Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 408732, 5 pages
doi:10.1155/2011/408732
Research Article
Global Well-Posedness for a Family of
MHD-Alpha-Like Models
Xiaowei He
College of Mathematics, Physics and Information Engineering, Zhejiang Normal University,
Jinhua 321004, China
Correspondence should be addressed to Xiaowei He, jhhxw@zjnu.cn
Received 17 July 2011; Accepted 12 August 2011
Academic Editor: J. C. Butcher
Copyright q 2011 Xiaowei He. 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.
Global well-posedness is proved for a family of n-dimensional MHD-alpha-like models.
1. Introduction
In this paper, we consider a family of MHD-alpha-like models:
1
∂t v −Δθ2 v u · ∇u ∇ p b2 b · ∇b,
2
1.1
∂t H −Δθ2 H u · ∇b − b · ∇u 0,
θ1 θ1 v 1 −α2 Δ
u, H 1 −α2M Δ
b, α > 0, αM > 0,
1.2
div v div u div H div b 0,
1.4
v, H0 v0 , H0 1.5
in Rn n ≥ 3,
1.3
where v is the fluid velocity field, u is the “filtered” fluid velocity, p is the pressure, H
is the magnetic field, and b is the “filtered” magnetic field. α > 0 and αM > 0 are the
length scales and for simplicity we will take α αM 1. The parameter θ1 ≥ 0 affects
2
Journal of Applied Mathematics
the strength of the nonlinear term and θ2 ≥ 0 represents the degree of viscous dissipation
satisfying
3θ1 2θ2 n2
.
2
1.6
When θ1 θ2 1 and n 3, a global well-posedness is proved in 1. The aim of this
paper is to prove a global well-posedness theorem under 1.6. We will prove the following
theorem.
Theorem 1.1. Let u0 , b0 ∈ H s with s ≥ 1, div v0 div u0 div H0 div b0 0 in Rn , and
1.6 holding true. Then for any T > 0, there exists a unique strong solution u, b satisfying
u, b ∈ L∞ 0, T ; H sθ1 ∩ L2 0, T ; H sθ1 θ2 .
1.7
Remark 1.2. For studies on some standard MHD-α or Leray-α models, we refer to 2–7 and
references therein.
2. Proof of Theorem 1.1
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.
Testing 1.1 by u, using 1.3 and 1.4, and letting Λ : −Δ1/2 , we see that
1 d
2 dt
θ1 2
θ2 2 θ1 θ2 2
u Λ u dx Λ u Λ
u dx b · ∇b · udx.
2
2.1
Testing 1.2 by b and using 1.3 and 1.4, we find that
1 d
2 dt
θ1 2
θ2 2 θ1 θ2 2
b Λ b dx Λ b Λ
b dx b · ∇u · bdx.
2
2.2
Summing up 2.1 and 2.2, thanks to the cancellation of the right-hand side of 2.1
and 2.2, we infer that
1 d
2 dt
2
2 2
u, b2 Λθ1 u, b dx Λθ2 u, b Λθ1 θ2 u, b dx 0,
2.3
whence
u, bL2 0,T ;H θ1 θ2 ≤ C.
2.4
Journal of Applied Mathematics
3
Case 1. θ1 θ2 > 1.
In the following calculations, we will use the following commutator estimates due to
Kato and Ponce 8:
s s−1 Λ fg − fΛs g p ≤ C ∇f p g
Λ
L1
L
Lq1
Λs f Lp2 g Lq2 ,
2.5
with s > 0 and 1/p 1/p1 1/q1 1/p2 1/q2 .
We will also use the Sobolev inequality:
n
n
,
∇uLp ≤ C
Λθ1 θ2 u
2 1 − θ1 θ2 −
L
p
2
2.6
and the Gagliardo-Nirenberg inequality:
Λs u2L2p/p−1 ≤ C
Λsθ1 u
2 Λsθ1 θ2 u
2 .
L
L
2.7
Taking Λs to 1.1, testing by Λs u, and using 1.3 and 1.4, we infer that
1 d
2 dt
2
2 2
|Λs u|2 Λsθ1 u dx Λsθ2 u Λsθ1 θ2 u dx
− Λs u · ∇u − u · ∇Λs uΛs udx Λs b · ∇b − b · ∇Λs bΛs udx
2.8
b · ∇Λs b · Λs udx.
