On local strong solutions to the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with vacuum and zero heat conduction

Nonlinear Analysis: Real World Applications 31 (2016) 409–430

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Nonlinear Analysis: Real World Applications www.elsevier.com/locate/nonrwa

On local strong solutions to the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with vacuum and zero heat conduction✩ Li Lu a , Bin Huang b,c,∗ a

College of Mathematics and Information Science, Nanchang Hangkong University, Nanchang 330063, PR China b School of Science, Beijing University of Chemical Technology, Beijing 100029, PR China c CEMA, Central University of Finance and Economics, Beijing 100081, PR China

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Article history: Received 17 July 2015 Received in revised form 19 February 2016 Accepted 20 February 2016

abstract This paper concerns the Cauchy problem of the two-dimensional full compressible magnetohydrodynamic equations with zero heat-conduction and vacuum as far field density. In particular, the initial density can have compact support. We prove that the Cauchy problem admits a local strong solution provided both the initial density and the initial magnetic field decay not too slow at infinity. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Compressible magnetohydrodynamic equations Two-dimensional space Vacuum Zero heat-conduction Strong solutions Cauchy problem

1. Introduction and main results We consider the two-dimensional full compressible magnetohydrodynamic (MHD) equations which read as follows:  ρt + div(ρu) = 0,     1  (ρu)t + div(ρu ⊗ u) + ∇P = µ△u + (µ + λ)∇(divu) + H · ∇H − ∇|H|2 , 2 (1.1) R 2 2   ((ρθ)t + div(ρuθ)) + P divu = κ∆θ + 2µ|D(u)| + λ(divu) + ν|∇ × H|2 ,    γ − 1 Ht − H · ∇u + u · ∇H + Hdivu = ν∆H, divH = 0. ✩ L. Lu is supported by NNSFC Tianyuan No. 11426131. B. Huang is supported by the China Scholarship Council and NNSFC Grant No. 11301020. ∗ Corresponding author at: School of Science, Beijing University of Chemical Technology, Beijing 100029, PR China. E-mail addresses: [email protected] (L. Lu), [email protected], [email protected] (B. Huang).

http://dx.doi.org/10.1016/j.nonrwa.2016.02.007 1468-1218/© 2016 Elsevier Ltd. All rights reserved.

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Here t ≥ 0 is time, x = (x1 , x2 ) ∈ Ω ⊂ R2 is the spatial coordinate, ρ = ρ(x, t), u = (u1 , u2 )(x, t), θ = θ(x, t) and H = (H 1 , H 2 )(x, t) represent, respectively, the fluid density, velocity, absolute temperature and magnetic, and the pressure P is given by P (ρ) = Rρθ,

(R > 0),

(1.2)

where R is a given constant. In addition, D(u) is the deformation tensor 1 (∇u + (∇u)tr ). 2 The constant viscosity coefficients µ and λ satisfy the following hypothesis: D(u) =

µ > 0,

µ + λ ≥ 0.

(1.3)

The adiabatic constant γ and the heat conductivity coefficient κ are assumed to be γ > 0,

κ = 0.

(1.4)

Indeed, we consider the viscous compressible magnetohydrodynamic flows without heat-conduction (κ = 0), which implies that the energy equation (1.1)3 can be rewritten equivalently as a hyperbolic equation for the pressure P as follows: Pt + div(P u) + (γ − 1)P divu = (γ − 1)Q(∇u) + ν(γ − 1)|∇ × H|2 ,

(1.5)

where Q(∇u) , 2µ|D(u)|2 + λ(divu)2 . The constant ν > 0 is the resistivity coefficient which is inversely proportional to the electrical conductivity constant and acts as the magnetic diffusivity of magnetic fields. Let Ω = R2 , we consider the Cauchy problem (1.1)–(1.5), that is,   ρt + div(ρu) = 0,    1  (ρu)t + div(ρu ⊗ u) + ∇P = µ△u + (µ + λ)∇(divu) + H · ∇H − ∇|H|2 , (1.6) 2 2   P + div(P u) + (γ − 1)P divu = (γ − 1)Q(∇u) + ν(γ − 1)|∇ × H| , t    Ht − H · ∇u + u · ∇H + Hdivu = ν∆H, divH = 0, with (ρ, u, P, H) vanishing at infinity (in some weak sense). For given initial data ρ0 , u0 , P0 and H0 , we require that ρ(x, 0) = ρ0 (x),

ρu(x, 0) = ρ0 u0 (x),

P (x, 0) = P0 (x),

H(x, 0) = H0 (x),

x ∈ R2 ,

(1.7)

where P0 = Rρ0 θ0 with θ0 (x) = θ(x, 0). Magnetohydrodynamics concerns the motion of conducting fluids in an electromagnetic field and has a very broad range of applications, whose rigorous derivation from the compressible Navier–Stokes–Maxwell system has been proved by Kawashima–Shizuta [1,2] and Jiang–Li [3], respectively, for 1D(2D) cases and 3D one. There have been huge literatures on the study of the compressible MHD problem (1.1) by many physicists and mathematicians due to its physical importance, complexity, rich phenomena and mathematical challenges, see for example, [1–22] and the references therein. Now, we briefly recall some results concerned with the multi-dimensional compressible MHD equations which are more relatively with our problem. In the absence of vacuum, Kawashima [9] established the local and global well-posedness of the solutions to the compressible MHD equations, see also Vol’pert–Khudiaev [12] and Strohmer[10] for the local existence results. For the presence of vacuum, Fan–Yu [6] and L¨ u–Huang [13] established the local well-posedness of strong solutions to the 3D nonisentropic MHD flow and 2D isentropic case, respectively. Hu–Wang [7,8] and Fan–Yu [5] proved the global existence of renormalized solutions for large initial data. Concerning the

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strong (classical) solutions to the isentropic MHD system, Li–Xu–Zhang [11] and L¨ u–Shi–Xu [14] established the global well-posedness of strong solutions with large oscillations and vacuum for 3D case and 2D one, respectively, provided the initial data be of small energy. However, the similar global existence for small data are still open for nonisentropic problem. Next, we also want to refer some interest and important topics on the full MHD system, such as study of the mechanism of blowup (see [15–19] and the references therein) and the low Mach number limit (see [20–23] and the references therein). For the Cauchy problem (1.6)–(1.7) with Ω = R2 , it is still open even for the local existence of strong solutions when the far field density is vacuum, in particular, the initial density may have compact support. Recently, Liang–Shi [24] obtained the local existence of strong solutions to the Cauchy problem (1.6)–(1.7) with H ≡ 0, which generalized the isentropic results in Li–Liang [25] to the full Navier–Stokes equations with zero heat-conduction. Noting that the local existence of strong solutions to the Cauchy problem of 2D isentropic MHD flow is proved by L¨ u–Huang [13], a natural question arise whether the isentropic results [13] still hold for the nonisentropic case (1.6)–(1.7). More precisely, the aim of this paper is to prove the local existence of strong solutions to the Cauchy problem (1.6)–(1.7), which generalized the local theory of [24] to the MHD system and simultaneously extended the isentropic results [13] to the nonisentropic case. Next, we give the definition of strong solution to (1.6)–(1.7) as follows: Definition 1.1. If all derivatives involved in (1.6)–(1.7) for (ρ, u, P, H) are regular distributions, and Eqs. (1.6)–(1.7) hold almost everywhere in R2 × (0, T ), then (ρ, u, P, H) is called a strong solution to (1.6)–(1.7). In this section, for 1 ≤ r ≤ ∞, we denote the standard Lebesgue and Sobolev spaces as follows: Lr = Lr (R2 ),

W s,r = W s,r (R2 ),

H s = W s,2 .

Theorem 1.1. Let η0 be a positive constant and x ¯ , (e + |x|2 )1/2 log1+η0 (e + |x|2 ). For constants q > 2 and a > 1, assume that the initial data (ρ0 , u0 , P0 , H0 ) satisfy  1/2 ρ0 ≥ 0, x ¯a ρ0 ∈ L1 ∩ H 1 ∩ W 1,q , ρ0 u0 ∈ L2 , ∇u0 ∈ H 1 , 1 1 1,q a/2 1 P0 ∈ L ∩ H ∩ W , x H0 ∈ H , ∇2 H0 ∈ L2 , divH0 = 0,

(1.8)

(1.9)

and the following compatibility condition 1 1/2 (1.10) − µ△u0 − (µ + λ)∇divu0 + ∇P0 − H0 · ∇H0 + ∇|H0 |2 = ρ0 g 2 for some g ∈ L2 (R2 ). Then there exists a positive time T0 > 0 such that the problem (1.6)–(1.7) admits a strong solution (ρ, u, P, H) on R2 × (0, T0 ] satisfying  ρ, P ∈ C([0, T0 ]; L1 ∩ H 1 ∩ W 1,q ),      x ¯a ρ ∈ L∞ (0, T0 ; L1 ∩ H 1 ∩ W 1,q ),   √ρu, √ρu, ˙ Ht , ∇2 H ∈ L∞ (0, T0 ; L2 ), (1.11) a/2  Hx ¯ , ∇u ∈ L∞ (0, T0 ; H 1 ),     ∇u, ˙ ∇Ht , x ¯a/2 ∆H ∈ L2 (R2 × (0, T0 )),    2 2 ∇ u, ∇ H ∈ L2 (0, T0 ; Lq ), where f˙ = ft + u · ∇f , and 

1 inf ρ(x, t)dx ≥ 0≤t≤T0 B 4 N    for some constant N > 0 and BN , x ∈ R2  |x| < N .

