Blow-up criterion for 3-D compressible magnetohydrodynamics with vacuum and zero resistivity

J. Math. Anal. Appl. 400 (2013) 174–186

Contents lists available at SciVerse ScienceDirect

Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa

Blow-up criterion for 3-D compressible magnetohydrodynamics with vacuum and zero resistivity Mingtao Chen a,∗ , Shengquan Liu b a

School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, PR China

b

School of Mathematics, Liaoning University, Shenyang 110036, PR China

article

info

Article history: Received 23 February 2012 Available online 1 November 2012 Submitted by Dehua Wang Keywords: Blow-up criterion Strong solution Compressible magnetohydrodynamics Vacuum Zero resistivity

abstract We study strong solutions of the equations of compressible magnetohydrodynamics with zero resistivity in a domain Ω ⊂ R3 . We establish a criterion for possible breakdown of such solutions at a finite time in terms of both ∥∇ u∥L1 (0,T ;L∞ ) and ∥θ∥L∞ (0,T ;L∞ ) . More precisely, if a solution of 3D compressible magnetohydrodynamics with zero resistivity is initially regular and loses its regularity at some later time, then the loss of regularity implies growth without bound of both ∥∇ u∥L1 (0,T ;L∞ ) and ∥θ ∥L∞ (0,T ;L∞ ) as the critical time approaches. In addition, initial vacuum states are allowed in our cases. © 2012 Elsevier Inc. All rights reserved.

1. Introduction Magnetohydrodynamics (MHD) concerns the dynamics of compressible quasineutrally ionized fluids under the influence of electromagnetic fields. It has a very broad range of applications from liquid metals to cosmic plasmas. It also finds applications in geophysics and astronomy. Due to its physical importance, complexity, rich phenomena and mathematical challenges, the compressible magnetohydrodynamic fluids have long attracted the attention of many physicists and mathematicians. For this problem, the mathematical analysis is much more complicated, due to the oscillation of the density, the concentration of the temperature, and the coupling interaction of hydrodynamics with the magnetic field. The full system of the three-dimensional magnetohydrodynamic equations in the Eulerian coordinates can be read as follows:

ρ + div(ρ u) = 0, t   (ρ u)t + div(ρ u ⊗ u) + ∇ P = (∇ × H ) × H + µ1u + (λ + µ)∇ div u,   µ (ρθ )t + div(ρθ u) + P div u = κ 1θ + |∇ u + ∇ uT |2 + λ(div u)2 + ν(∇ × H )2 , 2    Ht − ∇ × (u × H ) = −∇ × (ν∇ × H ),  div H = 0.

(1.1)

Here ρ = ρ(x, t ) denotes the density, u = (u1 (x, t ), u2 (x, t ), u3 (x, t )) the velocity, θ = θ (x, t ) the temperature, H = (H 1 (x, t ), H 2 (x, t ), H 3 (x, t )) the magnetic field, respectively. Moreover, P = Rρθ = (γ − 1)ρθ is the pressure with γ > 1 being the adiabatic exponent. The constants µ and λ are the shear and bulk viscosity coefficients satisfying the physical restrictions µ > 0, 2µ+3λ ≥ 0. The constants κ > 0 and ν ≥ 0 are the heat conductivity and the resistivity coefficient of the magnetics.

∗

Corresponding author. E-mail addresses: [email protected] (M. Chen), [email protected] (S. Liu).

0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.10.058

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

175

In the study of a highly conducting fluid, for example, the magnetic fusion, it is rational to ignore the magnetic diffusion term in the MHD equations since the resistivity coefficient is inversely proportional to the electrical conductivity coefficient, see [1]. In this situation, the system (1.1), describing the motion of the fluid in R3 , can be described by the following compressible nonisentropic MHD equations with zero magnetic diffusivity:

ρ + div(ρ u) = 0, t    (ρ u)t + div(ρ u ⊗ u) + ∇ P = (∇ × H ) × H + µ1u + (λ + µ)∇ div u,  µ (ρθ )t + div(ρθ u) + P div u = κ 1θ + |∇ u + ∇ uT |2 + λ(div u)2 , 2     − ∇ × ( u × H ) = 0 , H t  div H = 0.

(1.2)

The system (1.2) is also called the viscous and non-resistive magnetohydrodynamic equations. The Eq. (1.2)4 implies that the magnetic field in a highly conducting fluid line moves along exactly with the fluid, rather than simply diffusing out. The behavior is physically expressed as that the magnetic field lines are frozen into the field. The ‘‘frozen-in’’ nature of magnetic fields plays a very important role, and a typical illustration of the behavior is the phenomenon of sunspots. For more details of physical background, we refer the reader to [1,2]. The main goal of this paper is to give a blowup criterion of strong solutions for the non-resistive compressible MHD Eqs. (1.2) in a domain Ω ⊆ R3 . The system (1.2) must be complemented by a boundary condition for the velocity u and temperature θ in order to obtain, at least formally, a well-posed problem. In this paper, we will study the non-resistive compressible MHD Eqs. (1.2) with initial condition:

(ρ, u, θ , H )(x, 0) = (ρ0 , u0 , θ0 , H0 )(x),

(1.3)

and two kinds of boundary conditions: (1) Cauchy problem (Ω = R3 ):

(ρ, u, θ , H )(x, t ) → (0, 0, 0, 0) (in some weak sense),

as |x| → ∞;

(1.4)

(2) Dirichlet problem (Ω is a bounded smooth domain in R ): 3

(u, ∂θ /∂ n)(x, t ) = (0, 0),

on ∂ Ω ,

(1.5)

