# Blow-Up Criterion of Weak Solutions for the 3D Boussinesq Equations

Authors: Zhaohui Dai; Xiaosong Wang; Lingrui Zhang; Wei Hou
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

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 pressurep in the homogeneous Besov space B
̇
∞
,
∞
0.


---

## Body

## 1. Introduction

This paper is devoted to establish a blow-up criterion of weak solutions to the Cauchy problem for 3-dimensional Boussinesq equations:(1)
u
t
+
u
·
∇
u
-
η
Δ
u
+
∇
p
=
θ
e
3
,
(2)
θ
t
+
u
·
∇
θ
-
ν
Δ
θ
=
0
,
(3)
∇
·
u
=
0
,
(4)
t
=
0
:
u
=
u
0
x
,
θ
=
θ
0
x
,
where u is the velocity, p is the pressure, and θ is the small temperature deviations which depends on the density. η
≥
0 is the viscosity, ν
≥
0 is called the molecular diffusivity, and e
3
=
(
0,0
,
1
)
T. The above systems describe the evolution of the velocity field u 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 densityθ
0 is identically zero (or constant) and η
=
0, then 1–4 reduces to the classical incompressible Euler equation:
(5)
u
t
+
u
·
∇
u
+
∇
p
=
0
,
∇
·
u
=
0
,
u
x
,
t
t
=
0
=
u
0
x
.
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 ω
=
curl
⁡
u in L
1
(
0
,
T
;
L
∞
) is sufficient to get the global well posedness.The Boussinesq equations1–4 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 θ may for instance represent temperature variation in a gravity field and θ
e
3 the buoyancy force. For the regularity criteria of the Navier-Stokes equations, we can refer to Zhou et al. [2–9], 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 [14–19]. In contrast, in the case when η
=
ν
=
0, the Boussinesq system exhibits vorticity intensification and the global well-posedness issue remains an unsolved challenging open problem (except if θ
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 [20–24]). Hmidi and Rousset [23] proved the global well-posedness 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.Theorem 1.
Assume that(
u
0
,
θ
0
)
∈
H
3
(
R
3
) with div
⁡
u
0
=
0 in R
3. Assume that the pressure p satisfies the condition
(6)
∫
0
T
‍
∇
p
(
t
)
B
˙
∞
,
∞
0
2
/
3
1
+
ln
⁡
1
+
∇
p
t
B
˙
∞
,
∞
0
2
/
3
d
t
<
+
∞
;
then the solution (
u
,
θ
) can be extended smoothly only up to T.The paper is organized as follows. We first state some important inequalities in Section2. We will prove Theorem 1 in Section 3.


## 2. Preliminaries

Throughout this paper, we use the following usual notations.L
p
(
R
3
) denotes the Lebesgue space and H
m
(
R
3
) denotes the standard Sobolev space. BMO denotes the space of bounded mean oscillations. B
˙
m
,
n
0 is the homogeneous Besov space, where 0
≤
m, n
≤
+
∞.Lemma 2.
There exists a uniform positive constantC, such that
(7)
f
L
4
2
≤
C
f
L
2
f
B
M
O
,
(8)
f
B
˙
∞
,
2
0
≤
C
1
+
f
B
˙
∞
,
∞
0
ln
⁡
1
/
2
e
+
f
H
s
-
1
hold for all vectors f
∈
H
s
-
1
(
R
3
) with s
>
5
/
2.Proof.
See, for example, [19] or [25].Lemma 3.
From1, one has
(9)
∇
p
L
2
≤
C
u
·
∇
u
L
2
+
θ
L
2
,
∇
p
L
2
1
/
2
≤
C
u
·
∇
u
L
2
1
/
2
+
θ
L
2
1
/
2
.Lemma 4.
Assume thatΛ
=
(
-
Δ
)
1
/
2; one has the commutator estimate due to Kato and Ponce [24]:
(10)
Λ
s
f
g
-
f
Λ
s
g
L
p
≤
C
∇
f
L
p
1
λ
s
-
1
g
L
q
1
+
Λ
s
f
L
p
2
g
L
q
2
,
with s
>
0, 1
/
p
=
1
/
p
1
+
1
/
q
1
=
1
/
p
2
+
1
/
q
2.Lemma 5 (the Gagliardo-Nirenberg inequality).
Consider(11)
∇
f
L
4
≤
C
f
L
4
1
/
5
Δ
f
L
2
4
/
5
,
(12)
∇
f
L
3
≤
C
∇
f
L
2
3
/
4
Λ
3
f
L
2
1
/
4
,
(13)
Λ
3
f
L
3
≤
C
∇
f
L
2
1
/
6
Λ
4
f
L
2
5
/
6
.


