Low Mach number limit of the full compressible Navier–Stokes–Maxwell system

J. Math. Anal. Appl. 412 (2014) 334–344

Contents lists available at ScienceDirect

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

Low Mach number limit of the full compressible Navier–Stokes–Maxwell system Fucai Li, Yanmin Mu ∗ Department of Mathematics, Nanjing University, Nanjing 210093, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 24 August 2013 Available online 29 October 2013 Submitted by D. Wang

This paper deals with the low Mach number limit of the full compressible Navier–Stokes– Maxwell system. It is justified rigorously that, for the well-prepared initial data, the solutions of the full compressible Navier–Stokes–Maxwell system converge to that of the incompressible Navier–Stokes–Maxwell system as the Mach number tends to zero. © 2013 Elsevier Inc. All rights reserved.

Keywords: Full compressible Navier–Stokes–Maxwell system Incompressible Navier–Stokes–Maxwell system Low Mach number limit Nonlinear energy method

1. Introduction and main results In this paper we consider the low Mach number limit of the following full compressible Navier–Stokes–Maxwell system [10,18,19]:

∂t ρ + div(ρ u) = 0,







ρ (ut + u · ∇ u) + ∇ p = div 2μP + μ div uI3 + J × B, ρ

∂p ∂e div u = κ θ + Ψ + J · (E + u × B), (θt + u · ∇θ) + θ ∂θ ∂θ

ν Et −

1

μ0

curl B + J = 0,

Bt + curl E = 0,

(1.1) (1.2) (1.3) (1.4)

div B = 0.

(1.5)

Here the unknowns ρ > 0, u = (u 1 , u 2 , u 3 ), θ > 0, E = ( E 1 , E 2 , E 3 ), and B = ( B 1 , B 2 , B 3 ) represent the mass density, the velocity, the absolute temperature, the electric field, and the magnetic flux density, respectively. I3 is a 3 × 3 unit vector. P is the deformation tensor, whose entries are

Pij =

*

 ∂u j ∂ ui , + 2 ∂xj ∂ xi 1



i , j = 1, 2, 3.

Corresponding author. E-mail addresses: fl[email protected] (F. Li), [email protected] (Y. Mu).

0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmaa.2013.10.064

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

335

The current density J is given by Ohm’s law J = σ (E + u × B).

Ψ = 2μ

3 

( P i j )2 + μ (div u)2

i , j =1

is the viscous dissipation function. The pressure p = p (ρ , θ) and the internal energy e = e (ρ , θ) are smooth functions of the density ρ and the temperature θ of the flow and satisfy the Gibbs relation

  1 , θ dS = de + pd

ρ

for some smooth function S = S (ρ , θ) which expresses the first law of the thermodynamics. The viscosity coefficients μ and μ , the heat conductivity coefficient κ , and the electric conductivity coefficient σ are assumed to be constants such that μ > 0, 2μ + μ > 0, κ > 0, and σ > 0. The dielectric constant ν and the magnetic permeability μ0 are assumed to be positive constants. The system (1.1)–(1.5) describes the motion of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. By taking the dielectric constant ν = 0 in (1.4), then we obtain that J = curl B/μ0 . Thanks to Ohm’s law, we can eliminate the electric field E in (1.2), (1.3), and (1.5), and finally obtain the full compressible magnetohydrodynamic equations (see [11] for more details)

∂t ρ + div(ρ u) = 0,

(1.6)





ρ (ut + u · ∇ u) + ∇ p = div 2μP + μ div uI + ρ

1

μ0

curl B × B,

∂p ∂e 1 div u = κ θ + Ψ + | curl B|2 , (θt + u · ∇θ) + θ ∂θ ∂θ σ μ20

Bt − curl(u × B) +

1

σ μ0

curl curl B = 0,

div B = 0.

