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

Authors: Minglei Zang
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

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


---

## Body

## 1. Introduction and Main Results

In this paper, we consider the regularity of the following three-dimensional incompressible Boussinesq equations:(1.1)ut-μΔu+u·∇u+∇P=θe3,(x,t)∈ℝ3×(0,∞),θt-κΔθ+u·∇θ=0,∇·u=0,u(x,0)=u0,θ(x,0)=θ0,
where u=(u1(x,t),u2(x,t),u3(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 e3=(0,0,1)T, while u0 and θ0 are the given initial velocity and initial temperature, respectively, with ∇·u0=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:(1.2)curlu∈L1(0,T;B˙∞,∞0(ℝ3)),(1.3)∇u∈L1(0,T;L∞(ℝ3)).
Subsequently, Qiu et al. [4] obtained Serrin-type regularity condition for the three-dimensional 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∇·u0=0 in Triebel-Lizorkin spaces.Now we state our main results as follows.Theorem 1.1.
Let(u0,θ0)∈H1(ℝ3), (u(·,t),θ(·,t)) be the smooth solution to the problem (1.1) with the initial data (u0,θ0) for 0⩽t<T. If the solution u satisfies the following condition (1.4)∇u∈Lp(0,T;F˙q,(2q/3)(ℝ3)),2p+3q=2,32<q⩽∞,
then the solution (u,θ) can be extended smoothly beyond t=T.Corollary 1.2.
Let(u0,θ0)∈H1(ℝ3), (u(·,t),θ(·,t)) be the smooth solution to the problem (1.1) with the initial data (u0,θ0) for 0⩽t<T. If the solution u satisfies the following condition
(1.5)curlu∈L1(0,T;B˙∞,∞(ℝ3)),
then the solution (u,θ) can be extended smoothly beyond t=T.Remark 1.3.
By Corollary1.2, we can see that our main result is an improvement of (1.2).


## 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.Let𝒮(ℝ3) be the Schwartz class of rapidly decreasing function. Given f∈𝒮(ℝ3), its Fourier transform ℱf=f^ is definedby
(2.1)f^(ξ)=∫ℝ3e-ix·ξf(x)dx.
Let (χ,φ) be a couple of smooth functions valued in [0,1] such that χ is supported in the ball {ξ∈ℝ3:|ξ|⩽4/3}, φ is supported in the shell {ξ∈ℝ3:3/4⩽|ξ|⩽8/3}, and
(2.2)χ(ξ)+∑j⩾0‍φ(2-jξ)=1,∀ξ∈ℝ3,∑j∈Z‍φ(2-jξ=1),∀ξ∈ℝ3∖{0}.
Denoting φj=φ(2-jξ), h=F-1φ, and h~=F-1χ, we define the dyadic blocks as(2.3)Δ˙jf=φ(2-jD)=23j∫ℝ3h(2jy)f(x-y)dy,j∈ℤ,S˙jf=∑k⩽j-1‍Δkf=23j∫ℝ3h~(2jy)f(x-y)dy,j∈ℤ.Definition 2.1.
Let𝒮h′ be the space of temperate distribution u such that (2.4)limj→-∞S˙jf=0,in𝒮′.
The formal equality(2.5)f=∑j∈ℤ‍Δ˙jf
holds in Sh′ and is called the homogeneous Littlewood-Paley decomposition. It has nice properties of quasi-orthogonality (2.6)Δ˙jΔ˙qf≡0,|j-q|⩾2.Let us now define the homogeneous Besov spaces and Triebel-Lizorkin spaces; we refer to [6, 7] for more detailed properties.Definition 2.2.
Lettings∈ℝ,p,q∈[1,∞], the homogeneous Besov space B˙p,qs is defined by(2.7)B˙p,qs={f∈𝒵′(R3)∣∥f∥B˙p,qs<∞}.
Here
(2.8)∥f∥B˙p,qs={(∑j=-∞∞2jsq∥Δ˙jf∥pq)1/q,q<∞,supj∈ℤ2js∥Δ˙jf∥p,q=∞,
and 𝒵′(ℝ3) denotes the dual space of 𝒵(ℝ3)={f∈𝒮(ℝ3)∣Dαf^(0)=0,∀α∈ℕ3multi-index}.Definition 2.3.
Lets∈ℝ,  p∈[1,∞),  and  q∈[1,∞], and for s∈ℝ,  p=∞,  and  q=∞, the homogeneous Triebel-Lizorkin space F˙p,qs is defined by
(2.9)F˙p,qs={f∈𝒵′(ℝ3)∣∥f∥F˙p,qs<∞}.
Here
(2.10)∥f∥F˙p,qs={∥(∑j=-∞∞‍2jsq|Δ˙jf|q)1/q∥Lp,q<∞,∥supj∈ℤ2js|Δ˙jf|∥p,q=∞,
for p=∞ and q∈[1,∞), the space F˙p,qs 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)B˙p,qs⊂F˙p,qs,ifq⩽p,F˙p,qs⊂B˙p,qs,ifq⩾p.
Also it is well known that
(2.12)B˙p,ps=F˙p,ps,L∞⊂B˙∞,∞0=F˙∞,∞0,B˙2,2s=F˙2,2s=H˙s.Throughout the proof of Theorem1.1 in Section 3, we will use the following interpolation inequality frequently:
(2.13)∥f∥Lq⩽C∥f∥L23/q-1/2C∥∇f∥L23/2-3/q,2⩽q⩽6,f∈L2(ℝ3)∩H˙1(ℝ3).Lemma 2.4.
Letk∈N. Then there exists a constant C independent of f,j such that for 1⩽p⩽q⩽∞(2.14)sup|α|=k∥∂αΔ˙jf∥q⩽C2jk+3j(1/p-1/q)∥Δ˙jf∥p.Remark 2.5.
From the above Beinstein estimate, we easily know that(2.15)∥Δ˙jf∥q⩽C23j(1/p-1/q)∥Δ˙jf∥p.


