Convergence from an electromagnetic fluid system to the full compressible MHD equations

Acta Mathematica Scientia 2018,38B(3):805–818 http://actams.wipm.ac.cn

CONVERGENCE FROM AN ELECTROMAGNETIC FLUID SYSTEM TO THE FULL COMPRESSIBLE MHD EQUATIONS∗

M )

Xin XU (

Institute of Applied Physics and Computational Mathematics, Beijing 100088, China E-mail : [email protected] Abstract We are concerned with the zero dielectric constant limit for the full electromagneto-fluid dynamics in this article. This singular limit is justified rigorously for global smooth solution for both well-prepared and ill-prepared initial data. The explicit convergence rate is also obtained by a elaborate energy estimate. Moreover, we show that for the wellprepared initial data, there is no initial layer, and the electric field always converges strongly to the limit function. While for the ill-prepared data case, there will be an initial layer near t = 0. The strong convergence results only hold outside the initial layer. Key words

Zero dielectric constant limit; full compressible magnetohydrodynamic equation; initial layer

2010 MR Subject Classification

1

35B35; 35B40; 35Q30; 35Q61

Introduction

When we consider the motion of an electrically conducting fluid in the presence of an electromagnetic field, the following system of electro-magneto-fluid dynamics is a commonly used system (cf. [1–3])   ∂t ρ + ∇ · (ρu) = 0,         ρ(∂t u + u · ∇u) + ∇p = ∇ · Ψ(u) + J × B, ρeθ (∂t θ + u · ∇θ) + θpθ ∇ · u = κ△θ + Ψ(u) : ∇u + J(E + u × B),     ǫ∂t E − ∇ × B + J = 0,      ∂t B + ∇ × E = 0, ∇ · B = 0.

(1.1)

ρ2 eρ (ρ, θ) = p(ρ, θ) − θpθ (ρ, θ).

(1.2)

Here, ρ denotes the mass density, u = (u1 , u2 , u3 ) is the velocity, θ is the absolute temperature, E = (E1 , E2 , E3 ) is the electric field, and B = (B1 , B2 , B3 ) is the magnetic field. The pressure p and the internal energy e are expressed by the equations of state

∗ Received March 14, 2017; revised October 16, 2017. The research is supported by Postdoctoral Science Foundation of China through Grant 2017M610818.

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All these quantities are functions of time t ≥ 0 and position x ∈ R3 . The viscous stress tensor Ψ(u) is given by Ψ(u) = 2µS(u) + λ∇ · uI3 ,

S(u) = (∇u + ∇u⊤ )/2,

(1.3)

where I3 denotes the 3 × 3 identity matrix, and ∇u⊤ the transpose of the matrix ∇u. µ and λ are the viscosity coefficients of the fluid which satisfies µ > 0 and 2µ + 3λ > 0. The electric current density J is given by Ohm’s law, that is, J = σ(E + u × B),

(1.4)

where σ > 0 is the coefficient of electrical conductivity. The second term Ψ(u) : ∇u on the right-hand side of (1.1)3 denotes the viscous dissipation function  2 3 X µ ∂ui ∂uj Ψ(u) : ∇u = + λ|∇u|2 . (1.5) + 2 ∂x ∂x j i i,j=1 The coefficient of heat conductivity κ is assumed to be positive constant through this article. Finally, the positive number ǫ in the fourth equation of (1.1) is the dielectric constant. Generally, the dielectric constant is very small [2]. In this article, we want to study the behavior of the solution for (1.1) when dielectric constant ǫ goes to zero. Formally, if we let the dielectric constant ǫ = 0 in (1.1); that is, the displacement current for the flow is negligible. Thanks to the fourth equation in (1.1), we have J = ∇ × B. Combining this with (1.4), we immediately have E = σ1 ∇ × B − u × B. Thus, we can eliminate the electric field E by this equality in (1.1) and obtain the following system,   ∂t ρ + ∇ · (ρu) = 0,       ρ(∂t u + u · ∇u) + ∇p = ∇ · Ψ(u) + (∇ × B) × B,   (1.6) 1  ρcV (∂t θ + u · ∇θ) + p∇ · u = κ△θ + Ψ(u) : ∇u + |∇ × B|2 ,   σ      1  ∂ B − ∇ × (u × B) = − ∇ × (∇ × B), ∇ · B = 0. t σ Obviously, this system is the classical full compressible magnetohydrodynamic equations and it has attracted a lot of attention in recent years because of its physical importance and mathematical challenges. Here, we only list some references from the perspective of mathematics for the interested readers. Concerning the well-posedness of compressible magnetohydrodynamic equations (1.6), we refer to [4–6] for the global existence of smooth solutions when the initial data are small perturbations of some constant state, [7, 8] for the global existence of renormalized solutions for general large initial data with finite energy, and [9] for the existence of local strong solutions when the initial density contains vacuum. Singular limit problem (vanishing viscosity limit, low Mach number limit among others) for both isentropic and non-isentropic magnetohydrodynamic equation is another interesting question; see [10–12] for some recent result in this direction. There are also some other results concerning the blow-up criterions [13, 14]. The above formal derivation is usually referred as magnetohydrodynamic approximation. We can see that the magnetohydrodynamic equations (1.6) are obtained as the limit of (1.1) at the vanishing of the dielectric constant. The mathematical study of this singular limit problem

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goes back to the initial work by Kawashima and Shizuta [15, 16] who mainly considered the two spatial dimension case. More precisely, they rigorously justified this limit process for local large solution and global small solution. In a series of article [17–19], Jiang and Li proved local convergence for smooth solutions to the electromagnetic fluid system with and without viscosity in T3 under the assumption that the initial data are well-prepared. In this article, we want to show that similar convergence holds globally in time for both well-prepared and ill-prepared data case as long as the initial data is sufficiently small. For convenience of presentation, we call (1.1) the full compressible Maxwell-Navier-Stokes equations, because it is the NavierStokes equations coupled with the Maxwell equations through the Lorentz force. It should be mentioned that the similar terminology was used by Masmoudi [20] for a different incompressible modeling system. To emphasize the unknowns depending on the small parameter ǫ, we rewrite the electromagnetic fluid system (1.1) as   ∂t ρǫ + ∇ · (ρǫ uǫ ) = 0,      ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ    ρ (∂t u + u · ∇u ) + ∇p = ∇ · Ψ (u) + J × B ,