where n is the unit outer normal vector to ∂ Ω . Before stating our main result, we review some previous works related with the MHD equations. Indeed, there has been a lot of literature on the compressible MHD problems, see [1–5] and references therein. In particular, the onedimensional problem has been studied in many papers, for example, [6–8] and so on. For the three-dimensional compressible MHD equations, Umeda et al. obtained the global existence and the time decay of smooth solutions to the linearized MHD equations in [9]. Recently, Hu et al. [10,11] established the existence of global weak solutions to the compressible Magnetohydrodynamic flows with general initial data. The mathematical results on MHD problems mentioned above are mainly concerned with the viscous and resistive MHD system (1.1). To the author’s knowledge, there are very few results on the non-resistive MHD system (1.2) except the local existence result in [12] and a blow-up criterion in [13] due to the absence of the magnetic diffusion term ∇ × (ν∇ × H ). Moreover, the global existence of strong or weak solutions for (1.2) with large initial data is still an outstanding open problem. Without the magnetic field H, the system (1.1) (or (1.2)) becomes compressible Navier–Stokes equations. For initial data close to a non-vacuum equilibrium, Matsumura et al. [14], proved global existence of smooth solutions. Later after that, Hoff [15] proved the global existence of weak solutions of compressible Navier–Stokes equations with discontinuous initial data. The existence of solutions for arbitrary data in three dimensions, the major breakthrough is due to Lions [16], where he established global weak solutions for the whole space, periodic domain or bounded domains with Dirichlet boundary conditions. After that, Feireisl et al. improved Lions’ results in [17–19]. Significant progress has been made in the study of multi-dimensional compressible flows, however, there are still many physically important and mathematically fundamental problems unsolved, such as the global existence of classical solutions with general initial data. Recently, many works are devoted to studying the blow-up criterion of the compressible Navier–Stokes equations, we refer the reader to [20–24,13] and references therein. For the barotropic flows, Huang et al. [20] proved that if the strong solution blows up at some finite time t = T ∗ , then it is necessary T



∥D (u)∥L∞ dt = ∞,

lim

T →T ∗

0

where

D (u) =

1 2

(∇ u + ∇ uT ).

176

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

After that, they [21] deduced the following Serrin-type criterion:

  1 ∥ div u∥L1 (0,T ;L∞ ) + ∥ρ 2 u∥Ls (0,T ;Lr ) = ∞, ∗

lim T →T

and lim

T →T ∗

  1 ∥ρ∥L∞ (0,T ;L∞ ) + ∥ρ 2 u∥Ls (0,T ;Lr ) = ∞,

where r and s satisfy 2 s

+

3 r

≤ 1,

3 < r ≤ ∞.

Then, Sun et al. [23] established a Beale–Kato–Majda criterion: lim ∥ρ∥L∞ (0,T ;L∞ ) = ∞,

T →T ∗

under the assumption

λ < 7µ.

(1.6)

As for the compressible viscous heat-conductive flows, Fan et al. [25], proved that if the strong solution blows up at some finite time t = T ∗ , then lim (∥∇ u∥L1 (0,T ;L∞ ) + ∥θ ∥L∞ (0,T ;L∞ ) ) = ∞,

T →T ∗

with the assumption (1.6). Later, Sun et al. [24] proved the following Beale–Kato–Majda criterion under the condition (1.6), lim ∥ρ, ρ −1 , θ∥L∞ (0,T ;L∞ ) = ∞.

T →T ∗

For the MHD flows, Xu et al. [13] established the blow-up criterion for non-resistive compressible MHD as lim ∥∇ u∥L1 (0,T ;L∞ ) = ∞.

T →T ∗

After that, Lu et al. in [22] proved a blow-up criterion lim

T →T ∗

  ∥∇ u∥L1 (0,T ;L∞ ) + ∥θ ∥L∞ (0,T ;L∞ ) = ∞,

with 4λ < µ, and in [26] two blow-up criterions lim

  ∥div u∥L1 (0,T ;L∞ ) + ∥∇ u∥L4 (0,T ;L2 ) + ∥θ ∥L∞ (0,T ;L∞ ) = ∞,

lim

  ∥ρ∥L∞ (0,T ;L∞ ) + ∥∇ u∥L4 (0,T ;L2 ) + ∥θ∥L∞ (0,T ;L∞ ) = ∞.

T →T ∗

and T →T ∗

Later, Chen et al. [27] improved the blow-up criterion in [22] by removing the condition 4λ < µ. To proceed, we explain the notations used throughout the paper. We denote



 f dx = Ω

f dx.

For 1 ≤ r ≤ ∞, we denote the standard homogeneous and inhomogeneous Sobolev spaces as follows

 r r Dk,r = {u ∈ L1loc |∥∇ k u∥Lr < ∞}, ∥u∥Dk,r = ∥∇ k u∥Lr , L = L (Ω ), k,r r k,r H k = W k,2 , Dk = Dk,2 , W1 = L ∩6D , D0 = {u ∈ L |∥∇ u∥L2 < ∞, and Eq. (1.4) or Eq. (1.5) holds}, H01 = L2 ∩ D10 . To present our results, let us first recall the following local well-posedness result which was proved by Fan et al. [12]. Theorem 1.1. Assume that for some q ∈ (3, 6], and the initial data (ρ0 , u0 , θ0 , H0 ) satisfying

ρ0 ≥ 0,

(ρ0 , H0 ) ∈ W 1,q ,

div H0 = 0,

u0 ∈ H 2 ∩ H01 , θ0 ∈ H 2 ,

(1.7)

and 1

−µ1u0 − (λ + µ)∇ div u0 + ∇ P0 − (∇ × H0 ) × H0 = ρ02 g1 , −κ 1θ0 −

µ 2

1 2

|∇ u0 + ∇ uT0 |2 − λ(div u0 )2 = ρ0 g2 ,

(1.8) (1.9)

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

177

for some (g1 , g2 ) ∈ L2 . Then there exist a time T∗ > 0 and a unique strong solution (ρ, u, θ , H ) to (1.2)–(1.3) together with (1.4) or (1.5) in Ω × (0, T ), such that

 ρ ≥ 0, (ρ, H ) ∈ C ([0, T∗ ]; W 1,q ), (ρt , Ht ) ∈ C ([0, T∗ ]; Lq ),   u ∈ C ([0, T ]; D1 ∩ D2 ) ∩ L2 (0, T ; D2,q ), ∗

∗

0

1 2 2 2,q   θ ∈ C ([0, T2∗ ]; D ∩ D1 ) ∩ L (0√, T∗ ; D√ ), (ut , θt ) ∈ L (0, T∗ ; D ), ( ρ ut , ρθt ) ∈ L∞ (0, T∗ ; L2 ).

(1.10)

Motivated by [27,20,13], we consider the non-resistive compressible MHD flows, and the main result is stated as follows. Theorem 1.2. Let (ρ, u, θ , H ) be a strong solution of the initial boundary value problem (1.2)–(1.3) together with (1.4) or (1.5) satisfying (1.10). Assume that the initial data (ρ0 , u0 , θ0 , H0 ) satisfies (1.7)–(1.9). If T ∗ < ∞ is the maximal time of existence, then lim



T →T ∗

 ∥∇ u∥L1 (0,T ;L∞ ) + ∥θ ∥L∞ (0,T ;L∞ ) = ∞.