(1.7) (1.8) (1.9)

There is a lot of literature on the full compressible magnetohydrodynamic equations (1.6)–(1.9) due to their physical importance, complexity, rich phenomena, and mathematical challenges, see [2–4,6,14,16] and the references cited therein. In [14], Jiang, Ju, and Li studied the low Mach number limit to the system (1.6)–(1.9) in the framework of the local smooth solutions for small variational density and temperature. Later, the large variational density and temperature case was studied in [16]. For more results on low Mach number limit to the magnetohydrodynamic equations, please see [7,12,13] on isentropic case, [15] on the ideal non-isentropic case, [17] on full magnetohydrodynamics equations in framework of weak solutions, and [22] on magnetohydrodynamics equations with strong stratification. To the authors’ best knowledge, there are no results on low Mach number limit of the full compressible Navier–Stokes– Maxwell system (1.1)–(1.5). In this paper we extend the results in [14] to the system (1.1)–(1.5). We shall focus our study on the fluid obeying the perfect relations p = ρ θ and e = c V θ , where the constants  > 0 and c V > 0 are the gas constant and the heat capacity volume, respectively. In order to study the low Mach number limit of the system (1.1)–(1.5), similarly to [14], we use its appropriate dimensionless form as follows

∂t ρ + div(ρ u) = 0,   ∇(ρ θ) ρ ut + (u · ∇ u)u + = 2μ div P + μ div(div uI3 ) + J × B,

2   ρ (θt + u · ∇θ) + (γ − 1)ρ θ div u = κ θ + 2 Ψ + J · (E + u × B) ,

(1.11)

ν Et − (1/μ0 ) curl B + J = 0, J = σ (E + u × B),

(1.13)

Bt + curl E = 0,

(1.14)

div B = 0,

(1.10)

(1.12)

where is the Mach number and the coefficients μ, μ , ν , and κ are the scaled parameters. γ = 1 + R/c V is the ratio of specific heats. Note that we have used the same notations and assumed that the coefficients μ, μ , ν , and κ are independent of for simplicity. Setting

ρ = 1 + q,

θ = 1 + φ,

then we can rewrite (1.10)–(1.14) as

336

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

∂t q + u · ∇ q + 

1





1 + q div u = 0,

(1.15)



 1      ∂t u + u · ∇ u + 1 + q · ∇φ + 1 + φ · ∇ q

  = 2μ div P + μ div div u I3 + J × B ,   1 ν ∂t E − curl B = −J , J = σ E + u × B , 1 + q

μ0

∂t B + curl E = 0, div B = 0,    γ − 1      1 + q ∂t φ + u · ∇φ + 1 + q 1 + φ div u = Ψ + J · E + u × B + κ φ ,



(1.16) (1.17) (1.18) (1.19)

with

  ∂ u j   1 ∂ u i , P ij = + 2 ∂xj ∂ xi

Ψ = 2μ

3  

P

 2

i , j =1

ij

 2 + μ div u .

The system (1.15)–(1.19) is equipped with initial data

    q , u , E , B , φ t =0 = q 0 (x), u 0 (x), E 0 (x), B 0 (x), φ0 (x) ,

x ∈ Ω.

(1.20)

Here Ω = R3 or T3 , and we have added the superscript on the unknowns (q , u , E , B , φ ) to stress the dependence of the parameter . Formal if we take the limit → 0 in (1.15)–(1.19), we obtain the following incompressible Navier–Stokes–Maxwell equations (we suppose that the limits u → w, B → H, and E → D exist)

∂t w + w · ∇ w + ∇ π = μw + J × H,

ν ∂t D −

1

μ0

curl H + J = 0,

(1.21)

J = σ (D + w × H),

(1.22)

∂t H + curl D = 0,

(1.23)

div w = 0,

(1.24)

div H = 0.

The system (1.21)–(1.24) is supplemented with initial data

  (w, D, H) t =0 = w0 (x), D0 (x), H0 (x) ,

x ∈ Ω.

(1.25)

In this paper we shall establish the above limit rigorously in Ω . Moreover, we shall show that for sufficiently small Mach number, the compressible Navier–Stokes–Maxwell system (1.15)–(1.19) admits a smooth solution on the time interval where the smooth solution of the incompressible Navier–Stokes–Maxwell equations (1.21)–(1.24) exists. Before stating our main result we recall some known results on the incompressible Navier–Stokes–Maxwell equations (1.21)–(1.24). Masmoudi [21] obtained the existence and uniqueness of global strong solutions (1.21)–(1.24) for the initial data (w0 , D0 , H0 ) ∈ L 2 (R2 ) × H s (R2 ) × H s (R2 ) with s > 0. Ibrahim and Keraani [8] established the strong solutions to the 1/ 2 ˙ 1/2 (R3 ) × H˙ 1/2 (R3 ). Very recently, this result was improved by system (1.21)–(1.24) for the initial belonging to B˙ 2,1 (R3 ) × H

