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Abstract and Applied Analysis
Volume 2011, Article ID 380402, 9 pages
doi:10.1155/2011/380402
Research Article
Regularity Criteria for a Turbulent
Magnetohydrodynamic Model
Yong Zhou1 and Jishan Fan2
1
2
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
Correspondence should be addressed to Yong Zhou, yzhoumath@zjnu.edu.cn
Received 23 May 2011; Accepted 12 July 2011
Academic Editor: Simeon Reich
Copyright q 2011 Y. Zhou and J. Fan. 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 some regularity criteria for a turbulent magnetohydrodynamic model. As a corollary,
we prove that the smooth solution exists globally when the spatial dimension n satisfies 3 ≤ n ≤ 8.
1. Introduction
In this paper, we study the following simplified turbulent MHD model 1:
∂t v − Δv u · ∇u ∇π B · ∇B,
1.1
∂t H − ΔH u · ∇B − B · ∇u 0,
u 1 − α2 Δ u, H 1 − α2 Δ B, α > 0,
1.2
1.3
div v div u div H div B 0,
1.4
v, H0 v0 , H0 in Rn n ≥ 3.
1.5
Here v is the fluid velocity field, u is the “filtered” fluid velocity, π is the pressure, H is the
magnetic field, and B is the “filtered” magnetic field. α > 0 is the length scale parameter that
represents the width of the filter. For simplicity we will take α 1.
When n 3, the global well-posedness of the problem has been proved in 1. When
H B 0, 1.1 and 1.4 is the well-known Bardina model. Very recently, the authors
2 have proved that the Bardina model has a unique global-in-time weak solution when
2
Abstract and Applied Analysis
3 ≤ n ≤ 8. Here we wold like to point out that by the same arguments, we can prove the
following.
Theorem 1.1. Let 3 ≤ n ≤ 8. Let u0 , B0 ∈ H 1 Rn with div u0 div B0 0 in Rn . Then for any
T > 0, the problem 1.1–1.5 has a unique weak solution satisfying
1
2
u2 |∇u|2 B2 |∇B|2 dx 1
≤
2
T |∇u|2 |Δu|2 |∇B|2 |ΔB|2 dx dt
0
u20
2
|∇u0 | B02
1.6
2
|∇B0 | dx.
The proof for Theorem 1.1 is similar to that for the Bardina model in 2, so we omit it
here.
The aim of this paper is to study the regularity of the weak solutions. We will prove
Theorem 1.2. Let n ≥ 3. Let v0 , H0 ∈ H s Rn with s > 1 and div v0 div H0 0 in Rn . Let
v, H be a local smooth solution to the problem 1.1–1.5 satisfying
v, H ∈ L∞ 0, T ; H s ∩ L2 0, T ; H s1 ,
1.7
for any fixed T > 0. Then v, H can be extended beyond T > 0 provided that one of the following
condition is satisfied:
1 u, B ∈ Lp 0, T ; Lq with
n
2 n
3,
< q ≤ n,
p q
3
2 ∇u, ∇B ∈ Lp 0, T ; Lq Rn with
n
n
2 n
4,
<q≤ .
p q
4
2
1.8
1.9
By 1.6 and 1.8, as a corollary, we have the following
Corollary 1.3. Let 3 ≤ n ≤ 8. Let v0 , H0 ∈ H s Rn with s > 1 and div v0 div H0 0 in Rn .
Then for any T > 0, the problem 1.1–1.5 has a unique smooth solution v, H satisfying 1.7.
When n 9 or 10, we can get a better result as follows.
Theorem 1.4. Let n 9 or 10 and let v0 , H0 ∈ L2 Rn and div v0 div H0 0 in Rn . Let v, H
be a local smooth solution to the problem 1.1–1.5 satisfying
v, H ∈ L∞ 0, T ; L2 ∩ L2 0, T ; H 1 ,
1.10
Abstract and Applied Analysis
3
for any fixed T > 0. Then v, H can be extended beyond T > 0 if one of the following conditions is
satisfied:
1 u ∈ C 0, T ; Ln/3 ,
2 u ∈ Lp 0, T ; Lq with
1.11
n
2 n
3,
< q ≤ n,
p q
3
3 ∇u ∈ C 0, T ; Ln/4 ,
4 ∇u ∈ Lp 0, T ; Lq 1.12
1.13
with
n
n
2 n
4 with < q ≤ .