Compared with the previous results for Navier–Stokes equations as [20] and MHD flows as [13], the non-resistive compressible MHD flows (1.2) is more complicated and thus more delicate and new estimates are needed for the analysis of strong solutions. For example, there are some essential difficulties which will occur due to the highly nonlinear terms |∇ u + ∇ uT |2 and (div u)2 in the temperature equation, whose nonlinearity is stronger than that of div(ρ u ⊗ u) in the momentum equation. Chen et al. [27], assume the resistivity ν > 0, the magnetic diffusion term ∇ × (ν∇ × H ) exists, which helps to deal with the magnetic field H. Lu et al. [22], that the assumption µ > 4λ plays an important role in their analysis. Precisely, it is essential to bound the L2 norm of the convection term ρ ut + ρ u · ∇ u, and they also need the resistivity ν > 0. We also emphasize the vacuum states are also permitted in this paper. The detail of the proof is shown in the next section. 2. Proof of Theorem 1.2 Let (ρ, u, θ , H ) be a strong solution to the initial boundary value problem (1.2)–(1.3) and (1.4) or (1.5) as described in Theorem 1.2. The proof of Theorem 1.2 is due to the contradiction arguments, so we further assume that lim

T →T ∗



 ∥∇ u∥L1 (0,T ;L∞ ) + ∥θ ∥L∞ (0,T ;L∞ ) ≤ M0 < ∞.

(2.1)

First, with the assumption of (2.1), we derive the following L∞ estimates of ρ and H. Lemma 2.1. Under the assumption (2.1), one has for any 0 ≤ T < T ∗ , that sup (∥ρ∥L∞ + ∥H ∥L∞ ) ≤ C .

(2.2)

0≤t ≤T

Here and after, C will denote a generic constant depending only on M0 , T , and the initial data. Proof. It follows from (1.2)1 that for any p > 1, that

(ρ p )t + div(ρ p u) + (p − 1)ρ p div u = 0. Integrating the above equation over Ω leads to



d dt

ρ p dx ≤ (p − 1)∥div u∥L∞



ρ p dx,

that is, d dt

∥ρ∥Lp ≤

p−1 p

∥div u∥L∞ ∥ρ∥Lp .

This, together with (2.1), implies

∥ρ(t )∥Lp ≤ C , where C is independent of p. Then, letting p → ∞, we conclude the bound of ∥ρ∥L∞ . Similarly, for any q ≥ 2, multiplying (1.2)4 by q|H |q−2 H, integrating the resulting equation over Ω , we have d



q

|H | dx = q

dt



(H · ∇ u − u · ∇ H − Hdiv u)|H |q−2 Hdx.

Using integration by parts, the second term on the right-hand side could be written as

 −q

(u · ∇ H ) · |H |q−2 Hdx =



div u|H |q dx,

(2.3)

178

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

which, together with (2.3), we conclude d dt

∥H ∥Lq ≤

2q + 1 q

∥∇ u∥L∞ ∥H ∥Lq .

(2.4)

Hence, it follows from (2.1) and (2.4) that

∥H ∥Lq (t ) ≤ C , where C is independent of q. Letting q tend to ∞, we obtain the bound of ∥H ∥L∞ . Thus, we finish the proof of Lemma 2.1.



With the help of Lemma 2.1, we can now obtain the following basic energy estimate, which is very important in our paper. Lemma 2.2. Under the assumption (2.1), for any 0 ≤ T < T ∗ , we have

 √  sup ∥ ρ u∥2L2 + ∥H ∥2L2 +

0≤t ≤T

√

sup ∥ ρθ∥2L2 +

∥∇ u∥2L2 dt ≤ C ;

0

(2.5)

T



0≤t ≤T

T



0

∥∇θ∥2L2 dt ≤ C .

(2.6)

Proof. Multiplying (1.2)2 by u, and (1.2)4 by H, then integrating the resulting equations over Ω , and summing them together, we get d

 

1

1

2

ρu + H





  µ|∇ u|2 + (λ + µ)(div u)2 dx    = (∇ × H ) × H · udx + P div udx + ∇ × (u × H ) · Hdx.

dt

2

2

2

dx +

And using the facts that



(∇ × H ) × H · udx = −

 

1

T

H ∇ uH +

2



∇(|H | ) · u dx, 2

and



∇ × (u × H ) · Hdx =

 

T

H ∇ uH +

1 2



∇(|H | ) · u dx, 2

we conclude that 1 d



2 dt

(ρ u2 + H 2 )dx +



  µ|∇ u|2 + (λ + µ)(div u)2 dx =

 P div udx ≤

µ 2



(div u)2 dx + C



ρ 2 θ 2 dx,

which implies (2.5) by (2.1). Multiplying (1.2)3 by θ and integrating the resulting equation over Ω , we have 1 d



2 dt

ρθ 2 dx + κ



|∇θ |2 dx ≤ R



ρθ 2 |div u|dx + C



   |∇ u|2 |θ |dx ≤ C 1 + |∇ u|2 dx ,

which, together with (2.5), implies (2.6). Thus, we finish the proof of Lemma 2.2.



The following lemma is the key lemma in the proof of Theorem 1.2, which concerns the L2 -estimate of ∇ρ, ∇ H and ∇ u. Lemma 2.3. Under the assumption (2.1), for any 0 ≤ T < T ∗ , we have sup 0≤t ≤T



 ∥∇ρ∥2L2 + ∥∇ H ∥2L2 + ∥∇ u∥2L2 +

 0

T

∥∇ u∥2H 1 dt ≤ C .