˙ 1/2 (R3 ) × H˙ 1/2 (R3 ) × H˙ 1/2 (R3 ). Ibrahim and Yoneda [9] Germain, Ibrahim, and Masmoudi [5] where the initial data lie in H constructed local smooth solution for non-decaying initial data on the torus T3 and showed the loss of smoothness of solutions. We remark that although there are some results on the Navier–Stokes–Maxwell system (1.21)–(1.24), the global finite energy weak solution (Leray-type solution) to (1.21)–(1.24) for initial data lying in L 2 (Rd ) is still an interesting open problem for d = 2, 3. In this paper, for simplicity, we assume that the problem (1.21)–(1.25) has a local smooth solution as obtained in [9]. The main result of the present paper is the following. Theorem 1.1. Let s > 3/2 + 2. Suppose that the initial data (q 0 (x), u 0 (x), E 0 (x), B 0 (x), φ0 (x)) belong to H s and satisfy





q (x), u (x) − w0 (x), E (x) − D0 (x), B (x) − H0 (x) = O ( ) 0 0 0 0 s

(1.26)

for some smooth functions (w0 (x), D0 (x), H0 (x)). Let (w, D, H, π ) be a smooth solution to the system (1.21)–(1.24) with initial data (w0 (x), D0 (x), H0 (x)) on [0, T ∗ ] × Ω for T ∗ > 0 finite, then there exists a constant 0 > 0 such that, for all  0 , the problem (1.15)–(1.20) has a unique smooth solution (q , u , E , B , φ ) ∈ C ([0, T ∗ ], H s ). Moreover, there exists a positive constant K > 0, independent of , such that, for all  0 ,





sup

q −

t ∈[0, T ∗ ]





π , u − w, E − D, B − H, φ − π

 K . 2 2



s

(1.27)

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

337

The proof of Theorem 1.1 is based on the method developed in [14]. The key point is to obtain the uniform estimates of the error system and apply convergence-stability lemma [1,24] and Gronwall-type inequality. Compared with the full compressible magnetohydrodynamic equations (1.6)–(1.9), the full compressible Navier–Stokes–Maxwell system (1.15)–(1.19) is more complicated and more refined analysis is needed in our arguments, see the details presented in the next section. Before ending the introduction, we give the notations used throughout the current paper. We use the letter C to denote various positive constants independent of . For convenience, we denote by H l ≡ H l (Ω) (l ∈ R) the standard Sobolev spaces and write · l for the standard norm of H l and · for · 0 . The remainder of this paper is devoted to the proof of Theorem 1.1. 2. Proof of Theorem 1.1 This section is devoted to proving Theorem 1.1. Denoting U = (q , u , E , B , φ ) , we rewrite the system (1.15)–(1.19) in the vector form





A 0 U ∂t U +

3 









A j U ∂jU = Q U ,

(2.1)

j =1

where the matrices A j (U ) (0  j  3) are given by









A 0 U = diag 1, 1 + q , 1 + q , 1 + q , ν , ν , ν , 1, 1, 1, 1 + q ,

⎛

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜   ⎜ ⎜ A1 U = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜   ⎜ ⎜ A2 U = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜   ⎜ ⎜ A3 U = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0



1 + φ

0

0 0 0 0 0

0 0 0 0

u 1 (1 + q ) 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0



0

0

0 0 0 0 0

0 0

0 0

0 0 0 0 0 0 0 0 0 0

u (1 + q ) 0 0 0 0 0 0 0

0 u 2 (1 + q ) 0 0 0 0 0 0



0



0

⎛

1 + q

u 1

u 2 0

1 + φ

1 + q



u (1 + q ) 2

0 0 0 0 0 0 0 0

0

0

u 3 0 0

0

1 + φ

1

(γ −1)(1+ q )(1+ φ )

0 0 0 0 0 0 0



0 u (1 + q )

2

0 0 0

(γ −1)(1+ q )(1+ φ )

0 0

u 3 (1 + q ) 0 u 3 (1 + q )

0 0

0 0 u 1 (1 + q ) 0 0 0 0 0

1 + q



0 0

0

0 0

0 0

u 3 (1 + q ) 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0

0

0



(γ −1)(1+ q )(1+ φ )



