Global variational solutions to the compressible magnetohydrodynamic equations

Nonlinear Analysis 69 (2008) 3637–3660 www.elsevier.com/locate/na

Global variational solutions to the compressible magnetohydrodynamic equationsI Jishan Fan a,b,∗ , Wanghui Yu b a College of Information Sciences and Technology, Nanjing Forestry University, Nanjing 210037, PR China b Department of Mathematics, Suzhou University, Suzhou 215006, PR China

Received 29 May 2006; accepted 1 October 2007

Abstract We prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic (MHD) equations with the coefficients depending on the temperature. As a by-product, we give a simple proof for the nonexistence of nontrivial weak time-periodic solutions by the entropy principle of Clausius–Duhem and a new Poincar´e-type inequality. c 2007 Elsevier Ltd. All rights reserved.

MSC: 35B40; 76X05 Keywords: Magnetohydrodynamics (MHD); Weak solutions with a defect measure; Time-periodic solutions

1. Introduction Magnetohydrodynamics (MHD) is concerned with the study of the interaction between magnetic fields and fluid conductors of electricity. The applications of magnetohydrodynamics cover a very wide range of physical objects, from liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics, and plasma physics. The equations of magnetohydrodynamic flows have the following form [1–5]: ∂t ρ + div(ρu) = 0,

(1.1)

∂t (ρu) + div(ρu ⊗ u) + ∇ p − curl H × H = div S + ρ f, 1 ∂t (ρe) + div(ρeu) + div q = S : ∇u − p div u + (curl H )2 + ρg, σ   1 curl H , ∂t H − curl(u × H ) = −curl σ

(1.2)

div H = 0, I Supported by the NSFC Grant No. 10301014. ∗ Corresponding author at: College of Information Sciences and Technology, Nanjing Forestry University, Nanjing 210037, PR China.

E-mail addresses: [email protected] (J. Fan), [email protected] (W. Yu). c 2007 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2007.10.005

(1.3) (1.4) (1.5)

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

where the fluid occupies a bounded spatial domain Ω ⊆ R N . The symbol S denotes the viscous stress tensor, e is the specific internal energy, q stands for the heat flux, σ is the electronic conductivity, and f := f (t, x), g := g(t, x) are given functions representing the external force density and the heat source, respectively. We shall suppose that p := p(ρ, θ ),

e := e(ρ, θ ),

σ := σ (ρ, θ ),

where θ := θ (t, x) is the absolute temperature. Moreover, we shall assume that the fluid is Newtonian, that means, the viscous stress tensor S is given through formula S := µ(∇u + ∇u T ) + λ div uI,

(1.6)

where µ and λ are the viscosity coefficients depending, in general, on the absolute temperature θ. In accordance with the second law of thermodynamics, we assume that µ(θ ) ≥ µ > 0,

λ(θ ) +

2 µ(θ ) ≥ 0. N

(1.7)

For the sake of simplicity, we impose the following boundary conditions on the velocity u, the temperature θ, and the magnetic field H , respectively. u|∂ Ω = 0, q · n|∂ Ω = 0, H · n|∂ Ω = 0,

(1.8) curl H × n|∂ Ω = 0,

(1.9) (1.10)

where n is the outward normal vector to the boundary ∂Ω . As for the energy (heat) flux q, we adopt the standard Fourier law: q = −k(θ )∇θ, with the heat conductivity coefficient k > 0. The pressure p := p(ρ, θ ) will be determined through a general constitutive equation: p := p(ρ, θ ) = pe (ρ) + θ pθ (ρ)

(1.11)

for certain functions pe , pθ ∈ C[0, ∞)∩C 1 (0, ∞). In order words, the pressure is an affine function of the temperature for a given value of ρ. The constitutive relation (1.11) implies that the specific internal energy e can be decomposed as a sum: e(ρ, θ ) = Pe (ρ) + Q(θ );

(1.12)

that is, the elastic part Pe , attributed to the action of intermolecular forces, and the thermal part Q, related to the random translational motion of the molecules, are separated and contribute to e in an additive way. In accordance with the basic principles of classical thermodynamics, e and p are interrelated through Maxwell’s relationship:   ∂e 1 ∂p ∂e ∂Q = 2 p−θ , = = C V (θ ), ∂ρ ∂θ ∂θ ∂θ ρ that means, hypothesis (1.12) is in fact equivalent to (1.11). Here, the symbol C V denotes the specific heat at constant volume. Moreover, we have Z ρ pe (z) Pe (ρ) := dz. (1.13) z2 1 Multiplying Eq. (1.1) by (ρ Pe (ρ))0 yields ∂t (ρ Pe (ρ)) + div(ρ Pe (ρ)u) + pe (ρ)div u = 0,

(1.14)

