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Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient
Computers and Mathematics with Applications (
)
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Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient✩ Liying Chen a , Haifang Yu b , Yang Liu c, * a b c
College of Mathematics, Inner Mogolia University for Nationalities, Tongliao, 028000, China Department of Mathematics and Computer Science, Chaoyang Teachers College, Liaoning, 122000, China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
article
a b s t r a c t
info
Article history: Received 6 March 2017 Received in revised form 25 July 2017 Accepted 25 July 2017 Available online xxxx
This paper proves a regularity criterion for 3D density-dependent incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient. In particular, we build a blowup criterion just in terms of the gradient of density and velocity, and the initial density need not be strictly positive. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Regularity criterion Density-dependent viscosity Magnetohydrodynamic flows Vacuum
1. Introduction This paper is devoted the study of the incompressible MHD equations:
⎧ ρt + div(ρ u) = 0, ⎪ ⎪ ⎪ ⎨(ρ u) + div(ρ u ⊗ u) + ∇ P = div(µ(ρ )∇ u) + b · ∇ b, t ⎪ b + u · ∇ b = η 1 b + b · ∇ u, t ⎪ ⎪ ⎩ divu = 0, divb = 0,
(1.1)
in Ω × (0, ∞), where Ω is a bounded domain with smooth boundary in R3 . Here ρ, u, b, P denote the density, velocity, magnetic field and pressure of the fluid, respectively. The positive constants µ, η represent viscosity of fluid, resistivity coefficient respectively. The viscosity coefficient µ = µ(ρ ) is a general function of density, which is assumed to satisfy
µ ∈ C 1 [0, ∞) and 0 < µ ≤ µ ≤ µ < ∞ on [0, ∞),
(1.2)
for some positive constants µ, µ. Without loss of generality, η is normalized to 1. To complete Eqs. (1.1), we consider an initial boundary value problem for (1.1) with the following initial and boundary conditions (ρ, u, b)|t =0 = (ρ0 , u0 , b0 )(x),
in Ω ,
(1.3)
✩ Liying Chen’s research was supported by the Scientific Research Fund of Inner Mogolia University for Nationalities (NMDYB1473). Corresponding author. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Liu).
*
http://dx.doi.org/10.1016/j.camwa.2017.07.042 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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L. Chen et al. / Computers and Mathematics with Applications (
u = 0,
b · ν = 0,
curlb × ν = 0
on ∂ Ω × (0, T ),
)
–
(1.4)
where ν is the unit outward normal vector to ∂ Ω . When b = 0, then systems (1.1)–(1.4) are the well-known Navier–Stokes equations with density-dependent viscosity coefficient. Cho and Kim [1] proved local existence of unique strong solutions for all initial data satisfying a natural compatibility condition for the case of vacuum. They also built following blowup criterion: sup (∥∇ρ (t)∥Lq + ∥∇ u(t)∥L2 ) = ∞.
(1.5)
0≤t ≤T ∗
If T ∗ < ∞ is the maximal existence time of the local strong solutions. For related studies and problems, we refer to [2–5] and references therein. When µ is a constant, the systems (1.1)–(1.4) are density-dependent incompressible magnetohydrodynamic flows. Chen et al. [6] established local existence of unique strong solutions and a blowup criterion. Fan et al. [7] proved the following regularity criterion: u ∈ Ls (0, T ; Lrw (Ω ))
with
2 s
+
3 r
≤ 1, 3 < r ≤ ∞,
(1.6)
where Lrw denotes the weak Lr -space. For more recent results about the incompressible magnetohydrodynamic flows, the readers can refer to [8–11] and references therein. For the homogeneous viscous incompressible MHD equations (i.e. ρ (x, t) = constant), there are also many mathematical results. Duraut and Lions [12] constructed a class of weak solutions with finite energy and a class of local strong solutions. However, the regularity and uniqueness of such a given weak solution remain a challenging open problem. There are many sufficient conditions to guarantee the regularity of the weak solution (see [13–16]). The aim of this paper is to prove a new regularity criterion to the problem (1.1)–(1.4). Before stating our results, we first recall the following local well-posedness result by Wu [17]. Proposition 1.1. Let Ω be a bounded smooth domain in R3 and q ∈ (3, ∞) be a fixed constant. Suppose that the initial data (ρ0 , u0 , b0 , P0 ) satisfies the regularity conditions 0 ≤ ρ0 ≤ ρ, ¯
ρ0 ∈ W 1,q ,
u0 , b0 ∈ H01,σ ∩ H 2 ,
(1.7)
and the compatibility condition
− div(µ(ρ0 )∇ u0 ) + ∇ P0 − b0 ∇ b0 =
√ ρ0 g
in Ω
(1.8)
for some (P0 , g) ∈ H 1 × L2 . Then there exist a positive small time T ∗ and a unique strong solution (ρ, u, b, P) of (1.1)–(1.4) such that
⎧ 0 ≤ ρ (x, t) ≤ ρ, ¯ ρt ∈ C ([0, T ∗ ]; Lq ), ⎪ ⎪ ∗ 1 2 ⎪ ) ∩ L2 (0, T ∗ ; W 2,r ), ⎨u ∈ C ([0, T ]; H0 ∩ H√ 2 ∗ 1 ρ ut ∈ L∞ (0, T ∗ ; L2 ), ut ∈ L (0, T ; H0 ), ⎪ ∞ ∗ 1 2 1,r ⎪ ⎪ ⎩P ∈ L (0, T ;∗ H ) 1∩ L (0, ∞;∞W ),∗ 2 ∇ b ∈ C ([0, T ]; H ), bt ∈ L (0, T ; L ) ∩ L2 (0, T ∗ ; H01 )
(1.9)
for some r with 3 < r < min{6, q}. Motivated by [7] and [1], the main purpose of this paper is to extend result of Fan et al. [7] to the density-dependent viscosity coefficient case. More precisely, the main result in this paper can be stated as follows. Theorem 1.1. Assume that the initial data (ρ0 , u0 , b0 ) satisfies (1.7) and (1.8). Let (ρ, u, b) be a strong solution of the problem (1.1)–(1.4) satisfying (1.7). If 0 < T ∗ < ∞ is the maximal time of existence, then
(
)
lim ∥∇ρ∥L∞ (0,T ;Lq ) + ∥u∥Ls (0,T ;Lrw ) = ∞,
T →T ∗
(1.10)
for any r and s satisfying 2 s
+
3 r
≤ 1,
3 < r ≤ ∞,
where Lrw denotes the weak Lr -space.
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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2. Preliminaries Throughout this paper, we assume that Ω ⊂ R3 is a bounded domain with smooth boundary ∂ Ω . For simplicity, we denote
∫
∫ fdx = Ω
fdx.
For 1 ≤ r ≤ ∞ and k ∈ N, the Sobolev spaces are defined in a standard way. W k,r = {f ∈ Lr : ∇ k f ∈ Lr },
Lr = Lr (Ω ), k
H =W
k,2
,
H01 = C0∞ ,
∞ C0∞ ,σ = {f ∈ C0 : divf = 0}.
H01,σ = C0∞ ,σ ,
closure in the norm of H 1 .
Lemma 2.1 ([18]). Let Ω ⊂ R3 be a smooth bounded domain, let f : Ω → R3 be a smooth vector field, and let 1 < p < ∞. Then ∫ ∫ ∫ p 2 4(p − 2) p−2 p−2 2 − 1f · f |f | dx = |f | |∇ f | dx + |∇|f | 2 | dx p2
∫
p−2
|f |
− ∂Ω
(f · ∇ )f · ν dσ −
∫ ∂Ω
|f |p−2 (curlf × ν ) · fdσ .
(2.1)
Lemma 2.2 ([19,20]). Let Ω be a smooth and bounded open set, and let 1 < p < ∞. Then we have the estimate 1− 1
1
∥f ∥Lp (∂ Ω ) ≤ C ∥f ∥Lp p ∥f ∥Wp 1,p
(2.2)
holds for any f ∈ W 1,p . High-order a priori estimates rely on the following regularity results for density-dependent Stokes equations. Lemma 2.3 ([4,5]). Assume that ρ ∈ W 1,q for some 3 < q < ∞, and 0 ≤ ρ ≤ ρ¯ . Let (u, P) ∈ H01,σ × L2 be the unique weak solution to the boundary value problem:
− div(µ(ρ )∇ u) + ∇ P = F ,
in Ω ,
divu = 0
∫
Pdx = 0.