(2.7)

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

179

Proof. Multiplying (1.2)2 by ρ −1 [µ1u + (λ + µ)∇ div u − ∇ P + (∇ × H ) × H] and integrating the resulting equation over Ω , one has after integration by parts

µ

λ+µ

  (div u)2 dx + ρ −1 [µ1u + (λ + µ)∇ div u − ∇ P + (∇ × H ) × H]2 dx dt 2 2    = µ u · ∇ u · 1udx + (λ + µ) u · ∇ u · ∇ div udx − u · ∇ u · ∇ Pdx  

d

|∇ u|2 +



 −

u · ∇u ·

1 2



2



∇|H | − H · ∇ H dx −



 ut · ∇ Pdx −

ut ·

1 2



∇|H | − H · ∇ H dx, 2

(2.8)

due to div H = 0 and (∇ × H ) × H = − 12 ∇|H |2 + H · ∇ H. When u satisfies boundary condition (1.4) or (1.5), we deduce from the standard L2 -theory of elliptic systems that



ρ −1 [µ1u + (λ + µ)∇ div u − ∇ P + (∇ × H ) × H]2 dx   ≥ C −1 ∥µ1u + (λ + µ)∇ div u∥2L2 − C ∥∇ P ∥2L2 + ∥H ∥2L∞ ∥∇ H ∥2L2   ≥ C −1 ∥∇ 2 u∥2L2 − C ∥∇ u∥2L2 + ∥∇ P ∥2L2 + ∥H ∥2L∞ ∥∇ H ∥2L2 ,

(2.9)

due to the fact ρ −1 ≥ C −1 > 0. Now, we consider the terms on the right-hand side of (2.8). For the first term, we have

µ



u · ∇ u · 1udx = µ



2 j ui ∂i uj ∂kk u dx



∂k ui ∂i uj ∂k uj dx − µ





∂k ui ∂i uj ∂k uj dx +

µ



= −µ = −µ

≤ C ∥∇ u∥L∞ ∥∇ u∥2L2 ,

ui ∂ik2 uj ∂k uj dx

2

∂i ui (∂k uj )2 dx

i, j, k = 1, 2, 3.

(2.10)

Similarly, after direct computations, we get

(λ + µ)



u · ∇ u · ∇ div udx ≤ C ∥∇ u∥L∞ ∥∇ u∥2L2 .

(2.11)

Using (2.1), (2.2), Hölder, Young and Sobolev inequalities, we have

 −

 u · ∇ u · ∇ Pdx = −R

ρ u · ∇ u · ∇θ dx − R



θ u · ∇ u · ∇ρ dx

≤ C ∥ρ∥L∞ ∥u∥L6 ∥∇ u∥L3 ∥∇θ ∥L2 + C ∥θ∥L∞ ∥u∥L6 ∥∇ u∥L3 ∥∇ρ∥L2 2

1

1

2

≤ C ∥∇ u∥L2 ∥∇ u∥L3∞ ∥∇ u∥L32 ∥∇θ∥L2 + C ∥∇ u∥L2 ∥∇ u∥L3∞ ∥∇ u∥L32 ∥∇ρ∥L2   ≤ C ∥∇ u∥2L2 ∥∇ρ∥2L2 + ∥∇θ∥2L2 + C ∥∇ u∥L∞ ∥∇ u∥2L2 + C .

(2.12)

In a similar manner, we can get



 −

u · ∇u ·

1 2

2



∇|H | − H · ∇ H dx ≤ C ∥H ∥L∞ ∥u∥L6 ∥∇ u∥L3 ∥∇ H ∥L2 ≤ C ∥∇ H ∥2L2 ∥∇ u∥2L2 + C ∥∇ u∥L∞ ∥∇ u∥2L2 + C .

(2.13)

Now, we focus on the fifth term on the right-hand side of (2.8), let us first recall (1.2)1 and (1.2)3 can be rewritten as

ρt = −ρ div u − ∇ρ · u, and

ρθt = −ρ u · ∇θ − P div u + κ 1θ +

µ 2

|∇ u + ∇ uT |2 + λ(div u)2 .

180

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

Thus, we have

 −

 ut · ∇ Pdx =

P div ut dx = d



d

 P div udx −

dt





Pt div udx

dt  d

P div udx − R

ρt θ div udx − R



ρθt div udx  = P div udx + R ∇ρ · uθ div udx + R ρθ (div u)2 dx dt    + R ρ u · ∇θ div udx + R2 ρθ (div u)2 dx − κ R 1θ div udx   µ − R |∇ u + ∇ uT |2 div udx − λR (div u)3 dx

=



2

d

=

 P div udx +

dt

7 

Ii .

(2.14)

i=1

Similar to (2.12), we have I1 + I3 ≤ C ∥∇ρ∥2L2 + ∥∇θ ∥2L2 ∥∇ u∥2L2 + C ∥∇ u∥L∞ ∥∇ u∥2L2 + C .





Using (2.1) and (2.2), we deduce I2 + I4 ≤ C ∥ρ∥L∞ ∥θ ∥L∞ ∥∇ u∥2L2 ≤ C ∥∇ u∥2L2 . By integration by parts, and Young’s inequality, that I5 = κ R



∇θ · ∇ div udx ≤ ε∥∇ 2 u∥2L2 + C ∥∇θ∥2L2 .

As for the resident terms, we see I6 + I7 ≤ C ∥∇ u∥L∞ ∥∇ u∥2L2 + ∥∇ H ∥2L2 .





Substituting the above estimates into (2.14), we have

 −

d

ut · ∇ Pdx ≤

dt



P div udx + ε∥∇ 2 u∥2L2 + C ∥∇ u∥2L2 ∥∇ρ∥2L2 + ∥∇θ∥2L2 + ∥∇ u∥L∞ + 1





+ C ∥∇ u∥L∞ ∥∇ H ∥2L2 + C ∥∇θ∥2L2 + C .

(2.15)

Using integration by parts, the last term on the right-hand side of (2.8) can be written as



 −

ut ·

1 2

2



∇|H | − H · ∇ H dx =

1



2



|H | div ut dx − H · ∇ ut · Hdx   1 d d = |H |2 div udx − H · ∇ u · Hdx 2 dt dt    − (div u)H · Ht dx + Ht · ∇ u · Hdx + H · ∇ u · Ht dx, 2

(2.16)

where we have used the fact div H = 0. To handle the last three terms on the right-hand side of (2.16), we recall that ∇ × (u × H ) = H · ∇ u − u · ∇ H − Hdiv u and Ht = H · ∇ u − u · ∇ H − Hdiv u. Hence, after integration by parts, we deduce     − (div u)H · Ht dx = − (div u)H · H · ∇ u − (div u)H · u · ∇ H − (div u)2 H · H dx

≤ C ∥H ∥2L∞ ∥∇ u∥2L2 + C ∥H ∥L∞ ∥u∥L6 ∥∇ u∥L3 ∥∇ H ∥L2 ≤ C ∥∇ u∥2L2 + C ∥∇ H ∥2L2 ∥∇ u∥2L2 + ε∥∇ 2 u∥2L2 , and similarly,



 Ht · ∇ u · Hdx +

H · ∇ u · Ht dx ≤ 2ε∥∇ 2 u∥2L2 + C ∥∇ u∥2L2 + C ∥∇ H ∥2L2 ∥∇ u∥2L2 .