0 0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 −1

0 0 1 0

0 0 0 0

0 0 0 0 0 −1 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0

0 0 0 0 0 0 0 0 0 − μ1 0 1 0 μ0 0 0 0 0 0 0

0 0



0 0 0 0 0 0 0 0

⎞

0 0

1 + q

0 0 0 0 0 μ1 0 0 0 1 μ0 0 0 0 0 0 0 0



0 0 0 0 0 0 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 u 2 (1 + q ) 0 0 0

0 0 0 0 0 0 0 0 0 − μ1 0 0 0 0 μ1 0 0 0 0 0 0 0 0 1 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

1 + q

u 1 (1 + q )

0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

0

0

0 0 0 0 0 0 0 0 0 0

0 0 0

1 + q



0 0 0 0 0 0

0 u 3 (1 + q )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

338

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

and















Q U := 0, L U , −J , 0, 0, 0, κ φ + Ψ + J · E + u × B

with





J = σ E + u × B ,











L U = 2μ div P + μ div div u I3 + J × B

  = μu + μ + μ ∇ div u + J × B .

It is easy to see that the matrices A j (U ) (0  j  3) can be symmetrized by choosing  

  1 ˆA 0 U = diag 1 + φ , 1, 1, 1, μ0 , μ0 , μ0 , 1, 1, 1, . 1 + q

(γ − 1)(1 + φ )

˜ 0 (U ) is a positive definite symmetric matrix Moreover, for U ∈ G¯ 1  G with G being the state space for the system (2.1), A for sufficiently small . Assume that the initial data  

U (0, x) = U (x) := q (x), u (x), E (x)B (x), φ (x) ∈ H s 0

0

0

0

0

0

and U 0 (x) ∈ G 0 , G¯ 0  G. First, following the proof of the local existence theory for the initial value problem of symmetrizable hyperbolic–parabolic systems by Volpert and Hudjaev [23], we obtain that there exists a time interval [0, T ] with T > 0, so that the system (2.1) with initial data U 0 (x) has a unique classical solution U (t , x) ∈ C ([0, T ], H s ) and U (t , x) ∈ G 2 with G¯ 2  G. We remark that the crucial step in the proof of local existence result is to prove the uniform boundedness of the solutions. Now we define     T = sup T > 0: U (t , x) ∈ C [0, T ], H s , U (t , x) ∈ G 2 , ∀(t , x) ∈ [0, T ] × Ω . Note that T depends on and may tend to zero as goes to 0. To show that lim inf →0 T > 0, we shall make use of the convergence-stability lemma which was established in [1,24] for hyperbolic systems of balance laws. Similarly to [14], for the hyperbolic–parabolic system (2.1), we have the following convergence-stability lemma. Lemma 2.1. Let s > 3/2 + 2. Suppose that U 0 (x) ∈ G 0 , G¯ 0  G, and U 0 (x) ∈ H s , and the following convergence assumption (A) holds. (A) For each , there exist T  > 0 and U ∈ L ∞ (0, T  ; H s ) satisfying





U (t , x)  G ,

x,t ,

such that, for t ∈ [0, min{ T  , T }), sup U (t , x) − U (t , x) = o(1),





sup U (t , x) − U (t , x) s = O (1),

x,t

t

as → 0.

Then there exists an ¯ > 0 such that, for all ∈ (0, ¯ ], it holds that

T > T . To apply Lemma 2.1, we construct the approximation U = (q , v , D , H , φ ) to the original system (2.1) with q =

π /2, v = w, D = D, H = H, and φ = π /2, where (w, D, H, π ) is the smooth solution to the problem (1.21)–(1.25). It is easy to verify that U satisfies

1





2

∂t q + v · ∇ q + (1 + q ) div v = (1 + q )(∂t v + v · ∇ v ) + = μv +

ν ∂t D −

1

μ0

2 2

1



(πt + w · ∇ π ),

(1 + q )∇φ + (1 + φ )∇ q

π (wt + w · ∇ w + ∇ π ) + J × H ,

curl H = −J ,

∂t H + curl D = 0,

J = σ (D + w × H ),

div H = 0,

γ −1 (1 + q )(∂t φ + v · ∇φ ) + (1 + q )(1 + φ ) div v

 