(2.3)
Then the following regularity estimates hold for (u, P): (1) If F ∈ L2 , then (u, P) ∈ H 2 × H 1 and q
∥u∥H 2 + ∥P ∥H 1 ≤ C ∥F ∥L2 (1 + ∥∇ρ∥Lq ) q−3 . (2) If F ∈ L for some r ∈ (3, q), then (u, P) ∈ W r
2 ,r
∥u∥W 2,r + ∥P ∥W 1,r ≤ C ∥F ∥Lr (1 + ∥∇ρ∥Lq )
(2.4)
×W
qr 2(q−r)
1,r
such that
.
(2.5)
(3) If F ∈ H , then (u, P) ∈ H × H and 1
3
2
∥u∥H 3 + ∥P ∥H 2 ≤ C˜ ∥F ∥Lr (1 + ∥ρ∥W 2,q )N
(2.6)
for some N = N(3, q) > 0. The constant C˜ depends also on ∥∂ 2 µ/∂ρ 2 ∥C . The following lemma has been proved in [2,21], which plays an important role in the subsequent proof. Lemma 2.4. Assume g ∈ H 1 and f ∈ Lrw with r ∈ (3, ∞], then f · g ∈ L2 . Furthermore, for any ε > 0 and r ∈ (3, ∞], we have
( ) s ∥f · g ∥2L2 ≤ ε∥g ∥2H 1 + C (ε ) ∥f ∥L2r + 1 ∥g ∥2L2 w
(2.7)
where C is positive constant depending only on ε , r and the domain Ω . 3. Proof of Theorem 1.1 Let 0 < T ∗ < ∞ be the maximum time for the existence of strong solution (ρ, u, b, P) to (1.1)–(1.4). Suppose that (1.10) were false, that is M0 = lim ∥∇ρ∥L∞ (0,T ;Lq ) + ∥u∥Ls (0,T ;Lrw ) < ∞.
(
T →T ∗
)
(3.1)
Under the condition (3.1), one will extend existence time of the strong solutions to (1.1)–(1.4) beyond T ∗ , which contradicts the definition of maximum existence time. Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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Lemma 3.1. For any 0 ≤ T < T ∗ , it holds that
√
2
sup (∥ρ∥Lp + ∥ ρ u∥L2 + ∥b∥2L2 ) +
0≤t ≤T
T
∫
∫
(
) |∇ u|2 + |∇ b|2 dxds ≤ C .
(3.2)
0
Proof. Multiplying (1.1)1 by pρ p−1 and integrating the resulting equation over Ω , then it is easy to deduce that
∥ρ (t)∥Lp = ∥ρ0 ∥Lp (1 ≤ p ≤ ∞).
(3.3)
Multiplying (1.1)2 , (1.1)3 by u, b in L2 and integrating by parts, summing up the resulting equations, we infer from (1.1)4 that 1 d
√
2 dt
2
(∥ ρ u∥L2 + ∥b∥2L2 ) + µ∥∇ u∥2L2 + ∥∇ b∥2L2 ≤ 0.
(3.4)
Integrating (3.4) over (0, T ) yields 1 2
√
2
(∥ ρ u∥L2 + ∥b∥2L2 ) +
T
∫ 0
(µ∥∇ u∥2L2 + ∥∇ b∥2L2 ) ≤
1 2
√
2
(∥ ρ0 u0 ∥L2 + ∥b0 ∥2L2 ),
(3.5)
which completes the proof. □ Lemma 3.2. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ , sup (∥∇ u∥2L2 + ∥∇ b∥2L2 ) +
0≤t ≤T
T
∫ 0
√
2
(∥ ρ u˙ ∥L2 + ∥∇ b∥2H 1 + ∥bt ∥2L2 )dt ≤ C .
(3.6)
Proof. Multiplying (1.1)3 by |b|p−2 b(2 ≤ p < ∞), using (1.4), (2.1), (2.2), (3.1), (2.7) and Gagliardo–Nirenberg inequality, we derive 1 d
∫
|b|p dx +
p dt
∫
1
| b| ∂Ω
(b · ∇ )ν · bdσ +
∫
|b| dσ − p
≤ ∥∇ν∥
L∞
∫ ∂Ω
≤
2(p − 2)
∫ ∫
∫
(b · ∇ )u · |b|p−2 bdx
p
p
|u| |b| 2 |∇|b| 2 |dx p 2
∫
p 2
∫
|∇|b| 2 | dx + ε
p2
p 2
|∇|b| 2 | dx
p
bi u∂i (|b|p−2 b)dx
|∇|b| 2 | dx + C
p2 2(p − 2)
∑∫
∫
∫
i
|b|p dσ + C
≤C ≤
∂Ω
4(p − 2)
|b|p−2 |∇ b|2 dx +
2
p−2
=−
∫
|b|p dx + C
∫
p 2
|u|2 | |b| 2 | dx
p 2
|∇|b| 2 | dx + C (ε)(∥u∥sLrw + 1)
∫
|b|p dx.
(3.7)
Thus, choosing ε enough small, we have T
∫
∫
∥b∥L∞ (0,T ;Lq ) +
|b|q−2 |∇ b|2 dxdt ≤ C .
(3.8)
0
From (3.1), we get that
µ(ρ )t + u · ∇µ(ρ ) = 0.
(3.9)
Multiplying (1.1)2 by ut and integrating (by parts) over Ω , we have d dt
∫
µ(ρ )|∇ u|2 dx +
∫ =− ≤ε
∫
d
∫
dt
∫ (b · ∇ )u · bdx +
ρ|˙u|2 dx
u · ∇µ(ρ )|∇ u|2 + ρ u˙ · (u · ∇ u) dx +
(
ρ|˙u|2 dx + C (ε, δ )
)
∫ (
∫
|u| |∇ρ| |∇ u|2 dx +
((bt · ∇ )u · b + (b · ∇ )u · bt )dx
∫
∫ ∫ ) ρ|u|2 |∇ u|2 dx + δ |bt |2 dx + C (ε, δ ) |b|2 |∇ u|2 dx.
(3.10)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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Indeed, by (2.7), Young, Hölder and Sobolev inequality, we get
∫
|u| |∇ρ| |∇ u|2 dx +
∫
ρ|u|2 |∇ u|2 dx ≤ C (ε )
∫
|u|2 |∇ u|2 dx + ε
∫
≤ C (ε )
∫
|u|2 |∇ u|2 dx + ε
∫
≤ (δ + ε )∥∇ u∥
2 H1
|∇ρ|2 |∇ u|2 dx ∥∇ρ∥Lq ∥∇ u∥2 q−2 dx
+ C (ε, δ )(∥u∥
s Lrω
L 2q u 2L2
+ 1)∥∇ ∥
(3.11)
and
∫
|b|2 |∇ u|2 dx ≤ C ∥b∥2L6 ∥∇ u∥2L3 ≤ C ∥∇ u∥L2 ∥∇ u∥L6 ≤ ε∥∇ u∥2H 1 + C (ε)∥∇ u∥2L2 .
(3.12)
Similarly, taking L2 inner product of (1.1)3 with bt , we achieve d
∫
|∇ b|2 dx +
dt
∫
|bt |2 dx ≤ C
∫
( ) |b| |∇ u| |bt | + |u| |∇ b| |bt | dx ∫ ∫ ) ( 2 2 ≤ ε |bt | dx + C (ε ) |u| |∇ b|2 + |b|2 |∇ u|2 dx ∫ ≤ ε |bt |2 dx + δ (∥∇ u∥2H 1 + ∥∇ b∥2H 1 ) + C (ε, δ )(∥u∥sLrω + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ).