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3639

and consequently, (1.3) can be written as the thermal energy equation ∂t (ρ Q(θ )) + div(ρ Q(θ )u) − div(k(θ )∇θ ) = S : ∇u +

1 (curl H )2 − θ pθ (ρ)div u + ρg. σ

(1.15)

Furthermore, multiplying (1.15) by h(θ ) for a suitable function h, we obtain   1 ∂t (ρ Q h (θ )) + div(ρ Q h (θ )u) − 1Kh (θ ) = h(θ ) S : ∇u + (curl H )2 σ − h 0 (θ )k(θ )|∇θ|2 − h(θ )θ pθ (ρ)div u + h(θ )ρg,

(1.16)

with Q h (θ ) :=

Z

θ

Q 0 (z)h(z)dz,

Kh (θ ) :=

θ

Z

k(z)h(z)dz, 1

1

which can be considered as a renormalized version of (1.15). In order to deal, at least formally, with a well-posed problem, the systems (1.1), (1.2), (1.4), (1.5) and (1.15) with the boundary conditions (1.8)–(1.10) must be supplemented by the initial conditions: ρ(0, ·) = ρ0 ,

(1.17)

(ρu)(0, ·) = m 0 ,

(1.18)

(ρ Q(θ ))(0, ·) = ρ0 Q(θ0 ),

(1.19)

H (0, ·) = H0 ,

(1.20)

and

with given functions ρ0 , m 0 , θ0 , and H0 defined on Ω . The Eqs. (1.1), (1.2), (1.4), (1.5) and (1.15) together with the boundary conditions (1.8)–(1.10), and the initial conditions (1.17)–(1.20) represent the problem (MHD) we shall deal with. Similar to Chapter 4 in [6], the thermal energy equation (1.15) is replaced by two inequalities: ∂t (ρ Q(θ )) + div (ρ Q(θ )u) − 1K(θ ) ≥ Φ − θ pθ (ρ)div u + ρg,

(1.21)

where K(θ ) :=

θ

Z

Φ := S : ∇u +

k(z)dz, 0

1 (curl H )2 ; σ

and E[ρ, u, θ, H ](t) ≤ E[ρ, u, θ, H ](0) +

Z tZ 0

Ω

ρ f u + ρgdxdτ

for t ≥ 0,

(1.22)

with the total energy E[ρ, u, θ, H ] =

Z Ω

ρ



 1 2 1 u + Pe (ρ) + Q(θ ) + H 2 dx, 2 2

where the elastic potential Pe (ρ) is given by (1.13). As already observed in Chapter 4 in [6], inequalities (1.21) and (1.22) represent a suitable weak formulation of the thermal energy equation (1.15) in the sense that any smooth solution satisfying (1.21) and (1.22) together with (1.1), (1.2), (1.4), (1.5) and (1.8)–(1.10) solves (1.15). These ‘weak solutions with a defect measure’ were introduced by Diperna and Lions on the Fokker–Planck–Boltzmann equation [7]. Here we follow the presentation of Alexandre and Villani [8], Feireisl [6,9], Ducomet and Feireisl [10]. In accordance with our previous discussion, we shall say that (ρ, u, θ, H ) is a variational solution of the problem (MHD) on a time interval (0, T ) if the following conditions hold:

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

• the density ρ is a non-negative function satisfying the integral identity Z TZ ρ∂t φ + ρu · ∇φdxdt = 0, 0

for any φ ∈ C ∞ ([0, T ] × Ω ), φ(0) = φ(T ) = 0; • the velocity u ∈ L 2 (0, T ; H01 (Ω )), the momentum equation (1.2) holds in D0 ((0, T ) × Ω ), that means, Z TZ 1 ρu∂t φ + ρu ⊗ u : ∇φ + pdiv φ − H ⊗ H : ∇φ + H 2 div φdxdt 2 0 Ω Z TZ = S : ∇φ − ρ f φdxdt, 0

(1.24)

Ω

for any φ ∈ D((0, T ) × Ω ); • the temperature θ is a non-negative function satisfying Z TZ Z ρ Q(θ )∂t φ + ρ Q(θ )u · ∇φ + K(θ )1φdxdt ≤ 0

(1.23)

Ω

Ω

T 0

Z Ω

(θ pθ div u − Φ − ρg)φdxdt

(1.25)

for any φ ∈ C ∞ ([0, T ] × Ω ), φ ≥ 0, φ(0) = φ(T ) = 0, ∇φ · n|∂ Ω = 0; • the energy inequality (1.22) holds for a.e. t ∈ (0, T ), with Z 1 1 |m 0 |2 E[ρ, u, θ, H ](0) := + ρ0 Pe (ρ0 ) + ρ0 Q(θ0 ) + H02 dx; 2 ρ 2 0 Ω • we have S ∈ L 2 ((0, T ) × Ω ), ρS = ρ[µ(θ )(∇u + ∇u T ) + λ(θ )div uI],   1 2 ρΦ = ρ S : ∇u + (curl H ) , Φ = 0 on {ρ = 0}; σ