 ρ0 (x)dx, R2

(1.12)

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Remark 1.1. When H = 0, i.e., there is no electromagnetic field effect, (1.6)–(1.7) reduce to the full compressible Navier–Stokes equations without heat-conduction(κ = 0), and Theorem 1.1 is similar to the results in [24]. Moreover, it should be noted here that the requirement a ∈ (1, 2) in [24] is relaxed to a ∈ (1, ∞) in this paper. Roughly speaking, we generalize and improve the results of [24] to the compressible MHD equations. Remark 1.2. Similar as [25,26,24], by some standard arguments, we could also obtain some higher order estimates and then prove that the local strong solution obtained by Theorem 1.1 becomes classical one. We now comment on the analysis of this paper. For the two-dimensional case, it seems difficult to bound the Lp -norm of u just in terms of ∥ρ1/2 u∥L2 and ∥∇u∥L2 , i.e., the methods used in the three-dimensional case [27,6] cannot be applied directly to our case. In order to get the estimates on the Lp -norm of u, we will use the key ideas due to [25](see also [13]) where the 2D isentropic compressible Navier–Stokes (MHD) equations were considered. More precisely, combining a Hardy-type inequality due to Lions [28] (see (2.4)) with some careful estimates on the essential support of the density (see (3.40)), we first establish a key Hardy-type inequality (see (3.41)) to bound the Lp -norm of u¯ x−η instead of just the velocity u, and then obtain a crucial inequality (see (3.42)) which is used to control the Lp -norm of ρu. Comparing with the isentropic case [13], since the pressure P is not a function of the density, it is difficult to get some necessary lower order estimates of the solutions. Notice that the energy equation can be rewritten to a hyperbolic one (1.5) when κ = 0, using the basic estimates of the material derivatives of velocity u˙ (see [26,14]), we succeed in obtaining the desired a priori estimates of the solutions. However, the material derivatives of velocity u˙ will bring us some new difficulties for the strong coupling between the velocity field and the magnetic field. On the one hand, we establish some spatial weighted mean estimates of H and ∇H (i.e., x ¯a/2 H and x ¯a/2 ∇H, see (3.51) and (3.52)), which are crucial to control the coupling terms, such as ∥u · H∥ and ∥u · ∇H∥. On the other hand, the usual L2 -norms of Ht and ∇Ht cannot be estimated directly. Motivated by [14], multiplying the magnetic equations by ∆H and H∆|H|2 (see (3.12) and (3.31)), respectively, we can bound the L2 -norms of |H| |∇H| and |H| |∆H| instead of the L2 -norms of Ht and ∇Ht , and then get the estimates of ∥∇u∥L∞ and ∥∇H∥L∞ . Next, we construct the approximate solutions to (1.6)–(1.7), that is, for density strictly away from vacuum initially, we consider an initial–boundary value problem of (1.6)–(1.7) in any bounded ball BR with radius R > 0. Finally, combining all these ideas stated above with those due to [25,24,13], we derive some desired bounds which are independent of both the radius of the balls BR and the lower bound of the initial density. The rest of the paper is organized as follows: In Section 2, we collect some elementary facts and inequalities which will be used in later analysis. Section 3 is devoted to the a priori estimates which are needed to obtain the local existence of strong solutions. Finally, Theorem 1.1 is proved in Section 4. 2. Preliminaries First, the following local existence theory on bounded balls, where the initial density is strictly away from vacuum, can be shown by similar arguments as in [9,6]. Lemma 2.1. For R > 0 and BR = {x ∈ R2 | |x| < R}, assume that (ρ0 , u0 , P0 , H0 ) satisfies (1.10) and  inf ρ0 (x) > 0, (ρ0 , P0 , ) ∈ H 3 (BR ), (u0 , H0 , ) ∈ H01 (BR ) ∩ H 3 (BR ),  x∈B R (ρ, u, P, H)(x, t = 0) = (ρ0 , u0 , P0 , H0 )(x), divH0 = 0, x ∈ BR   u = 0, H = 0, x ∈ ∂BR .

(2.1)

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Then there exist a small time TR > 0 and a unique classical solution (ρ, u, P, H) to the initial–boundary-value problem (1.6)–(1.7) on BR × (0, TR ] such that      3 2  (ρ, P ) ∈ C [0, TR ]; H , ρt ∈ C [0, TR ]; H , (2.2) (u, H) ∈ C [0, TR ]; H 3 ∩ L2 0, TR ; H 4 ,  (u , H ) ∈ L∞ 0, T ; H 1  ∩ L2 0, T ; H 2  , t t R R where we denote H k = H k (BR ) for positive integer k. Next, for either Ω = R2 or Ω = BR with R ≥ 1, the following weighted Lp -bounds for elements of the ˜ 1,2 (Ω ) , {v ∈ H 1 (Ω )|∇v ∈ L2 (Ω )} can be found in [28, Theorem B.1]. Hilbert space D loc Lemma 2.2. For m ∈ [2, ∞) and β ∈ (1 + m/2, ∞), there exists a positive constant C such that for either ˜ 1,2 (Ω ), Ω = R2 or Ω = BR with R ≥ 1 and for any v ∈ D  Ω

1/m |v|m 2 −β (log(e + |x| )) dx ≤ C∥v∥L2 (B1 ) + C∥∇v∥L2 (Ω) . e + |x|2

(2.3)

˜ 1,2 (Ω ) A useful consequence of Lemma 2.2 is the following crucial weighted bounds for elements of D which has been proved in Lemma 2.4 of [25](see also [24,13]). Lemma 2.3. Let x ¯ and η0 be as in (1.8) and Ω as in Lemma 2.2. Assume that ρ ∈ L1 (Ω ) ∩ L∞ (Ω ) is a non-negative function such that  ρdx ≥ M1 , (2.4) BN 1

for positive constants M1 , and N1 ≥ 1 with BN1 ⊂ Ω . Then there is a positive constant C depending only on M1 , N1 , and η0 such that ∥v¯ x−1 ∥L2 (Ω) ≤ C∥ρ1/2 v∥L2 (Ω) + C(1 + ∥ρ∥L∞ )∥∇v∥L2 (Ω) ,

(2.5)

˜ 1,2 (Ω ). Moreover, for ε > 0 and η > 0, there is a positive constant C depending only on ε, η, M1 , N1 , for v ∈ D ˜ 1,2 (Ω ) satisfies and η0 such that every v ∈ D ∥v¯ x−η ∥L(2+ε)/η˜ (Ω) ≤ C∥ρ1/2 v∥L2 (Ω) + C(1 + ∥ρ∥L∞ )∥∇v∥L2 (Ω) ,

(2.6)

with η˜ = min{1, η}. Finally, we give some regularity results for the following Lam´e system with Dirichlet boundary condition  −µ∆v − (µ + λ)∇divv = f, x ∈ BR , (2.7) v = 0, x ∈ ∂BR . The following Lp -bound for the weak solution to (2.7), whose proof is shown in [27, Lemma 12], is a direct consequence of the combination of a well-known elliptic theory due to Agmon–Douglis–Nirenberg [29] with a standard scaling procedure. Lemma 2.4. For p > 1 and k ≥ 0, there exists a positive constant C depending only on p and k such that ∥∇k+2 v∥Lp (BR ) ≤ C∥f ∥W k,p (BR ) , for every solution v ∈ W0k+2,p (BR ) of (2.7).

(2.8)

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3. A priori estimates Throughout this section, for p ∈ [1, ∞] and k ≥ 0, we denote   f dx = f dx, Lp = Lp (BR ), W k,p = W k,p (BR ),

H k = W k,2 .

BR

Moreover, for R > 4N0 ≥ 4, assume that (ρ0 , u0 , P0 , H0 ) satisfies, in addition to (2.1) and (1.10), that   1/2 ≤ ρ0 (x)dx ≤ ρ0 (x)dx ≤ 3/2. (3.1) BN0

BR

Lemma 2.1 thus yields that there exists some TR > 0 such that the initial–boundary-value problem (1.6)–(1.7) has a unique classical solution (ρ, u, P, H) on BR × [0, TR ] satisfying (2.2). For x ¯, η0 , a and q as in Theorem 1.1, the main aim of this section is to derive the following key a priori estimate on ψ defined by ψ(t) , 1 + ∥ρ1/2 u∥L2 + ∥∇u∥L2 + ∥¯ xa/2 H∥H 1 + ∥¯ xa ρ∥L1 ∩H 1 ∩W 1,q + ∥P ∥L1 ∩H 1 ∩W 1,q .