ρǫ eǫθ (∂t θǫ + uǫ · ∇θǫ ) + θǫ pǫθ ∇ · uǫ = κ△θǫ + Ψǫ (u) : ∇uǫ + J ǫ (E ǫ + uǫ × B ǫ ),     ǫ∂t E ǫ − ∇ × B ǫ + J ǫ = 0,      ∂t B ǫ + ∇ × E ǫ = 0, ∇ · B ǫ = 0.

(1.7)

Initial data is given by

[ρǫ , uǫ , θǫ , E ǫ , B ǫ ]|t=0 = [ρǫ0 (x), uǫ0 (x), θ0ǫ (x), E0ǫ (x), B0ǫ (x)],

x ∈ R3 ,

(1.8)

with the compatible condition ∇ · B0ǫ (x) = 0.

(1.9)

¯ (¯ ¯ 0, 0) is a stationary solution of (1.7). For Obviously, for some positive constant ρ¯ and θ, ρ, 0, θ, the global existence of the above Cauchy problem, we have the following theorem, which has been established in our previous work [21]. ¯ Eǫ , Bǫ ] ∈ Theorem 1.1 ([21]) Assume that for each ǫ ∈ (0, 1] and s ≥ 3, [ρǫ0 − ρ¯, uǫ0 , θ0ǫ − θ, 0 0 H s and ∇ · B0ǫ = 0. Then, there exists a constant δ1 > 0 and C1 > 0, such that if ¯ ǫ 21 E ǫ , B ǫ )ks ≤ δ1 , Λ1 ≡ sup k(ρǫ0 − ρ¯, uǫ0 , θ0ǫ − θ, 0 0

(1.10)

ǫ

the Cauchy problem (1.7) admits a unique global solution U ǫ = [ρǫ , uǫ , θǫ , E ǫ , B ǫ ] which satisfies ¯ ǫ 21 E ǫ , B ǫ ](τ )k2 sup k[ρǫ − ρ¯, uǫ , θǫ − θ, s

0≤τ ≤t

+

Z

0

for any t ≥ 0.

t

(k∇(ρǫ , B ǫ )(τ )k2s−1 + k∇(uǫ , θǫ )(τ )k2s + kE ǫ (τ )k2s )dτ ≤ C1 Λ21 ,

(1.11)

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Similarly, we rewrite the limiting equations (1.6) as   ∂t ρ0 + ∇ · (ρ0 u0 ) = 0,       ρ0 (∂t u0 + u0 · ∇u0 ) + ∇p0 = ∇ · Ψ0 (u) + (∇ × B 0 ) × B 0 ,  

1  ρ0 e0θ (∂t θ0 + u0 · ∇θ0 ) + θp0θ ∇ · u0 = κ△θ0 + Ψ0 (u) : ∇u0 + |∇ × B 0 |2 ,   σ     1   ∂ B 0 − u0 × B 0 = − ∇ × (∇ × B 0 ), ∇ · B 0 = 0. t σ The above system is completed with the initial conditions

(1.12)

[ρ0 , u0 , θ0 , B 0 ]|t=0 = [ρ00 (x), u00 (x), θ00 (x), B00 (x)],

(1.13)

x ∈ R3 .

The compatible condition ∇ · B00 (x) = 0 is also needed here. We define 1 ∇ × B − u × B. σ Moreover, we introduce the following notation: E(ρ, u, θ, B) ≡

Λ2 ≡ sup ǫ−β kE0ǫ − E(ρǫ0 , uǫ0 , θ0ǫ , B0ǫ )ks−1 ,

(1.14)

(1.15)

ǫ

1

Λ3 ≡ sup ǫ 2 −γ k∇ × E0ǫ ks−1 ,

(1.16)

Λ4 ≡ sup ǫ−τ k(ρǫ0 − ρ00 , uǫ0 − u00 , θ0ǫ − θ00 , B0ǫ − B00 )ks−1 ,

(1.17)

ǫ ǫ

with β ≥ 0, γ ∈ [0, 21 ], and τ > 0. The first main result of this article can be stated as follows. Theorem 1.2 There exists a constant δ0 , such that if Λ ≡ Λ 1 + Λ 2 + Λ 3 + Λ 4 ≤ δ0 , the solution [ρǫ , uǫ , θǫ , E ǫ , B ǫ ] established in Theorem 1.1 converges to a function [ρ0 , u0 , θ0 , E 0 , B 0 ] on [0, T ] × R3 as ǫ goes to zero, where T is an arbitrary fixed time. The limit function [ρ0 , u0 , θ0 , B 0 ] is the unique solution of (1.12) with initial data [ρ0 , u0 , θ0 , B 0 ](0, x) = [ρ00 , u00 , θ00 , B00 ](x). And the following equality 1 ∇ × B 0 − u0 × B 0 (1.18) σ also holds for the limit function. Moreover, for t ∈ [0, T ] and ǫ ∈ (0, 1], we have the following estimate E0 =

k(ρǫ − ρ0 , uǫ − u0 , θǫ − θ0 , B ǫ − B 0 )(T )k2s−1 Z T + k(uǫ − u0 , θǫ − θ0 )(t)k2s + k(E ǫ − E 0 )(t)k2s−1 dt 0 i h ≤ CeT ǫ2τ + ǫ2−η ,

(1.19)

where η = max{1 − 2β, 1 − 2γ}. Furthermore, for both well-prepared data and ill-prepared data, we want to study the strong convergence of the electric field. The following theorem is our second main result.

