# A Remark on the Regularity Criterion for the MHD Equations via Two Components in Morrey-Campanato Spaces

Authors: Zujin Zhang; Tong Tang; Fumin Zhang
Journal: Journal of Difference Equations
 (2014)

---

## Abstract

We consider the regularity criterion for the 3D MHD equations. It is proved that if the horizontal components of the velocity and magnetic fields satisfyu
~
,
b
~
∈
L
2
/
(
1
-
r
)
(
0
,
T
;
M
˙
2
,
3
/
r
) with 0
<
r
<
1, then the solution smooth. This improves the result given by Gala (2012).


---

## Body

## 1. Introduction

In this paper, we consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations:
(1)
u
t
+
(
u
·
∇
)
u
-
(
b
·
∇
)
b
-
Δ
u
+
∇
π
=
0
,
b
t
+
(
u
·
∇
)
b
-
(
b
·
∇
)
u
-
Δ
b
=
0
,
∇
·
u
=
0
,
∇
·
b
=
0
,
u
(
0
)
=
u
0
,
b
(
0
)
=
b
0
,
where u
=
(
u
1
,
u
2
,
u
3
) is the fluid velocity field, b
=
(
b
1
,
b
2
,
b
3
) is the magnetic field, π is a scalar pressure, and (
u
0
,
b
0
) are the prescribed initial data satisfying ∇
·
u
0
=
∇
·
b
0
=
0 in the distributional sense. Physically, (1) governs the dynamics of the velocity and magnetic fields in electrically conducting fluids, such as plasmas, liquid metals, and salt water. Moreover, (1)1 reflects the conservation of momentum, (1)2 is the induction equation, and (1)3 specifies the conservation of mass.Besides its physical applications, the MHD system (1) is also mathematically significant. Duvaut and Lions [1] constructed a global weak solution to (1) for initial data with finite energy. However, the issue of regularity and uniqueness of such a given weak solution remains a challenging open problem. Many sufficient conditions (see, e.g., [2–14] and the references therein) were derived to guarantee the regularity of the weak solution. Among these results, we are interested in regularity criteria involving only partial components of the velocity u, the magnetic field b, the pressure gradient ∇
π, and so forth.Cao and Wu [2] proved the following regularity criterion:
(2)
∂
3
u
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
2
p
+
3
q
=
1
,
3
≤
q
≤
∞
.
Jia and Zhou showed that if
(3)
∂
3
u
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
2
p
+
3
q
=
3
4
+
2
2
q
,
q
>
2
,
then the solution is regular. These results were improved by Zhang [15] to be
(4)
∂
3
u
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
2
p
+
3
q
=
1
+
1
q
,
q
>
2
.Once only partial components of the velocity field are concerned, we have combinatoric regularity criterion involving partial components of the magnetic field also. This is due partially to the strong coupling of the velocity and magnetic fields. Let us list some recent progress. Gala and Lemarié-Rieusset [6] established the following two regularity conditions:
(5)
(
u
1
,
u
2
)
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
with
2
p
+
3
q
=
1
,
3
<
q
≤
∞
,
(
b
1
,
b
2
)
∈
L
r
(
0
,
T
;
L
s
(
R
3
)
)
,
with
2
r
+
3
s
=
1
,
3
<
s
≤
∞
,
(
u
1
,
u
2
)
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
b
3
∈
L
r
(
0
,
T
;
L
s
(
R
3
)
)
,
with
(
2
p
+
3
q
)
+
(
2
r
+
3
s
)
≤
2
,
2
p

+

3
q
≤
1
,
3
<
q
≤
∞
.
Recently, Ni et al. in [10] showed that each of the following three conditions
(6)
∇
h
u
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
with
2
p
+
3
q
=
2
,
3
2
<
q
≤
∞
,
∂
3
b
∈
L
r
(
0
,
T
;
L
s
(
R
3
)
)
,
with
2
r
+
3
s
=
2
,
3
2
<
s
≤
∞
;
or
(7)
∇
h
u
∈
L
p
(
0
,
T
;
L
q
(
R
3
)
)
,
with
2
p
+
3
q
=
2
,
3
2
<
q
≤
∞
,
∇
h
b
∈
L
r
(
0
,
T
;
L
s
(
R
3
)
)
,
with
2
r
+
3
s
=
2
,
3
2
<
s
≤
∞
;
or
(8)
u
3
∈
L
p
1
(
0
,
T
;
L
q
1
(
R
3
)
)
,
with
2
p
1