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

181

Substituting the above estimates into (2.16), we deduce



 −

ut ·

1 2



2

∇|H | − H · ∇ H dx ≤

1 d



2

|H | div udx −

2 dt

d dt



H · ∇ u · Hdx + 3ε∥∇ 2 u∥2L2

+ C ∥∇ u∥2L2 + C ∥∇ H ∥2L2 ∥∇ u∥2L2 .

(2.17)

Therefore, putting (2.9)–(2.13), (2.15) and (2.17) into (2.8), and choosing ε suitably small, we obtain



d

|∇ u|2 dx −

dt

  P+

dt

1 2

  |H |2 div u − H · ∇ u · H dx + ∥∇ 2 u∥2L2

≤ C ∥∇ρ∥ + ∥∇ H ∥ + ∥∇ u∥2L2 

2 L2

d

2 L2

  ∥∇ u∥2L2 + ∥∇θ ∥2L2 + ∥∇ u∥L∞ + 1 + C ∥∇θ∥2L2 + C .

(2.18)

To complete the proof of (2.7), it remains to bound the L2 -norms of ∇ρ and ∇ H. To this end, taking ‘‘∇ ’’ to both side of (1.2)1 and (1.2)4 and then multiplying the resultant equation by 2∇ρ and 2∇ H respectively, after summing up and integrating by parts we have d





dt

 |∇ρ|2 + |∇ H |2 dx ≤ C





 ρ|∇ρ| |∇ 2 u| + |∇ u| |∇ρ|2 + |H ∥∇ H ∥∇ 2 u| + |∇ u| |∇ H |2 dx

  ≤ ε∥∇ 2 u∥2L2 + C (∥∇ u∥L∞ + 1) ∥∇ρ∥2L2 + ∥∇ H ∥2L2 ,

(2.19)

where we have used (2.2) and Young’s inequality. By Lemmas 2.1 and 2.2, we have

        1 2  P div u + |H | div u − H · ∇ u · H dx ≤ ε∥∇ u∥2L2 + C ∥P ∥2L2 + ∥H ∥4L4 ≤ ε∥∇ u∥2L2 + C .  2

(2.20)

Therefore, choosing ε sufficiently small, using (2.18)–(2.20) and Gronwall’s inequality, we come to (2.7). This completes the proof of Lemma 2.3.  Using compatibility of (1.8) and L2 -estimate theory, we can improve the regularity of u. Lemma 2.4. Under the assumption (2.1), for any 0 ≤ T < T ∗ , we have

  √ sup ∥ ρ ut ∥2L2 + ∥∇θ ∥2L2 + ∥∇ u∥H 1 +

0 ≤t ≤T

T

 0

  √ ∥∇ ut ∥2L2 + ∥ ρθt ∥2L2 + ∥∇ 2 θ∥2L2 dt ≤ C .

(2.21)

Proof. Differentiating (1.2)2 with respect to t, and multiplying the resulting equation by ut , then integrating over Ω , we deduce



   ρ u2t dx + µ|∇ ut |2 + (λ + µ)(div ut )2 dx 2 dt    = −2 ρ u · ut ∇ ut dx − ρ u · ∇(u · ∇ u · ut )dx − ρ ut · ∇ u · ut dx   + Pt div ut dx + (H · Ht div ut − H · ∇ ut · Ht − Ht · ∇ ut · H ) dx.

1 d

Using Hölder, Sobolev and Young’s inequalities, we have

   1   −2 ρ u · ut · ∇ ut dx ≤ C ∥ρ∥ 2∞ ∥u∥L∞ ∥√ρ ut ∥L2 ∥∇ ut ∥L2 L   1 1 √ ≤ C ∥∇ u∥L22 ∥∇ u∥H2 1 ∥ ρ ut ∥L2 ∥∇ ut ∥L2 √ ≤ ε∥∇ ut ∥2L2 + C ∥∇ u∥H 1 ∥ ρ ut ∥2L2 .

After direct computation, we have

       2 2 2  ρ u · ∇ (u · ∇ u · ut ) dx ≤ ρ|u| |∇ u| |ut |dx + ρ u |∇ u| |ut |dx + ρ u2 |∇ u| |∇ ut |dx     ≤ C ∥u∥L6 ∥∇ u∥2L3 ∥ut ∥L6 + ∥u∥2L6 ∥∇ 2 u∥L2 ∥ut ∥L6 + ∥u∥2L6 ∥∇ u∥L6 ∥∇ ut ∥L2   ≤ C ∥∇ ut ∥L2 ∥∇ u∥2L2 ∥∇ u∥H 1 + ∥∇ u∥2L2 ∥∇ 2 u∥L2 + ∥∇ u∥2L2 ∥∇ u∥H 1 ≤ ε∥∇ ut ∥2L2 + C ∥∇ u∥2H 1 ,

(2.22)

182

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

where we have used (2.7), Sobolev, Hölder and Young’s inequalities. Directly, we see

     − ρ ut · ∇ u · ut dx ≤ C ∥∇ u∥L∞ ∥√ρ ut ∥22 .   L And then, similarly,

     Pt div ut dx =  

      R ρt θ div ut dx + R ρθt div ut dx     √ ≤ ε∥∇ ut ∥2L2 + C ∥ρt ∥2L2 + ∥ ρθt ∥2L2 .

And for the last term on the right-hand side of (2.22), we see

     (H · Ht div ut − H · ∇ ut · Ht − Ht · ∇ ut · H ) dx ≤ C ∥H ∥L∞ ∥Ht ∥L2 ∥∇ ut ∥L2   ≤ C (∥u∥L∞ ∥∇ H ∥L2 + ∥H ∥L∞ ∥∇ u∥L2 )∥∇ ut ∥L2   1 1 ≤ C ∥∇ u∥L22 ∥∇ u∥H2 1 + ∥∇ u∥L2 ∥∇ ut ∥L2 ≤ ε∥∇ ut ∥2L2 + C (∥∇ u∥H 1 + 1). Substituting all the above estimates into (2.22), we have 1 d



ρ

2 dt

u2t dx





+

 µ|∇ ut |2 + (λ + µ)(div ut )2 dx

  √ √ ≤ 4ε∥∇ ut ∥2L2 + C ∥∇ u∥H 1 + ∥∇ u∥L∞ ∥ ρ ut ∥2L2 + C (∥∇ u∥2H 1 + 1) + C ∥ ρθt ∥2L2 .