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Theorem 1.3 (i) When β > 0 in (1.15), that is the well-prepared data case. Then, we have E ǫ → E 0 strongly in L∞ ([0, T ]; H 2 (R3 )). (ii) When β = 0 in (1.15), that is the ill-prepared data case. Then, there exists a small constant κ > 0 , such that for t∗ = ǫ1−κ , we have E ǫ → E 0 strongly in L∞ ([t∗ , T ]; H 2 (R3 )). Remark 1.4 We can see from Theorem 1.3 that, in the ill-prepared initial data case, the convergence can not be uniformly in time; this is because of the presence of the initial layer near t = 0. And κ > 0 implies the thickness of this initial layer is O(ǫ). The rest of this article is arranged as follows. In Section 2, we will give some sharp estimate to the solution which will be useful to the proof of the main theorems. In Section 3, the limit process is rigorously proved by combining the crucial estimate in Section 2 and energy method. The last section is devoted to the analysis of the initial layer and strong convergence of the electric field. For the later use in this article, we give some notations. C denotes some positive genetic α l constant. ∇α x or simply ∇ with multi-index α stands for the usual spatial derivatives. ∇ with an integer l ≥ 0 stands for any spatial derivatives of order l. For any integer s ≥ 0, H s denote the inhomogeneous Sobolev space H s (R3 ). We denote k · kH s by k · ks for simplicity. Set L2 = H 0 and denote k · k = k · kL2 for simplicity. We use hf, gi denote the L2 inner product of f and g, namely, Z hf, gi =

f (x)g(x)dx.

R3

Moreover, for two operators A and B, [A, B] = AB − BA is the commutator.

2

Estimate for Time Derivative

In this section, we shall establish some sharp estimates of the solution and these estimates are useful in the next section. Our main purpose in this part is to establish the following theorem. Theorem 2.1 Under the assumptions of Theorem 1.1, then there exists a positive constant δ2 (≤ δ1 ) and C2 > 0, such that if ˜ ≡ Λ 1 + Λ 2 + Λ 3 ≤ δ2 , Λ

(2.1)

the solution established in Theorem 1.1 satisfies the estimate 1

sup kE ǫ (τ )k2s−1 + ǫη sup k∂t (ǫ 2 E ǫ , B ǫ )k2s−1

0≤τ ≤t

+

Z

0

0≤τ ≤t

t

k∂t (ρǫ , uǫ , θǫ , B ǫ )(τ )k2s−1 dτ + ǫη

Z

t

0

˜ 2, k∂t E ǫ (τ )k2s−1 dτ ≤ C2 Λ

(2.2)

for any t ≥ 0, where η = max{1 − 2β, 1 − 2γ}. Proof have

From estimate (1.11) in Theorem 1.1 and original system (1.7), we immediately k∂t ρǫ k2s−1 Z

+

Z

0

t

k∂t (ρǫ , uǫ , B ǫ )k2s−1 dτ ≤ CΛ21 ,

t 0

k∂t θǫ k2s−1 dτ ≤ CΛ21 + CΛ1 sup kE(τ )k2s−1 . 0≤τ ≤t

(2.3)

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Taking time-derivative to the last two equations in (1.7) yields   ǫ∂ E ǫ − ∇ × B ǫ + J ǫ = 0, tt t t  ∂tt B ǫ + ∇ × E ǫ = 0, ∇ · B ǫ = 0,

(2.4)

t

with

J ǫ = σ(E ǫ + uǫ × B ǫ ).

(2.5)

Then, by a direct energy estimate, we have 1 2

k∂t (ǫ E , B

1 2

ǫ

≤ k∂t (ǫ E , B

ǫ

ǫ

)k2s−1

Z

t

k∂t E ǫ k2s−1 dτ Z t 1 ≤ k∂t (ǫ 2 E ǫ , B ǫ )(0)k2s−1 + C k∂t (uǫ × B ǫ )k2s−1 dτ ǫ

+

0

)(0)k2s−1

+

0 2 CΛ1 ,

(2.6)

where estimate (2.3) is used. From the fourth equation in (1.7) and assumption (2.1), we have 1

1

kǫ 2 ∂t E ǫ (0)ks−1 = kǫ− 2 (∇ × B ǫ − J ǫ )(0)ks−1 1

= kǫ− 2 (∇ × B0ǫ − σ(E0ǫ + uǫ0 × B0ǫ ))ks−1 1

≤ ǫβ− 2 Λ2 ,

(2.7)

and 1

k∂t B ǫ (0)ks−1 = k∇ × E0ǫ ks−1 ≤ ǫγ− 2 Λ3 . Combining (2.6) and (2.7)-(2.8) gives Z t 1 k∂t (ǫ 2 E ǫ , B ǫ )k2s−1 + k∂t E ǫ k2s−1 dτ ≤ ǫ2β−1 Λ22 + ǫ2γ−1 Λ23 + CΛ21 .

(2.8)

(2.9)

0

˜ ≡ Λ1 + Λ2 + Λ3 , then Denoting η = max{1 − 2β, 1 − 2γ}, clearly, we have 1 ≥ η ≥ 0. Recall Λ (2.9) leads to Z t 1 ˜ 2. k∂t E ǫ k2s−1 dτ ≤ C Λ (2.10) ǫη k∂t (ǫ 2 E ǫ , B ǫ )k2s−1 + ǫη 0

Furthermore, because of the fact η ≤ 1, we get

1

ǫ



+ ∇ × Bǫ kE ǫ ks−1 = kuǫ × B ǫ ks−1 + ∂t E ǫ σ σ s−1 s−1 η

1

≤ CΛ1 + Ckǫ 2 + 2 ∂t E ǫ ks−1 ˜ ≤ C Λ.

As a result, substituting (2.11) into (2.3) gives Z t ˜ 2. k∂t θǫ k2s−1 dτ ≤ CΛ21 + CΛ1 sup kE(τ )k2s−1 ≤ C Λ 0

(2.11)

(2.12)

0≤τ ≤t

Finally, (2.2) is proved by combining (2.3), (2.10), and (2.12). This complete the proof of Theorem 2.1. 