## 3. Proof of Theorem1

Proof of Theorem1.
Multiplying1 by u, using 3, and integrating in R
3, we derive
(14)
1
2
d
d
t
u
L
2
2
+
η
∇
u
L
2
2
=
∫
R
3
‍
θ
e
3
·
u
d
x
≤
θ
L
2
u
L
2
≤
1
2
θ
L
2
2
+
1
2
u
L
2
2
.
Multiplying2 by θ, using 3, and integrating in R
3, we obtain
(15)
1
2
d
d
t
θ
L
2
2
+
ν
∇
θ
L
2
=
0
.
Combining14 and 15, using the Gronwall inequality, we deduce that
(16)
u
L
∞
(
0
,
T
;
L
2
)
+
u
L
2
(
0
,
T
;
H
1
)
≤
C
,
θ
L
∞
(
0
,
T
;
L
2
)
+
θ
L
2
(
0
,
T
;
H
1
)
≤
C
.
Multiplying1 by |
u
|
2
u, using 3 and 7, and integrating in R
3, we derive
(17)
∫
u
2
·
u
u
t
+
u
·
∇
u
-
η
Δ
u
+
∇
p
=
1
4
d
d
t
∫
‍
u
4
d
x
+
∫
‍
u
2
·
u
2
·
∇
u
d
x
+
η
2
∫
‍
∇
u
2
2
d
x
+
η
∫
‍
u
2
∇
u
2
d
x
+
∫
‍
u
·
∇
p
u
2
d
x
=
∫
‍
u
2
·
u
·
θ
e
3
d
x
≤
C
∫
‍
u
4
+
θ
4
d
x
;
that is,
(18)
1
4
d
d
t
∫
‍
u
4
d
x
+
η
2
∫
∇
u
2
2
d
x
+
η
∫
‍
u
2
∇
u
2
d
x
≤
-
∫
‍
(
u
·
∇
p
)
u
2
d
x
+
C
∫
‍
u
4
+
θ
4
d
x
≤
u
L
4
3
∇
p
L
4
+
C
u
L
4
4
+
C
θ
L
4
4
≤
C
u
L
4
3
∇
p
L
2
1
/
2
∇
p
BMO
1
/
2
+
C
u
L
4
4
+
C
θ
L
4
4
.
Multiplying2 by |
θ
|
2
θ, using 3, and integrating in R
3, we arrive at
(19)
∫
θ
2
·
θ
·
θ
t
+
θ
2
θ
·
u
·
∇
θ
-
ν
θ
2
·
θ
·
Δ
θ
d
x
=
1
4
d
d
t
∫
‍
θ
4
d
x
+
ν
∫
‍
θ
2
∇
θ
2
d
x
+
ν
2
∫
‍
θ
2
div
⁡
θ
2
d
x
.
Combining18 and 19, using 9 and 16, we derive that
(20)
1
4
d
d
t
∫
‍
θ
4
+
u
4
d
x
+
η
2
∫
‍
∇
u
2
2
d
x
+
η
∫
‍
u
2
∇
u
2
d
x
+
ν
∫
‍
θ
2
∇
θ
2
d
x
+
ν
2
∫
‍
θ
2
div
⁡
θ
2
d
x
,
≤
C
u
L
4
3
∇
p
L
2
1
/
2
∇
p
BMO
1
/
2
+
C
u
L
4
4
+
C
θ
L
4
4
≤
C
u
L
4
3
(
u
·
∇
u
L
2
1
/
2
+
θ
L
2
1
/
2
)
∇
p
BMO
1
/
2
+
C
u
L
4
4
+
C
θ
L
4
4
≤
2
C
u
L
4
4
∇
p
BMO
2
/
3
+
η
2
u
∇
u
L
2
2
+
C
θ
L
2
2
+
C
u
L
4
4
+
C
θ
L
4