(2.23)

Multiplying (1.2)3 by θt , then integrating over Ω , after integration by parts, we deduce

κ d



2 dt

|∇θ|2 dx +



ρθt2 dx = −



ρ u · ∇θ θt dx −



P div ut θt dx +

  µ 2

 |∇ u + ∇ uT |2 + λ(div u)2 θt dx. (2.24)

By Sobolev, Hölder and Young’s inequalities, that

   1   − ρ u · ∇θ θt dx ≤ C ∥ρ∥ 2∞ ∥u∥L∞ ∥√ρθt ∥L2 ∥∇θ∥L2 L   1 √

≤

4

∥ ρθt ∥2L2 + C ∥∇ u∥L2 ∥∇ u∥H 1 ∥∇θ∥2L2 .

Similarly,

          − P div uθt dx = R ρθ div uθt dx     1 √ ≤ C ∥ρ∥L2∞ ∥ ρθt ∥L2 ∥θ ∥L∞ ∥∇ u∥L2 1 √ ≤ ∥ ρθt ∥2L2 + C ∥∇ u∥2L2 .

4

Integrating by parts with respect to t, we have

  µ 2

|∇ u + ∇ u | + λ(div u) T 2

2



θt dx =

d

  µ

dt

2

 |∇ u + ∇ uT |2 θ + λ(div u)2 θ dx



      − µ ∇ u + ∇ uT : ∇ ut + ∇ uTt θ + 2λdiv udiv ut θ dx  d ≤C |∇ u|2 |θ |dx + C ∥∇ u∥2L2 + δ∥∇ ut ∥2L2 . dt

Substituting all the above estimates into (2.24), we obtain

κ d 2 dt



2

|∇θ| dx +

1 2



ρθ

2 t dx

  d ≤ C ∥∇ u∥H 1 + 1 + C



dt

+ δ∥∇ ut ∥2L2 + δ∥∇ Ht ∥2L2 .

2

|∇ u| |θ |dx + C

d dt



|∇ H |2 |θ|dx (2.25)

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

183

Thus, we deduce from (2.23) and (2.25), that d



dt

Ht2

+ρ



  + |∇θ| + |∇ Ht |2 + |∇ ut |2 + ρθt2 dx    √ ≤ C ∥Ht ∥2L2 + ∥ ρ ut ∥2L2 + ∥∇θ∥2L2 ∥∇ u∥L∞ + ∥∇ u∥H 1 + 1     d d +C |∇ u|2 |θ|dx + C |∇ H |2 |θ|dx + C ∥∇ u∥H 1 + 1 , 

u2t

2



dt

dt

only if we choose ε and δ sufficiently small, then Gronwall’s inequality implies that

 √  sup ∥ ρ ut ∥2L2 + ∥∇θ ∥2L2 +

0≤t ≤T

T





0

 √ ∥∇ ut ∥2L2 + ∥ ρθt ∥2L2 dt ≤ C .

(2.26)

Moreover, we see u and θ satisfy

 1 2   µ1u + (λ + µ)∇ div u = ρ ut + ρ u · ∇ u + ∇ P − H · ∇ H + 2 ∇|H | , µ κ 1θ = ρθt + ρ u · ∇θ + P div u − |∇ u + ∇ uT |2 − λ(div u)2 ,   2  (1.4) or (1.5) holds.

(2.27)

Then, from the standard L2 -theory of elliptic system, that

 √  ∥∇ u∥H 1 ≤ C ∥ ρ ut ∥L2 + ∥u∥L∞ ∥∇ u∥L2 + ∥∇ρ∥L2 + ∥∇θ ∥L2 + ∥H ∥L∞ ∥∇ H ∥L2 + ∥∇ u∥L2 1

∥∇θ ∥H 1

≤ C ∥∇ u∥H2 1 + C ;   √ ≤ C ∥ ρθt ∥L2 + ∥u∥L∞ ∥∇θ∥L2 + ∥∇ u∥L2 + ∥∇ u∥2L4 + ∥∇θ∥L2   1 1 3 √ 2 2 2 ≤ C ∥ ρθt ∥L2 + ∥∇ u∥H 1 + ∥∇ u∥L2 ∥∇ u∥H 1 + 1 .

(2.28)

Then from the above two estimates and (2.26), we conclude (2.21). Thus, we complete the proof of Lemma 2.4.



Using compatibility of (1.9) and L -estimate theory, we can improve the regularity of θ . 2

Lemma 2.5. Under the assumption (2.1), for any 0 ≤ T < T ∗ , we have sup 0≤t ≤T

 √  ∥ ρθt ∥2L2 + ∥∇θ ∥2H 1 +

T



∥∇θt ∥2L2 dt ≤ C .

0

(2.29)

Proof. Differentiating (1.2)3 with respect to t, then multiplying the resulting equation by θt , and using integration by parts over Ω , one gets 1 d 2 dt



ρθ

2 t dx

+κ





|∇θt | dx = R

ρθ 



2 t div

ρt θ div uθt dx + R



ρθ div ut θt dx     µ  + ∇ u + ∇ uT : ∇ ut + ∇ uTt θt dx + 2λ div udiv ut θt dx 2   − ρt uθ θt dx − ρt θt2 dx.

2

udx + R

Obviously, we have

 R

√ ρθt2 div udx ≤ C ∥∇ u∥L∞ ∥ ρθt ∥2L2 .