Proposition 3.1. Assume that (ρ0 , u0 , P0 , H0 ) satisfies (1.10), (2.1), and (3.1). Let (ρ, u, P, H) be the solution to the initial–boundary-value problem (1.6)–(1.7) on BR × (0, TR ] obtained by Lemma 2.1. Then there exist positive constants T0 and M both depending only on µ, λ, ν, γ, q, a, η0 , N0 , and E0 such that   sup ψ(t) + ∥ρ1/2 u∥ ˙ L2 + ∥ |H| |∇H| ∥L2 + ∥Ht ∥L2 + ∥∇2 u∥L2 + ∥∇2 H∥L2 0≤t≤T0



T0



+

 ∥∇u∥ ˙ 2L2 + ∥∇Ht ∥2L2 + ∥ |H| |∆H| ∥2L2 dt

0



T0

+



 ∥¯ xa/2 ∆H∥2L2 + ∥∇2 u∥2L2 ∩Lq + ∥∇2 H∥2L2 ∩Lq dt ≤ M,

(3.2)

0

where 1/2

E0 , ∥ρ0 u0 ∥L2 + ∥∇u0 ∥H 1 + ∥¯ xa/2 H0 ∥H 1 + ∥∇2 H0 ∥L2 + ∥¯ xa ρ0 ∥L1 ∩H 1 ∩W 1,q + ∥P0 ∥L1 ∩H 1 ∩W 1,q + ∥g∥L2 . To prove Proposition 3.1, we begin with the following standard energy estimate for (ρ, u, P, H) and preliminary L2 -bounds for both ∇u and ∇H. Lemma 3.2. Under the conditions of Proposition 3.1, then there exists a positive constant α > 1 such that for all t > 0,  t    ∥ρ1/2 u∥ ˙ 2L2 + ∥∆H∥2L2 + ∥ |H| |∇H| ∥2L2 ds sup ∥∇u∥2L2 + ∥∇H∥2L2 + ∥H∥4L4 + 0≤s≤t

0

 ≤C +C

t

ψ α (s)ds,

(3.3)

0

where (and in what follows) C denotes a generic positive constant depending only on µ, λ, ν, γ, q, a, η0 , N0 , and E0 . Proof. First, applying standard energy estimate to (1.6)–(1.7) gives   1 1 1 1/2 2 2 ∥ρ u∥L2 + ∥H∥L2 + ∥P ∥L1 ≤ C. sup 2 γ−1 0≤s≤t 2

(3.4)

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Next, multiplying equations (1.6)2 by u˙ and integrating by parts yield     2 ρ|u| ˙ dx = − u˙ · ∇P dx + µ △u · udx ˙ + (µ + λ) ∇divu · udx ˙ 1 − 2



2



u˙ · ∇|H| dx +

H · ∇H · udx ˙ ,

5 

Ri .

(3.5)

i=1

We estimate each term on the right-hand side of (3.5) as follows: First, integration by parts together with (1.6)3 and Gagliardo–Nirenberg inequality (see [30]) leads to   R1 = divut P dx + div(u · ∇u)P dx 

    divuP dx − divuPt dx + u · ∇(divu)P dx + ∂i uj ∂j ui P dx



   divuP dx − divu(Pt + u · ∇P )dx + C P |∇u|2 dx



     (γ − 1)Q(∇u) + ν(γ − 1)|∇ × H|2 − γP divu divudx divuP dx + C P |∇u|2 dx −



    divuP dx + C |∇u|3 dx + C |∇H|3 dx + C P |∇u|2 dx



  ν divuP dx + C∥∇u∥3L3 + C∥∇H∥4L2 + C + ∥∇2 H∥2L2 + C P |∇u|2 dx. 4 t

=

t

≤

t

≤

t

≤

t

≤

(3.6)

Similar to the proof of [26, Lemma 3.2](see also [14, Lemma 3.2]), we have 3 

 Ri ≤ − µ2 ∥∇u∥2L2 −

i=2

λ+µ 2 2 ∥divu∥L2

 t

+ C∥∇u∥3L3 .

(3.7)

Using (1.6)4 , (3.4), and Gagliardo–Nirenberg inequality, we get   1 1 R4 = |H|2 divut dx + |H|2 div(u · ∇u)dx 2 2      1 1 1 2 2 2 |H| divudx − |H| (divu) dx + |H|2 ∂i uj ∂j ui dx = 2 2 2 t  − (H · ∇u + ν∆H − Hdivu) · Hdivudx     1 ν |H|2 divudx + C |H|2 |∇u|2 dx + ∥∆H∥2L2 2 8 t    1 ν ≤ |H|2 divudx + C∥∇u∥3L3 + C∥H∥6L6 + ∥∆H∥2L2 2 8 t    ν 1 |H|2 divudx + C∥∇u∥3L3 + C∥∇H∥4L2 + ∥∆H∥2L2 . ≤ 2 4 t

≤

(3.8)

Similarly, we have d R5 ≤ − dt



H · ∇u · Hdx + C∥∇u∥3L3 + C∥∇H∥4L2 +

ν ∥∆H∥2L2 . 4

(3.9)

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Putting (3.6)–(3.9) into (3.5) yields    d µ λ+µ ∥∇u∥2L2 + ∥divu∥2L2 + ρ|u| ˙ 2 dx dt 2 2  3ν ∥∆H∥2L2 , ≤ B ′ (t) + C P |∇u|2 dx + C∥∇u∥3L3 + C∥∇H∥4L2 + C + 4 where    B(t) , divuP dx + 12 divu|H|2 dx − H · ∇u · Hdx.

(3.10)

(3.11)

Next, multiplying (1.6)4 by ∆H and integrating by parts, we have     d |∇H|2 dx + 2ν |∆H|2 dx ≤ C |∇u| |∇H|2 dx + C |∇u| |H| |∆H|dx dt 4/3

2/3

≤ C∥∇u∥L3 ∥∇H∥L2 ∥∇H∥H 1 + C∥∇u∥L3 ∥H∥L6 ∥∆H∥L2 ν ≤ C∥∇u∥3L3 + C∥∇H∥4L2 + C + ∥∆H∥2L2 , 4 which together with (3.10) and Gagliardo–Nirenberg inequality leads to     d µ λ+µ 2 2 2 2 ∥∇u∥L2 + ∥divu∥L2 + ∥∇H∥L2 + ρ|u| ˙ dx + ν |△H|2 dx dt 2 2

(3.12)

≤ B ′ (t) + C∥P ∥L∞ ∥∇u∥2L2 + C∥∇u∥2L2 ∥∇u∥H 1 + C∥∇H∥4L2 + C ≤ B ′ (t) + Cψ α + ψ 2 ∥∇2 u∥L2 ,

(3.13)

where α > 1 is a generic constant which may be different from line to line. Furthermore, it follows from (2.8) and (1.6)2 that for any p > 1, ∥∇2 u∥Lp ≤ C∥ρu∥ ˙ Lp + C∥∇P ∥Lp + C∥ |H| |∇H| ∥Lp .

(3.14)

Choosing p = 2 in (3.14), it follows from (3.13) that     λ+µ d µ 2 2 2 ∥∇u∥L2 + ∥divu∥L2 + ∥∇H∥L2 + ρ|u| ˙ 2 dx + ν |△H|2 dx dt 2 2 1/2

≤ B ′ (t) + Cψ α + Cψ 2 (∥ρ∥L∞ ∥ρ1/2 u∥ ˙ L2 + C∥∇P ∥L2 + C∥ |H| |∇H| ∥L2 ) 1 ˙ 2L2 + C∥ |H| |∇H| ∥2L2 . ≤ B ′ (t) + Cψ α + ∥ρ1/2 u∥ 2 Integrating (3.15) over (0, t), we have  t    sup ∥∇u∥2L2 + ∥∇H∥2L2 + ∥ρ1/2 u∥ ˙ 2L2 + ∥∆H∥2L2 ds 0≤s≤t

(3.15)

0



t



t

≤C +C ψ α ds + C ∥ |H| |∇H| ∥2L2 ds 0 0    1 2 divu|H| dx − H · ∇u · Hdx + divuP dx + 2  t  t 1 ψ α ds + C ∥ |H| |∇H| ∥2L2 ds + ∥∇u∥2L2 + C∥P ∥2L2 + C∥H∥4L4 ≤C +C 2 0 0  t 1 ≤C +C ψ α ds + ∥∇u∥2L2 + C∥P ∥2L2 , 2 0 where in the last inequality one has used the following estimate  t  t sup ∥ |H|2 ∥2L2 + ∥ |∇H| |H| ∥2L2 ds ≤ C + C ψ α ds. 0≤s≤t

0

0

(3.16)

(3.17)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

417

Indeed, multiplying (1.6)4 by H|H|2 and integrating by parts give  ν 1 ∥ |H|2 ∥2L2 t + ν∥ |∇H| |H| ∥2 + ∥∇|H|2 ∥2 ≤ C∥∇u∥L2 ∥ |H|2 ∥2L4 ≤ C∥∇u∥L2 ∥ |H|2 ∥L2 ∥∇|H|2 ∥L2 4 2 ν ≤ ∥∇|H|2 ∥2L2 + C∥∇u∥2L2 ∥H∥2L2 ∥∇H∥2L2 4 ν (3.18) ≤ ∥∇|H|2 ∥2L2 + Cψ α , 4 where one has used (3.4) and Gagliardo–Nirenberg inequality. Integrating (3.18) over (0, t) yields (3.17). Finally, in order to estimate the last term on the right hand of (3.16), multiplying (1.6)3 by P , we have   1 d ∥P ∥2L2 ≤ C P 2 divudx + C P (|∇u|2 + |∇H|2 )dx 2 dt ≤ C∥P ∥L∞ ∥P ∥L2 ∥∇u∥L2 + C∥P ∥L∞ (∥∇u∥2L2 + ∥∇H∥2L2 ) ≤ Cψ α ,

(3.19)

which yields that sup ∥P ∥2L2 ≤ C + C

0≤s≤t

t 0

ψ α ds.

(3.20)

Hence, (3.3) is deduced directly from (3.16), (3.17) and (3.20). This completes the proof of Lemma 3.2.