4
,
which implies
(21)
d
d
t
∫
‍
θ
4
+
u
4
d
x
+
η
∫
‍
∇
u
2
2
d
x
+
η
∫
‍
u
2
∇
u
2
d
x
+
ν
∫
‍
θ
2
∇
θ
2
d
x
+
ν
∫
‍
θ
2
div
⁡
θ
2
d
x
≤
8
C
u
L
4
4
∇
p
BMO
2
/
3
+
4
C
θ
L
2
2
+
4
C
u
L
4
4
+
4
C
θ
L
4
4
≤
8
C
u
L
4
4
∇
p
B
˙
∞
,
∞
0
2
/
3
ln
⁡
1
/
3
1
+
∇
p
H
2
+
4
C
θ
L
2
2
+
4
C
u
L
4
4
+
4
C
θ
L
4
4
≤
8
C
u
L
4
4
∇
p
B
˙
∞
,
∞
0
2
/
3
×
ln
⁡
1
/
3
1
+
∇
Δ
u
L
2
+
Δ
θ
L
2
+
4
C
θ
L
2
2
+
4
C
u
L
4
4
+
4
C
θ
L
4
4
≤
8
C
u
L
4
4
∇
p
B
˙
∞
,
∞
0
2
/
3
1
+
ln
⁡
1
+
∇
p
B
˙
∞
,
∞
0
2
/
3
×
ln
⁡
1
+
∇
Δ
u
L
2
+
Δ
θ
L
2
+
4
C
θ
L
2
2
+
4
C
u
L
4
4
+
4
C
θ
L
4
4
.
Choosingt
∈
[
T
*
,
T
] and setting
(22)
y
(
t
)
=
sup
⁡
t
∈
T
*
,
T
∇
·
Δ
u
t
L
2
+
Δ
θ
L
2
,
we have
(23)
sup
⁡
t
∈
[
T
*
,
T
]
(
u
L
4
+
θ
L
4
)
≤
C
*
1
+
y
t
C
ε
,
where ε is a small enough constant, such that
(24)
∫
T
*
T
‍
∇
p
B
˙
∞
,
∞
0
2
/
3
1
+
ln
⁡
1
+
∇
p
B
˙
∞
,
∞
0
2
/
3
d
t
<
ε
.
Next, we want to estimate theL
2-norm of ∇
u and ∇
θ.
Multiplying1 by -
Δ
u, integrating in R
3, and using 3 and 11, we derive that
(25)
∫
‍
u
t
·
-
Δ
u
d
x
+
∫
‍
u
·
∇
u
-
Δ
u
d
x
+
η
Δ
u
L
2
2
+
∫
‍
∇
p
·
-
Δ
u
d
x
=
-
∫
‍
θ
e
3
·
Δ
u
d
x
;
that is,
(26)
1
2
d
d
t
∫
‍
∇
u
2
d
x
+
η
∫
‍
Δ
u
2
d
x
=
∫
‍
u
·
∇
u
Δ
u
d
x
-
∫
‍
θ
e
3
Δ
u
d
x
≤
u
L
4
∇
u
L
4
Δ
u
L
2
+
Δ
u
L
2
θ
L
2
≤
C
u
L
4
u
L
4
1
/
5
Δ
u
L
2
4
/
5
Δ
u
L
2
+
η
4
Δ
u
L
2
2
+
C
θ
L
2
2
≤
η
4
Δ
u
L
2
2
+
C
u
L
4
12
+
η
4
Δ
u
L
2
2
+
C
θ
L
2
Δ
θ
L
2
≤
η
2
Δ
u
L
2
2
+
C
u
L
4
12
+
ν
4
Δ
θ
L
2
2
+
C
θ
L
2
2
.
Multiplying2 by -
Δ
θ, integrating in R
3, and using 3 and 11, we derive that
(27)
∫
‍
θ
t
·
-
Δ
θ
d
x
+
∫
‍
u
·
∇
θ
-
Δ
θ
d
x
+
ν
Δ
θ
L
2
2
=
0
;
that is,
(28)
1
2
d
d
t
∫
‍
∇
θ
2
d
x
+
ν
∫