(3.13)
Applying Lemma 2.3 with F ≜ −ρ ut − ρ (u · ∇ )u + curlb × b, we derive that q
∥u∥H 2 + ∥P ∥H 1 ≤ C ∥F ∥L2 (1 + ∥∇ρ∥Lq ) q−3 ≤ C ∥ − ρ ut − ρ u · ∇ u + curlb × b∥L2 ) ( √ ≤ C ∥ ρ u˙ ∥L2 + ∥ |b| |∇ b|∥L2 ( √ ) ≤ C ∥ ρ u˙ ∥L2 + C ∥b∥L6 ∥∇ b∥L3 ( √ 1 1 ) ≤ C ∥ ρ u˙ ∥L2 + ∥∇ b∥L22 ∥∇ b∥H2 1 √ ≤ C ∥ ρ u˙ ∥L2 + κ∥∇ b∥H 1 + C (κ )∥∇ b∥L2 .
(3.14)
In a similar way that
( ) ∥b∥H 2 ≤ C ∥bt ∥L2 + ∥(b · ∇ )u∥L2 + ∥(u · ∇ )b∥L2 ≤ C ∥bt ∥L2 + ϵ (∥∇ u∥H 1 + ∥∇ b∥H 1 ) + C (ϵ )(∥u∥sLrω + 1)(∥∇ u∥L2 + ∥∇ b∥L2 ),
(3.15)
then, we have by taking κ , ϵ small enough
√ 2 ∥u∥2H 2 + ∥b∥2H 2 ≤ C (∥bt ∥2L2 + ∥ ρ u˙ ∥L2 ) + C (∥u∥sLrω + 1)(∥∇ u∥L2 + ∥∇ b∥L2 ).
(3.16)
Collecting (3.10), (3.11), (3.13) and (3.16) deduces
∫ ( ) ( 2) µ(ρ )|∇ u|2 + |∇ b|2 dx + ρ|˙u|2 + |bt |2 + |∇ 2 b| dx dt ∫ ∫ ∫ d ≤− (b · ∇ )u · bdx + ε ρ|˙u|2 dx + δ |bt |2 dx d
∫
dt
+ (ε + δ )(∥∇ u∥2H 1 + ∥∇ b∥2H 1 ) + C (ε, δ )(∥u∥sLrω + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ) ∫ d √ 2 ≤− (b · ∇ )u · bdx + (ε + δ )(∥ ρ u˙ ∥L2 + ∥bt ∥2L2 ) + C (ε, δ )(∥u∥sLr + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ). ω dt
(3.17)
By (3.1), (3.2), (3.8) and Cauchy–Schwarz inequality, it is easily seen that
∫ C
|b|2 |∇ u|dx ≤
µ 4
∥∇ u∥2L2 + C .
(3.18)
Taking this into account, by choosing ε , δ small enough, we then conclude from (3.17) and the Gronwall inequality that (3.6) holds for any 0 ≤ T < T ∗ . The proof of Lemma 3.2 is therefore complete. □ Lemma 3.3. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ ,
( √
2
sup ∥ ρ ut ∥L2 + ∥bt ∥2L2 +
0≤t ≤T
)
T
∫ 0
( ) ∥∇ ut ∥2L2 + ∥∇ bt ∥2L2 dt ≤ C .
(3.19)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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Proof. Differentiating (1.1)2 with respect to t, we see that
[ (1 )] ρ utt + ρ u · ∇ ut − div(µ(ρ )∇ ut ) + ∇ Pt + |b|2 2
t
= div(b ⊗ b)t + div(µ′ ρt ∇ u) − ρt (ut + u · ∇ u) − ρ ut · ∇ u.
(3.20)
Multiplying (3.20) by ut and integrating (by parts) over Ω , one arrives at 1 d 2 dt
∫
ρ|ut |2 dx + µ
∫
∫ ∫ ρ|u| |ut | |∇ ut |dx + C ρ|u| |∇ u|2 |ut |dx + C ρ|u|2 |∇ 2 u| |ut |dx ∫ ∫ ∫ + C ρ|u|2 |∇ u| |∇ ut |dx + C ρ|ut |2 |∇ u|dx + C |b| |bt | |∇ ut |dx
|∇ ut |2 dx ≤ C
∫
∫ +C
|u| |∇ρ| |∇ u| |∇ ut |dx ≜
7 ∑
Ii .
(3.21)
i=1
By (3.6), Hölder, interpolation, Sobolev’s and Young’s inequality, we get
√
√
I1 ≤ C ∥ ρ∥L∞ ∥u∥L6 ∥ ρ ut ∥L3 ∥∇ ut ∥L2 1 √ 1 √ ≤ C ∥∇ρ∥Lq ∥∇ u∥L2 ∥ ρ ut ∥L22 ∥ ρ ut ∥L26 ∥∇ ut ∥L2 1 3 √ ≤ C ∥ ρ ut ∥L22 ∥∇ ut ∥L22 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥ ρ ut ∥L2 , √ I2 ≤ C ∥ ρ∥L∞ ∥u∥L6 ∥∇ u∥L6 ∥ut ∥L6 ∥∇ u∥L2
(3.22)
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ u∥H 1 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ u∥2H 1 , 2 L6
(3.23)
2
I3 ≤ C ∥ρ∥L∞ ∥u∥ ∥∇ u∥L2 ∥∇ ut ∥L6
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ 2 u∥L2 ∥∇ ut ∥L2 2
≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ 2 u∥L2 , √ I4 ≤ C ∥ ρ∥L∞ ∥u∥2L6 ∥∇ u∥L6 ∥∇ ut ∥L2
(3.24)
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ u∥H 1 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ u∥2H 1 , √ √ I5 ≤ C ∥ ρ∥L∞ ∥ut ∥L6 ∥ ρ ut ∥L3 ∥∇ u∥L2 1 √ 1 √ ≤ C ∥∇ρ∥Lq ∥∇ ut ∥L2 ∥ ρ ut ∥L22 ∥ ρ ut ∥L26
(3.25)
3 √ 1 ≤ C ∥∇ ut ∥L22 ∥ ρ ut ∥L22 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥ ρ ut ∥L2 ,
(3.26)
I6 ≤ C ∥b∥L6 ∥bt ∥L3 ∥∇ ut ∥L2 1
1
≤ C ∥∇ b∥L2 ∥bt ∥L22 ∥bt ∥L26 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + δ∥∇ bt ∥2L2 + C (ε, δ )∥bt ∥2L2 .
(3.27)
It remains to estimate the term I7 . Indeed, on the one hand, using (3.8), Cauchy and interpolation inequality, we deduce
√ ∥u∥H 2 + ∥P ∥H 1 ≤ C (∥ ρ ut ∥L2 + ∥ |u| |∇ u|∥L2 + ∥ |b| |∇ b|∥L2 ) √ ≤ C ∥ ρ ut ∥L2 + C ∥u∥L6 ∥∇ u∥L3 + C ∥b∥L6 ∥∇ b∥L3 3 1 3 1 √ ≤ C ∥ ρ ut ∥L2 + C ∥∇ u∥L22 ∥∇ u∥H2 1 + C ∥∇ b∥L22 ∥∇ b∥H2 1 ( √ ) 1 1 ≤ C ∥ ρ ut ∥L2 + 1 + ∥∇ b∥H 1 + ∥∇ u∥H 1 , 2
2
(3.28)
and
∥b∥H 1 ≤ C ∥bt ∥L2 + C ∥ |u| |∇ b|∥L2 + C ∥|b| |∇ u|∥L2 ≤ C ∥bt ∥L2 + C ∥u∥L6 ∥∇ b∥L3 + C ∥b∥L6 ∥∇ u∥L3 Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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7
1
≤ C ∥bt ∥L2 + C ∥∇ u∥L2 ∥∇ b∥L22 ∥∇ b∥H2 1 + C ∥∇ b∥L2 ∥∇ u∥L22 ∥∇ u∥H2 1 1
1
2
2
≤ C ∥bt ∥L2 + ∥∇ b∥H 1 + ∥∇ u∥H 1 + C
(3.29)
which implies
) ( √ ∥u∥H 2 + ∥b∥H 1 ≤ C ∥ ρ ut ∥L2 + ∥bt ∥L2 + 1 ,
(3.30)
On the other hand, by Lemma 2.3, we get pr
∥u∥W 2,r + ∥P ∥W 1,r ≤ C ∥F ∥Lr (1 + ∥∇µ(ρ )∥Lp ) 2(p−r) ≤ C (∥ρ ut ∥Lr + ∥ρ u · ∇ u∥Lr + ∥ |b| |∇ b|∥Lr ) 6−r 3r −6 √ ≤ C ∥ ρ ut ∥L22r ∥∇ ut ∥L22r + C ∥u∥L6 ∥∇ u∥ 6r + C ∥b∥L∞ ∥∇ b∥Lr L 6−r
√
6−r 2r L2
≤ C ∥ ρ ut ∥
3r −6 2r L2
∥∇ ut ∥
+ C ∥∇ u∥
6(r −1) 5r −6 L2
4r −6
1
3
,
∥∇ u∥W5r 1−,6r + C ∥b∥L22 ∥∇ b∥H2 1
(3.31)
which deduces
(
∥u∥W 2,r + ∥P ∥W 1,r
6(r −1) 6 −r 3r −6 1 3 √ ≤ C ∥ ρ ut ∥L22r ∥∇ ut ∥L22r + ∥∇ u∥L2 r + ∥∇ b∥L22 ∥∇ b∥H2 1 ( ) 3 √ ≤ C ∥ ρ ut ∥L2 + ∥∇ ut ∥L2 + ∥∇ b∥H2 1 + 1 .