Lemma 3.3. Let α be as in Lemma 3.2. Then, for all t > 0,   t     t 1/2 2 2 2 2 α sup ∥ρ u∥ ˙ L2 + ∥ |H| |∇H| ∥L2 + (∥∇u∥ ˙ L2 + ∥ |∆H| |H| ∥L2 )ds ≤ C exp C ψ ds . (3.21) 0≤s≤t

0

0

j

Proof. Operating ∂/∂t + div(u·) to (1.6)2 , one gets by some direct calculations that ρ(u˙ j )t + ρu · ∇u˙ j − µ∆u˙ j − (µ + λ)∂j (divu) ˙ = µ∂i (−∂i u · ∇uj + divu∂i uj ) − µdiv(∂i u∂i uj ) − (µ + λ)∂j (∂i u · ∇ui − (divu)2 ) − (µ + λ)div(∂j udivu) − [∂j (Pt + div(P u)) − div(P ∂j u)]  1 ∂t ∂j |H|2 + div(u∂j |H|2 ) − 2  + ∂t (H · ∇H j ) + div(u(H · ∇H j )) ,

7 

Ki ,

(3.22)

i=1

which multiplied by u˙ j leads to 1 2



ρ|u| ˙ 2 dx



+ µ∥∇u∥ ˙ 2L2 + (µ + λ)∥divu∥ ˙ 2L2 = t

7  

Ki u˙ j dx.

(3.23)

i=1

First, with the same arguments as in [26,14], we have 4   i=1

Ki u˙ j dx ≤ ε∥∇u∥ ˙ 2L2 + C∥∇u∥4L4 .

(3.24)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

418

Integration by parts together with (1.6)3 yields that    K5 u˙ j dx = divu(P ˙ t + div(P u))dx − ∂i u˙ j ∂j ui P dx   = (γ − 1) divu(−P ˙ divu + Q(∇u) + ν|∇ × H|2 )dx − ∂i u˙ j ∂j ui P dx ≤ ε∥∇u∥ ˙ 2L2 + C∥∇u∥4L4 + C∥∇H∥4L4 + C∥P ∥4L4 .

(3.25)

Next, it follows from (1.6)4 and Gagliardo–Nirenberg inequality that    1 j j K6 u˙ dx = ∂i u˙ j ui ∂j H 2 dx ∂j u˙ H · Ht dx + 2    1 1 j i 2 j i 2 = ∂j u˙ ∂i u H dx − ∂i u˙ ∂j u H dx + ∂j u˙ j H · (H · ∇u + ν∆H − Hdivu)dx 2 2  ≤ C |∇u| ˙ |∇u| |H|2 dx + C |∇u| ˙ |∆H · H|dx     ≤ ε |∇u| ˙ 2 dx + C |∇u|4 dx + C |H|8 dx + C |∆H|2 |H|2 dx. (3.26) Similar to (3.26), it holds      j 2 4 8 K7 u˙ dx ≤ ε |∇u| ˙ dx + C |∇u| dx + C |H| dx + C |∆H|2 |H|2 dx.

(3.27)

Submitting (3.24)–(3.27) into (3.23) and choosing ε suitably small yield    2 ˙ 2 dx ≤ C∥∇u∥4L4 + C∥P ∥4L4 + C∥∇H∥4L4 + C∥ |H|2 ∥4L4 ρ|u| ˙ dx + µ |∇u| t

+ C1 ∥ |∆H||H| ∥2L2 .

(3.28)

Next, we will use the methods of [13,14] to control the last term of (3.28). More precisely, for a1 , a2 ∈ {−1, 0, 1}, denoting ˜ 1 , a2 ) = a1 H 1 + a2 H 2 , H(a

u ˜(a1 , a2 ) = a1 u1 + a2 u2 ,

(3.29)

it thus follows from (1.6)4 that ˜ t − ν∆H ˜ = H · ∇˜ ˜ − Hdivu. ˜ H u − u · ∇H

(3.30)

˜ H| ˜ 2 and integrating the resulting equations by parts lead to Multiplying (3.30) by 4ν −1 H△|     ˜ 2 ∥2 2 + 2∥∆|H| ˜ 2 ∥2 2 = 4 |∇H| ˜ 2 ∆|H| ˜ 2 dx − 4ν −1 H · ∇˜ ˜ H| ˜ 2 dx ν −1 ∥∇|H| uH∆| L t L   ˜ 2 ∆|H| ˜ 2 dx + 2ν −1 u · ∇|H| ˜ 2 ∆|H| ˜ 2 dx + 4ν −1 divu|H| ˜ 2 ∥2 2 , ≤ C∥∇u∥4L4 + C∥∇H∥4L4 + C∥ |H|2 ∥4L4 + ∥∆|H| L

(3.31)

where in the last inequality we have used the following estimate    ˜ 2 |2 dx ˜ 2 ∆|H| ˜ 2 dx = − ∇u · ∇|H| ˜ 2 · ∇|H| ˜ 2 dx + 1 divu|∇|H| u · ∇|H| 2 ≤ C∥∇u∥4L4 + C∥∇H∥4L4 + C∥ |H|2 ∥4L4 . Noticing that ∥ |∇H| |H|∥2L2 ˜ 0)|2 ∥2 2 + ∥∇|H(0, ˜ 1)|2 ∥2 2 + ∥∇|H(1, ˜ 1)|2 ∥2 2 + ∥∇|H(1, ˜ −1)|2 ∥2 2 , G(t) ≤ ∥∇|H(1, L L L L

(3.32)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

419

and ∥ |∆H| |H|∥2L2 ˜ 0)|2 ∥2 2 + ∥∆|H(0, ˜ 1)|2 ∥2 2 + ∥∆|H(1, ˜ 1)|2 ∥2 2 + ∥∆|H(1, ˜ −1)|2 ∥2 2 , ≤ C∥∇H∥4L4 + ∥∆|H(1, L L L L then multiplying (3.31) by (C1 + 1) leads to  −1  ν (C1 + 1)G(t) t + (C1 + 1)∥ |∆H| |H|∥2L2 ≤ C∥∇u∥4L4 + C∥∇H∥4L4 + C∥ |H|2 ∥4L4 .

(3.33)

(3.34)

Adding (3.34) to (3.28), we obtain after using Gagliardo–Nirenberg inequality, (3.14), and (3.32) that  d  1/2 2 ∥ρ u∥ ˙ L2 + ν −1 (C1 + 1)G(t) + ∥∇u∥ ˙ 2L2 + ∥ |∆H| |H| ∥2L2 dt ≤ C∥∇u∥4L4 + C∥P ∥4L4 + C∥∇H∥4L4 + C∥ |H|2 ∥4L4 ≤ C∥∇u∥2L2 ∥∇u∥2H 1 + Cψ α + C∥∇H∥2L2 ∥∇H∥2H 1 + C∥H| ∥4L4 ∥ |H| |∇H| ∥2L2 ≤ Cψ α ∥∇2 u∥2L2 + Cψ α ∥ |H| |∇H| ∥2L2 + Cψ α + C∥∇H∥2L2 ∥∇2 H∥2L2 ≤ Cψ α (∥ρ1/2 u∥ ˙ 2L2 + G(t)) + Cψ α + C∥∇H∥2L2 ∥∇2 H∥2L2 .

(3.35)

Finally, Gronwall’s inequality together with (3.35), (3.32), (3.3), and (1.10) gives    t 1/2 2 2 sup ∥ρ u∥ ˙ L2 + ∥ |H| |∇H| ∥L2 + (∥∇u∥ ˙ 2L2 + ∥ |∆H| |H| ∥2L2 )ds 0≤s≤t 0   t    t  t α α 2 2 2 ≤ C exp C ψ ds 1+ ψ ds + sup ∥∇H∥L2 ∥∇ H∥L2 ds 0

0≤s≤t

0

0

  t  ≤ C exp C ψ α ds .

(3.36)

0

The proof of Lemma 3.3 is finished.



Lemma 3.4. Let α be as in Lemma 3.2. Then, there exists a T1 = T1 (E0 ) such that for all t ∈ (0, T1 ],    t  a α 1 1 1,q sup ∥¯ x ρ∥L ∩H ∩W ≤ exp C exp C ψ ds . (3.37) 0≤s≤t

0

Proof. First, for N > 1 and ϕN ∈ C0∞ (BN ) such that 0 ≤ ϕN ≤ 1,

ϕN (x) = 1, if |x| ≤ N/2,

|∇ϕN | ≤ CN −1 ,

it follows from (3.4) and (3.1) that   d ρϕ2N0 dx = ρu · ∇ϕ2N0 dx dt  1/2  1/2 ˜ 0 ), ≥ −CN0−1 ρdx ρ|u|2 dx ≥ −C(E

(3.38)

(3.39)

where in the last inequality we have used 

 ρdx =

due to (1.6)1 . Integrating (3.39) gives  inf ρdx ≥ 0≤t≤T1

B2N0

inf

0≤t≤T1



ρ0 dx,

ρϕ2N0 dx ≥



˜ 1 ≥ 1/4, ρ0 ϕ2N0 dx − CT

(3.40)

420

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

˜ −1 }. From now on, we will always assume that t ≤ T1 . The combination of (3.40), where T1 , min{1, (4C) ˜ 1,2 (BR ) satisfies (3.4), and (2.6) yields that for ε > 0 and η > 0, every v ∈ D ∥v¯ x−η ∥2L(2+ε)/η˜ ≤ C(ε, η)∥ρ1/2 v∥2L2 + C(ε, η)(1 + ∥ρ∥L∞ )∥∇v∥2L2 ,

(3.41)

with η˜ = min{1, η}. In particular, we have ∥ρη u∥L(2+ε)/η˜ + ∥u¯ x−η ∥L(2+ε)/η˜ ≤ C(ε, η)ψ 1+η .