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811

Zero Dielectric Constant Limit

We will study the zero dielectric constant limit of the solution constructed in Theorem 1.1 in this section. In other word, we need to prove the convergence of the sequence (ρǫ , uǫ , θǫ , E ǫ , B ǫ ) as ǫ → 0. Let 0 ≤ δ ≤ ǫ, and denote ˆ E, ˆ B) ˆ = (ρδ − ρǫ , uδ − uǫ , θδ − θǫ , E δ − E ǫ , B δ − B ǫ ). (ˆ ρ, u ˆ, θ,

(3.1)

ˆ E, ˆ B) ˆ satisfy the following equation Then, from (1.7), we can see that (ˆ ρ, u ˆ, θ,   ∂t ρˆ + uδ · ∇ˆ ρ + ρδ ∇ · uˆ = f1 ,        pδρ pδθ ˆ 1  δ  ∂ u ˆ + u · ∇ˆ u + ∇ˆ ρ + ∇θ − δ △ˆ u = f2 ,  t  ρδ ρδ ρ   θ δ pδ κ  ∂t θˆ + uδ · ∇θˆ + δ θδ ∇ · u ˆ − δ δ △θˆ = f3 ,   ρ e ρ eθ  θ      ˆ−∇×B ˆ + σE ˆ = f4 , δ∂t E      ˆ ˆ = 0, ˆ = 0, ∂t B + ∇ × E ∇·B

(3.2)

where the source terms are given by

f1 = − u ˆ · ∇ρǫ − ρˆ∇ · uǫ ,  pδ 1  pδ pǫρ  pǫ  1 ρ f2 = − u ˆ · ∇uǫ − δ − ǫ ∇ρǫ − θδ − θǫ ∇θǫ + δ − ǫ ∇ · Ψǫ (uǫ ) ρ ρ ρ ρ ρ ρ 1  1 ǫ δ δ ǫ + δJ ×B − ǫJ ×B , ρ ρ  pδ  κ pǫ  κ  f3 = − u ˆ · ∇θǫ − δ δ − ǫ ǫ ∇ · uǫ + δ δ − ǫ ǫ △θǫ ρ eθ ρ eθ ρ eθ ρ eθ  Ψδ (uδ ) : ∇uδ ǫ ǫ ǫ Ψ (u ) : ∇u − + δ δ ρǫ eǫθ ρ eθ  J δ (E δ + uδ × B δ ) J ǫ (E ǫ + uǫ × B ǫ )  + − ρǫ eǫθ ρδ eδθ f4 = − (δ − ǫ)∂t E ǫ − σ(uδ × B δ − uǫ × B ǫ ).

(3.3)

(3.4)

(3.5) (3.6)

ˆ E, ˆ B), ˆ we have the following proposition. For the estimate of (ˆ ρ, u ˆ, θ, Proposition 3.1 If the assumptions of Theorem 1.2 hold, then we have the following estimate Z T i h 1 T 2 2 2 ˆ ˆ ˆ ˆ ˆ 2 k(ˆ u, θ)(t)k Λ2 δ + Λ24 ǫ2τ + ǫ2−η . (3.7) k(ˆ ρ, u ˆ, θ, δ E, B)(T )ks−1 + s + kE(t)ks−1 dt ≤ Ce 0

Applying ∇α with 0 6 |α| 6 s − 1 to equations (3.2) and multiplying them by pδρ α ρδ eδ ˆ ∇α E, ˆ ∇α B ˆ respectively, integrating over R3 , then adding them up, ˆ, ρδ ∇α uˆ, θδ θ ∇α θ, ρδ ∇ ρ we have Z n δ o pρ α 2 1 d ρδ eδθ α ˆ 2 δ α 2 α ˆ 2 α ˆ 2 (∇ ρ ˆ ) + ρ (∇ u ˆ ) + (∇ θ) + δµ (∇ E) + (∇ B) dx 0 2 dt ρδ θδ Z n o κ ˆ 2 + σ(∇α E) ˆ 2 dx + µ(∇α+1 u ˆ)2 + (µ + λ)(∇α ∇ · u ˆ)2 + δ (∇α+1 θ) θ Proof

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=

3 X i=1

Ti +

5 X i=1

Ri +

3 X i=1

Gi +

4 X

Vol.38 Ser.B

(3.8)

Fi ,

i=1

with Z  δ Z Z  δ δ pρ ρ eθ α 2 δ α 2 ˆ 2 dx; (∇ ρ ˆ ) dx, T = ρ (∇ u ˆ ) dx, T = (∇α θ) 2 3 t δ ρ t θδ t Z Z pδρ = ∇α (uδ · ∇ˆ ρ) δ ∇α ρˆdx, R2 = ∇α (uδ · ∇ˆ u)ρδ ∇α u ˆdx, ρ Z δ δ ˆ ρ eθ ∇α θdx, ˆ = ∇α (uδ · ∇θ) θδ Z  pδ  pδρ ρ ρ ρδ ∇α u ˆdx, = ∇α (ρδ ∇ · uˆ) δ ∇α ρˆdx + ∇α δ ∇ˆ ρ ρ Z  pδ  θ δ pδ   ρδ e δ θ αˆ = ∇α θδ ∇θˆ ρδ ∇α uˆ + ∇α δ θδ ∇ · u ∇ θdx; ˆ ρ θδ ρ eθ Z h 1i = ∇α , δ (µ∆ˆ u + (µ + λ)∇∇ · u ˆ)ρδ ∇α uˆdx, ρ Z h Z   κ i ρδ e δ κ ˆ ˆ α+1 θdx, ˆ = ∇α , δ δ ∆θˆ δ θ ∇α θdx, G3 = ∇ δ ∇α θ∇ θ θ ρ eθ Z Z pδρ α α = ∇ f1 δ ∇ ρˆdx, F2 = ∇α f2 ρδ ∇α uˆdx, ρ Z Z ρδ eδθ α ˆ α ˆ = ∇ f3 δ ∇ θdx, F4 = µ0 ∇α f4 ∇α Edx, θ

