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Discrete Dynamics in Nature and Society
Volume 2012, Article ID 539278, 9 pages
doi:10.1155/2012/539278
Research Article
Blow-Up Criteria for Three-Dimensional
Boussinesq Equations in Triebel-Lizorkin Spaces
Minglei Zang
School of Mathematics and Information Science, Yantai University, Yantai 264005, China
Correspondence should be addressed to Minglei Zang, mingleizang@126.com
Received 28 September 2012; Accepted 30 October 2012
Academic Editor: Hua Su
Copyright q 2012 Minglei Zang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We establish a new blow-up criteria for solution of the three-dimensional Boussinesq equations in
Triebel-Lizorkin spaces by using Littlewood-Paley decomposition.
1. Introduction and Main Results
In this paper, we consider the regularity of the following three-dimensional incompressible
Boussinesq equations:
ut − μΔu u · ∇u ∇P θe3 ,
x, t ∈ R3 × 0, ∞,
θt − κΔθ u · ∇θ 0,
∇ · u 0,
ux, 0 u0 ,
1.1
θx, 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, θx, t is the scalar temperature, μ > 0 is the constant kinematic viscosity, κ > 0
is the thermal diffusivity, and e3 0, 0, 1T , 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
2
Discrete Dynamics in Nature and Society
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:
0
R3 ,
curl u ∈ L1 0, T ; Ḃ∞,∞
1.2
∇u ∈ L1 0, T ; L∞ R3 .
1.3
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 ∇ · u0 0 in
Triebel-Lizorkin spaces.
Now we state our main results as follows.
Theorem 1.1. Let u0 , θ0 ∈ H 1 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
∇u ∈ Lp 0, T ; Ḟq,2q/3 R3 ,
2 3
2,
p q
3
< q ∞,
2
1.4
then the solution u, θ can be extended smoothly beyond t T .
Corollary 1.2. Let u0 , θ0 ∈ H 1 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
curl u ∈ L1 0, T ; Ḃ∞,∞ R3 ,
1.5
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.
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.
Discrete Dynamics in Nature and Society
3
Let SR3 be the Schwartz class of rapidly decreasing function. Given f ∈ SR3 , its
Fourier transform Ff f is defined by
fξ
e−ix·ξ fxdx.
R3
2.1
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
χξ ϕ 2−j ξ 1,
∀ξ ∈ R3 ,
j0
2.2
ϕ 2−j ξ 1 ,
∀ξ ∈ R3 \ {0}.
j∈Z
F −1 χ, we define the dyadic blocks as
Denoting ϕj ϕ2−j ξ, h F −1 ϕ, and h
Δ̇j f ϕ 2−j D 23j
Ṡj f Δk f 23j
kj−1
R3
R3
h 2j y f x − y dy,
j ∈ Z,
2j y f x − y dy,
h
j ∈ Z.
2.3
Definition 2.1. Let Sh be the space of temperate distribution u such that
lim Ṡj f 0,
j → −∞
in S .
2.4
The formal equality
f
Δ̇j f
j∈Z
2.5
holds in Sh and is called the homogeneous Littlewood-Paley decomposition. It has nice
properties of quasi-orthogonality
Δ̇j Δ̇q f ≡ 0,
j − q
2.
2.6
Let us now define the homogeneous Besov spaces and Triebel-Lizorkin spaces; we
refer to 6, 7 for more detailed properties.
s
Definition 2.2. Letting s ∈ R, p, q ∈ 1, ∞, the homogeneous Besov space Ḃp,q
is defined by
s
Ḃp,q
f ∈ Z R3 | f Ḃp,q
<∞ .
s
2.7
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Here
f s Ḃp,q
⎧⎛
⎞1/q
⎪
∞
⎪
⎪
q
⎪
⎨⎝
2jsq Δ̇j f p ⎠ ,
q < ∞,
⎪
⎪
js ⎪
⎪
⎩sup2 Δ̇j f p ,
q ∞,
j−∞
2.8
j∈Z
0, ∀α ∈ N3
and Z R3 denotes the dual space of ZR3 {f ∈ SR3 | Dα f0
multi-index}.