(3.42)

Multiplying Eq. (1.6)1 by x ¯a and integrating by parts give  d xa−1 ln1+η0 (e + |x|2 )dx ∥ρ¯ xa ∥L1 ≤ C ρ|u|¯ dt    8 4 4 = C ρ¯ xa−1+ 8+a |u|¯ x− 8+a x ¯− 8+a ln1+η0 (e + |x|2 ) dx 8

≤ C∥ρ¯ xa−1+ 8+a ∥

4

8+a

L 7+a

∥u¯ x− 8+a ∥L8+a

≤ Cψ α

(3.43)

owing to (3.42). This leads to   t  sup ∥ρ¯ xa ∥L1 ≤ C exp C ψ α ds . 0≤s≤t

(3.44)

0

Next, it follows from the Sobolev inequality and (3.42) that for 0 < δ < 1,   ∥u¯ x−δ ∥L∞ ≤ C(δ) ∥u¯ x−δ ∥L4/δ + ∥∇(u¯ x−δ )∥L3   ≤ C(δ) ∥u¯ x−δ ∥L4/δ + ∥∇u∥L3 + ∥u¯ x−δ ∥L4/δ ∥¯ x−1 ∇¯ x∥L12/(4−3δ)   ≤ C(δ) ψ α + ∥∇2 u∥L2 .

(3.45)

Now, one derives from (1.6)1 that ρ¯ xa satisfies ∂t (ρ¯ xa ) + u · ∇(ρ¯ xa ) − aρ¯ xa u · ∇ log x ¯ + ρ¯ xa divu = 0.

(3.46)

Taking the xi -derivative on the both sides of (3.46), we get ∂t ∂i (ρ¯ xa ) + u · ∇∂i (ρ¯ xa ) + ∂i u · ∇(ρ¯ xa ) − a∂i (ρ¯ xa )u · ∇ log x ¯ − aρ¯ xa ∂i u · ∇ log x ¯ − aρ¯ xa u · ∂i ∇ log x ¯ + ∂i (ρ¯ xa )divu + ρ¯ xa ∂i divu = 0.

(3.47)

xa )|r ), multiplying (3.47) by For any r ∈ [2, q], noting that |∇(ρ¯ xa )|r−2 ∂i (ρ¯ xa )∇∂i (ρ¯ xa ) = 1r ∇(|∂i (ρ¯ |∇(ρ¯ xa )|r−2 ∂i (ρ¯ xa ) and integrating the resulting equality by parts, we obtain that d ∥∇(ρ¯ xa )∥Lr ≤ C (1 + ∥∇u∥L∞ + ∥u · ∇ log x ¯∥L∞ ) ∥∇(ρ¯ xa )∥Lr dt   + C∥ρ¯ xa ∥L∞ ∥ |∇u| |∇ log x ¯| ∥Lr + ∥ |u| |∇2 log x ¯| ∥Lr  2 3 ≤ C (ψ α + ∥∇u∥W 1,q ) ∥∇(ρ¯ xa )∥Lr + C∥ρ¯ xa ∥L∞ ∥∇u∥Lr + ∥u¯ x− 5 ∥L4r ∥¯ x− 2 ∥



4r L 3

≤ C (ψ α + ∥∇u∥W 1,q ) (ψ α + ∥∇(ρ¯ xa )∥Lr + ∥∇(ρ¯ xa )∥Lq ) .

(3.48)

Now, we claim that  0

t

  t  ∥∇2 u∥2L2 ∩Lq ds ≤ C exp C ψ α (s)ds , 0

which together with (3.44), (3.48), and Gronwall’s inequality yields (3.37).

(3.49)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

421

Finally, to finish the proof of Lemma 3.4, it remains to prove (3.49). Indeed, for any r ∈ [2, q], it deduces from (3.14), (3.42) and Gagliardo–Nirenberg inequality that ∥∇2 u∥2Lr ≤ C∥ρu∥ ˙ 2Lr + C∥∇P ∥2Lr + C∥ |H| |∇H| ∥2Lr 2(r 2 −2r) 2

4(r−1) 2

+ Cψ α + C∥H∥2L2r ∥∇H∥2L2r

˙ Lrr2 −2 ≤ C∥ρu∥ ˙ Lr2 −2 ∥ρu∥

 r22−2r    2(r−1) 2r−2 r 2 −2 r −2 ∥ρ1/2 u∥ ˙ 2L2 + (1 + ∥ρ∥L∞ )2 ∥∇u∥ ˙ 2L2 + Cψ α + Cψ α ∥∇2 H∥L2r ≤ Cψ α ∥ρ1/2 u∥ ˙ 2L2 ≤ Cψ α ∥ρ1/2 u∥ ˙ 2L2 + Cψ α + C∥∇u∥ ˙ 2L2 + C∥∇2 H∥2L2 ,

(3.50)

which along with (3.3) and (3.21) yields (3.49). The proof of Lemma 3.4 is finished.



Next, motivated by [13,14], we will also show some useful spatial weighted estimates on both H and ∇H.

Lemma 3.5. Let α and T1 be as in Lemmas 3.2 and 3.4, respectively. Then, for all t ∈ (0, T1 ], sup 0≤s≤t

sup 0≤s≤t

∥H x ¯a/2 ∥2L2

t



∥∇H x ¯a/2 ∥2L2 ds

+

  t  α ≤ C exp C ψ ds ,

0

∥∇H x ¯a/2 ∥2L2

(3.51)

0

 +

t

∥∆H x ¯a/2 ∥2L2 ds

  t  α ≤ exp C exp C ψ ds . 

0

(3.52)

0

Proof. First, multiplying (1.6)4 by H x ¯a and integrating by parts lead to 1 2



    ν |H|2 x ¯a dx + ν |∇H|2 x ¯a dx = |H|2 ∆¯ xa dx + H · ∇u · H x ¯a dx 2 t   1 1 − divu|H|2 x ¯a dx + |H|2 u · ∇¯ xa dx 2 2 ,

4 

I¯i .

(3.53)

i=1

Direct calculations yield that |I¯1 | ≤ C



|I¯2 | + |I¯3 | ≤ C



|H|2 x ¯a x ¯−2 log2(1−η0 ) (e + |x|2 )dx ≤ C



|H|2 x ¯a dx,

|∇u| |H|2 x ¯a dx ≤ C∥∇u∥L2 ∥H x ¯a/2 ∥2L4

≤ C∥∇u∥L2 ∥H x ¯a/2 ∥L2 (∥∇H x ¯a/2 ∥L2 + ∥H∇¯ xa/2 ∥L2 ) ν ≤ Cψ α ∥H x ¯a/2 ∥2L2 + ∥∇H x ¯a/2 ∥2L2 ,  4 |I¯4 | ≤ C x ¯a |H|2 x ¯−3/4 |u|¯ x−1/4 log(1−η0 ) (e + |x|2 )dx ≤ C∥H x ¯a/2 ∥L4 ∥H x ¯a/2 ∥L2 ∥u¯ x−3/4 ∥L4 a/2 2 α a/2 2 ≤ C∥H x ¯ ∥L4 + Cψ ∥H x ¯ ∥L2 ν α a/2 2 ¯a/2 ∥2L2 . ≤ Cψ ∥H x ¯ ∥L2 + ∥∇H x 4 Putting (3.54) into (3.53), using Gronwall’s inequality and (3.4), we thus get (3.51).

(3.54)

422

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

Next, multiplying (1.6)4 by ∆H x ¯a and integrating the resultant equations by parts yield that    1 2 a |∇H| x ¯ dx + ν |∆H|2 x ¯a dx 2 t    ≤ C |∇H||H| |∇u| |∇¯ xa |dx + C |∇H|2 |u| |∇¯ xa |dx + C |∇H| |∆H|¯ xa dx  +C



a

|H| |∇u| |∆H|¯ x dx + C

2 a

|∇u| |∇H| x ¯ dx ,

5 

Ji .

(3.55)

i=1

Using Gagliardo–Nirenberg inequality and (3.42), it holds  J1 ≤ C |∇H||H| |∇u|¯ xa (¯ x−1 |∇¯ x|)dx ≤ C∥H x ¯a/2 ∥4L4  + C∥∇u∥4L4 + C∥∇H x ¯a/2 ∥2L2

≤ C∥H x ¯a/2 ∥2L2 ∥∇H x ¯a/2 ∥2L2 + ∥H x ¯a/2 ∥2L2 + C∥∇u∥4L4 + C∥∇H x ¯a/2 ∥2L2

≤ Cψ α + C∥∇u∥2L2 ∥∇2 u∥2L2 + Cψ α ∥∇H x ¯a/2 ∥2L2 , 2 2 a− 13 − 13 2− 3a 6a ∥u¯ x ¯ ∥ 6a−2 x ∥L6a ∥ |∇H| 3a ∥L6a J2 ≤ C∥ |∇H| 6a−2

≤ ≤ J3 + J4 ≤ ≤ J5 ≤ ≤

L

2

Cψ α ∥∇H x ¯a/2 ∥L23a ∥∇H∥L3a4 ≤ Cψ α ∥∇H x ¯a/2 ∥2L2 + C∥∇H∥2L4 α a/2 2 a/2 2 Cψ ∥∇H x ¯ ∥L2 + ε∥∆H x ¯ ∥L2 , a/2 2 a/2 2 ε∥∆H x ¯ ∥L2 + C∥∇H x ¯ ∥L2 + C∥H x ¯a/2 ∥4L4 + C∥∇u∥4L4 a/2 2 α a/2 2 ε∥∆H x ¯ ∥L2 + Cψ ∥∇H x ¯ ∥L2 + C∥∇u∥2L2 ∥∇2 u∥2L2 + Cψ α , a/2 2 C∥∇u∥L∞ ∥∇H x ¯ ∥L2 C (ψ α + ∥∇u∥W 1,q ) ∥∇H x ¯a/2 ∥2L2 .