T1 = R1 R3 R4 R5 G1 G2 F1 F3

where [, ] represent the commutator. Now, we need to estimate each term in the right-hand side of (3.8). For the estimate of Ti , we have

 δ δ     δ 

pρ

ρ eθ δ α ˆ 2

+ kρt kL∞ + |Ti | ≤ δ ρ, u ˆ, θ)k

θδ t ∞ k∇ (ˆ ρ t L∞ L ˆ 2. ≤ Ck(ρδ , θδ )t kL∞ k∇α (ˆ ρ, u ˆ, θ)k (3.9) Next, we will give the estimate of Ri . Actually, we have Z pδρ |R1 | = ∇α (uδ · ∇ˆ ρ) δ ∇α ρˆdx ρ Z Z  pδ   pδρ α  ρ α δ α α δ ρ) δ ∇ ρˆ dx ≤ [∇ , u ·]∇ˆ ρ δ ∇ ρˆ dx + (u · ∇ ∇ˆ ρ ρ

δ



pρ α pδ 

+ ∇ uδ ρ ≤ (k∇uδ kL∞ k∇α ρˆk + k∇ˆ ρkL∞ k∇α uδ k) ∇ ρ ˆ

ρδ

ρδ

k∇α ρˆk2

L∞

≤ Ckuδ ks−1 kρˆk2s−1 .

(3.10)

The estimate of R2 and R3 is similar, so we omit the details. ˆ 2 . |R2 | + |R3 | ≤ Ck(ρδ , uδ , θδ )ks−1 k(ˆ u, θ)k s−1

(3.11)

For the estimate of R4 and R5 , we need to use its symmetric property. First, R4 can be estimated as Z  δ  pδρ pρ |R4 | = ∇α (ρδ ∇ · u ˆ) δ ∇α ρˆdx + ∇α δ ∇ˆ ρ ρδ ∇α uˆdx ρ ρ

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Z     δ pδρ pρ α α δ α α δ ρ(ρ ∇ u ˆ)dx = [∇ , ρ ]∇ · u ˆ δ ∇ ρˆ dx + ∇ , δ ∇ˆ ρ ρ Z δ α α δ α α + pρ ∇ ∇ · u ˆ∇ ρˆdx + pρ ∇ ∇ˆ ρ∇ u ˆdx Z δ δ 2 δ α α ≤ k(ρ , θ )ks−1 k(ˆ ρ, u ˆ)ks−1 + (∇pρ )∇ uˆ∇ ρˆdx

≤ Ck(ρδ , θδ )ks−1 k(ˆ ρ, uˆ)k2s−1 .

813

(3.12)

Similarly, we have ˆ 2 . |R5 | ≤ Ck(ρδ , θδ )ks−1 k(ˆ u, θ)k s−1

(3.13)

Now, we need to estimate G1 , G2 , and G3 , respectively. For G1 , we have Z   α 1 δ α |G1 | = ∇ , δ (µ∆ˆ u + (µ + λ)∇∇ · u ˆ)ρ ∇ uˆdx ρ

     

α 1 2 1 α



≤ C ∇ δ k∇ ∇ˆ uk + ∇ k∇ uˆkL∞ k∇α u ˆk ρ ρδ L∞ ≤ εk∇α ∇ˆ uk2 + Cε k∇ρδ k2L∞ k∇α u ˆk2 + kρδ ks−1 kˆ uk2s−1 ≤ εk∇α ∇ˆ uk2 + +Ckρδ ks−1 kˆ uk2s−1 ,

(3.14)

where ε is a small constant and the Cauchy-Schwartz inequality is used. Similar to the estimate of G1 , we can estimate G2 as follows ˆ 2 + Ck(ρδ , θδ )ks−1 kθk ˆ 2 . |G2 | ≤ εk∇α ∇θk s−1 We can also estimate G3 as Z   κ α ˆ α+1 ˆ ˆ 2 + +Ckθδ ks−1 kθk ˆ 2 . θdx ≤ εk∇α ∇θk |G3 | = ∇ δ ∇ θ∇ s−1 θ

(3.15)

(3.16)

Finally, we only need to give the estimate of Fi . To this end, we need the following lemma which concerns the estimate of fi . Lemma 3.2 For 0 ≤ |α| ≤ s − 1, we have the estimates k∇α f1 k ≤ k(ρǫ , uǫ )ks k(ˆ ρ, u ˆ)ks−1 ,

   

α ˆ s−1

∇ f2 − 1 − 1 ∇ · Ψǫ (uǫ ) ≤ k(ρǫ , uǫ , θǫ )ks k(ˆ ρ, u ˆ, θ)k

δ ǫ ρ ρ

ˆ B)k ˆ s−1 , + k(ρǫ,δ , uǫ,δ , E ǫ,δ , B ǫ,δ )ks−1 k(ˆ ρ, uˆ, E,

   

α

κ κ ǫ ǫ ǫ ˆ s−1 + k(ρǫ , uǫ , θǫ )ks k(ˆ ˆ s−1

∇ f3 − △θ ρ, u ˆ, θ)k ρ, ∇ˆ u, θ)k − ǫ

≤ k(u , θ )ks k(ˆ δ ǫ ρ eθ ρδ e θ ˆ E, ˆ B)k ˆ s−1 , + k(ρǫ,δ , uǫ,δ , θǫ,δ , E ǫ,δ , B ǫ,δ )ks−1 k(ˆ ρ, uˆ, θ, ˆ s−1 , k∇α f4 k ≤ ǫk∂t E ǫ ks−1 + k(uǫ,δ , B ǫ,δ )ks−1 k(ˆ u, B)k

where Aǫ,δ represents (Aǫ , Aδ ). We postpone the proof of this lemma to the end of this section, and we continue the proof of Proposition 3.1. By Lemma 3.2, we can estimate Fi as Z pδρ α α |F1 | = ∇ f1 δ ∇ ρˆdx ≤ Ck∇α f1 kk∇α ρˆk ≤ Ck(ρǫ , uǫ )ks k(ˆ ρ, uˆ)k2s−1 , (3.17) ρ