)
(3.32)
Thus, by using (3.1), Hölder, Gagliardo–Nirenberg and Young inequalities, we obtain I7 ≤ C ∥u∥L6 ∥∇ρ∥Lq ∥∇ u∥
3q
L q −3
∥∇ ut ∥L2
2q−6
q+6
3q ≤ C ∥∇ u∥L2 ∥∇ρ∥Lq ∥∇ u∥L23q ∥∇ u∥L∞ ∥∇ ut ∥L2 q+6
≤ C ∥∇ u∥W3q1,r ∥∇ ut ∥L2 +6 ( ) q3q 3 √ 2 ≤ C ∥ ρ ut ∥L2 + ∥∇ ut ∥L2 + ∥∇ b∥H 1 + 1 ∥∇ ut ∥L2 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )(∥ ρ ut ∥L2 + ∥∇ b∥4H 1 + 1).
(3.33)
Differentiating (1.1)3 with respect to t, and multiplying the resulting equation by bt in L2 , we deduce after integrating by parts 1 d
∫
2 dt
|bt |2 dx +
∫
|∇ bt |2 dx = −
∫ (ut · ∇ b − bt · ∇ u − b · ∇ ut ) · bt dx
≤ C ∥ut ∥L6 ∥∇ b∥L2 ∥bt ∥L3 + C ∥∇ u∥L2 ∥bt ∥2L4 + C ∥∇ ut ∥L2 ∥b∥L6 ∥bt ∥L3 1
1
1
3
≤ C ∥∇ ut ∥L2 ∥∇ b∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 + C ∥∇ u∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 1
1
+ C ∥∇ ut ∥L2 ∥∇ b∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 ≤ ε∥∇ ut ∥2L2 + δ∥∇ bt ∥2L2 + C (ε, δ )∥bt ∥2L2 .
(3.34)
Combining (3.21)– and choosing ε , δ suitably small, we obtain 1 d
∫
2 dt
( ) ρ|ut |2 + |bt |2 dx +
∫
(
) √ 2 µ|∇ ut |2 + |∇ bt |2 dx ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + ∥∇ b∥4H 1 ) √ √ 2 4 ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + ∥ ρ ut ∥L2 + ∥bt ∥4L2 ) √ √ 2 2 ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + 1)(∥ ρ ut ∥L2 + ∥bt ∥2L2 ),
(3.35)
which, together with the Gronwall inequality and (3.6), completes the proof. □ Lemma 3.4. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ ,
(
)
∫
T
sup ∥ρt ∥Lq + ∥P ∥H 1 + ∥u∥H 2 + ∥b∥H 2 +
0≤t ≤T
0
( 2 ) ∥u∥W 2,r + ∥P ∥2W 1,r dt ≤ C .
(3.36)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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Proof. By (3.30) and (3.32), it is easy to deduce
(
)
∫
T
sup ∥P ∥H 1 + ∥u∥H 2 + ∥b∥H 2 +
0≤t ≤T
0
) ( 2 ∥u∥W 2,r + ∥P ∥2W 1,r dt ≤ C ,
(3.37)
which, together with (1.1)1 yields
∥ρt ∥Lq ≤ ∥u∥L∞ ∥∇ρ∥Lq ≤ C ∥u∥H 2 ∥∇ρ∥Lq ≤ C .
(3.38)
This completes the proof. □ After having Lemmas 3.1–3.4 at hand, it is easy to apply Proposition 1.1 to extend the strong solution (ρ, u, P , b) beyond time T ∗ . Therefore, we complete the proof of Theorem 1.1. References [1] Y. Cho, H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal. 59 (2004) 465–489. [2] H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal. 37 (2006) 1417–1434. [3] X.D. Huang, Y. Wang, Global strong solution with vacuum to the two dimensional density dependent Navier-Stokes system, SIAM J. Math. Anal. 46 (2014) 1771–1788. [4] X.D. Huang, Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations 259 (2015) 1606–1627. [5] J. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density dependent viscosity coefficient, J. Differential Equations 259 (2015) 1722–1742. [6] Q. Chen, Z. Tan, Y.J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci. 34 (2011) 94–107. [7] J. Fan, F. Li, G. Nakamura, Z. Tan, Regularity criteria for the three-dimensional magneto-hydrodynamic equations, J. Differential Equations 256 (2014) 2858–2875. [8] Y. Zhou, J. Fan, A regularity criterion for the density-dependent magnetohydrodynamic equations, Math. Methods Appl. Sci. 33 (2010) 1350–1355. [9] J. Fan, W. Sun, J. Yin, Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain, Bound. Value Probl. 90 (2016) 19. [10] X. Si, X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys. 67 (2016) 15. [11] F. Chen, Y. Li, Y. Zhao, Global well-posedness for the incompressible MHD equations with variable viscosity and conductivity, J. Math. Anal. Appl. 447 (2017) 1051–1071. [12] G. Duraut, J.L. Lions, Inéquations en thermoéasticit et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972) 241–279. [13] C. Cao, J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations 248 (2010) 2263–2274. [14] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equation in the multiplier space, Z. Angew. Math. Phys. 61 (2010) 193–199. [15] X. Chen, S. Gala, Z. Guo, A new regularity criterion in terms of the direction of the velocity for the MHD equations, Acta Appl. Math. 113 (2011) 207–213. [16] L. Ni, Z. Guo, Y. Zhou, Some new regularity criteria for the 3D MHD equations, J. Math. Anal. Appl. 396 (2012) 108–118. [17] H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl. 61 (2011) 2742–2753. [18] H. Beirão da Veiga, F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An Lp theory, J. Math. Fluid Mech. 12 (2010) 397–411. [19] R.A. Adams, J.F. Fournier, Sobolev Spaces, second ed, in: Pure and Appl. Math. (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. [20] A. Lunardi, Interpolation Theory, second ed, in: Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009. [21] X. Xu, J. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci. 22 (2012) 1150010 23pp.