3 = + π (πt + w · ∇ π ). 2

4

(2.2)

 (2.3) (2.4) (2.5)

(2.6)

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

339

We rewrite the system (2.2)–(2.6) in the following vector form 3 

A 0 (U )∂t U +

A j (U )∂ j U = S (U ) + R

(2.7)

j =1

with S (U ) = (0, μv + J × H , −J , 0, 0, 0, 0) and

⎛

(π + w · ∇ π ) t 2

⎞

⎜ π (wt + w · ∇ w + ∇ π ) ⎟ ⎟ ⎜ 2 ⎟. 0 R =⎜ ⎟ ⎜ ⎠ ⎝ 0  3  + 4 π (πt + w · ∇ π ) 2 2

Due to the regularity assumptions on (w, π ) in Theorem 1.1, we have





max R (t ) s  C .

(2.8)

t ∈[0, T ∗ ]

In order to prove Theorem 1.1, thanks to Lemma 2.1, it suffices to establish the error estimate in (1.27) for t ∈

[0, min{ T ∗ , T }). To this end, introducing F = U − U

and

1 A j (U ) = A − 0 (U ) A j (U ),

and utilizing (2.1) and (2.7), we arrive at

Ft +

3 

3             1 1 A j U Fxj = A j (U ) − A j U U x j + A − U Q U − A− 0 0 (U ) S (U ) + R .

j =1

(2.9)

j =1

α satisfying |α |  s, we apply the operator D α to (2.9) to obtain that

For any multi-index

∂t D α F +

3 

  A j U ∂x j D α F = P 1α + P 2α + Q α + R α

(2.10)

j =1

with

P 1α =

3  

      A j U ∂x j D α F − D α A j U ∂x j F ,

j =1

P 2α =

3 

Dα



   A j (U ) − A j U U x j ,

j =1

 1     1 Y α = D α A− U Q U − A− 0 0 (U ) S (U ) ,  1  R α = D α A− 0 (U ) R . As in [14], we define the canonical energy by



F 2e :=





Define







˜ 0 U F , F dx. A



˜ 0 U = diag A



 1 . , 1 , 1 , 1 , νμ , νμ , νμ , 1 , 1 , 1 , 0 0 0 (γ − 1)(1 + φ ) (1 + q )2 1 + φ

˜ 0 (U ) is a positive definite symmetric matrix and A˜ 0 (U )A j (U ) is symmetric. Now, if we multiply (2.10) by Note that A ˜ 0 (U ) and take the inner product between the resulting system and D α F , we arrive at A d α 2

D F = e dt where





 Γ D α F , D α F dx + 2



  α T   α ˜ 0 U P + P α + Y α + R α dx, D F A 1 2

             Γ = (∂t , ∇) · A˜ 0 , A˜ 0 U A1 U , A˜ 0 U A2 U , A˜ 0 U A3 U .

(2.11)

340

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

Next, we estimate every term on the right-hand side of (2.10). Note that our estimates only need to be done for t ∈ [0, min{ T ∗ , T }) where both U and U are regular enough and take values in a convex compact subset of the state space. Thus, we have

C −1



α 2

D F dx  D α F 2  C e



α 2 D F dx

(2.12)

for some C > 0 and

  α    α   D F A˜ 0 U P + P α + R α  C D α F 2 + P α 2 + P α 2 + R α 2 . 1

2

1

2

(2.13)

To estimate Γ , we rewrite A j (U ) = B j + A¯ j (U ) with

  B j = diag u j , u j , u j , u j , 0, 0, 0, 0, 0, 0, u j .

Notice that A¯ j (U ) depends only on q and φ . Thus we have

 ∂ ∂ ˜ A0 + ( A˜ 0 B j + A˜ 0 A¯ j ) ∂t ∂xj 3

Γ =

j =1

= A˜ 0 div u +  2

3      ∂ ˜

 div u − κ 1 + q −1 φ

˜  1 + q div u − A˜  A 0 A¯ j − A 0η1 0η2 (γ − 1) 1 + φ ∂xj j =1

− 1 + q

− 1 

  Ψ + J · E + u × B .