Using the estimates we have obtained, Sobolev, Hölder and Young’s inequalities lead to

 R

ρt θ div uθt dx = −R



ρ∇ uθ div uθt dx − R



∇ρ · uθ div uθt dx

≤ C ∥ρ∥L∞ ∥θ ∥L∞ ∥∇ u∥L2 ∥∇ u∥L3 ∥θt ∥L6 + C ∥u∥L∞ ∥θ ∥L∞ ∥∇ρ∥L2 ∥∇ u∥L3 ∥θt ∥L6 1

1

1

1

1

1

≤ C ∥∇ u∥L22 ∥∇ u∥H2 1 ∥∇θt ∥L2 + C ∥∇ u∥L22 ∥∇ u∥H2 1 ∥∇ u∥L22 ∥∇ u∥H2 1 ∥∇θt ∥L2 ≤ ε∥∇θt ∥2L2 + C .

(2.30)

184

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

Similarly,



2

1

ρθ div ut θt dx ≤ C ∥θ∥L∞ ∥ρ∥L3∞ ∥ρ∥L31 ∥∇ ut ∥L2 ∥θt ∥L6 ≤ ε∥∇θt ∥2L2 + C ,

R and

µ



    ∇ u + ∇ uT : ∇ ut + ∇ uTt θt dx + 2λ

2

1



div udiv ut θt dx 1

≤ C ∥∇ u∥L3 ∥∇ ut ∥L2 ∥θt ∥L6 ≤ C ∥∇ u∥L22 ∥∇ u∥H2 1 ∥∇ ut ∥L2 ∥∇θt ∥L2 ≤ ε∥∇θt ∥2L2 + C . That

 −

ρt uθ θt dx =



ρ∇ u · uθ θt dx +



∇ρ u2 θ θt dx

1 √ ≤ C ∥ρ∥L2∞ ∥θ ∥L∞ ∥ ρ u∥L2 ∥∇ u∥L3 ∥θt ∥L6 + C ∥θ ∥L∞ ∥∇ρ∥L2 ∥u∥2L6 ∥θt ∥L6

≤ 2ε∥∇θt ∥2L2 + C , and using (1.2)1 and integrating by parts,

 −

ρθ

2 t t dx



ρ u · ∇θt θt dx

=−

1 √ ≤ C ∥ρ∥L2∞ ∥u∥L∞ ∥ ρθt ∥L2 ∥∇θt ∥L2 √ ≤ ε∥∇θt ∥2L2 + C ∥ ρθt ∥2L2 .

Thus, substituting all the above estimates into (2.30), choosing ε suitably small, we have d



dt

ρθt2 dx +



√ |∇θt |2 dx ≤ C (∥∇ u∥L∞ + 1) ∥ ρθt ∥2L2 + C .

Then Gronwall’s inequality implies

√

sup ∥ ρθt ∥2L2 +

T



0≤t ≤T

0

∥∇θt ∥2L2 dt ≤ C ,

which together with (2.28) gives (2.29). This is the end of the proof of Lemma 2.5.



Remark 2.1. In the proof of Lemma 2.5 and the following Lemma 2.6, we have only dealt with the Cauchy problem. More precisely, for the Dirichlet problem, the estimate ∥θt ∥L6 ≤ C ∥∇θt ∥L2 is incorrect due to the lack of necessary estimates of ∥θt ∥L2 . Instead, we could use the following Poincaré type inequality (see [16]) to overcome the difficulty:

√ ∥θt ∥L6 ≤ C ∥ ρθt ∥L2 + C ∥∇θt ∥L2 . The other estimates can be obtained in the same manner. The final step is to obtain the bounds of the first derivative of ρ and H as follows. Lemma 2.6. Under the assumption (2.1) and for any q ∈ (3, 6], for any 0 ≤ T < T ∗ , we have sup 0≤t ≤T



 ∥ρt ∥Lq + ∥ρ∥W 1,q + ∥Ht ∥Lq + ∥H ∥W 1,q +



T



 ∥∇ 2 u∥2Lq + ∥∇ 2 θ ∥2Lq dt ≤ C .

(2.31)

0

Proof. From (1.2)1 , for any q ∈ (3, 6], one deduces d dt



|∇ρ|q dx ≤ C



  |∇ u| |∇ρ|q + |∇ 2 u| |∇ρ|q−1 dx

≤ C ∥∇ u∥L∞ ∥∇ρ∥qLq + C ∥∇ 2 u∥Lq ∥∇ρ∥Lqq−1 , which immediately implies d dt

∥∇ρ∥Lq ≤ C (∥∇ u∥L∞ + 1) ∥∇ρ∥Lq + C ∥∇ 2 u∥Lq .

(2.32)

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

185

Similarly, d dt

∥∇ H ∥Lq ≤ C ∥∇ u∥L∞ ∥∇ H ∥Lq + C ∥∇ 2 u∥Lq ,

which, together with (2.32), immediately leads to d dt

(∥∇ρ∥Lq + ∥∇ H ∥Lq ) ≤ C (∥∇ u∥L∞ + 1) (∥∇ρ∥Lq + ∥∇ H ∥Lq ) + C ∥∇ 2 u∥Lq .

(2.33)

Now, it is the position to estimate ∥∇ 2 u∥Lq . Using Lp -theory for elliptic systems (2.27) leads to

∥∇ 2 u∥Lq ≤ C (∥ρ ut ∥Lq + ∥u · ∇ u∥Lq + ∥∇ρ∥Lq + ∥∇θ ∥Lq + ∥∇ H ∥Lq )   3q−6 6−q 3q−6 6−q √ 2q 2q 2q 2q 2 ≤ C ∥ ρ ut ∥L2 ∥ut ∥L6 + ∥u∥L∞ ∥∇ u∥Lq + ∥∇ρ∥Lq + ∥∇θ ∥L2 ∥∇ θ ∥L2 + ∥∇ H ∥Lq   ≤ C ∥∇ ut ∥L2 + ∥∇ρ∥Lq + ∥∇ H ∥Lq + 1 ,

(2.34)

due to (2.21) and (2.29). Substituting (2.34) into (2.33) and using Gronwall’s inequality to the resulting inequality, we obtain sup (∥∇ρ∥Lq + ∥∇ H ∥Lq ) + 0≤t ≤T

T



∥∇ 2 u∥2Lq dt ≤ C .