(3.56)

Submitting (3.56) into (3.55) and choosing ε suitably small, we have   ∥∇H x ¯a/2 ∥2L2 + ν∥∆H x ¯a/2 ∥2L2 t

≤ C (ψ α + ∥∇u∥W 1,q ) ∥∇H x ¯a/2 ∥2L2 + C∥∇u∥2L2 ∥∇2 u∥2L2 + Cψ α ,

(3.57)

which together with Gronwall’s inequality, (3.3) and (3.49) yields (3.52). This completes the proof of Lemma 3.5.  Next, we will show some estimates of Ht and ∇Ht . Lemma 3.6. Let α and T1 be as in Lemmas 3.2 and 3.4, respectively. Then, for all t ∈ (0, T1 ],    t   t 2 2 α sup ∥Ht ∥L2 + ∥∇Ht ∥L2 ds ≤ exp C exp C ψ ds . 0≤s≤t

0

(3.58)

0

Proof. Differentiating (1.6)4 with respect to t shows Htt − Ht · ∇u − H · ∇ut + ut · ∇H + u · ∇Ht + Ht divu + Hdivut = ν∆Ht .

(3.59)

Multiplying (3.59) by Ht and integrating the resulting equations over BR yield that     1 d 2 2 |Ht | dx + ν |∇Ht | dx = H · ∇ut · Ht dx + ut · ∇Ht · Hdx 2 dt   1 + Ht · ∇u · Ht dx − |Ht |2 divudx 2 ,

4  i=1

Si .

(3.60)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

423

On the one hand, it follows from (3.41), (3.14) and Gagliardo–Nirenberg inequality that     2  Si = H · ∇u˙ · Ht dx + u˙ · ∇Ht · Hdx − H · ∇(u · ∇u) · Ht dx − u · ∇u · ∇Ht · Hdx i=1

ε ˙ |H| ∥2L2 + C(ε) ˙ 2L2 + ∥∇Ht ∥2L2 + C(ε)∥ |u| ≤ Cψ α ∥Ht ∥2L4 + C∥∇u∥ 2



|u|2 |∇u|2 |H|2 dx

≤ ε∥∇Ht ∥2L2 + Cψ α ∥Ht ∥2L2 + C∥∇u∥ ˙ 2L2 + C∥u¯ ˙ x−a/4 ∥2L8 ∥H x ¯a/2 ∥L2 ∥H∥L4 + C∥∇u∥4L4 + C∥u¯ x−a/8 ∥4L16 ∥H x ¯a/2 ∥L2 ∥H∥3L12 2  ˙ L2 + (1 + ∥ρ∥L∞ )∥∇u∥ ˙ L2 ≤ ε∥∇Ht ∥2L2 + Cψ α ∥Ht ∥2L2 + C∥∇u∥ ˙ 2L2 + C∥H x ¯a/2 ∥L2 ∥H∥L4 ∥ρ1/2 u∥   + C∥∇u∥4L2 + C∥∇u∥2L2 ∥ρ∥L∞ ∥ρ1/2 u∥ ˙ 2L2 + ψ α + ∥ |H| |∇H| ∥2L2  4 + Cψ α ∥ρ1/2 u∥L2 + (1 + ∥ρ∥L∞ )∥∇u∥L2   ≤ ε∥∇Ht ∥2L2 + Cψ α ∥Ht ∥2L2 + Cψ α ∥ρ1/2 u∥ ˙ 2L2 + ∥ |H| |∇H| ∥2L2 + 1   + C ∥H x ¯a/2 ∥L2 ∥H∥L4 (1 + ∥ρ∥2L∞ ) + 1 ∥∇u∥ ˙ 2L2 . (3.61) On the other hand, Gagliardo–Nirenberg inequality together with Young’s inequality gives 4 

 Si ≤ C

|Ht |2 |∇u|dx ≤ C∥Ht ∥L2 ∥∇Ht ∥L2 ∥∇u∥L2 ≤ ε∥∇Ht ∥2L2 + C(ε)ψ α ∥Ht ∥2L2 .

(3.62)

i=3

Now, putting (3.61)–(3.62) into (3.60) and choosing ε suitably small, we have   d ∥Ht ∥2L2 + ∥∇Ht ∥2L2 ≤ Cψ α ∥Ht ∥2L2 + Cψ α ∥ρ1/2 u∥ ˙ 2L2 + ∥ |H| |∇H| ∥2L2 + 1 dt   + C ∥H x ¯a/2 ∥L2 ∥H∥L4 (1 + ∥ρ∥2L∞ ) + 1 ∥∇u∥ ˙ 2L2

(3.63)

which together with Gronwall’s inequality, (3.3), (3.21), (3.17), (3.37), and (3.51) yields that  t sup ∥Ht ∥2L2 + ∥∇Ht ∥2L2 ds 0≤s≤t

0

  t    t α 1/2 2 2 ≤ exp C ψ ds 1 + sup ∥ρ u∥ ˙ L2 + ∥ |H| |∇H| ∥L2 + 1 ψ α ds 0≤s≤t

0



+ sup 0≤s≤t

∥H x ¯a/2 ∥L2 ∥H∥L4 (1 + ∥ρ∥2L∞ ) + 1

0



t

 ∥∇u∥ ˙ 2L2 ds

0

  t  ≤ exp C exp C ψ α ds . 

(3.64)

0

The proof of Lemma 3.6 is finished.



Lemma 3.7. Let α and T1 be as in Lemma 3.2 and Lemma 3.4, respectively. Then, for all t ∈ (0, T1 ],    t  sup ∥P ∥L1 ∩H 1 ∩W 1,q ≤ exp C exp C ψ α ds . (3.65) 0≤s≤t

0

Proof. Operating ∇ to (1.6)3 , it holds (∇P )t + ∇(u · ∇P ) + γ∇(P divu) = (γ − 1)∇Q(∇u) + ν(γ − 1)∇(|∇ × H|2 ),

(3.66)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

424

which multiplied by |∇P |r−2 ∇P for r ∈ [2, q] gives   1 d ∥∇P ∥rLr = − ∇(u · ∇P ) · ∇P |∇P |r−2 dx − λ ∇(P divu) · ∇P |∇P |r−2 dx r dt  + (γ − 1) ∇Q(∇u) · ∇P |∇P |r−2 dx  + ν(γ − 1)

r−2

2

∇(|∇ × H| ) · ∇P |∇P |

dx ,

4 

J¯i ,

(3.67)

i=1

where J¯1 =

 

=

∇u · ∇P · ∇P |∇P |r−2 dx +



u · ∇(∇P ) · ∇P |∇P |r−2 dx  1 u · ∇(|∇P |r )dx ∇u · ∇P · ∇P |∇P |r−2 dx + r 

|∇u| |∇P |r dx ≤ C∥∇u∥L∞ ∥∇P ∥rLr ,   J¯2 ≤ C |∇u| |∇P |r dx + C P |∇2 u| |∇P |r−1 dx ≤ C

(3.68)

r−1 r 2 ≤ C∥∇u∥ L∞ ∥∇P ∥Lr + C∥P ∥L∞ ∥∇ u∥Lr ∥∇P ∥Lr ,  J¯3 ≤ C |∇u| |∇2 u| |∇P |r−1 dx ≤ C∥∇u∥L∞ ∥∇2 u∥Lr ∥∇P ∥r−1 Lr ,  r−1 J¯4 ≤ C |∇H| |∇2 H| |∇P |r−1 dx ≤ C∥∇H∥L∞ ∥∇2 H∥Lr ∥∇P ∥L r .

Then, one obtains after submitting (3.67) into (3.68) that d ∥∇P ∥L2 ∩Lq ≤ C(ψ α + ∥∇2 u∥L2 + ∥∇2 u∥Lq )(1 + ∥∇P ∥L2 + ∥∇P ∥Lq ) dt + C∥∇u∥L∞ (∥∇2 u∥L2 + ∥∇2 u∥Lq ) + C∥∇H∥L∞ (∥∇2 H∥L2 + ∥∇2 H∥Lq ).

(3.69)

Now, we claim that  t ∥∇u∥L∞ (∥∇2 u∥L2 + ∥∇2 u∥Lq )ds 0



t

∥∇H∥

+

L∞

2

(∥∇ H∥

L2

  t  α + ∥∇ H∥ )ds ≤ exp C exp C ψ ds , 2



(3.70)

Lq

0

0

which together with Gronwall’s inequality, (3.69), (3.49), and (3.4) gives (3.65). Finally, it remains to prove (3.70). Indeed, for any r ∈ [2, q], it follows from (1.6)4 , (3.41), and Gagliardo–Nirenberg inequality that ∥∇2 H∥Lr ≤ C∥Ht ∥Lr + C∥ |H| |∇u| ∥Lr + C∥ |u| |∇H| ∥Lr 2/r

(r−2)/r

≤ C∥Ht ∥L2 ∥∇Ht ∥L2

a

a

1/r

(r−1)/r

¯ 2 ∥L2 ∥∇H∥L4(r−1) + C∥H∥L2r ∥∇u∥L2r + C∥u¯ x− 2r ∥L4r ∥∇H x

≤ C∥Ht ∥L2 + C∥∇Ht ∥L2 + ∥∇2 u∥L2 + ∥∇2 H∥L2 + Cψ α .