p q
4
2
1.14
Remark 1.5. If we delete the harmless lower order terms ∂t u − Δu and ∂t B − ΔB in 1.1 and
1.2, then we have
−∂t Δu Δ2 u u · ∇u ∇π B · ∇B,
−∂t ΔB Δ2 B u · ∇B − B · ∇u 0,
1.15
then the system 1.15 has the following property: if u, B, π is a solution of 1.15, then for
all λ > 0,
uλ , Bλ , πλ x, t λ3 u, λ3 B, λ6 π λx, λ2 t
1.16
is also a solution. In this sense, our conditions 1.8 and 1.11–1.14 are scaling invariant
optimal. Equations 1.8 and 1.12 do not hold true for q > n. But we also can establish
regularity criteria for q > n in nonscaling invariant forms.
In Section 2, we will prove Theorem 1.2. In Section 3, we will prove Theorem 1.4.
2. Proof of Theorem 1.2
Since it is easy to prove that the problem 1.1–1.5 has a unique local smooth solution, we
only need to establish the a priori estimates. The proof of the case n ≤ 4 is easier and similar
and thus we omit the details here, we only deal with the case n ≥ 5.
Testing 1.1 by u, using 1.3 and 1.4, we find that
1 d
2 dt
2
u |∇u| dx 2
2
2
|∇u| |Δu| dx B · ∇B · u dx.
2.1
4
Abstract and Applied Analysis
Testing 1.2 by B, using 1.3 and 1.4, we see that
1 d
2 dt
B2 |∇B|2 dx |∇B|2 |ΔB|2 dx B · ∇u · B dx − B · ∇B · u dx.
2.2
Summing up 2.1 and 2.2, we easily get 1.6.
I Let 1.8 hold true.
In the following calculations, we will use the product estimates due to Kato and Ponce
3:
s Λ fg p ≤ C Λs f p g q f p Λs g q ,
L1
L1
L2
L2
L
2.3
with s > 0, Λ : −Δ1/2 and 1/p 1/p1 1/q1 1/p2 1/q2 .
The proof of the case q n is easier and similar, we omit the details here. Now we
assume n/3 < q < n.
Applying Λs to 1.1, testing by Λs v, using 1.4, we deduce that
1 d
2 dt
|Λs v|2 dx s1 2
v
dx
Λs B · ∇B − Λs u · ∇uΛs v dx
Λ
2.4
Λ divB ⊗ B − u ⊗ uΛ v dx.
s
s
Similarly, applying Λs to 1.2, testing by Λs H, using 1.4, we infer that
1 d
2 dt
s1 2
|Λ H| dx Λ H dx Λs curlu × B · Λs H dx.
s
2
Summing up 2.4 and 2.5, using 2.3, we get
1 d
2 dt
2 2
|Λ v| |Λ H| dx Λs1 v Λs1 H dx
s
2
s
2
Λs divB ⊗ B − u ⊗ uΛs v dx ≤ C BLq Λs1 B
Lt1
C uLq Λs1 B
Λs curlu × B · Λs H dx
uLq Λs1 u
Lt1
Lt1
Λs vLt2
BLq Λs1 u
Lt1
Λs HLt2
2.5
Abstract and Applied Analysis
≤ CuLq
5
BLq Λs1 u
Cu, BLq Λs1 u, B
Lt1
Lt1
Λs1 B
Λs vLt2 Λs HLt2 Lt1
Λs v, HLt2
1
2n
2n
1 1
1, 2 ≤ t2 ≤
< t1 ≤
q t1 t2
n−2
n−4
≤ Cu, BLq Λs−1 v, H t Λs v, HLt2
L1
θ1
θ2
1−θ2 s1
s1
s
1
≤ Cu, BLq Λs v, H1−θ
H
H
H
v,
v,
v,
Λ
2
Λ
Λ
2
2
L
L
2
L
L
θ1 θ2
s1
1 −θ2 Cu, BLq Λs v, H2−θ
H
v,
2
Λ
2
L
L
2
1
2/2−θ1 −θ2 ≤ Λs1 v, H 2 Cu, BLq
Λs v, H2L2 ,
L
2
2.6
which implies
v, HL∞ 0,T ;H s v, HL2 0,T ;H s1 ≤ C.
