
Let S R 3 be the Schwartz class of rapidly decreasing function.Given f ∈ S R 3 , its Fourier transform Ff f is defined by Let χ, ϕ be a couple of smooth functions valued in 0, 1 such that χ is supported in the ball {ξ ∈ R 3 : |ξ| 4/3}, ϕ is supported in the shell {ξ ∈ R 3 : 3/4 |ξ| 8/3}, and

2.3
Definition 2.1.Let S h be the space of temperate distribution u such that lim j → −∞ Ṡj f 0, in S .

2.4
The formal equality holds in S h and is called the homogeneous Littlewood-Paley decomposition.It has nice properties of quasi-orthogonality

2.6
Let us now define the homogeneous Besov spaces and Triebel-Lizorkin spaces; we refer to 6, 7 for more detailed properties.Definition 2.2.Letting s ∈ R, p, q ∈ 1, ∞ , the homogeneous Besov space Ḃs p,q is defined by Here ∞ , and q ∈ 1, ∞ , and for s ∈ R, p ∞, and q ∞, the homogeneous Triebel-Lizorkin space Ḟs p,q is defined by Here for p ∞ and q ∈ 1, ∞ , the space Ḟs p,q is defined by means of Carleson measures which is not treated in this paper.Notice that by Minkowski's inequality, we have the following inclusions:

2.11
Also it is well known that Ḃs Ḣs .

2.12
Throughout the proof of Theorem 1.1 in Section 3, we will use the following interpolation inequality frequently:

Proofs of the Main Results
In this section, we prove Theorem 1.1.For simplicity, without loss of generality, we assume μ κ 1.
Proof of Theorem 1.1.Differentiating the first equation and the second equation of 1.1 with respect to x k 1 k 3 , and multiplying the resulting equations by ∂u/∂x k ∂ k u and ∂θ/∂x k ∂ k θ, respectively, then by integrating by parts over R 3 we get 1 2

3.1
Noting the incompressibility condition ∇ • u 0, since then the above equations 3.1 can be rewritten as

3.4
Firstly, for the third term I 3 , by Hölder's inequality and Young's inequality, we get The other terms are bounded similarly.For simplicity, we detail the term I 2 .Using the Littlewood-Paley decomposition 2.5 , we decompose ∇u as follows: Here N is a positive integer to be chosen later.Plugging 3.6 into I 2 produces that

3.9
Here q denotes the conjugate exponent of q.Since 2q > 3 by the Gagliardo-Nirenberg inequality and the Young inequality, we have q, 2q/3 .

3.12
Similarly, we also obtain the estimate
Proof of Corollary 1.2.In Theorem 1.1, taking p 1, and combining 2.12 with the classical Riesz transformation is bounded in Ḃ∞,∞ R 3 , we can prove it.
for all t ∈ 0, T .This completes the proof of Theorem 1.1.