# A Remark on the Regularity Criterion for the 3D Boussinesq Equations Involving the Pressure Gradient

Authors: Zujin Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider the three-dimensional Boussinesq equations and obtain a regularity criterion involving the pressure gradient in the Morrey-Companato spaceM
p
,
q. This extends and improves the result of Gala (Gala 2013) for the Navier-Stokes equations.


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## Body

## 1. Introduction

This paper concerns itself with the following three-dimensional (3D) Boussinesq equations:(1)
u
t
+
(
u
·
∇
)
u
-
Δ
u
+
∇
π
=
θ
e
3
,
in
ℝ
3
×
(
0
,
T
)
,
θ
t
+
(
u
·
∇
)
θ
-
Δ
θ
=
0
,
in
ℝ
3
×
(
0
,
T
)
,
∇
·
u
=
0
,
in
ℝ
3
×
(
0
,
T
)
,
u
(
0
)
=
u
0
,
θ
(
0
)
=
θ
0
,
on
⁡
ℝ
3
,
where T
>
0 is given time, u
=
(
u
1
(
x
,
t
)
,
u
2
(
x
,
t
)
,
u
3
(
x
,
t
)
) is the fluid velocity, π
=
π
(
x
,
t
) is a scalar pressure, and θ
=
θ
(
x
,
t
) is the temperature, while u
0 and θ
0 are the prescribed initial velocity field and temperature, respectively.Whenθ
=
0, (1) reduces to the incompressible Navier-Stokes equations. The regularity of its weak solutions and the existence of global strong solutions are challenging open problems; see [1–3]. Starting with [4, 5], there have been a lot of literature devoted to finding sufficient conditions to ensure the smoothness of the solutions; see [6–15] and the references cited therein. Since the convective terms are similar in the Navier-Stokes equations and Boussinesq equations, the authors also consider the regularity conditions for (1); see [16–20] and so forth.In [6], Gala uses intricate decomposition technique to obtain the following regularity criterion for the Navier-Stokes equations:
(2)
∇
π
∈
L
2
/
(
3
-
r
)
(
0
,
T
;
X
˙
r
)
with
0
≤
r
≤
1
.
Here, X
˙
r is the point-wise multiplier space from H
˙
r to L
2, which is strictly larger than L
3
/
r
(
ℝ
3
) (see [6, Lemma 1.2]).In this paper, we will extend and improve the regularity condition (2) to the Boussineq equations (1).Before stating the precise result, let us recall the weak formulation of (1).Definition 1.
Letu
0
∈
L
2
(
ℝ
3
), θ
0
∈
L
1
∩
L
∞
(
ℝ
3
). A measurable pair (
u
,
θ
) is said to be a weak solution of (1) in (
0
,
T
), provided that(1)
(
u
,
θ
)
∈
L
∞
(
0
,
T
;
L
2
(
ℝ
3
)
)
∩
L
2
(
0
,
T
;
H
1
(
ℝ
3
)
), θ
∈
L
∞
(
0
,
T
;
L
1
∩
L
∞
(
ℝ
3
)
);
(2)
(1)1,2,3 are satisfied in the sense of distributions;
(3)
the energy inequality(3)
∥
(
u
,
θ
)
∥
L
2
2
+
2
∫
0
t
‍
∥
∇
(
u
,
θ
)
∥
L
2
2
d
s
≤
∥
(
u
0
,
θ
0
)
∥
L
2
2
+
2
∫
0
t
‍
∫
ℝ
3
‍
θ
u
3
d
x
d
s
,for all 0
≤
t
≤
T.Now, our main result reads the following.Theorem 2.
Letu
0
∈
L
2
(
ℝ
3
) with ∇
·
u
0
=
0 in the sense of distributions, θ
0
∈
L
1
∩
L
∞
(
ℝ
3
). Supposing that (
u
,
θ
) is a weak solution of (1) in [
0
,
T
), and the pressure gradient ∇
π satisfies
(4)
∇
π
∈
L
2
/
(
3
-
r
)
(
0
,
T
;
M
˙
2
,
3
/
r
)
w
i
t
h
0
<
r
≤
1
,
then the solution (
u
,
θ
)
∈
C
∞
(
(
0
,
T
)
×
ℝ
3
).Here,M
˙
p
,
q is the Morrey-Campanato space, which will be introduced in Section 2. And Section 3 is devoted to the proof of Theorem 2.Remark 3.
Noticing thatX
˙
r
⊂
M
˙
2
,
3
/
r for 0
<
r
<
1 (see (10)), we indeed improve the result of [6] for the Navier-Stokes equations.