2.7
Here we have used the following Gagliardo-Nirenberg inequalities:
s−1
Λ v, H
Lt1
θ1
s1
1
≤ CΛs v, H1−θ
H
v,
2,
Λ
2
L
L
n
n
n
θ1 2 −
− 1 − θ1 1 −
t1
2
2
θ2
s1
2
H
v,
Λs v, HLt2 ≤ CΛs v, H1−θ
2,
Λ
2
L
n
n
n
θ2 1 −
.
− 1 − θ2 −
t2
2
2
2.8
L
II Let 1.9 hold true.
In the following calculations, we will use the following commutator estimates due to
Kato and Ponce 3:
s s−1 Λ fg − fΛs g p ≤ C ∇f p g
Λ
L1
L
Lq1
Λs f Lp2 g Lq2 ,
2.9
with s > 0 and 1/p 1/p1 1/q1 1/p2 1/q2 .
The proof of the case q n/2 is easier and similar, we omit the details here. Now we
assume n/4 < q < n/2.
6
Abstract and Applied Analysis
Applying Λs to 1.1, testing by Λs u, and using 1.3 and 1.4, we deduce that
1 d
2 dt
2 2
s1 2
|Λ u| Λ u dx Λs1 u Λs2 u dx
s
2
− Λ u · ∇u − u · ∇Λ uΛ u dx Λs B · ∇B − B · ∇Λs BΛs u dx
s
s
s
2.10
B · ∇Λs B · Λs u dx.
Applying Λs to 1.2, testing by Λs B, using 1.3 and 1.4, we infer that
1 d
2 dt
2
2 2
|Λs B|2 Λs1 B dx Λs1 B Λs2 B dx
− Λs u · ∇B − u∇Λs BΛs B dx Λs B · ∇u − B · ∇Λs uΛs B dx
2.11
B · ∇Λs u · Λs B dx.
Summing up 2.10 and 2.11, noting that the last terms of 2.10 and 2.11
disappeared, and using 2.9, we obtain
1 d
2 dt
2
2
|Λs u|2 Λs1 u |Λs B|2 Λs1 B dx
2 2 2 2
Λs1 u Λs2 u Λs1 B Λs2 B dx
− Λ u · ∇u − u · ∇Λ uΛ u dx Λs B · ∇B − B · ∇Λs BΛs u dx
s
s
s
− Λs u · ∇B − u · ∇Λs BΛs B dx Λs B · ∇u − B · ∇Λs uΛs B dx
≤ C∇uLq Λs u2L2q/q−1 C∇BLq Λs BL2q/q−1 Λs uL2q/q−1
C∇BLq Λs B2L2q/q−1 C∇uLq Λs B2L2q/q−1
≤ C∇u, ∇BLq Λs u, B2L2q/q−1
21−θ 2θ
≤ C∇u, ∇BLq Λs1 u, B 2 Λs2 u, B 2
L
L
2
2
1
1/1−θ s1
≤ Λs2 u, B 2 C∇u, ∇BLq
Λ u, B 2 ,
L
L
2
2.12
Abstract and Applied Analysis
7
which yields
u, BL∞ 0,T ;H s1 u, BL2 0,T ;H s2 ≤ C.
2.13
Here we have used the following Gagliardo-Nirenberg inequality:
1−θ θ
Λs u, BL2q/q−1 ≤ CΛs1 u, B 2 Λs2 u, B 2 ,
L
L
2.14
with
−
q−1
n
n
n 1 − θ 1 −
θ 2−
,
2q
2
2
2.15
2q
2n
2n
≤
≤
.
n−2 q−1 n−4
This completes the proof.
3. Proof of Theorem 1.4
We only need to prove the a priori estimates.
Testing 1.1 and 1.2 by v, H, using 1.3 and 1.4, and summing up the results,
we have
1 d
2 dt
v H dx 2
2
|∇v|2 |∇H|2 dx
u · ∇u · Δu dx − B · ∇B · Δu dx u · ∇B · ΔB dx − B · ∇u · ΔB dx
I1 I2 I3 I4 .