+

3
q
1
=
1
,
3
<
q
≤
∞
,
∂
3
u
∈
L
p
2
(
0
,
T
;
L
q
2
(
R
3
)
)
,
with
2
p
2
+
3
q
2
=
2
,
i
3
2
<
q
≤
∞
,
b
∈
L
p
3
(
0
,
T
;
L
q
3
(
R
3
)
)
,
with
2
p
3
+
3
q
3
=
1
,
3
<
q
≤
∞
,
∂
3
b
∈
L
p
4
(
0
,
T
;
L
q
4
(
R
3
)
)
,
with
2
p
4
+
3
q
4
=
2
,
i
3
2
<
q
≤
∞
,
ensures the smoothness of the solution. Here, ∇
h
=
(
∂
1
,
∂
2
) is the horizontal gradient operator.The motivation of this paper is to give another contribution in this direction. Motivated by [8], we would like to show the following regularity condition for (1):
(9)
u
~
,
b
~
∈
L
2
/
(
1
-
r
)
(
0
,
T
;
M
˙
2
,
3
/
r
)
with
0
<
r
<
1
.
Here, and in what follows, we denote by
(10)
u
~
=
(
u
1
,
u
2
,
0
)
,
b
~
=
(
b
1
,
b
2
,
0
)
the horizontal components of the velocity and magnetic fields, respectively, and by
(11)
∇
~
=
(
∂
1
,
∂
2
,
0
)
with
∂
i
=
∂
∂
x
i
the horizontal gradient operator.Before stating the precise result, let us recall the weak formulation of the MHD equations (1).Definition 1.
Let(
u
0
,
b
0
)
∈
L
2
(
R
3
) with ∇
·
u
0
=
∇
·
b
0
=
0 and T
>
0. A measurable R
3-valued pair (
u
,
b
) is called a weak solution to (1) with initial data (
u
0
,
b
0
), provided that the following three conditions hold:(1)
u
∈
L
∞
(
0
,
T
;
L
2
(
R
3
)
)
∩
L
2
(
0
,
T
;
H
1
(
R
3
)
), b
∈
L
∞
(
0
,
T
;
L
2
(
R
3
)
)
∩
L
2
(
0
,
T
;
H
1
(
R
3
)
);
(2)
(1)1,2,3,4 are satisfied in the distributional sense;
(3)
the energy inequality(12)
∥
(
u
,
b
)
∥
L
2
2
(
t
)
+
2
∫
0
t
‍
∥
∇
(
u
,
b
)
∥
L
2
2
(
s
)
d
s
≤
∥
(
u
0
,
b
0
)
∥
L
2
2
.Now, our main result reads as follows.Theorem 2.
Let(
u
0
,
b
0
)
∈
H
1
(
R
3
) with ∇
·
u
0
=
∇
·
b
0
=
0 and T
>
0. Assume that (
u
,
b
) is a given weak solution pair of the MHD system (1) with initial data (
u
0
,
b
0
) on (
0
,
T
). If
(13)
u
~
,
b
~
∈
L
2
/
(
1
-
r
)
(
0
,
T
;
M
˙
2
,
3
/
r
)
w
i
t
h
0
<
r
<
1
,
then the solution is smooth on (
0
,
T
).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 (19)), we indeed improve the result of [16].