Therefore,

2 2 4 4   |Γ |  C + C ∇ q + ∇φ + φ + ∇ u + E + u + B          C + C |∇ F | + |∇ F |2 + C  φ − φ + C | F |2 + | F |4 + C |∇ v | + |∇ v |2 + 1    C + C F s + F 4s    C 1 + F 4s .

(2.14)

˜  and A˜  denote the Here we have used Sobolev’s embedding theorem and the fact that s > 3/2 + 2, and the symbols A 0η1 0η2 ˜ 0 with respect to ρ and θ , respectively. differentiation of A Since P 1α =

3 

      A j U ∂x j D α F − D α A j U ∂x j F

j =1

3   α    ∂ β A j U ∂ α −β F x j =− β j =1 0<βα

3   α     ∂ β B j + A¯ j U ∂ α −β F x j , =− β j =1 0<βα

we obtain, with the help of the Moser-type calculus inequalities in Sobolev spaces [20], that

β   α −β





∂ B j U ∂ F x j  C u s F x j |α |−1  C u s F |α | ,

β   α −β −1 β   α −β

∂ A¯ j U ∂ F x j =

∂ f q , φ ∂ Fxj



s

s 





 C −1 ∇ ρ s−1 + ∇θ s−1 + ∇ ρ s−1 + ∇θ s−1 F α

   

 s   C q , φ s + q , φ s F α    C 1 + F ss F α where f (q , φ ) := (1 + q ) + (γ − 1)(1 + φ ) + (1 + q )−1 (1 + φ ). Thus, we have

α

 

P  C 1 + F s F |α | . 1

s

Similarly, by the uniform boundedness of U s+1 , the term P 2α can be bounded as follows.

(2.15)

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

341

α

 

P  C U x s A j ( U ) − A j U

j 2 |α |



 



 C u j − v j |α | + A¯ j U − A¯ j (U ) |α |







     C 1 + u − v |α | + C

−1 f q , φ − f (q , φ ) |α |

     s   C 1 + q + η3 q − q + φ + η4 φ − φ s F |α |    C 1 + F ss F |α | ,

(2.16)

where 0  η3 , η4  1 are constants.  ˜ 0 (U )Y α dx is more complex and delicate, and we give the details below. Since The estimate of the term ( D α E ) A

⎛

⎜ ⎜ ⎜ Yα = ⎜ ⎜ ⎜ ⎝

D

 α L (U )

1 + q

D

⎞

0

μv +J ×H 

− 1 + q

  α J −J

ν

0

 κ φ + (Ψ +J (E +u ×H ))

Dα 1 + q

 α

˜ 0 (U )Y α dx as we can rewrite ( D E ) A



⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

 α    α ˜ 0 U Y dx D E A           L (U ) μv + J × H

= D α u − v D α − dx + μ0 D α E − D D α J − J dx



1 + q 1 + q  







   1 α φ − φ D α κ φ + (Ψ + J · (E + u × H )) dx + D

(γ − 1)(1 + φ ) 1 + q

  





 α μu + (μ + μ )∇ div u μv

α = D u − v D − dx 1 + q

1 + q



         J × B

J × H

+ D α u − v D α − dx + μ0 D α E − D D α J − J dx

1 + q 1 + q

 

  1 α φ − φ  D α κ φ + D dx

(γ − 1)(1 + φ ) 1 + q

   Ψ

1 α φ − φ  D α +

D dx

(γ − 1)(1 + φ ) 1 + q



  

 α J (E + u × B ) 1 α +

D φ − φ D dx (γ − 1)(1 + φ ) 1 + q

:=

6 

Ii .

(2.17)

i =1

By the Moser-type calculus inequalities, Sobolev’s embedding theorem, Holder’s inequality, and the regularity of

(q , v , D , H , φ ), the first four terms Ii (i = 1, 2, 3, 4) can be bounded as follows.