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Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient✩ Liying Chen a , Haifang Yu b , Yang Liu c, * a b c
College of Mathematics, Inner Mogolia University for Nationalities, Tongliao, 028000, China Department of Mathematics and Computer Science, Chaoyang Teachers College, Liaoning, 122000, China School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
article
a b s t r a c t
info
Article history: Received 6 March 2017 Received in revised form 25 July 2017 Accepted 25 July 2017 Available online xxxx
This paper proves a regularity criterion for 3D density-dependent incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient. In particular, we build a blowup criterion just in terms of the gradient of density and velocity, and the initial density need not be strictly positive. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Regularity criterion Density-dependent viscosity Magnetohydrodynamic flows Vacuum
1. Introduction This paper is devoted the study of the incompressible MHD equations:
⎧ ρt + div(ρ u) = 0, ⎪ ⎪ ⎪ ⎨(ρ u) + div(ρ u ⊗ u) + ∇ P = div(µ(ρ )∇ u) + b · ∇ b, t ⎪ b + u · ∇ b = η 1 b + b · ∇ u, t ⎪ ⎪ ⎩ divu = 0, divb = 0,
(1.1)
in Ω × (0, ∞), where Ω is a bounded domain with smooth boundary in R3 . Here ρ, u, b, P denote the density, velocity, magnetic field and pressure of the fluid, respectively. The positive constants µ, η represent viscosity of fluid, resistivity coefficient respectively. The viscosity coefficient µ = µ(ρ ) is a general function of density, which is assumed to satisfy
µ ∈ C 1 [0, ∞) and 0 < µ ≤ µ ≤ µ < ∞ on [0, ∞),
(1.2)
for some positive constants µ, µ. Without loss of generality, η is normalized to 1. To complete Eqs. (1.1), we consider an initial boundary value problem for (1.1) with the following initial and boundary conditions (ρ, u, b)|t =0 = (ρ0 , u0 , b0 )(x),
in Ω ,
(1.3)
✩ Liying Chen’s research was supported by the Scientific Research Fund of Inner Mogolia University for Nationalities (NMDYB1473). Corresponding author. E-mail addresses: [email protected] (L. Chen), [email protected] (Y. Liu).
*
http://dx.doi.org/10.1016/j.camwa.2017.07.042 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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L. Chen et al. / Computers and Mathematics with Applications (
u = 0,
b · ν = 0,
curlb × ν = 0
on ∂ Ω × (0, T ),
)
–
(1.4)
where ν is the unit outward normal vector to ∂ Ω . When b = 0, then systems (1.1)–(1.4) are the well-known Navier–Stokes equations with density-dependent viscosity coefficient. Cho and Kim [1] proved local existence of unique strong solutions for all initial data satisfying a natural compatibility condition for the case of vacuum. They also built following blowup criterion: sup (∥∇ρ (t)∥Lq + ∥∇ u(t)∥L2 ) = ∞.
(1.5)
0≤t ≤T ∗
If T ∗ < ∞ is the maximal existence time of the local strong solutions. For related studies and problems, we refer to [2–5] and references therein. When µ is a constant, the systems (1.1)–(1.4) are density-dependent incompressible magnetohydrodynamic flows. Chen et al. [6] established local existence of unique strong solutions and a blowup criterion. Fan et al. [7] proved the following regularity criterion: u ∈ Ls (0, T ; Lrw (Ω ))
with
2 s
+
3 r
≤ 1, 3 < r ≤ ∞,
(1.6)
where Lrw denotes the weak Lr -space. For more recent results about the incompressible magnetohydrodynamic flows, the readers can refer to [8–11] and references therein. For the homogeneous viscous incompressible MHD equations (i.e. ρ (x, t) = constant), there are also many mathematical results. Duraut and Lions [12] constructed a class of weak solutions with finite energy and a class of local strong solutions. However, the regularity and uniqueness of such a given weak solution remain a challenging open problem. There are many sufficient conditions to guarantee the regularity of the weak solution (see [13–16]). The aim of this paper is to prove a new regularity criterion to the problem (1.1)–(1.4). Before stating our results, we first recall the following local well-posedness result by Wu [17]. Proposition 1.1. Let Ω be a bounded smooth domain in R3 and q ∈ (3, ∞) be a fixed constant. Suppose that the initial data (ρ0 , u0 , b0 , P0 ) satisfies the regularity conditions 0 ≤ ρ0 ≤ ρ, ¯
ρ0 ∈ W 1,q ,
u0 , b0 ∈ H01,σ ∩ H 2 ,
(1.7)
and the compatibility condition
− div(µ(ρ0 )∇ u0 ) + ∇ P0 − b0 ∇ b0 =
√ ρ0 g
in Ω
(1.8)
for some (P0 , g) ∈ H 1 × L2 . Then there exist a positive small time T ∗ and a unique strong solution (ρ, u, b, P) of (1.1)–(1.4) such that
⎧ 0 ≤ ρ (x, t) ≤ ρ, ¯ ρt ∈ C ([0, T ∗ ]; Lq ), ⎪ ⎪ ∗ 1 2 ⎪ ) ∩ L2 (0, T ∗ ; W 2,r ), ⎨u ∈ C ([0, T ]; H0 ∩ H√ 2 ∗ 1 ρ ut ∈ L∞ (0, T ∗ ; L2 ), ut ∈ L (0, T ; H0 ), ⎪ ∞ ∗ 1 2 1,r ⎪ ⎪ ⎩P ∈ L (0, T ;∗ H ) 1∩ L (0, ∞;∞W ),∗ 2 ∇ b ∈ C ([0, T ]; H ), bt ∈ L (0, T ; L ) ∩ L2 (0, T ∗ ; H01 )
(1.9)
for some r with 3 < r < min{6, q}. Motivated by [7] and [1], the main purpose of this paper is to extend result of Fan et al. [7] to the density-dependent viscosity coefficient case. More precisely, the main result in this paper can be stated as follows. Theorem 1.1. Assume that the initial data (ρ0 , u0 , b0 ) satisfies (1.7) and (1.8). Let (ρ, u, b) be a strong solution of the problem (1.1)–(1.4) satisfying (1.7). If 0 < T ∗ < ∞ is the maximal time of existence, then
(
)
lim ∥∇ρ∥L∞ (0,T ;Lq ) + ∥u∥Ls (0,T ;Lrw ) = ∞,
T →T ∗
(1.10)
for any r and s satisfying 2 s
+
3 r
≤ 1,
3 < r ≤ ∞,
where Lrw denotes the weak Lr -space.
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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3
2. Preliminaries Throughout this paper, we assume that Ω ⊂ R3 is a bounded domain with smooth boundary ∂ Ω . For simplicity, we denote
∫
∫ fdx = Ω
fdx.
For 1 ≤ r ≤ ∞ and k ∈ N, the Sobolev spaces are defined in a standard way. W k,r = {f ∈ Lr : ∇ k f ∈ Lr },
Lr = Lr (Ω ), k
H =W
k,2
,
H01 = C0∞ ,
∞ C0∞ ,σ = {f ∈ C0 : divf = 0}.
H01,σ = C0∞ ,σ ,
closure in the norm of H 1 .
Lemma 2.1 ([18]). Let Ω ⊂ R3 be a smooth bounded domain, let f : Ω → R3 be a smooth vector field, and let 1 < p < ∞. Then ∫ ∫ ∫ p 2 4(p − 2) p−2 p−2 2 − 1f · f |f | dx = |f | |∇ f | dx + |∇|f | 2 | dx p2
∫
p−2
|f |
− ∂Ω
(f · ∇ )f · ν dσ −
∫ ∂Ω
|f |p−2 (curlf × ν ) · fdσ .
(2.1)
Lemma 2.2 ([19,20]). Let Ω be a smooth and bounded open set, and let 1 < p < ∞. Then we have the estimate 1− 1
1
∥f ∥Lp (∂ Ω ) ≤ C ∥f ∥Lp p ∥f ∥Wp 1,p
(2.2)
holds for any f ∈ W 1,p . High-order a priori estimates rely on the following regularity results for density-dependent Stokes equations. Lemma 2.3 ([4,5]). Assume that ρ ∈ W 1,q for some 3 < q < ∞, and 0 ≤ ρ ≤ ρ¯ . Let (u, P) ∈ H01,σ × L2 be the unique weak solution to the boundary value problem:
− div(µ(ρ )∇ u) + ∇ P = F ,
in Ω ,
divu = 0
∫
Pdx = 0.
(2.3)
Then the following regularity estimates hold for (u, P): (1) If F ∈ L2 , then (u, P) ∈ H 2 × H 1 and q
∥u∥H 2 + ∥P ∥H 1 ≤ C ∥F ∥L2 (1 + ∥∇ρ∥Lq ) q−3 . (2) If F ∈ L for some r ∈ (3, q), then (u, P) ∈ W r
2 ,r
∥u∥W 2,r + ∥P ∥W 1,r ≤ C ∥F ∥Lr (1 + ∥∇ρ∥Lq )
(2.4)
×W
qr 2(q−r)
1,r
such that
.