3.1
Using 1.4, we see that
I1 ui ∂i u∂2j u dx −
i,j
I2 −
Bi ∂i B∂2j u dx Bi ∂i u∂2j B dx
Bi ∂j B∂i ∂j u dx,
i,j
ui ∂i B∂2j B dx
i,j
i,j
∂j Bi ∂i B∂j u dx −
i,j
I3 I4 −
∂j ui ∂i u∂j u dx,
i,j
i,j
−
3.2
∂j ui ∂i B∂j B dx,
i,j
i,j
∂j Bi ∂i u∂j B dx i,j
Bi ∂j B∂i ∂j u dx.
8
Abstract and Applied Analysis
Inserting the above estimates into 3.1, noting that the last term of I2 and I4 disappeared, we have
1 d
2 dt
v H dx 2
2
2
2
|∇v| |∇H| dx −
∂i u∂j u∂j ui dx i,j
−
∂j ui ∂i B∂j Bi dx
i,j
∂j ui ∂i B∂j B dx i,j
∂i u∂j B∂j Bi dx
i,j
J.
3.3
The proofs of the cases 1.11 and 1.13 are similar, we omit the details here.
I Let 1.14 hold true,
J ≤ C∇uLq ∇u, B2L2q/q−1
21−θ
≤ C∇uLq Δu, BL2
21−θ
≤ C∇uLq v, HL2
≤
Δ∇u, B2θ
L2
∇v, H2θ
L2
3.4
1
1/1−θ
v, H2L2 .
∇v, H2L2 C∇uLq
2
Inserting the above estimates into 3.3 and using the Gronwall inequality yields
v, HL∞ 0,T ;L2 v, HL2 0,T ;H 1 ≤ C.
3.5
Here we have used the Gagliardo-Nirenberg inequality:
θ
wL2q/q−1 ≤ C∇w1−θ
L2 ΔwL2
3.6
with
−
q−1
n
n
n 1 − θ 1 −
θ 2−
,
2q
2
2
2q
2n
2n
≤
≤
.
n−2 q−1 n−4
II Let 1.12 hold true.
3.7
Abstract and Applied Analysis
9
Integrating by parts, using 1.4 we have
J
u∂i ∂j u∂j ui dx −
i,j
ui ∂j ∂i B∂j Bi dx i,j
ui ∂j ∂i B∂j B dx −
i,j
≤ CuLq ∇u, BLt1 Δu, BLt2
u∂i ∂j B∂j Bi dx
i,j
1
1 1
1
q t1 t2
1
2
≤ CuLq Δu, B1−θ
∇Δu, BθL12 Δu, B1−θ
∇Δu, BθL22
L2
L2
1 −θ2
CuLq Δu, B2−θ
∇Δu, BθL12θ2
L2
1 −θ2
≤ CuLq v, H2−θ
∇v, HθL12θ2
L2
≤
1
2/2−θ1 −θ2 v, H2L2 ,
∇v, H2L2 CuLq
2
3.8
which yields 3.5.
Here we have used the Gagliardo-Nirenberg inequalities:
1
∇Δu, BθL12 ,
∇u, BLt1 ≤ CΔu, B1−θ
L2
n
n
n
θ1 2 −
− 1 − θ1 1 −
t1
2
2
2
Δu, BLt2 ≤ CΔu, B1−θ
∇Δu, BθL22 ,
L2
n
n
n
− 1 − θ2 −
θ2 1 −
.
t2
2
2
3.9
This completes the proof.
Acknowledgments
This paper is partially supported by Zhejiang Innovation Project Grant no. T200905, ZJNSF
Grant no. R6090109, and NSFC Grant no. 10971197.
References
1 A. Labovsky and C. Trenchea, “Large eddy simulation for turbulent magnetohydrodynamic flows,”
Journal of Mathematical Analysis and Applications, vol. 377, no. 2, pp. 516–533, 2011.
2 Y. Zhou and J. Fan, “Global well-posedness of a Bardina model,” Applied Mathematics Letters, vol. 24,
no. 5, pp. 605–607, 2011.
3 T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891–907, 1988.
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