Definition 2.3. Let s ∈ R, p ∈ 1, ∞, and q ∈ 1, ∞, and for s ∈ R, p ∞, and q ∞, the
s
is defined by
homogeneous Triebel-Lizorkin space Ḟp,q
s
Ḟp,q
f ∈ Z R3 | f Ḟp,q
<∞ .
s
2.9
Here
f s
Ḟp,q
⎧
⎞1/q ⎛
⎪
∞
⎪
⎪
q
jsq
⎪⎝
⎪
2 Δ̇j f ⎠ ,
⎪
⎨
j−∞
p
L
⎪
⎪
⎪
⎪
js
⎪
⎪sup2 Δ̇j f ,
⎩
j∈Z
q < ∞,
2.10
q ∞,
p
s
for p ∞ and q ∈ 1, ∞, the space Ḟp,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:
s
s
⊂ Ḟp,q
,
Ḃp,q
if q p,
s
s
Ḟp,q
⊂ Ḃp,q
,
if q p.
2.11
Also it is well known that
s
s
Ḃp,p
Ḟp,p
,
0
0
L∞ ⊂ Ḃ∞,∞
Ḟ∞,∞
,
s
s
Ḃ2,2
Ḟ2,2
Ḣ s .
2.12
Throughout the proof of Theorem 1.1 in Section 3, we will use the following interpolation inequality frequently:
3/2−3/q
f q Cf 3/q−1/2
C∇f L2
,
L
L2
2 q 6, f ∈ L2 R3 ∩ Ḣ 1 R3 .
2.13
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5
Lemma 2.4. Let k ∈ N. Then there exists a constant C independent of f, j such that for 1 p q∞
sup∂α Δ̇j f q C2jk3j1/p−1/q Δ̇j f p .
2.14
|α|k
Remark 2.5. From the above Beinstein estimate, we easily know that
Δ̇j f C23j1/p−1/q Δ̇j f .
q
p
2.15
3. 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 xk 1 k 3, and multiplying the resulting equations by ∂u/∂xk ∂k u and
∂θ/∂xk ∂k θ, respectively, then by integrating by parts over R3 we get
1 d
∂k u2L2 ∇∂k u2L2 −
2 dt
∂k u · ∇u · ∂k u dx −
1 d
∂k θ2L2 ∇∂k θ2L2 −
2 dt
∂k ∇P · ∂k u dx ∂k θe3 ∂k u dx,
∂k u · ∇θ · ∂k θ dx.
3.1
Noting the incompressibility condition ∇ · u 0, since
∂k u · ∇u · ∂k u dx ∂k u · ∇u · ∂k u dx,
∂k ∇P · ∂k u dx 0,
3.2
∂k u · ∇θ · ∂k θ dx ∂k u · ∇θ · ∂k θ dx,
then the above equations 3.1 can be rewritten as
1 d
2
2
∂k uL2 ∇∂k uL2 − ∂k u · ∇u · ∂k u dx ∂k θe3 ∂k u dx,
2 dt
1 d
2
2
∂k θL2 ∇∂k θL2 − ∂k u · ∇θ · ∂k θ dx.
2 dt
3.3
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Adding up 3.3, then we have
1 d
∂k u2L2 ∂k θ2L2 ∇∂k u2L2 ∇∂k θ2L2
2 dt
− ∂k u · ∇u · ∂k u dx − ∂k u · ∇θ · ∂k θ dx ∂k θe3 · ∂k u dx
3.4
I1 I 2 I 3 .
Firstly, for the third term I3 , by Hölder’s inequality and Young’s inequality, we get
I3 ∂k θe3 · ∂k u dx 1
1
∇θ2L2 ∇u2L2 .
2
2
3.5
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:
∇u Δ̇j ∇u j∈Z
Δ̇j ∇u j<−N
jN
Δ̇j ∇u j−N
Δ̇j ∇u.
3.6
j>N
Here N is a positive integer to be chosen later. Plugging 3.6 into I2 produces that
I2 Δ̇j ∇u
|∇θ|2 dx
R3
j<−N
jN
R3
j−N
R3
j>N
Δ̇j ∇u
|∇θ|2 dx
3.7
Δ̇j ∇u
|∇θ|2 dx
≡ I21 I22 I23 .