(2.5)
(3) If F ∈ H , then (u, P) ∈ H × H and 1
3
2
∥u∥H 3 + ∥P ∥H 2 ≤ C˜ ∥F ∥Lr (1 + ∥ρ∥W 2,q )N
(2.6)
for some N = N(3, q) > 0. The constant C˜ depends also on ∥∂ 2 µ/∂ρ 2 ∥C . The following lemma has been proved in [2,21], which plays an important role in the subsequent proof. Lemma 2.4. Assume g ∈ H 1 and f ∈ Lrw with r ∈ (3, ∞], then f · g ∈ L2 . Furthermore, for any ε > 0 and r ∈ (3, ∞], we have
( ) s ∥f · g ∥2L2 ≤ ε∥g ∥2H 1 + C (ε ) ∥f ∥L2r + 1 ∥g ∥2L2 w
(2.7)
where C is positive constant depending only on ε , r and the domain Ω . 3. Proof of Theorem 1.1 Let 0 < T ∗ < ∞ be the maximum time for the existence of strong solution (ρ, u, b, P) to (1.1)–(1.4). Suppose that (1.10) were false, that is M0 = lim ∥∇ρ∥L∞ (0,T ;Lq ) + ∥u∥Ls (0,T ;Lrw ) < ∞.
(
T →T ∗
)
(3.1)
Under the condition (3.1), one will extend existence time of the strong solutions to (1.1)–(1.4) beyond T ∗ , which contradicts the definition of maximum existence time. Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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Lemma 3.1. For any 0 ≤ T < T ∗ , it holds that
√
2
sup (∥ρ∥Lp + ∥ ρ u∥L2 + ∥b∥2L2 ) +
0≤t ≤T
T
∫
∫
(
) |∇ u|2 + |∇ b|2 dxds ≤ C .
(3.2)
0
Proof. Multiplying (1.1)1 by pρ p−1 and integrating the resulting equation over Ω , then it is easy to deduce that
∥ρ (t)∥Lp = ∥ρ0 ∥Lp (1 ≤ p ≤ ∞).
(3.3)
Multiplying (1.1)2 , (1.1)3 by u, b in L2 and integrating by parts, summing up the resulting equations, we infer from (1.1)4 that 1 d
√
2 dt
2
(∥ ρ u∥L2 + ∥b∥2L2 ) + µ∥∇ u∥2L2 + ∥∇ b∥2L2 ≤ 0.
(3.4)
Integrating (3.4) over (0, T ) yields 1 2
√
2
(∥ ρ u∥L2 + ∥b∥2L2 ) +
T
∫ 0
(µ∥∇ u∥2L2 + ∥∇ b∥2L2 ) ≤
1 2
√
2
(∥ ρ0 u0 ∥L2 + ∥b0 ∥2L2 ),
(3.5)
which completes the proof. □ Lemma 3.2. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ , sup (∥∇ u∥2L2 + ∥∇ b∥2L2 ) +
0≤t ≤T
T
∫ 0
√
2
(∥ ρ u˙ ∥L2 + ∥∇ b∥2H 1 + ∥bt ∥2L2 )dt ≤ C .
(3.6)
Proof. Multiplying (1.1)3 by |b|p−2 b(2 ≤ p < ∞), using (1.4), (2.1), (2.2), (3.1), (2.7) and Gagliardo–Nirenberg inequality, we derive 1 d
∫
|b|p dx +
p dt
∫
1
| b| ∂Ω
(b · ∇ )ν · bdσ +
∫
|b| dσ − p
≤ ∥∇ν∥
L∞
∫ ∂Ω
≤
2(p − 2)
∫ ∫
∫
(b · ∇ )u · |b|p−2 bdx
p
p
|u| |b| 2 |∇|b| 2 |dx p 2
∫
p 2
∫
|∇|b| 2 | dx + ε
p2
p 2
|∇|b| 2 | dx
p
bi u∂i (|b|p−2 b)dx
|∇|b| 2 | dx + C
p2 2(p − 2)
∑∫
∫
∫
i
|b|p dσ + C
≤C ≤
∂Ω
4(p − 2)
|b|p−2 |∇ b|2 dx +
2
p−2
=−
∫
|b|p dx + C
∫
p 2
|u|2 | |b| 2 | dx
p 2
|∇|b| 2 | dx + C (ε)(∥u∥sLrw + 1)
∫
|b|p dx.
(3.7)
Thus, choosing ε enough small, we have T
∫
∫
∥b∥L∞ (0,T ;Lq ) +
|b|q−2 |∇ b|2 dxdt ≤ C .
(3.8)
0
From (3.1), we get that
µ(ρ )t + u · ∇µ(ρ ) = 0.
(3.9)
Multiplying (1.1)2 by ut and integrating (by parts) over Ω , we have d dt
∫
µ(ρ )|∇ u|2 dx +
∫ =− ≤ε
∫
d
∫
dt
∫ (b · ∇ )u · bdx +
ρ|˙u|2 dx
u · ∇µ(ρ )|∇ u|2 + ρ u˙ · (u · ∇ u) dx +
(
ρ|˙u|2 dx + C (ε, δ )
)
∫ (
∫
|u| |∇ρ| |∇ u|2 dx +
((bt · ∇ )u · b + (b · ∇ )u · bt )dx
∫
∫ ∫ ) ρ|u|2 |∇ u|2 dx + δ |bt |2 dx + C (ε, δ ) |b|2 |∇ u|2 dx.
(3.10)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
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)
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5
Indeed, by (2.7), Young, Hölder and Sobolev inequality, we get
∫
|u| |∇ρ| |∇ u|2 dx +
∫
ρ|u|2 |∇ u|2 dx ≤ C (ε )
∫
|u|2 |∇ u|2 dx + ε
∫
≤ C (ε )
∫
|u|2 |∇ u|2 dx + ε
∫
≤ (δ + ε )∥∇ u∥
2 H1
|∇ρ|2 |∇ u|2 dx ∥∇ρ∥Lq ∥∇ u∥2 q−2 dx
+ C (ε, δ )(∥u∥
s Lrω
L 2q u 2L2
+ 1)∥∇ ∥
(3.11)
and
∫
|b|2 |∇ u|2 dx ≤ C ∥b∥2L6 ∥∇ u∥2L3 ≤ C ∥∇ u∥L2 ∥∇ u∥L6 ≤ ε∥∇ u∥2H 1 + C (ε)∥∇ u∥2L2 .
(3.12)
Similarly, taking L2 inner product of (1.1)3 with bt , we achieve d
∫
|∇ b|2 dx +
dt
∫
|bt |2 dx ≤ C
∫
( ) |b| |∇ u| |bt | + |u| |∇ b| |bt | dx ∫ ∫ ) ( 2 2 ≤ ε |bt | dx + C (ε ) |u| |∇ b|2 + |b|2 |∇ u|2 dx ∫ ≤ ε |bt |2 dx + δ (∥∇ u∥2H 1 + ∥∇ b∥2H 1 ) + C (ε, δ )(∥u∥sLrω + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ).
(3.13)
Applying Lemma 2.3 with F ≜ −ρ ut − ρ (u · ∇ )u + curlb × b, we derive that q
∥u∥H 2 + ∥P ∥H 1 ≤ C ∥F ∥L2 (1 + ∥∇ρ∥Lq ) q−3 ≤ C ∥ − ρ ut − ρ u · ∇ u + curlb × b∥L2 ) ( √ ≤ C ∥ ρ u˙ ∥L2 + ∥ |b| |∇ b|∥L2 ( √ ) ≤ C ∥ ρ u˙ ∥L2 + C ∥b∥L6 ∥∇ b∥L3 ( √ 1 1 ) ≤ C ∥ ρ u˙ ∥L2 + ∥∇ b∥L22 ∥∇ b∥H2 1 √ ≤ C ∥ ρ u˙ ∥L2 + κ∥∇ b∥H 1 + C (κ )∥∇ b∥L2 .