 

=









D α u − v D α

I1 =

D α u − v D α





μu + (μ + μ )∇ div u

μv

− dx

1 + q 1 + q



1 + q

− 1













μ u − v + μ + μ ∇ div u − v dx

    − 1   + μ D α u − v D α 1 + q

− (1 + q )−1 v dx    2  2 μ α 

μ + μ α 

− D ∇ u − v dx − D div u − v dx



1 + q 1 + q       −1  α −β   + D α u − v · D β 1 + q

D μ u − v

0<βα





  2  + μ + μ ∇ div u − v dx + C F 2|α | + C F 4s + C D α ∇ u − v



 

342

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

  −C







2 μ D α ∇ u − v dx − C











2 μ + μ D α div u − v + C F 2|α | dx



  2   + C D α ∇ u − v + C F 4s + C F 2s F 2|α | + D α u − v

·



D

1<βα



 −C



β





 −1  α −β       1 + q

D μ u − v + μ + μ ∇ div u − v dx 



2 μ D α ∇ u − v dx − C











2 μ + μ D α div u − v dx

 2  + C D α ∇ u − v + C F 4s + C F 2|α | + C F 2|α | F ss , (2.18)  



  J ×B J × H

|I2 | = D α u − v D α − dx 1 + q

1 + q





J × B

J × H

−  F |α |

1 + q

1 + q |α |







1 



 



J − J × B + 1 J × B − H

 F |α |

1 + q

1 + q

|α | |α | s s





1

1

+

1 + q − 1 + q J × H |α |

s   (2.19)  C F |α | 1 + F ss+2 F s ,      |I3 | = μ0 D α E − D D α J − J dx



 μ0 E − D |α | J − J |α |    C 1 + F s F 2|α | , (2.20)   

  1 κ φ |I4 | = D α φ − φ D α dx (γ − 1)(1 + φ ) 1 + q

     

φ

κ 1 1 α φ − φ  D α (φ − φ ) dx + κ α φ − φ  D α D D dx 









γ −1 1 + φ 1 + q γ −1 1 + φ 1 + q      κ 1 1  D α φ − φ

D α  φ − φ dx γ − 1 1 + φ

1 + q

  

 β  −1  α −β 

 κ α

D φ − φ D 1 + q D  φ − φ dx + γ − 1 0<βα  

 α φ

κ 1 α + D φ − φ D dx (γ − 1) 1 + φ

1 + q 

  2   2  −C κ ∇ D α φ − φ dx + C ∇ D α φ − φ + C F 2|α | + C F 4s + C F 2|α | F ss + C 2 . (2.21) Similarly, we have

 |I5 | =

1







Ψ

D α φ − φ D α − 1)(1 + φ ) 1 + q





dx

(γ

α 

 2

 C D ∇ φ − φ + C F 2|α | + C F 4s + C F 2|α | F ss + C 2 ,    

 α J · (E + u × B ) 1 α |I 6 | = D φ − φ D dx (γ − 1)(1 + φ ) 1 + q



1 1 α φ − φ  D α J · E + u × B  dx D 





γ −1 1 + φ 1 + q    

1 α −β J · E + u × B  dx D α φ − φ D β D +

γ − 1 0<βα 1 + q    C F 2s + C F 8s + C F 2α 1 + F 2s + C 2. s

(2.22)

(2.23)

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

Putting the estimates (2.8) and (2.13)–(2.23) into (2.11) and taking

d α 2

D F + ξ e

dt



α 

 D ∇ u − v 2 dx + κ



343

small enough, we obtain that

α 

 D ∇ φ − φ 2 dx

2     2 s +2 F s F α + F 2s + F 8s + C 2 ,  C R α + C 1 + F 2s s F |α | + 1 + F s

where we have used the following estimate



μ

α 

   D ∇ u − v 2 + μ + μ



α   D div u − v 2  ξ



(2.24)

α 

 D ∇ u − v 2

for some positive constant ξ > 0. Thanks to (2.12), we can integrate the inequality (2.24) over (0, t ) with t < min{ T , T ∗ } to obtain that

α





D F (t ) 2  C D α F (0) 2 + C

t

α 2

R (τ ) dτ

0

t +C











2 2 s +2 F s F α + F 8s + C 2 (τ ) dτ . 1 + F 2s s F |α | + F s + 1 + F s

0

α with |α |  s, we get

Summing up this inequality for all







F (t ) 2  C F (0) 2 + C s

T∗



s



R (τ ) 2 dτ + C s

0

t







+2 F 2s (τ ) dτ . 1 + F 2s s

0

With the help of Gronwall’s lemma and the fact that



F (0) 2 + s

T ∗



 