‍
Δ
θ
2
d
x
=
∫
‍
u
·
∇
θ
Δ
θ
+
η
4
Δ
u
L
2
2
+
C
θ
L
2
2
d
x
≤
u
L
4
∇
θ
L
4
Δ
θ
L
2
≤
C
u
L
4
θ
L
4
1
/
5
Δ
θ
L
2
4
/
5
Δ
θ
L
2
≤
ν
4
Δ
θ
L
2
2
+
C
u
L
4
10
θ
L
4
2
.
Combining26 and 28, using 16, we deduce
(29)
1
2
d
d
t
∫
‍
∇
u
2
+
∇
θ
2
d
x
+
η
∫
‍
Δ
u
2
d
x
+
ν
∫
‍
Δ
θ
2
d
x
≤
η
2
Δ
u
L
2
2
+
C
u
L
4
12
+
ν
4
Δ
θ
L
2
2
+
C
θ
L
2
2
+
ν
4
Δ
θ
L
2
2
+
C
u
L
4
10
θ
L
4
2
=
η
2
Δ
u
L
2
2
+
ν
2
Δ
θ
L
2
2
+
C
u
L
4
12
+
C
θ
L
2
2
+
C
u
L
4
10
θ
L
4
2
;
that is,
(30)
d
d
t
∫
‍
∇
u
2
+
∇
θ
2
d
x
+
η
∫
‍
Δ
u
2
d
x
+
ν
∫
‍
Δ
θ
2
d
x
≤
2
C
u
L
4
12
+
2
C
θ
L
2
2
+
2
C
u
L
4
10
θ
L
4
2
,
which implies that
(31)
∇
u
t
,
·
L
2
2
+
∇
θ
t
,
·
L
2
2
≤
C
1
+
y
t
C
ε
.
Last, we will estimate theH
3-norm and H
4-norm of u and θ and use the operator Λ to derive our goal.
ApplyingΛ
3
=
(
-
Δ
)
3
/
2 to 1 and then multiplying 1 with Λ
3
u, we deduce
(32)
∫
‍
Λ
3
u
t
·
Λ
3
u
d
x
+
∫
‍
Λ
3
u
·
∇
u
·
Λ
3
u
d
x
-
η
∫
‍
Λ
3
Δ
u
·
Λ
3
u
d
x
+
∫
‍
Λ
3
∇
p
Λ
3
u
d
x
=
∫
‍
Λ
3
θ
e
3
·
Λ
3
u
d
x
;
that is,
(33)
1
2
d
d
t
∫
‍
Λ
3
u
2
d
x
+
η
∫
‍
Λ
4
u
2
d
x
=
-
∫
‍
Λ
3
u
·
∇
u
-
u
·
∇
Λ
3
u
·
Λ
3
u
d
x
+
∫
‍
Λ
3
θ
e
3
Λ
3
u
d
x
≤
C
(
∇
u
L
3
Λ
3
u
L
3
2
+
Λ
3
u
L
2
Λ
4
u
L
2
)
+
Λ
3
u
L
2
θ
L
2
≤
C
∇
u
L
2
3
/
4
Λ
3
u
L
2
1
/
4
∇
u
L
2
1
/
3
Λ
4
u
L
2
5
/
3
+
Λ
3
u
L
2
θ
L
2
+
C
Λ
3
u
L
2
2
+
η
16
Λ
4
u
L
2
≤
C
∇
u
L
2
13
/
12
Λ
3
u
L
2
1
/
4
Λ
4
u
L
2
5
/
3
+
C
Λ
3
u
L
2
2
+
C
θ
L
2
+
C
Λ
3
u
L
2
2
+
η
16
Λ
4
u
L
2
≤
η
16
Λ
4
u
L
2
+
C
∇
u
L
2
13
/
10
Λ
3
u
L
2
1
/
2
+
2
C
Λ
3
u
L
2
2
+
C
θ
L
2
+
η
16
Λ
4
u
L
2
≤
C
∇
u
L
2
13
/
12
Λ
3
u
L
2
1
/
4
Λ
4
u
L
2
5
/
3
+
C
Λ
3
u
L
2
2
+
C
θ
L
2
+
C
Λ
3
u
L
2
2
+
η
8
Λ
4
u
L
2
=
η
4
Λ
4
u