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and Z α δ α |F2 | = ∇ f2 ρ ∇ u ˆdx Z     1 1 ǫ ǫ δ α α − ǫ ∇ · Ψ (u ) ρ ∇ uˆdx ≤ ∇ f2 − δ ρ ρ Z    1 1 ǫ ǫ δ α α − ǫ ∇ · Ψ (u ) ρ ∇ uˆdx + ∇ δ ρ ρ ǫ,δ ǫ,δ ǫ,δ ǫ,δ ˆ B)k ˆ 2s−1 ≤ k(ρ , u , E , B )ks−1 k(ˆ ρ, u ˆ, E, Z    1 1 ǫ ǫ δ α + ∇α−1 − ∇ · Ψ (u ) ∇[ρ ∇ u ˆ ]dx δ ǫ ρ ρ ˆ B)k ˆ 2s−1 + k(ρǫ,δ , uǫ,δ )kk∇ˆ ≤ k(ρǫ,δ , uǫ,δ , E ǫ,δ , B ǫ,δ )ks−1 k(ˆ ρ, u ˆ, E, uk2s−1 .

(3.18)

Similarly, the following estimate on F3 holds ˆ 2 . ˆ B)k ˆ 2 + k(ρǫ,δ , uǫ,δ )kk∇θk |F3 | ≤ k(ρǫ,δ , uǫ,δ , E ǫ,δ , B ǫ,δ )ks−1 k(ˆ ρ, u ˆ, E, s−1 s−1

(3.19)

For F4 , one has Z ˆ |F4 | = µ0 ∇α f4 ∇α Edx

ˆ E)k ˆ 2 + CkEk ˆ 2 . ≤ ǫk∂t E ǫ ks−1 + Ck(uǫ,δ , B ǫ,δ )ks−1 k(ˆ u, B, s−1 s−1

(3.20)

Denote  X Z  pδρ ρδ eδθ α ˆ 2 α 2 δ α 2 α ˆ 2 α ˆ 2 (∇ ρˆ) + ρ (∇ u ˆ) + δ (∇ θ) + δ(∇ E) + (∇ B) dx, M (t) = ρδ θ |α|≤s−1   Z X κ α+1 ˆ 2 α ˆ 2 α+1 2 α 2 θ) + σ(∇ E) dx. (3.21) µ(∇ u ˆ) + (µ + λ)(∇ ∇ · u ˆ) + δ (∇ D(t) = θ |α|≤s−1

As a result, 1 d 2 ˆ M (t) + D(t) + k(ˆ u, θ)(t)k ≤ C(Λ + 1)M (t) + ǫ2 k∂t E ǫ k2s−1 . dt 2 By Gronwall inequality, we have Z T Z T h i 2 ˆ M (T ) + D(t) + k(ˆ u, θ)(t)k dt ≤ Ce(Λ+1)T M (0) + ǫ2 k∂t E ǫ k2s−1 ds 0 0 h i (Λ+1)T 2 2 2τ ≤ Ce Λ δ + Λ4 ǫ + ǫ2−η ,

(3.22)

(3.23)

where we have used the fact that 1 ˆ0 )k2s−1 ≤ CΛ2 δ + CΛ24 ǫ2τ . M (0) ≤ Ck(ˆ ρ0 , uˆ0 , θˆ0 , δ 2 Eˆ0 , B

Thus, this completes the proof of Proposition 3.1.

(3.24) 

On the basis of the estimate in Proposition 3.1, we can give the proof of our main theorem. Proof of Theorem 1.2 For any fixed time T > 0, let ǫ → 0 in estimate (3.7) of Proposition 3.1, we can see Z T 2 2 ˆ B)(T ˆ ˆ ˆ k(ˆ ρ, u ˆ, θ, )k2s−1 + k(ˆ u, θ)(t)k (3.25) s + kE(t)ks−1 dt → 0. 0

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As a result, there exists a unique function (ρ0 , u0 , θ0 , E 0 , B 0 ) satisfying (ρ0 − ρ¯, B 0 ) ∈ C 0 (0, T ; H s−1), ¯ ∈ C 0 (0, T ; H s−1) ∩ L2 (0, T ; H s ), (u0 , θ0 − θ) E 0 ∈ L2 (0, T ; H s−1 ).

(3.26)

And as ǫ → 0, we have the following convergence result: (ρǫ − ρ¯, B ǫ ) → (ρ0 − ρ¯, B 0 ), ¯ (uǫ , θǫ ) → (u0 , θ0 − θ), Eǫ → E0,

(3.27)

strongly in C 0 (0, T ; H s−1 ), C 0 (0, T ; H s−1 )∩L2 (0, T ; H s ), L2 (0, T ; H s−1 ), respectively. Clearly, we can see that the limit function (ρ0 , u0 , θ0 , B 0 ) is the solution of (1.12) with the initial data (1.13). Moreover, 1 ∇ × B 0 − u0 × B 0 (3.28) σ also holds. Finally, the convergence rate can be easily obtained from (3.7) by letting δ goes to zero. That is E0 =

k(ρǫ − ρ0 , uǫ − u0 , θǫ − θ0 , B ǫ − B 0 )(T )k2s−1 Z T   + k(uǫ − u0 , θǫ − θ0 )(t)k2s + k(E ǫ − E 0 )(t)k2s−1 dt ≤ CeT ǫ2τ + ǫ2−η .

(3.29)

0

Thus, we complete the proof of Theorem 1.2.