## 2. Preliminaries

In this section, we will introduce the definition of Morrey-Campanato spaceM
˙
p
,
q and recall its fundamental properties. The space plays an important role in studying the regularity of solutions to partial differential equations (see [21–23], e.g.).Definition 4.
For1
<
p
≤
q
≤
+
∞, the Morrey-Campanato space M
˙
p
,
q is defined as
(5)
M
˙
p
,
q
=
{
=
sup
⁡
x
∈
ℝ
3
,
R
>
0
1
R
(
3
/
p
)
-
(
3
/
q
)
(
∫
B
(
x
,
R
)
‍
|
f
(
y
)
|
p
d
y
)
1
/
p
f
∈
L
loc
⁡
p
(
ℝ
3
)
;
∥
f
∥
M
˙
p
,
q
=
sup
⁡
x
∈
ℝ
3
,
R
>
0
1
R
(
3
/
p
)
-
(
3
/
q
)
(
∫
B
(
x
,
R
)
‍
|
f
(
y
)
|
p
d
y
)
1
/
p
<
+
∞
=
sup
⁡
x
∈
ℝ
3
,
R
>
0
1
R
(
3
/
p
)
-
(
3
/
q
)
(
∫
B
(
x
,
R
)
‍
|
f
(
y
)
|
p
d
y
)
1
/
p
}
,
where B
(
x
,
R
)
⊂
ℝ
3 is the ball with center x and radius R.One sees readily thatM
˙
p
,
q is a Banach space under the norm ∥
·
∥
M
˙
p
,
q and contains the classical Lebesgue space as a subspace:
(6)
L
q
=
M
˙
q
,
q
⊂
M
˙
p
,
q
.Moreover, the following scaling property holds:(7)
∥
f
(
λ
·
)
∥
M
˙
p
,
q
=
1
λ
3
/
q
∥
f
∥
M
˙
p
,
q
,
for
λ
>
0
.Due to the following characterization in [24].Lemma 5.
For0
≤
r
<
3
/
2, the space Z
˙
r is defined as the space of all functions f
∈
L
loc
⁡
2
(
ℝ
3
) such that
(8)
∥
f
∥
Z
˙
r
=
sup
⁡
∥
g
∥
B
˙
2,1
r
≤
1
∥
f
g
∥
L
2
<
+
∞
.
Thenf
∈
M
˙
2
,
3
/
r if and only if f
∈
Z
˙
r with equivalent norm.And with the fact that(9)
L
2
∩
H
˙
r
⊂
B
˙
2,1
r
⊂
H
˙
r
for
0
<
r
<
1
,
we have
(10)
X
˙
r
⊂
M
˙
2
,
3
/
r
.HereB
˙
2,1
r is the Besov space, which is intermediate between L
2 and H
˙
1 (see [25]):
(11)
∥
f
∥
B
˙
2,1
r
≤
C
∥
f
∥
L
2
1
-
r
∥
∇
f
∥
L
2
r
,
for
0
<
r
<
1
.