## 3. Proofs of the Main Results

In this section, we prove Theorem1.1. For simplicity, without loss of generality, we assume μ=κ=1.Proof of Theorem1.1.
Differentiating the first equation and the second equation of (1.1) with respect to xk(1⩽k⩽3), and multiplying the resulting equations by ∂u/∂xk=∂ku and ∂θ/∂xk=∂kθ, respectively, then by integrating by parts over ℝ3 we get
(3.1)12ddt∥∂ku∥L22+∥∇∂ku∥L22=-∫∂k[(u·∇)u]·∂kudx-∫∂k∇P·∂kudx+∫∂k(θe3)∂kudx,12ddt∥∂kθ∥L22+∥∇∂kθ∥L22=-∫∂k[(u·∇)θ]·∂kθdx.
Noting the incompressibility condition ∇·u=0, since
(3.2)∫∂k[(u·∇)u]·∂kudx=∫(∂ku·∇)u·∂kudx,∫∂k∇P·∂kudx=0,∫∂k[(u·∇)θ]·∂kθdx=∫(∂ku·∇)θ·∂kθdx,
then the above equations (3.1) can be rewritten as(3.3)12ddt∥∂ku∥L22+∥∇∂ku∥L22=-∫(∂ku·∇)u·∂kudx+∫∂k(θe3)∂kudx,12ddt∥∂kθ∥L22+∥∇∂kθ∥L22=-∫(∂ku·∇)θ·∂kθdx.
Adding up (3.3), then we have
(3.4)12ddt(∥∂ku∥L22+∥∂kθ∥L22)+∥∇∂ku∥L22+∥∇∂kθ∥L22=-∫(∂ku·∇)u·∂kudx-∫(∂ku·∇)θ·∂kθdx+∫∂k(θe3)·∂kudx≜I1+I2+I3.
Firstly, for the third term I3, by Hölder’s inequality and Young’s inequality, we get(3.5)I3=∫∂k(θe3)·∂kudx⩽12∥∇θ∥L22+12∥∇u∥L22.
The other terms are bounded similarly. For simplicity, we detail the term I2. Using the Littlewood-Paley decomposition (2.5), we decompose ∇u as follows:
(3.6)∇u=∑j∈ℤ‍Δ˙j(∇u)=∑j<-N‍Δ˙j(∇u)+∑j=-Nj=N‍Δ˙j(∇u)+∑j>N‍Δ˙j(∇u).
Here N is a positive integer to be chosen later. Plugging (3.6) into I2 produces that
(3.7)I2=∑j<-N‍∫ℝ3‍|Δ˙j(∇u)||∇θ|2dx+∑j=-Nj=N‍∫ℝ3‍|Δ˙j(∇u)||∇θ|2dx+∑j>N‍∫ℝ3‍|Δ˙j(∇u)||∇θ|2dx≡I21+I22+I23.
For I21, using the Hölder inequality, (2.12), and (2.15), we obtain that
(3.8)I21⩽∥∇θ∥L22∑j<-N‍∥Δ˙j∇u∥L∞⩽C∥∇θ∥L22∑j<-N‍2(3/2)j∥Δ˙j∇u∥L2⩽C2-(3/2)N∥∇u∥L2∥∇θ∥L22⩽C2-(3/2)N(∥∇u∥L22+∥∇θ∥L22)3/2.
For I22, from the Hölder inequality and (2.15), it follows that