(2.35)

0

Similar to (2.34), we have

  ∥∇ 2 θ∥Lq ≤ C ∥ρθt ∥Lq + ∥ρ u · ∇θ∥Lq + ∥P div u∥Lq + ∥∇ u∥2L2q   3q−6 6−q √ 2q 2q ≤ C ∥ ρθt ∥L2 ∥θt ∥L6 + ∥u∥L∞ ∥∇θ∥Lq + ∥∇ u∥Lq + ∥∇ u∥L∞ ∥∇ u∥Lq 

1

3q−6

6−q

1

6−q

3q−6

≤ C ∥∇θt ∥L2 + ∥∇ u∥L22 ∥∇ 2 u∥L22 ∥∇θ ∥L22q ∥∇ 2 θ∥L22q + ∥∇ u∥L22q ∥∇ 2 u∥L22q + ∥∇ u∥

2q−6 5q−6 L2

2

∥∇ u∥

3q 5q−6 Lq

∥∇ u∥

6−q 2q L2

2

∥∇ u∥

3q−6 2q L2



  ≤ C ∥∇ 2 u∥Lq + ∥∇θt ∥L2 + 1 . Thus, due to the above estimates and (2.35), we conclude that (2.31) except the estimates ∥ρt ∥Lq and ∥Ht ∥Lq . Using this, together with (2.21), and Sobolev embedding inequality, we can also deduce the estimates of ∥ρt ∥Lq and ∥Ht ∥Lq from (1.2)1 and (1.2)4 respectively. Therefore, we finish the proof of Lemma 2.6.  The combination of Lemmas 2.4–2.6 is enough to extend the strong solution (ρ, u, θ , H ) beyond t ≥ T ∗ . Actually, due to (2.21), (2.29) and (2.31), the functions (ρ, u, θ , H )|t =T ∗ = limt →T ∗ (ρ, u, θ , H ) satisfy the condition imposed on the initial data (1.3) at the time t = T ∗ . Furthermore, 1

−µ1u − (λ + µ)∇ div u + P − (∇ × H ) × H |t =T ∗ = lim∗ (ρ ut + ρ u · ∇ u) , ρ 2 g1 |t =T ∗ , t →T

and

−κ 1θ −

µ 2

1

|∇ u + ∇ uT |2 + λ(div u)2 |t =T ∗ = lim∗ − (ρθt + ρ u · ∇θ + P div u) , ρ 2 g2 |t =T ∗ , t →T

with (g1 , g2 ) ∈ L2 . Thus, (ρ, u, θ , H )|t =T ∗ also satisfies (1.8) and (1.9). Therefore, we can take (ρ, u, θ , H )|t =T ∗ as the initial data and apply the local existence theorem [12] to extend the local strong solution beyond T ∗ . This contradicts the assumption on T ∗ . References [1] [2] [3] [4] [5] [6] [7]

J.P. Freidberg, Ideal magnetohydrodynamic theory of magnetic fusion systems, Rev. Modern Phys. 54 (1982). The American Physical Society. D.A. Gurnett, A. Bhattacharjee, Introduction to Plasma Physics, Cambridge Univ. Press, Cambridge, 2005. A.F. Kulikovskiy, G.A. Lyubimov, Magnetohydrodynamics, Addison-Wasley, Reading, MA, 1965. L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, second ed., Butterworth,Heinemann, London, 1999. L.C. Woods, Priciples of Magnetoplasma Dynamics, Oxford University Press, Oxford, 1987. G.Q. Chen, D. Wang, Global solution of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182 (2002) 344–376. G.Q. Chen, D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamic equations, Z. Angew. Math. Phys. 54 (2003) 608–632. [8] D.H. Wang, Large solutions to the initial–boundary value problem for planar magnetohydrodynamics, SIAM J. Appl. Math. 63 (2003) 1424–1441.

186

M. Chen, S. Liu / J. Math. Anal. Appl. 400 (2013) 174–186

[9] T. Umeda, S. Kawashima, Y. Shizuta, On the decay of solutions to the linearized equations of electromagneto uid dynamics, Japan. J. Appl. Math. 1 (1984) 435–457. [10] X.P. Hu, D.H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 283 (2008) 255–284. [11] X.P. Hu, D.H. Wang, Global existence and large-time behavior of solutions to the three-dimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010) 203–238. [12] J.S. Fan, W.H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. RWA 10 (2009) 392–409. [13] X.Y. Xu, J.W. Zhang, A blow-up criterion for 3D non-resistive compressible magnetohydrodynamic equations with initial vacuum, Nonlinear Anal. RWA 12 (2011) 3442–3451. [14] A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heatconductive gases, J. Math. Kyoto Univ. 20 (1980) 67–104. [15] D. Hoff, Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data, Arch. Ration. Mech. Anal. 132 (1995) 1–14. [16] P.L. Lions, Mathematical Topics in Fluid Mechanics, in: Compressible Model., vol. 2, Oxford University Press, New York, 1998. [17] E. Feireisl, A. Novotny, H. Petzeltová, On the existence of globally defined weak solutions to the Navier–Stokes equations, J. Math. Fluid Mech. 3 (2001) 358–392. [18] E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid, Indiana Univ. Math. J. 53 (2004) 1705–1738. [19] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. [20] X.D. Huang, J. Li, Z.P. Xin, Blowup criterion for the compressible flows with vacuum states, Comm. Math. Phys. 301 (2010) 23–35. [21] X.D. Huang, J. Li, Z.P. Xin, Serrin type criterion for the three-dimensional viscous compressible flows, SIAM J. Math. Anal. 43 (2011) 1872–1886. [22] M. Lu, Y. Du, Z.A. Yao, Blow-up criterion for compressible MHD equations, J. Math. Anal. Appl. 379 (2011) 425–438. [23] Y.Z. Sun, C. Wang, Z.F. Zhang, A Beale–Kato–Majda blow-up criterion for 3-D compressible Navier–Stokes equations, J. Math. Pures Appl. 95 (2011) 36–47. [24] Y.Z. Sun, C. Wang, Z.F. Zhang, A Beale–Kato–Majda creterion for three dimensional compressible viscous heat-conductive flows, Arch. Ration. Mech. Anal. 201 (2011) 727–742. [25] J.S. Fan, S. Jiang, Y.B. Ou, A blow-up criterion for three-dimensional compressible viscous flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010) 337–350. [26] M. Lu, Y. Du, Z.A. Yao, Z.J. Zhang, A blow-up criterion for compressible MHD equations, Commun. Pure Appl. Anal. 11 (2012) 1167–1183. [27] M.T. Chen, S.Q. Liu, Blowup criterion for 3D viscous-resistive compressible MHD equations, Mathematical Methods in the Applied Sciences, http://dx.doi.org/10.1002/mma.2674.