## 3. Proof of Theorem2

In this section, we will prove Theorem2.Due to the Serrin type regularity criterion(12)
u
∈
L
p
(
0
,
T
;
L
q
(
ℝ
3
)
)
with
2
p
+
3
q
=
1
,
3
<
q
≤
+
∞
in [19], we need only to prove
(13)
u
∈
L
∞
(
0
,
T
;
L
4
(
ℝ
4
)
)
⊂
L
8
(
0
,
T
;
L
4
(
ℝ
3
)
)
.We just do a priori estimates, with the justification being from passage to limits for the Galerkin approximated solutions.Taking the inner product of (1)2 with 2
θ in L
2
(
ℝ
3
), we find
(14)
d
d
t
∥
θ
∥
L
2
2
+
2
∥
∇
θ
∥
L
2
2
=
0
.Thus,(15)
∥
θ
∥
L
2
≤
∥
θ
0
∥
L
2
.One can also take the inner product of (1)2 with p
θ
p
-
1 (1
≤
p
<
∞) in L
2
(
ℝ
3
) to derive the estimate of θ in L
p-norm and invoke the maximum principle to bound the L
∞-norm of θ, as stated in Definition 1.Taking the divergence of (1)1, we get
(16)
-
Δ
π
=
∑
i
,
j
=
1
3
‍
∂
i
∂
j
(
u
i
u
j
)
-
∂
3
θ
.Consequently,(17)
∥
∇
π
∥
L
2
≤
C
∥
|
u
|
·
|
∇
u
|
∥
L
2
+
∥
θ
∥
L
2
≤
C
(
∥
|
u
|
·
|
∇
u
|
∥
L
2
+
1
)
.Taking the inner product of (1)1 with 4
|
u
|
2
u in L
2
(
ℝ
3
), we get
(18)
d
d
t
∥
|
u
|
2
∥
L
2
2
+
4
∫
ℝ
3
‍
|
u
|
2
|
∇
u
|
2
d
x
+
2
∫
ℝ
3
‍
|
∇
|
u
|
2
|
d
x
≤
4
∫
ℝ
3
‍
|
∇
π
|
·
|
u
|
3
d
x
+
4
∫
ℝ
3
‍
|
θ
|
·
|
u
|
3
d
x
≡
I
1
+
I
2
.ForI
1, we estimate as(19)
I
1
=
∫
ℝ
3
‍
|
∇
π
|
1
/
2
·
|
∇
π
|
1
/
2
|
u
|
·
|
u
|
2
d
x
≤
∥
|
∇
π
|
1
/
2
∥
L
4
·
∥
|
∇
π
|
1
/
2
|
u
|
∥
L
4
·
∥
|
u
|
2
∥
L
2
(
by
H
o
¨
lder
inequality
)
=
∥
∇
π
∥
L
2
1
/
2
·
∥
|
∇
π
|
·
|
u
|
2
∥
L
2
1
/
2
·
∥
|
u
|
2
∥
L
2
≤
∥
∇
π
∥
L
2
1
/
2
·
∥
∇
π
∥
M
˙
2
,
3
/
r
1
/
2
∥
|
u
|
2
∥
B
˙
2,1
r
1
/
2
·
∥
|
u
|
2
∥
L
2
(
by
Lemma
5
)
≤
C
(
∥
|
u
|
·
|
∇
u
|
∥
L
2
+
1
)
1
/
2
·
∥
∇
π
∥
M
˙
2
,
3
/
r
1
/
2
·
∥
|
u
|
2
∥
L
2
(
1
-
r
)
/
2
∥
∇
|
u
|
2
∥
L
2
r
/
2
·
∥
|
u
|
2
∥
L
2
(
by
(
17
)
and
(
11
)
)
≤
C
∥
|
u
|
·
|
∇
u
|
∥
L
2
1
/
2
·
∥
∇
|
u
|
2
∥
L
2
r
/
2
·
∥
∇
π
∥
M
˙
2
,
3
/
r
1
/
2
∥
|
u
|
2
∥
L
2
(
3
-
r
)
/
2
+
C
∥
∇
|
u
|
2
∥
L
2
r
/
2
·
1
·
∥
∇
π
∥
M
˙
2
,
3
/
r
1
/
2
∥
|
u
|
2
∥
L
2
(
3
-
r
)
/
2
≤
3
∥
|
u
|
·
|
∇
u
|
∥
L
2
2
+
1
2
∥
∇
|
u
|
2
∥
L
2
2
+
C
+
C
∥
∇
π
∥
M
˙
2
,
3
/
r
2
/
(
3
-
r
)
∥
|
u
|
2
∥
L
2
2
(
Young
inequality
a
b
c
≤
ε
a
p
+
δ
b
q
+
C
ε
δ
c
r
,
1
p
+
1
q
+
1
r
=
1
with
p
=
4
,
q
=
4
r
,
r
=
4
3
-
r
)
.The termI
2 can be dominated as
(20)
I
2
≤
4
∥
θ
∥
L
2
∥
|
u
|
3
∥
L
2
≤
C
∥
|
u
|
2
∥
L
3
3
/
2
(
by
(
15
)
)
≤
C
∥
|
u
|
2
∥
L
2
3
/
4
∥
∇
|
u
|
2
∥
L
2
3
/
4
(
by
interpolation
inequality
)
≤
C
∥
|
u
|
2
∥
L
2
6
/
5
+
1
2
∥
∇
|
u
|
2
∥
L
2
2
≤
C
+
C
∥
|
u
|
2
∥
L
2
2
+
1
2
∥
∇
|
u
|
2
∥
L
2
2
.Plugging (19) and (20) into (18), we deduce that
(21)
d
d
t
∥
|
u
|
2
∥
L
2
2
+
∫
ℝ
3
‍
|
u
|
2
|
∇
u
|
2
d
x
+
∫
ℝ
3
‍
|
∇
|
u
|
2
|
d
x
≤
C
+
C
(
∥
∇
π
∥
M
˙
2
,
3
/
r
2
/
(
3
-
r
)
+
1
)
∥
|
u
|
2
∥
L
2
2
.Applying Gronwall inequality, we see that(22)
∥
u
(
t
)
∥
L
4
4
≤
(
∥
u
0
∥
L
4
4
+
C
T
)
·
exp
⁡
{
C
∫
0
T
‍
(
∥
∇
π
∥
M
˙
2
,
3
/
r
2
/
(
3
-
r
)
+
1
)
d
s
}
,
for every t
∈
[
0
,
T
]. Recalling (13), we complete the proof of Theorem 2.



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*Source: 510924-2014-01-21.xml*