Finally, let us give the proof of Lemma 3.2. Proof of Lemma 3.2 For the estimate of ∇α f1 , by direct calculation, we can see that k∇α f1 k ≤k∇α (ˆ u · ∇ρǫ )k + k∇α (ˆ ρ∇ · uǫ )k ≤k∇α uˆkk∇ρǫ kL∞ + kˆ ukL∞ k∇α ∇ρǫ k + k∇α ρˆkk∇ · uǫ kL∞ + kρˆkL∞ k∇α ∇ · uǫ k ≤kˆ uks−1 kρǫ ks + kρˆks−1 kuǫ ks ≤k(ρǫ , uǫ )ks k(ˆ ρ, u ˆ)ks−1 . For ∇α f2 , we have

   

α

∇ f2 − 1 − 1 ∇ · Ψǫ (uǫ )

δ ǫ ρ ρ

 δ    δ  

α pρ

α pθ

pǫρ pǫθ α ǫ ǫ ǫ

≤ k∇ (ˆ u · ∇u )k + ∇ − ǫ ∇ρ + ∇ − ǫ ∇θ ρδ ρ ρδ ρ

 

α 1 δ

1 ǫ δ ǫ

+ ∇ J × B − ǫ J × B . ρδ ρ

(3.30)

(3.31)

Now, denote the right-hand side term of (3.31) by gi with i = 1, 2, 3, 4. Let us give the estimate of each term. Firstly, we have kg1 k ≤ k∇α u ˆkk∇uǫ kL∞ + kˆ ukL∞ k∇α ∇uǫ k ≤ kˆ uks−1 k∇uǫ ks−1 .

(3.32)

For g2 , we can use the Taylor formula to get kg2 k ≤ k∇α [C(pρρ , ρδ , ρǫ )(ρδ − ρǫ )∇ρǫ ]k ≤ Ckρˆks−1 kρǫ ks ,

(3.33)

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where C(pρρ , ρδ , ρǫ ) is bounded from below and above by some constants. Similarly, it holds that kg3 k ≤Ckρˆks−1 kθǫ ks . For the estimate of g4 , we need to rewrite it into 1 δ 1 J × Bδ − ǫ J ǫ × Bǫ δ ρ ρ 1 δ 1 1 1 δ = δ J × B − ǫ J δ × Bδ + ǫ J δ × Bδ − ǫ J ǫ × Bǫ ρ ρ ρ ρ 1 1 δ 1 = − J × B δ + ǫ (J δ × B δ − J ǫ × B ǫ ) δ ǫ ρ ρ ρ 1 1 δ 1 δ = − ǫ J × B + ǫ (J δ × B δ − J δ × B ǫ + J δ × B ǫ − J ǫ × B ǫ ) ρδ ρ ρ 1 1 δ 1 1 = − J × B δ + ǫ J δ × (B δ − B ǫ ) + ǫ (J δ − J ǫ ) × B ǫ . δ ǫ ρ ρ ρ ρ

(3.34)

(3.35)

Notice that J δ − J ǫ = σ(E δ + uδ × B δ ) − σ(E ǫ + uǫ × B ǫ ) = σ(E δ − E ǫ ) + σ(uδ × B δ − uδ × B ǫ + uδ × B ǫ − uǫ × B ǫ ) = σ(E δ − E ǫ ) + σuδ × (B δ − B ǫ ) + σ(uδ − uǫ ) × B ǫ .

(3.36)

Thus, from (3.35)–(3.36), we can estimate g4 as ˆ s−1 + k(ρǫ,δ , uǫ,δ , E ǫ,δ , B ǫ,δ )ks−1 k(ˆ ˆ B)k ˆ s−1 . kg4 k ≤ k(ρǫ , uǫ , θǫ )ks k(ˆ ρ, uˆ, θ)k ρ, u ˆ, E,

(3.37)

Collecting the estimate of (3.32)–(3.37), we get the estimate of f2 . And the estimate for f3 is similar, so we omit it for simplicity. Finally, the estimate of f4 is easy because k∇α f4 k ≤(δ − ǫ)k∂t E ǫ k + σk(uδ × B δ − uǫ × B ǫ )k ˆ s−1 . ≤ǫk∂t E ǫ ks−1 + k(uǫ,δ , B ǫ,δ )ks−1 k(ˆ u, B)k Thus, we complete the proof of Lemma 3.2.

4



Initial Layer Analysis

In this section, we want to give a detailed analysis to the initial layer when the initial data is ill-prepared. We show that when the initial data is well-prepared, there is no initial layer, hence the solution will converge strongly to the limit function. While for ill-prepared initial data, an initial layer will appear, and the strong convergence for electric filed holds only outside the initial layer. Now, we introduce ξ = ∇ × B ǫ − J ǫ = ∇ × B ǫ − σ(E ǫ + uǫ × B ǫ ),

(4.1)

then it follows from equation (1.7) that ξ = ǫ∂t E ǫ .

(4.2)

Taking one derivative to (4.1) with respect to t yields ξt = ∇ × Btǫ − Jtǫ = ∇ × Btǫ − σ(Etǫ + uǫt × B ǫ + uǫ × Btǫ ).

(4.3)

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Thus, ξ = ∇ × Btǫ − σ(uǫt × B ǫ + uǫ × Btǫ ). (4.4) ǫ Denote the right-hand side of (4.4) by M . Then, a direct energy estimate to (4.4) gives X Z d 2σ 2 2 α α kξk2 + kξk2 ≤ C ∇ M ∇ ξdx . (4.5) dt ǫ 3 ξt + σ

|α|≤2

R

Using Cauchy-Schwarz inequality to (4.5), we get

2σ σ d kξk22 + kξk22 ≤ CkM k22 + kξk22 . dt ǫ ǫ Combining estimate (2.2) in Theorem 2.1 and equation (1.7), we can see that kM k22 ≤ k∇ × Btǫ k22 + kσ(uǫt × B ǫ + uǫ × Btǫ )k22 ≤ CΛ2 .