Taking Λs to 1.2, testing by Λs b, and using 1.3 and 1.4, we deduce that
1 d
2 dt
2 2
sθ1 2
|Λ b| Λ b dx Λsθ2 b Λsθ1 θ2 b dx
s
2
− Λ u · ∇b − u · ∇Λ bΛ bdx Λs b · ∇u − b · ∇Λs uΛs bdx
s
b · ∇Λs u · Λs bdx.
s
s
2.9
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Journal of Applied Mathematics
Summing up 2.8 and 2.9, thanks to the cancellation of the right-hand side of 2.8
and 2.9, and using 2.5, 2.6 and 2.7, we conclude that
1 d
2 dt
2
2 2
sθ1
|Λ u, b| Λ u, b dx Λsθ2 u, b Λsθ1 θ2 u, b dx
2
s
≤ C∇uLp Λs u2L2p/p−1 C∇bLp Λs bL2p/p−1 Λs uL2p/p−1 C∇uLp Λs b2L2p/p−1
≤ C∇u, bLp Λs u, b2L2p/p−1
≤ C
Λθ1 θ2 u, b
2 Λsθ1 u, b
2 Λsθ1 θ2 u, b
L
≤
2.10
L2
L
2
2 2
1
sθ1 θ2
u, b
2 C
Λθ1 θ2 u, b
2 Λsθ1 u, b
2 ,
Λ
L
L
L
2
which implies 1.7.
Case 2. 0 < θ1 θ2 ≤ 1 only when n 3.
Testing 1.1 by v, using 1.4, we see that
1 d
2 dt
θ2 2
v dx Λ v dx b · ∇b − u · ∇uvdx
2
≤ bLp1 ∇bL2p1 /p1 −2 uLp1 ∇uL2p1 /p1 −2 vL2
≤ u, bLp1 ∇u, bL2p1 /p1 −2 vL2
≤ Cu, bH θ1 θ2 Λθ2 v, H
2 vL2 .
2.11
L
Here we have used the Sobolev inequalities
3
3
,
u, bLp1 ≤ Cu, bH θ1 θ2 − θ1 θ2 −
p1
2
3 p1 − 2
3
θ2
.
θ2 2θ1 −
∇u, bL2p1 /p1 −2 ≤ C
Λ v, H
2 1 −
L
2p1
2
2.12
Similarly, testing 1.2 by H and using 1.4 and 2.12, we find that
1 d
2 dt
H 2 dx θ2 2
Λ H dx b · ∇u − u · ∇bHdx
≤ u, bLp1 ∇u, bL2p1 /p1 −2 HL2
≤ Cu, bH θ1 θ2 Λθ2 v, H
2 HL2 .
2.13
L
Combining 2.11 and 2.13 and using 2.4 and the Gronwall inequality, we have
u, bL2 0,T ;H θ2 2θ1 ≤ C.
2.14
Journal of Applied Mathematics
5
Similarly to 2.10, we have
1 d
2 dt
2
2 2
|Λs u, b|2 Λsθ1 u, b dx Λsθ2 u, b Λsθ1 θ2 u, b dx
≤ C∇u, bLp2 Λs u, b2L2p2 /p2 −1
2.15
21−α1 2α1
sθ1 θ2
≤ Cu, bH θ2 2θ1 Λsθ1 u, b
2
u, b
2
Λ
L
L
2
2
1
sθ1
1 ≤ Λsθ1 θ2 u, b
2 Cu, b1/1−α
b
u,
2,
Λ
θ
2θ
H 2 1
L
L
2
which implies 1.7 by 1/1 − α1 ≤ 2. Here we have used the Sobolev inequality:
n
n
< θ2 2θ1 −
∇u, bLp2 ≤ Cu, bH θ2 2θ1 1 −
p2
2
2.16
and the Gagliardo-Nirenberg inequality:
1−α1 α1
Λs u, bL2p2 /p2 −1 ≤ C
Λsθ1 u, b
2 Λsθ1 θ2 u, b
2 ,
L
L
2.17
with −p2 − 1/2p2 n α1 θ2 θ1 − n/2 and p2 ≥ 2 ≥ 3/2θ1 θ2 . This completes the proof.
References
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