L
2
2
+
C
∇
u
L
2
13
/
10
Λ
3
u
L
2
1
/
2
+
2
C
Λ
3
u
L
2
2
+
C
θ
L
2
.
Similarly, applyingΛ
3 to 2 and multiplying 2 by Λ
3
θ, we derive
(34)
∫
‍
Λ
3
θ
t
Λ
3
θ
d
x
+
∫
‍
Λ
3
u
·
∇
θ
Λ
3
θ
d
x
-
ν
∫
‍
Λ
3
Δ
θ
·
Λ
3
θ
d
x
=
0
;
that is,
(35)
1
2
d
d
t
∫
‍
Λ
3
θ
2
d
x
+
ν
∫
‍
Λ
4
θ
2
d
x
=
-
∫
‍
Λ
3
u
·
∇
θ
Λ
3
θ
d
x
=
C
∇
u
L
3
Λ
3
θ
L
3
2
+
Λ
3
u
L
3
∇
θ
L
3
Λ
3
θ
L
3
≤
C
∇
u
L
2
3
/
4
Λ
3
u
L
2
1
/
4
∇
θ
L
2
1
/
3
Λ
4
θ
L
2
5
/
3
+
C
∇
u
L
2
1
/
6
Λ
4
u
L
2
5
/
6
∇
θ
L
2
3
/
4
Λ
3
θ
L
2
1
/
4
×
∇
θ
L
2
1
/
6
Λ
4
θ
L
2
5
/
6
≤
C
∇
u
L
2
9
/
2
Λ
3
u
L
2
3
/
2
∇
θ
L
2
2
+
ν
4
Λ
4
θ
L
2
2
+
C
∇
u
L
2
1
/
3
Λ
4
u
L
2
5
/
3
+
C
∇
θ
L
2
3
/
2
Λ
3
θ
L
2
1
/
2
∇
θ
L
2
1
/
3
Λ
4
θ
L
2
5
/
3
≤
C
∇
u
L
2
9
/
2
Λ
3
u
L
2
3
/
2
∇
θ
L
2
2
+
ν
4
Λ
4
θ
L
2
2
+
C
∇
u
L
2
2
+
η
4
Λ
4
u
L
2
2
+
C
∇
θ
L
2
9
Λ
3
θ
L
2
3
∇
θ
L
2
2
+
ν
4
Λ
4
θ
L
2
2
.
Combining33 and 35, we have
(36)
1
2
d
d
t
∫
‍
Λ
3
u
2
Λ
3
θ
2
d
x
+
η
∫
‍
Λ
4
u
2
d
x
+
ν
∫
‍
Λ
4
θ
2
d
x
≤
η
2
Λ
4
u
L
2
2
+
C
∇
u
L
2
13
/
10
Λ
3
u
L
2
1
/
2
+
2
C
Λ
3
u
L
2
2
+
C
θ
L
2
+
C
∇
u
L
2
9
/
2
Λ
3
u
L
2
3
/
2
∇
θ
L
2
2
+
ν
2
Λ
4
θ
L
2
2
+
C
∇
u
L
2
2
+
C
∇
θ
L
2
9
Λ
3
θ
L
2
3
∇
θ
L
2
2
;
that is,
(37)
d
d
t
∫
‍
Λ
3
u
2
Λ
3
θ
2
d
x
+
η
∫
‍
Λ
4
u
2
d
x
+
ν
∫
‍
Λ
4
θ
2
d
x
≤
2
C
∇
u
L
2
13
/
10
Λ
3
u
L
2
1
/
2
+
4
C
Λ
3
u
L
2
2
+
2
C
θ
L
2
+
2
C
∇
u
L
2
9
/
2
Λ
3
u
L
2
3
/
2
∇
θ
L
2
2
+
2
C
∇
u
L
2
2
+
2
C
∇
θ
L
2
9
Λ
3
θ
L
2
3
∇
θ
L
2
2
.
Choosingε small enough, using 16, 23, and 24, we conclude that
(38)
u
L
∞
(
0
,
T
;
H
3
)
+
u
L
2
(
0
,
T
;
H
4
)
≤
C
,
θ
L
∞
(
0
,
T
;
H
3
)
+
θ
L
2
(
0
,
T
;
H
4
)
≤
C
.
We complete the proof.



---

*Source: 303025-2015-04-06.xml*