(4.6)

(4.7)

From (4.6) and (4.7), we get d σ kξk22 + kξk22 ≤ CΛ2 . dt ǫ

(4.8)

σ   σ  d exp t kξk22 ≤ C exp t Λ2 . dt ǫ ǫ

(4.9)

Solving (4.8)

That is σ   σ Z t  σ  exp τ dτ kξ(t)k22 ≤ exp − t kξ(0)k22 + CΛ2 exp − t ǫ ǫ ǫ 0  σ  ≤ exp − t kξ(0)k22 + CΛ2 ǫ. ǫ Now, for the well-prepared initial data, this is, β > 0 in (1.15), we have kξ(0)k22 = k∇ × B0ǫ − σ(E0ǫ + uǫ0 × B0ǫ )k22 ≤ Cǫ2β .

(4.10)

(4.11)

Putting (4.11) into (4.10) gives kξ(t)k22 ≤ C(ǫ2β + ǫ). 1 σ∇

× B 0 − u0 × B 0 . So,

1 2 1

2 1



kE ǫ − E 0 k22 ≤ ξ + ∇ × B ǫ − (uǫ × B ǫ ) − ∇ × B 0 − u0 × B 0 σ 2 σ σ 2

1 2 1

2

2 1



≤ ξ + ∇ × B ǫ − ∇ × B 0 + (uǫ × B ǫ ) − u0 × B 0 2 σ 2 σ σ 2   ≤ C ǫ2β + ǫ + ǫ2τ + ǫ2−η .

Recall E 0 =

(4.12)

(4.13)

That is E ǫ converge to E 0 strongly in L∞ (0, T ; H 2 ). While for the well-prepared initial data, that is, β = 0 in (1.15), we only have kξ(0)k22 = k∇ × B0ǫ − σ(E0ǫ + uǫ0 × B0ǫ )k22 ≤ C.

(4.14)

Obviously, the similar strong convergence results can not be expected to hold near t = 0. However, we can prove that after a small time, we will have the strong convergence. Actually, from (4.10), we have  σ  kξ(t)k22 ≤ C exp − t + CΛ2 ǫ. (4.15) ǫ

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Now, we choose t∗ = Cǫ1−κ for a small constant κ > 0. Then, for t ≥ t∗ , we can get  σ   σ  exp − t ≤ exp − t∗ = exp(−Cσǫ−κ ) ≤ Cǫ. ǫ ǫ Substituting (4.16) into (4.15), we get kξ(t)k22 ≤ Cǫ.

(4.16)

(4.17)

Similar to the estimate of (4.13), we can show that E ǫ converge to E 0 strongly in L∞ (t∗ , T ; H 2 ). Collecting the argument in this section, we have proved Theorem 1.3. References [1] Cabannes H. Theoretical Magnetohydrodynamics. New York: Academic Press, 1970 [2] Imai I. General principles of magneto-fluid dynamics//Magneto-Fulid Dynamics. Suppl Prog Theor Phys, 1962, 1–34 [3] Jang J, Masmoudi N. Derivation of Ohm’s law from the kinetic equations. SIAM J Math Anal, 2012, 44(5): 3649–3669 [4] Kawashima S. System of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Manetohydrodynamics[D]. Kyoto: Kyoto University, 1983 [5] Li F C, Yu H J. Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations. Proc R Soc Edinb, 2011, 141A: 109–126 [6] Zhang J W, Zhao J N. Some decay estimates of solutions for the 3-D compressible isentropic magnetohydrodynamics. Commun Math Sci, 2010, 8: 835–850 [7] Hu X P, Wang D H. Global solutions to the three-dimensional full compressible magnetohydrodynamic flows. Commun Math Phys, 2008, 283: 255–284 [8] Hu X P, Wang D H. Global existence and large-time behavior of solutions to the three dimensional equations of compressible magnetohydrodynamic flows. Arch Ration Mech Anal, 2010, 197: 203–238 [9] Fan J, Yu W. Strong solution to the compressible MHD equations with vacuum. Nonlinear Anal Real World Appl, 2009, 10: 392–409 [10] Hu X P, Wang D H. Low mach number limit of viscous compressible magnetohydrodynamic flows. SIAM J Math Anal, 2009, 41: 127–1294 [11] Jiang S, Ju Q C, Li F C. Incompressible limit of the compressible magnetohydrodynamic equations with periodic boundary conditions. Commun Math Phys, 2010, 297: 371–400 [12] Xiao Y L, Xin Z P, Wu J H. Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition. J Funct Anal, 2009, 257(11): 3375–3394 [13] Cannone M, Chen Q L, Miao C X. A losing estimate for the ideal MHD equations with application to blow-up criterion[J/OL]. SIAM J Math Anal, 2007, 38(6): 1847–1859 [14] Lei Z, Zhou Y. BKM’s criterion and global weak solutions for magnetohydrodynamics with zero viscosity. Discrete Contin Dyn Syst, 2009, 25(2): 575–583 [15] Kawashima S, Shizuta Y. Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. Tsukuba J Math, 1986, 10(1): 131–149 [16] Kawashima S, Shizuta Y. Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid II. Proc Jpn Acad Ser A, 1986, 62: 181–184 [17] Jiang S, Li F C. Rigorous derivation of the compressible magnetohydrodynamic equations from the electromagnetic fluid system. Nonlinearity, 2012, 25: 1735–1752 [18] Jiang S, Li F C. Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations. Asymptotic Analysis, 2015, 95: 161–185 [19] Jiang S, Li F C. Zero dielectric constant limit to the non-isentropic compressible Euler-Maxwell system. Sci China Math, 2015, 58(1): 61–76 [20] Masmoudi M. Global well posedness for the Maxwell-Navier-Stokes system in 2D. J Math Pures Appl, 2010, 93: 559–571 [21] Xu X. On the large time behavior of the electromagnetic fluid system in R3 . Nonlinear Analysis: Real World Applications, 2017, 33: 83–99