(3.71)

This along with Sobolev inequality leads to ∥∇H∥L∞ (∥∇2 H∥L2 + ∥∇2 H∥Lq ) ≤ C(ψ α + ∥∇2 H∥L2 + ∥∇2 H∥Lq )(∥∇2 H∥L2 + ∥∇2 H∥Lq ) ≤ Cψ α + C∥∇2 H∥2L2 + C∥∇2 H∥2Lq ≤ C∥Ht ∥2L2 + C∥∇Ht ∥2L2 + ∥∇2 u∥2L2 + ∥∇2 H∥2L2 + Cψ α , (3.72)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

which together with (3.3), (3.49), and (3.58) implies that    t   t  t ∥∇2 H∥2L2 ∩Lq ds + ∥∇H∥L∞ (∥∇2 H∥L2 + ∥∇2 H∥Lq )ds ≤ exp C exp C ψ α ds . 0

0

425

(3.73)

0

With the similar arguments as (3.72)–(3.73), one deduces directly from Sobolev inequality and (3.50) that   ∥∇u∥L∞ (∥∇2 u∥L2 + ∥∇2 u∥Lq )ds ≤ C (ψ α + ∥∇2 u∥2L2 + ∥∇2 u∥2Lq )ds   t  α ≤ C exp C ψ ds ,

(3.74)

0

which together with (3.73) leads to (3.70). The proof of Lemma 3.7 is finished.



Now, Proposition 3.1 is a direct consequence of Lemmas 3.2–3.7. Proof of Proposition 3.1. It follows from (3.4), (3.3), (3.51), and (3.37) that     t ψ(t) ≤ exp C exp C 0 ψ α ds . Standard arguments thus yield that for M , eCe and T0 , min{T1 , (CM α )−1 }, sup ψ(t) ≤ M.

(3.75)

0≤t≤T0

This combines with (3.14), (3.71), (3.41), (3.21), (3.52), and (3.58) gives ∥∇2 u∥2L2 + ∥∇2 H∥2L2 ≤ C∥ρu∥ ˙ 2L2 + C∥∇P ∥2L2 + C∥ |H| |∇H| ∥2L2 + C∥Ht ∥2L2 + C∥ |u||∇H| ∥2L2 + C∥ |H| |∇u| ∥2L2 ≤ C(M )∥ρ1/2 u∥ ˙ 2L2 + C(M ) + C∥ |H| |∇H| ∥2L2 + C∥Ht ∥2L2 + C∥u¯ x−a/4 ∥2L8 ∥∇H x ¯a/2 ∥L2 ∥∇H∥L4 + C∥H∥L2 ∥∇2 H∥L2 ∥∇u∥2L2 1 ≤ C(M ) + C(M )∥ρ1/2 u∥ ˙ 2L2 + C∥ |H| |∇H| ∥2L2 + C∥Ht ∥2L2 + C∥∇H∥2L4 + ∥∇2 H∥2L2 4 1 ≤ C(M ) + ∥∇2 H∥2L2 , (3.76) 2 which together with (3.21), (3.52), (3.58), (3.49), and (3.73) gives (3.2). The proof of Proposition 3.1 is thus completed.  4. Proofs of Theorem 1.1 Proof of Theorem 1.1. Let (ρ0 , u0 , P0 , H0 ) be as in Theorem 1.1. Without loss of generality, assume that  ρ0 dx = 1, R2

which implies that there exists a positive constant N0 such that   3 3 ρ0 dx ≥ ρ0 dx = . 4 R2 4 BN0

(4.1)

2

−1 −|x| ∞ 2 We construct ρR ˆR e where 0 ≤ ρˆR 0 =ρ 0 +R 0 ∈ C0 (R ) satisfies   ρˆR 0 dx ≥ 1/2, BN0  a R x ¯ ρˆ0 → x ¯a ρ0 in L1 (R2 ) ∩ H 1 (R2 ) ∩ W 1,q (R2 ), as R → ∞.

(4.2)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

426

Similarly, we can also choose P0R ∈ C0∞ (R2 ) such that P0R → P0

in L1 (R2 ) ∩ H 1 (R2 ) ∩ W 1,q (R2 ), as R → ∞.

(4.3)

Notice that H0 x ¯a/2 ∈ H 1 (R2 ) and ∇2 H0 ∈ L2 (R2 ), choosing H0R ∈ C0∞ (BR ) such that H0R x ¯a/2 → H0 x ¯a/2 ,

∇2 H0R → ∇2 H0

in H 1 (R2 ),

in L2 (R2 ), as R → ∞.

(4.4)

Now, consider the unique smooth solution uR 0 of the following elliptic problem in BR :  

1 R 2 R R R R −µ△uR 0 − (µ + λ)∇divu0 + ∇P0 = H0 · ∇H0 − ∇|H0 | + 2 uR | 0 ∂BR = 0,

 R R R ρR 0 h − ρ0 u0 ,

(4.5)

√ where hR = ( ρ0 u0 + g) ∗ j1/R with jδ being the standard mollifying kernel of width δ. 2 Extending uR ˜R 0 to R by defining 0 outside BR and denoting it by u 0 , we claim that

lim

R→∞

   √ R Ru 2 2 1 2 ρ ρ u ∥ = 0. ∥∇(˜ uR − u )∥ + ∥ ˜ − 0 0 0 L (R ) H (R ) 0 0 0

(4.6)

2 In fact, it is easy to find that u ˜R ˜R 0 is also a solution of (4.5) in R . Multiplying (4.5) by u 0 and integrating 2 the resulting equation by parts over R lead to    R 2 R 2 2 ρR |˜ u | dx + µ |∇˜ u | dx + (µ + λ) |div˜ uR 0 0 0 0 | dx R2 R2 R2  R R ˜R uR ≤ ∥ ρR 0 ∥L2 (BR ) ∥h ∥L2 (BR ) + C∥P0 ∥L2 (BR ) ∥∇˜ 0 ∥L2 (BR ) 0u R R + C∥H0R ∥2L4 (BR ) ∥∇˜ uR uR 0 ∥L2 (BR ) + C∥ |H0 | |∇H0 | |˜ 0 | ∥L1 (BR )  1 µ 2 uR ρR |˜ uR |2 dx + C∥hR ∥2L2 (BR ) + C∥P0R ∥2L2 (BR ) ≤ ∥∇˜ 0 ∥L2 (BR ) + 4 4 BR 0 0

+ C∥H0R ∥4L4 (BR ) + C∥H0R ∥L4 (BR ) ∥¯ x−a/2 u ˜R xa/2 ∇H0R ∥L2 (BR ) 0 ∥L4 (BR ) ∥¯   µ 1 2 uR ≤ ∥∇˜ ρR |˜ uR |2 dx + C + C(∥ ρˆR ˜R ρR uR 0 ∥L2 (BR ) + 0 ∥L2 (BR ) + (1 + ∥ˆ 0 ∥L∞ )∥∇˜ 0 ∥L2 (BR ) ) 0u 4 4 BR 0 0  1 µ 2 uR ∥ + ρR |˜ uR |2 dx + C, ≤ ∥∇˜ 2 0 L (BR ) 2 2 BR 0 0 owing to (4.2), (4.3), (4.4) and (3.41). This implies  R2

2 ρR uR 0 |˜ 0 | dx +

 R2

2 |∇˜ uR 0 | dx ≤ C,

(4.7)

for some C independent of R. This together with (4.2) implies that there exist a subsequence Rj → ∞ and √ 1 a function u ˜0 ∈ {˜ u0 ∈ Hloc (R2 )| ρ0 u ˜0 ∈ L2 (R2 ), ∇˜ u0 ∈ L2 (R2 )} such that √ R Rj ρ0 j u ˜ 0 ⇀ ρ0 u ˜0 weakly in L2 (R2 ), Rj ∇˜ u0 ⇀ ∇˜ u0 weakly in L2 (R2 ).



(4.8)

Next, we will prove that u ˜0 = u0 .

(4.9)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

427

Indeed, subtracting (1.10) from (4.5) gives       R R R − µ△ u ˜0 j − u0 − (µ + λ)∇div u ˜0 j − u0 + ∇ P0 j − P0  1  R R R = H0 j · ∇H0 j − H0 · ∇H0 − ∇ |H0 j |2 − |H0 |2 2      √ √ R R R Rj + ρ0 j g ∗ j1 /Rj − ρ0 g − ρ0 j ρ0 j u ˜0 − ρ0 u0 ∗ j1 /Rj .

(4.10)

Multiplying (4.10) by a test function π ∈ C0∞ (R2 ), it holds that   R R µ ∂i (˜ u0 j − u0 ) · ∂i πdx + (µ + λ) div(˜ u0 j − u0 )divπdx R2 R2    √ R R Rj + ρ0 j ( ρ0 j u ˜0 − ρ0 u0 ∗ j1 /Rj ) · πdx 2     R  √ Rj Rj P0 − P0 divπdx + ρ0 g ∗ j1 /Rj − ρ0 g πdx = R2       R R R R + H0 j − H0 · ∇H0 j πdx + H0 j · ∇ H0 j − H0 πdx   1  Rj 2 + |H0 | − |H0 |2 divπdx. 2

(4.11)

Let Rj → ∞ in (4.11), it follows from (4.2), (4.3), (4.4) and (4.8) that   ∂i (˜ u0 − u0 ) · ∂i πdx + ρ0 (˜ u0 − u0 ) · πdx = 0, R2

(4.12)

R2

which implies that (4.9) for the arbitrary π. Furthermore, with the same arguments as (4.12), multiplying R (4.5) by u ˜0 j , integrating the resulting equation over R2 and letting Rj → ∞ imply that    R R R R lim µ|∇˜ u0 j |2 + (µ + λ)(div˜ u0 j )2 + ρ0 j |˜ u0 j |2 dx Rj →∞

R2





=

 µ|∇u0 |2 + (µ + λ)(divu0 )2 + ρ0 |u0 |2 dx,

(4.13)

R2

which combined with (4.8) implies   Rj 2 |∇˜ u0 | dx = lim Rj →∞

R2



2

|∇˜ u0 | dx,

lim

Rj →∞

R2

R2

R R u0 j |2 dx ρ0 j |˜

 =

ρ0 |˜ u0 |2 dx.