(3.14)
In a similar way that
( ) ∥b∥H 2 ≤ C ∥bt ∥L2 + ∥(b · ∇ )u∥L2 + ∥(u · ∇ )b∥L2 ≤ C ∥bt ∥L2 + ϵ (∥∇ u∥H 1 + ∥∇ b∥H 1 ) + C (ϵ )(∥u∥sLrω + 1)(∥∇ u∥L2 + ∥∇ b∥L2 ),
(3.15)
then, we have by taking κ , ϵ small enough
√ 2 ∥u∥2H 2 + ∥b∥2H 2 ≤ C (∥bt ∥2L2 + ∥ ρ u˙ ∥L2 ) + C (∥u∥sLrω + 1)(∥∇ u∥L2 + ∥∇ b∥L2 ).
(3.16)
Collecting (3.10), (3.11), (3.13) and (3.16) deduces
∫ ( ) ( 2) µ(ρ )|∇ u|2 + |∇ b|2 dx + ρ|˙u|2 + |bt |2 + |∇ 2 b| dx dt ∫ ∫ ∫ d ≤− (b · ∇ )u · bdx + ε ρ|˙u|2 dx + δ |bt |2 dx d
∫
dt
+ (ε + δ )(∥∇ u∥2H 1 + ∥∇ b∥2H 1 ) + C (ε, δ )(∥u∥sLrω + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ) ∫ d √ 2 ≤− (b · ∇ )u · bdx + (ε + δ )(∥ ρ u˙ ∥L2 + ∥bt ∥2L2 ) + C (ε, δ )(∥u∥sLr + 1)(∥∇ u∥2L2 + ∥∇ b∥2L2 ). ω dt
(3.17)
By (3.1), (3.2), (3.8) and Cauchy–Schwarz inequality, it is easily seen that
∫ C
|b|2 |∇ u|dx ≤
µ 4
∥∇ u∥2L2 + C .
(3.18)
Taking this into account, by choosing ε , δ small enough, we then conclude from (3.17) and the Gronwall inequality that (3.6) holds for any 0 ≤ T < T ∗ . The proof of Lemma 3.2 is therefore complete. □ Lemma 3.3. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ ,
( √
2
sup ∥ ρ ut ∥L2 + ∥bt ∥2L2 +
0≤t ≤T
)
T
∫ 0
( ) ∥∇ ut ∥2L2 + ∥∇ bt ∥2L2 dt ≤ C .
(3.19)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
6
L. Chen et al. / Computers and Mathematics with Applications (
)
–
Proof. Differentiating (1.1)2 with respect to t, we see that
[ (1 )] ρ utt + ρ u · ∇ ut − div(µ(ρ )∇ ut ) + ∇ Pt + |b|2 2
t
= div(b ⊗ b)t + div(µ′ ρt ∇ u) − ρt (ut + u · ∇ u) − ρ ut · ∇ u.
(3.20)
Multiplying (3.20) by ut and integrating (by parts) over Ω , one arrives at 1 d 2 dt
∫
ρ|ut |2 dx + µ
∫
∫ ∫ ρ|u| |ut | |∇ ut |dx + C ρ|u| |∇ u|2 |ut |dx + C ρ|u|2 |∇ 2 u| |ut |dx ∫ ∫ ∫ + C ρ|u|2 |∇ u| |∇ ut |dx + C ρ|ut |2 |∇ u|dx + C |b| |bt | |∇ ut |dx
|∇ ut |2 dx ≤ C
∫
∫ +C
|u| |∇ρ| |∇ u| |∇ ut |dx ≜
7 ∑
Ii .
(3.21)
i=1
By (3.6), Hölder, interpolation, Sobolev’s and Young’s inequality, we get
√
√
I1 ≤ C ∥ ρ∥L∞ ∥u∥L6 ∥ ρ ut ∥L3 ∥∇ ut ∥L2 1 √ 1 √ ≤ C ∥∇ρ∥Lq ∥∇ u∥L2 ∥ ρ ut ∥L22 ∥ ρ ut ∥L26 ∥∇ ut ∥L2 1 3 √ ≤ C ∥ ρ ut ∥L22 ∥∇ ut ∥L22 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥ ρ ut ∥L2 , √ I2 ≤ C ∥ ρ∥L∞ ∥u∥L6 ∥∇ u∥L6 ∥ut ∥L6 ∥∇ u∥L2
(3.22)
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ u∥H 1 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ u∥2H 1 , 2 L6
(3.23)
2
I3 ≤ C ∥ρ∥L∞ ∥u∥ ∥∇ u∥L2 ∥∇ ut ∥L6
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ 2 u∥L2 ∥∇ ut ∥L2 2
≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ 2 u∥L2 , √ I4 ≤ C ∥ ρ∥L∞ ∥u∥2L6 ∥∇ u∥L6 ∥∇ ut ∥L2
(3.24)
≤ C ∥∇ρ∥Lq ∥∇ u∥2L2 ∥∇ u∥H 1 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥∇ u∥2H 1 , √ √ I5 ≤ C ∥ ρ∥L∞ ∥ut ∥L6 ∥ ρ ut ∥L3 ∥∇ u∥L2 1 √ 1 √ ≤ C ∥∇ρ∥Lq ∥∇ ut ∥L2 ∥ ρ ut ∥L22 ∥ ρ ut ∥L26
(3.25)
3 √ 1 ≤ C ∥∇ ut ∥L22 ∥ ρ ut ∥L22 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )∥ ρ ut ∥L2 ,
(3.26)
I6 ≤ C ∥b∥L6 ∥bt ∥L3 ∥∇ ut ∥L2 1
1
≤ C ∥∇ b∥L2 ∥bt ∥L22 ∥bt ∥L26 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + δ∥∇ bt ∥2L2 + C (ε, δ )∥bt ∥2L2 .
(3.27)
It remains to estimate the term I7 . Indeed, on the one hand, using (3.8), Cauchy and interpolation inequality, we deduce
√ ∥u∥H 2 + ∥P ∥H 1 ≤ C (∥ ρ ut ∥L2 + ∥ |u| |∇ u|∥L2 + ∥ |b| |∇ b|∥L2 ) √ ≤ C ∥ ρ ut ∥L2 + C ∥u∥L6 ∥∇ u∥L3 + C ∥b∥L6 ∥∇ b∥L3 3 1 3 1 √ ≤ C ∥ ρ ut ∥L2 + C ∥∇ u∥L22 ∥∇ u∥H2 1 + C ∥∇ b∥L22 ∥∇ b∥H2 1 ( √ ) 1 1 ≤ C ∥ ρ ut ∥L2 + 1 + ∥∇ b∥H 1 + ∥∇ u∥H 1 , 2
2
(3.28)
and
∥b∥H 1 ≤ C ∥bt ∥L2 + C ∥ |u| |∇ b|∥L2 + C ∥|b| |∇ u|∥L2 ≤ C ∥bt ∥L2 + C ∥u∥L6 ∥∇ b∥L3 + C ∥b∥L6 ∥∇ u∥L3 Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
L. Chen et al. / Computers and Mathematics with Applications ( 1
1
1
)
–
7
1
≤ C ∥bt ∥L2 + C ∥∇ u∥L2 ∥∇ b∥L22 ∥∇ b∥H2 1 + C ∥∇ b∥L2 ∥∇ u∥L22 ∥∇ u∥H2 1 1
1
2
2
≤ C ∥bt ∥L2 + ∥∇ b∥H 1 + ∥∇ u∥H 1 + C
(3.29)
which implies
) ( √ ∥u∥H 2 + ∥b∥H 1 ≤ C ∥ ρ ut ∥L2 + ∥bt ∥L2 + 1 ,
(3.30)
On the other hand, by Lemma 2.3, we get pr
∥u∥W 2,r + ∥P ∥W 1,r ≤ C ∥F ∥Lr (1 + ∥∇µ(ρ )∥Lp ) 2(p−r) ≤ C (∥ρ ut ∥Lr + ∥ρ u · ∇ u∥Lr + ∥ |b| |∇ b|∥Lr ) 6−r 3r −6 √ ≤ C ∥ ρ ut ∥L22r ∥∇ ut ∥L22r + C ∥u∥L6 ∥∇ u∥ 6r + C ∥b∥L∞ ∥∇ b∥Lr L 6−r
√
6−r 2r L2
≤ C ∥ ρ ut ∥
3r −6 2r L2
∥∇ ut ∥
+ C ∥∇ u∥
6(r −1) 5r −6 L2
4r −6
1
3
,
∥∇ u∥W5r 1−,6r + C ∥b∥L22 ∥∇ b∥H2 1
(3.31)
which deduces
(
∥u∥W 2,r + ∥P ∥W 1,r
6(r −1) 6 −r 3r −6 1 3 √ ≤ C ∥ ρ ut ∥L22r ∥∇ ut ∥L22r + ∥∇ u∥L2 r + ∥∇ b∥L22 ∥∇ b∥H2 1 ( ) 3 √ ≤ C ∥ ρ ut ∥L2 + ∥∇ ut ∥L2 + ∥∇ b∥H2 1 + 1 .