(3.9)I22=∑j=-Nj=N‍∫ℝ3‍|∇θ|2|Δ˙j(∇u)|dx=∫ℝ3‍|∇θ|2∑j=-Nj=N‍|Δ˙j(∇u)|dx⩽∫ℝ3‍|∇θ|2(∑j=-Nj=N‍|Δ˙j(∇u)|2q/3)3/2qN1-3/2qdx⩽CN(2q-3)/2q∫ℝ3‍|∇θ|2(∑j=-Nj=N‍|Δ˙j(∇u)|2q/3)3/2qdx⩽CN(2q-3)/2q∥∇θ∥L2q′2∥∇u∥F˙q,(2q/3)0.
Here q′ denotes the conjugate exponent of q. Since 2q>3 by the Gagliardo-Nirenberg inequality and the Young inequality, we have
(3.10)I22⩽CN(2q-3)/2q∥∇θ∥L2(2q-3)/q∥∇2θ∥L23/q∥∇u∥F˙q,(2q/3)0⩽12∥∇2θ∥L22+CN∥∇θ∥L22∥∇u∥F˙q,(2q/3)0p.
For I23, from the Hölder and Young inequalities, (2.12), (2.15), and Gagliardo-Nirenberg inequality, we have
(3.11)I23=∑j>N‍∫ℝ3‍|Δ˙j(∇u)||∇θ|2dx⩽∥∇θ∥L32∑j>N‍∥Δ˙j(∇u)∥L3⩽C∥∇θ∥L32∑j>N‍2(j/2)∥Δ˙j(∇u)∥L2,C⩽∥∇θ∥L2∥∇2θ∥L2(∑j>N‍2-j)1/2(∑j>N‍22j∥Δ˙j(∇u)∥22)1/2⩽C2-(N/2)∥∇θ∥L2∥∇2θ∥L2∥∇2u∥2⩽C2-(N/2)∥∇θ∥L2(∥∇2θ∥L22+∥∇2u∥L22).
Plugging (3.8), (3.10), and (3.11) into (3.7) yields
(3.12)I2⩽C2-(3/2)N(∥∇u∥22+∥∇θ∥22)3/2+12∥∇2θ∥22+CN∥∇θ∥L22∥∇u∥F˙q,(2q/3)0p+C2-N/2∥∇θ∥L2(∥∇2θ∥L22+∥∇2u∥L22).
Similarly, we also obtain the estimate
(3.13)I1⩽C2-(3/2)N(∥∇u∥22+∥∇θ∥22)3/2+12∥∇2u∥22+CN∥∇u∥L22∥∇u∥F˙q,(2q/3)0p+C2-N/2∥∇u∥L2(∥∇2θ∥L22+∥∇2u∥L22).
Putting (3.5), (3.12), and (3.13) into (3.4) yields
(3.14)12ddt∥∇u,∇θ∥L22+12∥(∇2u,∇2θ)∥L22⩽{C2-N∥(∇u,∇θ)∥22}3/2+CN∥(∇u,∇θ)∥L22∥∇u∥F˙q,(2q/3)0p+{C2-N∥(∇u,∇θ)∥L22}1/2∥(∇2u,∇2θ)∥L22.
Now we take N in (3.14) such that (3.15)C2-N∥(∇u,∇θ)∥L22⩽116,
that is,(3.16)N⩾Clog(e+∥(∇u,∇θ)∥L22)log2+4.
Then (3.14) implies that (3.17)ddt∥(∇u,∇θ)∥L22⩽C+Clog(e+∥(∇u,∇θ)∥L22)∥(∇u,∇θ)∥L22∥∇u∥F˙q,(2q/3)0p.
Applying the Gronwall inequality twice, we have(3.18)∥(∇u,∇θ)∥L22⩽Cexp{exp(C∫0T‍∥∇u∥F˙q,(2q/3)0p(s)ds)},
for all t∈(0,T). This completes the proof of Theorem 1.1.Proof of Corollary1.2.
In Theorem1.1, taking p=1, and combining (2.12) with the classical Riesz transformation is bounded in B˙∞,∞(ℝ3), we can prove it.



---

*Source: 539278-2012-11-25.xml*