R2

This along with (4.9) and (4.8) gives    √ R Ru 2 2 2 2 lim ∥∇(˜ uR − u )∥ + ∥ ρ ˜ − ρ u ∥ = 0. 0 L (R ) 0 0 L (R ) 0 0 0

(4.14)

R→∞

Furthermore, using the Lp regularity of elliptic equations together with Lemma 2.4 and (4.10), we have           √  2 Rj    R Rj  ˜0 − u0  2 ≤ C ∇ P0 j − P0  2 + C  ρ g ∗ j /R − ρ g ∇ u 1 j 0 0   2 L L L      R R R + C∥H0 j − H0 ∥L4 ∥∇H0 j ∥L4 + C∥H0 ∥L∞ ∇ H0 j − H0  2 L      √ R R R j ρ0 j u ˜0 j − ρ0 u0 ∗ j1 /Rj  +C   2  ρ0 L        √   Rj Rj  ≤ C ∇ P0 − P0  + C  ρ0 g ∗ j1 /Rj − ρ0 g   L2

R

+ C∥H0 j

L2

  1/2    R R R 1/2  − H0 ∥L2 ∇ H0 j − H0  2 ∥∇H0 j ∥L2 + ∥∇2 H0 j ∥L2 L

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

428

    R 1/2 1/2  + C∥H0 ∥L2 ∥∇2 H0 ∥L2 ∇ H0 j − H0  2 L      √ R R R j ρ0 j u ˜0 j − ρ0 u0 ∗ j1 /Rj  +C    ρ0

.

(4.15)

L2

Let Rj → ∞ in (4.15), it follows from (4.2), (4.3), (4.4) and (4.14) that lim ∥∇2 (˜ uR 0 − u0 )∥L2 (R2 ) = 0,

(4.16)

R→∞

which together with (4.14) yields (4.6). Then, in terms of Lemma 2.1, the initial–boundary-value problem (1.6)–(1.7) with the initial data R R R R R R R (ρR 0 , u0 , P0 , H0 ) has a classical solution (ρ , u , P , H ) on BR × [0, TR ]. Moreover, Proposition 3.1 shows that there exists a T0 independent of R such that (3.2) holds for (ρR , uR , P R , H R ). For simplicity, in what follows, we denote Lp = Lp (R2 ),

W k,p = W k,p (R2 ).

˜ R ) where Extending (ρR , uR , P R , H R ) by zero on R2 \ BR and denoting it by (˜ ρR , u ˜R , P˜ R , H P˜ R , ϕR P R ,

ρ˜R , ϕR ρR ,

with ϕR as in (3.38). First, we have from (3.2) that     ˜˙ R ∥L2 + ∥˜ ρR x ¯a ∥L1 ∩Lq + ∥P˜ R ∥L1 ∩Lq ˜R ∥L2 + ∥ ρ˜R u sup ∥ ρ˜R u 0≤t≤T0

≤ sup 0≤t≤T0

      ∥ ρR uR ∥L2 (BR ) + ∥ ρR u˙ R ∥L2 (BR ) + sup ∥ρR x ¯a ∥L1 (BR )∩Lq (BR ) + ∥P R ∥L1 (BR )∩Lq (BR ) 0≤t≤T0

≤ C.

(4.17)

Similarly, it follows from (3.2) that   ˜ R ∥H 1 + ∥ |H ˜ R | |∇H ˜ R | ∥L2 + ∥H ˜ tR ∥L2 + ∥∇2 H ˜ R ∥L2 sup ∥∇˜ uR ∥H 1 + ∥¯ xa/2 H 0≤t≤T0



T0

+



 ˜ R ∥2 2 + ∥ |H ˜ R | |∆H ˜ R | ∥2 2 dt ˜˙ R ∥2L2 + ∥∇H ∥∇u t L L



 R 2 2 ˜R 2 ˜ R ∥2 2 + ∥∇2 u ∥¯ xa/2 ∆H ˜ ∥ + ∥∇ H ∥ dt ≤ C. 2 q 2 q L L ∩L L ∩L

0

 +

T0

(4.18)

0

Next, for p ∈ [2, q], it follows from (3.2), (3.38) and (3.37) that     sup ∥∇(˜ ρR x ¯a )∥Lp + ∥∇(P˜ R )∥Lp ≤ C sup ∥∇(ρR x ¯a )∥Lp (BR ) + R−1 ∥ρR x ¯a ∥Lp (BR ) 0≤t≤T0

0≤t≤T0



+ C sup 0≤t≤T0

≤ C sup 0≤t≤T0



∥∇(P R )∥Lp (BR ) + R−1 ∥P R ∥Lp (BR )



∥ρR x ¯a ∥H 1 ∩W 1,p (BR ) + ∥P R ∥H 1 ∩W 1,p (BR )

≤ C.

 (4.19)

This together with (3.45) and (3.2) yields that  T0  T0 R 2 ∥¯ xρ˜t ∥Lp dt ≤ C ∥¯ x|uR | |∇ρR |∥2Lp (BR ) dt 0

0

 ≤C 0

≤ C.

T0

∥¯ x1−a uR ∥2L∞ (BR ) ∥¯ xa ∇ρR ∥2Lp (BR ) dt (4.20)

L. Lu, B. Huang / Nonlinear Analysis: Real World Applications 31 (2016) 409–430

429

Moreover, one derives from (1.5), (4.18), (4.19) and Gagliardo–Nirenberg inequality that  T0  T0  T0  T0 R 2 R 4 R 4 ˜ ˜ ∥Pt ∥Lp dt ≤ C ∥∇H ∥L2p dt + C ∥P˜ R div˜ ∥∇˜ u ∥L2p dt + C uR ∥2Lp dt 0

0

0

0

T0





≤C

∥∇˜ uR ∥4L2 + ∥∇˜ uR ∥2L2 ∥∇2 u ˜R ∥2L2 dt 

0



T0



+C 0 T0

 +

 ˜ R ∥4 2 + ∥∇H ˜ R ∥2 2 ∥∇2 H ˜ R ∥2 2 dt ∥∇H L L L

  ∥P˜ R ∥2L∞ ∥∇˜ uR ∥2L2 + ∥∇2 u ˜R ∥2L2 dt

0

≤ C.

(4.21)

˜ R ) converges, up to With the estimates (4.17)–(4.21) at hand, we find that the sequence (˜ ρR , u ˜R , P˜ R , H the extraction of subsequences, to some limit (ρ, u, P, H) in the obvious weak sense, that is, as R → ∞, we have x ¯ρ˜R → x ¯ρ, a R

a

x ¯ ρ˜ ⇀ x ¯ ρ,

P˜ R → P, in C(BN × [0, T0 ]), for any N > 0, P˜ R ⇀ P, weakly * in L∞ (0, T0 ; H 1 ∩ W 1,q ),

˜ Rx H ¯a/2 ⇀ H x ¯a/2 ,  √ ρ˜R u ˜R ⇀ ρu, 2

˜R

∇ H

2

⇀ ∇ H,

R

˜˙ ⇀ ∇u, ∇u ˙ 2 R

2

∇ u ˜ ⇀ ∇ u,

∞

R

(4.22) (4.23)

1

∇˜ u ⇀ ∇u, weakly * in L (0, T0 ; H ),  ˜˙ R ⇀ √ρu, ρ˜R u ˙ weakly * in L∞ (0, T0 ; L2 ), ˜ tR H

⇀ Ht ,

∞

(4.24) (4.25)

2

weakly * in L (0, T0 ; L ),

(4.26)

˜ tR ⇀ ∇Ht , ˜ Rx ∇H ∆H ¯a/2 ⇀ ∆H x ¯a/2 , weakly in L2 (R2 × (0, T0 )), ˜ R ⇀ ∇2 H R , weakly in L2 (0, T0 ; Lq ), ∇2 H

(4.27) (4.28)

with x ¯a ρ, P ∈ L∞ (0, T0 ; L1 ),

 ρ(x, t)dx ≥

inf

0≤t≤T0

B2N0

1 . 4

(4.29)

Then letting R → ∞, some standard arguments together with (4.22)–(4.29) show that (ρ, u, P, H) is a strong solution of (1.6)–(1.7) on R2 × (0, T0 ] satisfying (1.11) and (1.12). The proof of Theorem 1.1 is completed.  References [1] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid. II, Proc. Japan Acad. Ser. A Math. Sci. 62 (5) (1986) 181–184. [2] S. Kawashima, Y. Shizuta, Magnetohydrodynamic approximation of the complete equations for an electromagnetic fluid, Tsukuba J. Math. 10 (1) (1986) 131–149. [3] S. Jiang, F.C. Li, Convergence of the complete electromagnetic fluid system to the full compressible magnetohydrodynamic equations, arXiv:1309.3668. [4] 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. [5] J.S. Fan, W.H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Anal. 69 (2008) 3637–3660. [6] J.S. Fan, W.H. Yu, Strong solution to the compressible MHD equations with vacuum, Nonlinear Anal. RWA 10 (2009) 392–409. [7] X.P. Hu, D.H. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. 283 (2008) 255–284. [8] X.P. Hu, D.H. Wang, Global existence and large-time behavior of solutions to the threedimensional equations of compressible magnetohydrodynamic flows, Arch. Ration. Mech. Anal. 197 (2010) 203–238. [9] S. Kawashima, Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics (Ph.D. thesis), Kyoto University, 1984.

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