)
(3.32)
Thus, by using (3.1), Hölder, Gagliardo–Nirenberg and Young inequalities, we obtain I7 ≤ C ∥u∥L6 ∥∇ρ∥Lq ∥∇ u∥
3q
L q −3
∥∇ ut ∥L2
2q−6
q+6
3q ≤ C ∥∇ u∥L2 ∥∇ρ∥Lq ∥∇ u∥L23q ∥∇ u∥L∞ ∥∇ ut ∥L2 q+6
≤ C ∥∇ u∥W3q1,r ∥∇ ut ∥L2 +6 ( ) q3q 3 √ 2 ≤ C ∥ ρ ut ∥L2 + ∥∇ ut ∥L2 + ∥∇ b∥H 1 + 1 ∥∇ ut ∥L2 √ 2 ≤ ε∥∇ ut ∥2L2 + C (ε )(∥ ρ ut ∥L2 + ∥∇ b∥4H 1 + 1).
(3.33)
Differentiating (1.1)3 with respect to t, and multiplying the resulting equation by bt in L2 , we deduce after integrating by parts 1 d
∫
2 dt
|bt |2 dx +
∫
|∇ bt |2 dx = −
∫ (ut · ∇ b − bt · ∇ u − b · ∇ ut ) · bt dx
≤ C ∥ut ∥L6 ∥∇ b∥L2 ∥bt ∥L3 + C ∥∇ u∥L2 ∥bt ∥2L4 + C ∥∇ ut ∥L2 ∥b∥L6 ∥bt ∥L3 1
1
1
3
≤ C ∥∇ ut ∥L2 ∥∇ b∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 + C ∥∇ u∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 1
1
+ C ∥∇ ut ∥L2 ∥∇ b∥L2 ∥bt ∥L22 ∥∇ bt ∥L22 ≤ ε∥∇ ut ∥2L2 + δ∥∇ bt ∥2L2 + C (ε, δ )∥bt ∥2L2 .
(3.34)
Combining (3.21)– and choosing ε , δ suitably small, we obtain 1 d
∫
2 dt
( ) ρ|ut |2 + |bt |2 dx +
∫
(
) √ 2 µ|∇ ut |2 + |∇ bt |2 dx ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + ∥∇ b∥4H 1 ) √ √ 2 4 ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + ∥ ρ ut ∥L2 + ∥bt ∥4L2 ) √ √ 2 2 ≤ C + C (∥ ρ ut ∥L2 + ∥bt ∥2L2 + 1)(∥ ρ ut ∥L2 + ∥bt ∥2L2 ),
(3.35)
which, together with the Gronwall inequality and (3.6), completes the proof. □ Lemma 3.4. Under the condition (3.1), it holds that for 0 ≤ T < T ∗ ,
(
)
∫
T
sup ∥ρt ∥Lq + ∥P ∥H 1 + ∥u∥H 2 + ∥b∥H 2 +
0≤t ≤T
0
( 2 ) ∥u∥W 2,r + ∥P ∥2W 1,r dt ≤ C .
(3.36)
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.
8
L. Chen et al. / Computers and Mathematics with Applications (
)
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Proof. By (3.30) and (3.32), it is easy to deduce
(
)
∫
T
sup ∥P ∥H 1 + ∥u∥H 2 + ∥b∥H 2 +
0≤t ≤T
0
) ( 2 ∥u∥W 2,r + ∥P ∥2W 1,r dt ≤ C ,
(3.37)
which, together with (1.1)1 yields
∥ρt ∥Lq ≤ ∥u∥L∞ ∥∇ρ∥Lq ≤ C ∥u∥H 2 ∥∇ρ∥Lq ≤ C .
(3.38)
This completes the proof. □ After having Lemmas 3.1–3.4 at hand, it is easy to apply Proposition 1.1 to extend the strong solution (ρ, u, P , b) beyond time T ∗ . Therefore, we complete the proof of Theorem 1.1. References [1] Y. Cho, H. Kim, Unique solvability for the density-dependent Navier-Stokes equations, Nonlinear Anal. 59 (2004) 465–489. [2] H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal. 37 (2006) 1417–1434. [3] X.D. Huang, Y. Wang, Global strong solution with vacuum to the two dimensional density dependent Navier-Stokes system, SIAM J. Math. Anal. 46 (2014) 1771–1788. [4] X.D. Huang, Y. Wang, Global strong solution of 3D inhomogeneous Navier-Stokes equations with density-dependent viscosity, J. Differential Equations 259 (2015) 1606–1627. [5] J. Zhang, Global well-posedness for the incompressible Navier-Stokes equations with density dependent viscosity coefficient, J. Differential Equations 259 (2015) 1722–1742. [6] Q. Chen, Z. Tan, Y.J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci. 34 (2011) 94–107. [7] J. Fan, F. Li, G. Nakamura, Z. Tan, Regularity criteria for the three-dimensional magneto-hydrodynamic equations, J. Differential Equations 256 (2014) 2858–2875. [8] Y. Zhou, J. Fan, A regularity criterion for the density-dependent magnetohydrodynamic equations, Math. Methods Appl. Sci. 33 (2010) 1350–1355. [9] J. Fan, W. Sun, J. Yin, Blow-up criteria for Boussinesq system and MHD system and Landau-Lifshitz equations in a bounded domain, Bound. Value Probl. 90 (2016) 19. [10] X. Si, X. Ye, Global well-posedness for the incompressible MHD equations with density-dependent viscosity and resistivity coefficients, Z. Angew. Math. Phys. 67 (2016) 15. [11] F. Chen, Y. Li, Y. Zhao, Global well-posedness for the incompressible MHD equations with variable viscosity and conductivity, J. Math. Anal. Appl. 447 (2017) 1051–1071. [12] G. Duraut, J.L. Lions, Inéquations en thermoéasticit et magnétohydrodynamique, Arch. Ration. Mech. Anal. 46 (1972) 241–279. [13] C. Cao, J. Wu, Two regularity criteria for the 3D MHD equations, J. Differential Equations 248 (2010) 2263–2274. [14] Y. Zhou, S. Gala, Regularity criteria for the solutions to the 3D MHD equation in the multiplier space, Z. Angew. Math. Phys. 61 (2010) 193–199. [15] X. Chen, S. Gala, Z. Guo, A new regularity criterion in terms of the direction of the velocity for the MHD equations, Acta Appl. Math. 113 (2011) 207–213. [16] L. Ni, Z. Guo, Y. Zhou, Some new regularity criteria for the 3D MHD equations, J. Math. Anal. Appl. 396 (2012) 108–118. [17] H. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl. 61 (2011) 2742–2753. [18] H. Beirão da Veiga, F. Crispo, Sharp inviscid limit results under Navier type boundary conditions. An Lp theory, J. Math. Fluid Mech. 12 (2010) 397–411. [19] R.A. Adams, J.F. Fournier, Sobolev Spaces, second ed, in: Pure and Appl. Math. (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. [20] A. Lunardi, Interpolation Theory, second ed, in: Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, 2009. [21] X. Xu, J. Zhang, A blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum, Math. Models Methods Appl. Sci. 22 (2012) 1150010 23pp.
Please cite this article in press as: L. Chen, et al., Regularity criteria for the incompressible magnetohydrodynamic flows with density-dependent viscosity coefficient, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.07.042.