## 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 [11, 17, 18], e.g.).Definition 4.
For1
<
p
≤
q
≤
+
∞, the Morrey-Campanato space M
˙
p
,
q is defined as
(14)
M
˙
p
,
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
)
i
i
×
(
∫
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:
(15)
L
q
=
M
˙
q
,
q
⊂
M
˙
p
,
q
.
Moreover, the following scaling property holds:
(16)
∥
f
(
λ
·
)
∥
M
˙
p
,
q
=
1
λ
(
3
/
q
)
∥
f
∥
M
˙
p
,
q
,
for
λ
>
0
.
Due to the following characterization in [19].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
(17)
∥
f
∥
Z
˙
r
=
sup
⁡
∥
g
∥
B
˙
2,1
r
≤
1
∥
f
g
∥
L
2
<
+
∞
.
Then, f
∈
M
˙
2,3
/
r if and only if f
∈
Z
˙
r with equivalent norm.And, with the fact that(18)
L
2
∩
H
˙
r
⊂
B
˙
2,1
r
⊂
H
˙
r
for
0
<
r
<
1
,
we have
(19)
X
˙
r
⊂
M
˙
2
,
3
/
r
.
Here, B
˙
2,1
r is the Besov space, which is intermediate between L
2 and H
˙
1 (see [20]):
(20)
∥
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.It is well known (see [21], e.g.) that, for u
0
∈
H
1
(
R
3
) with ∇
·
u
0
=
0, (1) possesses a local strong solution
(21)
(
u
,
b
)
∈
L
∞
(
0
,
Γ
*
;
H
1
(
R
3
)
)
∩
L
2
(
0
,
Γ
*
;
H
2
(
R
3
)
)
,
where Γ
* is the maximal existence of the strong solution. Moreover, this strong solution is smooth and unique in the class of weak solutions. Thus, if Γ
*
≥
T, we have nothing to prove. Otherwise, we will show that the H
1 norm of this strong solution remains bounded as t
∈
[
0
,
Γ
*
). The standard continuation argument then yields that Γ
* could not be the maximal existence of the strong solution. This contradiction concludes that Γ
*
≥
T, and we complete the proof.Taking the inner product of (1)1 with -
Δ
u, (1)2 with -
Δ
b in L
2
(
R
3
), respectively, and adding the resulting equations together, we obtain
(22)
1
2
d
d
t
∥
∇
(
u
,
b
)
∥
L
2
2
+
∥
Δ
(
u
,
b
)
∥
L
2
2
=
∫
R
3
‍
[
(
u
·
∇
)
u
]
·
Δ
u
d
x
-
∫
R
3
‍
[
(
b
·
∇
)
u
]
·
Δ
u
d
x
+
∫
R
3
‍
[
(
u
·
∇
)
b
]
·
Δ
b
d
x
-
∫
R
3
‍
[
(
b
·
∇
)
u
]
·
Δ
b
d
x
=
-
∑
k
=
1
3
‍
∫
R
3
‍
[
(
∂
k
u
·
∇
)
u
]
·
∂
k
u
d
x
+
∑
k
=
1
3
‍
∫
R
3
‍
[
(
∂
k
b
·
∇
)
b
]
·
∂
k
u
d
x
-
∑
k
=
1
3
‍
∫
R
3
‍
[
(
∂
k
u
·
∇
)
b
]
·
∂
k
b
d
x
+
∑
k
=
1
3
‍
∫
R
3
‍
[
(
∂
k
b
·
∇
)
u
]
·
∂
k
b
d
x
≡
I
1
+
I
2
+
I
3
+
I
4
,
where we use integration by parts formula, the fact that
(23)
∇
·
u
=
∇
·
b
=
0
and its consequence
(24)
∫
R
3
‍
[
(
b
·
∇
)
∂
k
b
]
·
∂
k
u
+
[
(
b
·
∇
)
∂
k
u
]
·
∂
k
b
d
x
=
0
.
For I
1,
(25)
I
1
=
-
∑
k
=
1
3
‍
∫
R
3
‍
[
(
∂
k
u
~
·
∇
~
)
u
]
·
∂
k
u
d
x
-
∑
k
=
1
3
‍
∫
R
3
‍
∂
k
u
3
∂
3
u
~
·
∂
k
u
~
d
x
-
∑
k
=
1
3
‍
∫
R
3
‍
∂
k
u
3
∂
3
u
3
∂
k
u
3
d
x
.
Integrating by parts and noticing that ∂
3
u
3
=
-
∂
1
u
1
-
∂
2
u
2, we get
(26)
I
1
≤
C
∫
R
3
‍
|
u
~
|
·
|
∇
u
|
·
|
∇
2
u
|
d
x
.
The remaining terms I
2
,
I
3
,
I
4 can be similarly decomposed and bounded so that
(27)
I
2
,
I
3
,
I
4
≤
C
∫
R
3
‍
|
(
u
~
,
b
~
)
|
·
|
∇
(
u
,
b
)
|
·
|
∇
2
(
u
,
b
)
|
d
x
.
Plugging (26) and (27) into (22), we gather
(28)
1
2
d
d
t
∥
∇
(
u
,
b
)
∥
L
2
2
+
∥
Δ
(
u
,
b
)
∥
L
2
2
≤
C
∫
R
3
‍
|
(
u
~
,
b
~
)
|
·
|
∇
(
u
,
b
)
|
·
|
∇
2
(
u
,
b
)
|
d
x
≤
C
∥
|
(
u
~
,
b
~
)
|
·
|
∇
(
u
,
b
)
|
∥
L
2
∥
∇
2
(
u
,
b
)
∥
L
2
≤
C
∥
(
u
~
,
b
~
)
∥
M
˙
2
,
3
/
r
∥
∇
(
u
,
b
)
∥
B
˙
2,1
r
∥
Δ
(
u
,
b
)
∥
L
2
(
by
Lemma
5
)
≤
C
∥
(
u
~
,
b
~
)
∥
M
˙
2
,
3
/
r
∥
∇
(
u
,
b
)
∥
L
2
1
-
r
∥
Δ
(
u
,
b
)
∥
L
2
1
+
r
(
by
(
20
)
)
≤
C
∥
(
u
~
,
b
~
)
∥
M
˙
2
,
3
/
r
2
/
(
1
-
r
)
∥
∇
(
u
,
b
)
∥
L
2
2
+
1
2
∥
Δ
(
u
,
b
)
∥
L
2
2
.
Consequently,
(29)
d
d
t
∥
∇
(
u
,
b
)
∥
L
2
2
+
∥
Δ
(
u
,
b
)
∥
L
2
2
≤
C
∥
(
u
~
,
b
~
)
∥
M
˙
2
,
3
/
r
2
/
(
1
-
r
)
∥
∇
(
u
,
b
)
∥
L
2
2
.
Applying Gronwall inequality, we deduce that
(30)
∥
∇
(
u
,
b
)
∥
L
2
2
(
t
)
≤
∥
∇
(
u
0
,
b
0
)
∥
L
2
2
·
exp
⁡
{
C
∫
0
T
‍
∥
(
u
~
,
b
~
)
∥
M
˙
2
,
3
/
r
2
/
(
1
-
r
)
(
s
)
d
s
}
<
+
∞
.As soon as the estimates of∥
∇
(
u
,
b
)
∥
L
2 are obtained, we can invoke the standard energy method to bootstrap the solution to be in W
k
,
2
(
R
3
) for all k
∈
Z
+. The Sobolev imbedding theorem then implies that the solution is smooth.The proof of Theorem2 is completed.



---

*Source: 364269-2014-06-25.xml*