R (t ) 2 dt = O 2 , s

0

we conclude that

 t 





2s+2  

F (t ) 2  C 2 exp C



1 + F (τ ) s dτ ≡ Φ(t ). s 0

It is easy to see that Φ(t ) satisfies



2s+2   Φ(t )  C Φ(t ) + C Φ s+2 (t ). Φ  (t ) = C 1 + F (t ) s Thus, by employing the nonlinear Gronwall-type inequality, we conclude that there exists a constant K , independent of such that

,



F (t )  K

s

for all t ∈ [0, min{ T , T ∗ }) provided Φ(0) = C 2 < exp{−C T ∗ }. Hence the proof of Theorem 1.1 is complete. Acknowledgments Li was supported by NSFC (Grant No. 11271184), NCET-11-0227, PAPD, and the Fundamental Research Funds for the Central Universities. Mu was supported NSFC (Grant No. 11271184), and the Fundamental Research Funds for the Central Universities. References [1] Y. Brenier, W.-A. Yong, Derivation of particle, string, and membrane motions from the Born–Infeld electromagnetism, J. Math. Phys. 46 (2005) 062305, 17 pp. [2] B. Ducomet, E. Feireisl, The equations of magnetohydrodynamics: on the interaction between matter and radiation in the evolution of gaseous stars, Comm. Math. Phys. 266 (2006) 595–629. [3] J.-S. Fan, W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. 69 (10) (2008) 3637–3660. [4] J.-S. Fan, W. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl. 10 (1) (2009) 392–409. [5] P. Germain, S. Ibrahim, N. Masmoudi, Wellposedness of the Navier–Stokes–Maxwell equations, arXiv:1207.6187v1 [math.AP].

344

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

F. Li, Y. Mu / J. Math. Anal. Appl. 412 (2014) 334–344

X.-P. Hu, D.-H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 283 (2008) 255–284. X.-P. Hu, D.-H. Wang, Low Mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal. 41 (2009) 1272–1294. S. Ibrahim, S. Keraani, Global small solutions for the Navier–Stokes–Maxwell system, SIAM J. Math. Anal. 43 (5) (2011) 2275–2295. S. Ibrahim, T. Yoneda, Local solvability and loss of smoothness of the Navier–Stokes–Maxwell equations with large initial data, J. Math. Anal. Appl. 396 (2) (2012) 555–561. I. Imai, General principles of magneto-fluid dynamics, in: Magneto-Fluid Dynamics, in: Suppl. Prog. Theor. Phys., vol. 24, Kyoto University, 1962, pp. 1–34. S. Jiang, F.-C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, available at arXiv:1309.3668 [math.AP]. S. Jiang, Q.-C. Ju, F.-C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys. 297 (2010) 371–400. S. Jiang, Q.-C. Ju, F.-C. Li, Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal. 42 (2010) 2539–2553. S. Jiang, Q.-C. Ju, F.-C. Li, Low Mach number limit for the multi-dimensional full magnetohydrodynamic equations, Nonlinearity 25 (2012) 1351–1365. S. Jiang, Q.-C. Ju, F.-C. Li, Incompressible limit of the non-isentropic ideal magnetohydrodynamic equations, available at arXiv:1301.5126v1 [math.AP]. S. Jiang, Q.-C. Ju, F.-C. Li, Z.-P. Xin, Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data, available at arXiv:1111.2925v2 [math.AP]. Q.-C. Ju, F.-C. Li, Y. Li, Asymptotic limits of the full compressible magnetohydrodynamic equations, SIAM J. Math. Aanal. 45 (5) (2013) 2597–2624. S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math. 10 (1) (1986) 131–149. S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci. 62 (1986) 181–184. S. Klainerman, A. Majda, Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981) 481–524. N. Masmoudi, Global well posedness for the Maxwell–Navier–Stokes system in 2D, J. Math. Pures Appl. (9) 93 (6) (2010) 559–571. A. Novotny, M. Ruzicka, G. Thäter, Singular limit of the equations of magnetohydrodynamics in the presence of strong stratification, Math. Models Methods Appl. Sci. 21 (2011) 115–147. A.I. Volpert, S.I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Mat. Sb. 87 (1972) 504–528 (in Russian); English transl.: Math. USSR Sb. 16 (1973) 517–544. W.A. Yong, Basic aspects of hyperbolic relaxation systems, in: H. Freistühler, A. Szepessy (Eds.), Advances in the Theory of Shock Waves, in: Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser, Boston, 2001, pp. 259–305.