For I21 , using the Hölder inequality, 2.12, and 2.15, we obtain that
I21 ∇θ2L2
Δ̇j ∇u
j<−N
C∇θ2L2
j<−N
L∞
23/2j Δ̇j ∇uL2
C2−3/2N ∇uL2 ∇θ2L2
3/2
C2−3/2N ∇u2L2 ∇θ2L2
.
3.8
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For I22 , from the Hölder inequality and 2.15, it follows that
jN
I22 j−N
R3
|∇θ|2 Δ̇j ∇u
dx ⎛
R3
|∇θ|
CN
jN
2⎝
R3
|∇θ|2
jN
Δ̇j ∇u
dx
j−N
⎞3/2q
2q/3
Δ̇j ∇u
⎠
N 1−3/2q dx
3.9
j−N
2q−3/2q
⎛
R3
jN
|∇θ|2 ⎝
⎞3/2q
Δ̇j ∇u
2q/3 ⎠
dx
j−N
CN 2q−3/2q ∇θ2L2q ∇uḞ 0
q,2q/3
.
Here q denotes the conjugate exponent of q. Since 2q > 3 by the Gagliardo-Nirenberg
inequality and the Young inequality, we have
3/q
0
∇2 θ 2 ∇uḞq,2q/3
2q−3/q I22 CN 2q−3/2q ∇θL2
L
2
1
p
∇2 θ 2 CN∇θ2L2 ∇uḞ 0
.
L
2
q,2q/3
3.10
For I23 , from the Hölder and Young inequalities, 2.12, 2.15, and Gagliardo-Nirenberg
inequality, we have
I23
Δ̇j ∇u
|∇θ|2 dx
j>N
R3
∇θ2L3
Δ̇j ∇u 3
L
j>N
C∇θ2L3
j>N
2j/2 Δ̇j ∇uL2 ,
⎛
⎞1/2 ⎛
⎞1/2
2
2 ⎝ −j ⎠ ⎝ 2j C ∇θL2 ∇ θ 2
2
2 Δ̇j ∇u2 ⎠
L
j>N
j>N
C2−N/2 ∇θL2 ∇2 θ 2 ∇2 u
L
2
2 2 2 2
−N/2
C2
∇θL2 ∇ θ 2 ∇ u 2 .
L
L
3.11
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Plugging 3.8, 3.10, and 3.11 into 3.7 yields
2
3/2 1 p
I2 C2−3/2N ∇u22 ∇θ22
∇2 θ CN∇θ2L2 ∇uḞ 0
2
2
q,2q/3
2 2 C2−N/2 ∇θL2 ∇2 θ 2 ∇2 u 2 .
L
3.12
L
Similarly, we also obtain the estimate
1
p
2 2
∇ u CN∇u2L2 ∇uḞ 0
2
2
q,2q/3
2
∇2 u 2 .
3/2
I1 C2−3/2N ∇u22 ∇θ22
−N/2
C2
2
∇uL2 ∇2 θ 2
L
3.13
L
Putting 3.5, 3.12, and 3.13 into 3.4 yields
2
1 d
1
∇u, ∇θ2L2 ∇2 u, ∇2 θ 2
L
2 dt
2
3/2
p
C2−N ∇u, ∇θ22
CN∇u, ∇θ2L2 ∇uḞ 0
q,2q/3
3.14
1/2 2
2
C2−N ∇u, ∇θ2L2
∇ u, ∇2 θ 2 .
L
Now we take N in 3.14 such that
C2−N ∇u, ∇θ2L2 1
,
16
3.15
that is,
NC
log e ∇u, ∇θ2L2
log 2
3.16
4.
Then 3.14 implies that
d
p
.
∇u, ∇θ2L2 C C log e ∇u, ∇θ2L2 ∇u, ∇θ2L2 ∇uḞ 0
dt
q,2q/3
Applying the Gronwall inequality twice, we have
T
∇u, ∇θ2L2
C exp exp C
0
3.17
p
∇uḞ 0
for all t ∈ 0, T . This completes the proof of Theorem 1.1.
q,2q/3
sds
,
3.18
Discrete Dynamics in Nature and Society
9
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∞,∞ R3 , we can prove it.
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