(1.26) (1.27)

• the magnetic field H ∈ V2 ((0, T ) × Ω ) := L ∞ (0, T ; L 2 (Ω )) ∩ L 2 (0, T ; H 1 (Ω )) satisfying Z TZ H ∇φdxdt = 0 0

Ω

for any φ ∈ C ∞ ([0, T ] × Ω ) and Z TZ 1 H ∂t φ + (u × H ) · curl φ − curl H · curl φdxdt = 0 σ 0 Ω for any φ ∈ C ∞ ([0, T ] × Ω ), φ(0) = φ(T ) = 0, φ · n|∂ Ω = curl φ × n|∂ Ω = 0; • the functions ρ, ρu, ρ Q(θ ) and H satisfy the initial conditions (1.17)–(1.20), more specifically, Z Z ess lim ρ(t)ηdx = ρ0 ηdx, t→0+ Ω Ω Z Z ess lim (ρu)(t) · ηdx = m 0 · ηdx, t→0+ Ω ΩZ Z ess lim (ρ Q(θ ))(t)ηdx = ρ0 Q(θ0 )ηdx, t→0+ Ω Ω Z Z ess lim H (t)ηdx = H0 ηdx, t→0+ Ω

Ω

for any η ∈ D(Ω ). Now we are in a position to state our MAIN RESULT of this paper.

(1.28)

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

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Theorem 1.1. Let Ω ∈ R N , N = 2, 3 be a bounded domain of class C 2+ν , ν > 0. Assume that the pressure p is given by the constitutive equation (1.11), where the components pe , pθ are continuously differentiable on (0, ∞), and ) pe (0) = 0, pe0 (ρ) ≥ a1 ρ γ −1 − C for all ρ > 0, (1.29) pe (ρ) ≤ a2 ρ γ + C for all ρ ≥ 0, ) pθ (0) = 0, pθ0 (ρ) ≥ 0 for all ρ > 0, (1.30) pθ (ρ) ≤ C(1 + ρ Γ ), where N , 2 γ γ Γ < for N = 2, Γ = 2 3 and a1 , a2 , C are positive constants. γ >

(1.31) for N = 3,

(1.32)

Let k := k(θ ) be continuously differentiable on [0, ∞) such that k(1 + θ α ) ≤ k(θ ) ≤ k(1 + θ α ), k > 0

(1.33)

for a certain α ≥ 2. Furthermore, let the viscosity coefficients µ and λ be continuously differentiable functions of θ satisfying (1.7) together with the growth restriction 0 < µ ≤ µ(θ ) ≤ µ, µ00 µ ≥ 3(µ0 )2 ,



|λ(θ )| ≤ λ, µ, λ globally Lipschitz on [0, ∞),  00  2 2 2 2 λ + µ ≥ 3 λ0 + µ0 . λ+ µ N N N

(1.34)

Similarly, we require the existence of two constants C V , C V so that Q(0) = 0,

0 < C V ≤ Q 0 (θ ) ≤ C V ,

and σ := σ (ρ, θ ) is continuously differentiable on [0, ∞) × [0, ∞) such that  0 < σ ≤ σ (ρ, θ ) ≤ σ .   and the following matrix issemi-positive definite, i.e.,           h  h h      σ ρρ  σ ρθ σ ρ           h h h      ≥ 0,    σ θθ σ θ   σ ρθ            h h 1h     σ ρ σ θ 4σ

(1.35)

(1.36)

where h := h(θ ) satisfying (1.46) below. Finally, assume that f and g are bounded measurable functions on (0, T ) × Ω , g ≥ 0, and that the initial data satisfy  ρ0 ∈ L γ (Ω ), ρ0 ≥ 0 on Ω ,    |m 0 |2  1 ∈ L (Ω ), (1.37) ρ0   θ0 ∈ L ∞ (Ω ), θ0 ≥ θ > 0,    H0 ∈ L 2 (Ω ).

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

The problem (MHD) possesses at least one variational solution ρ, u, θ, H on the time interval (0, T ) such that ρ ∈ L ∞ (0, T ; L γ (Ω )) ∩ C([0, T ]; L 1 (Ω )), u ∈ L (0, T ; 2

H01 (Ω )),

ρu ∈

(1.38)

2γ /(γ +1) C([0, T ]; L weak (Ω )),

(1.39)

S ∈ L 2 ((0, T ) × Ω ), θ∈L

α+1

(1.40)

((0, T ) × Ω ),

ρ Q(θ ) ∈ L (0, T ; L (Ω )), 1

∞

(1.41)

H ∈ L ∞ (0, T ; L 2 (Ω )) ∩ L 2 (0, T ; H 1 (Ω )) ∩ C([0, T ]; L 2weak (Ω )),

(1.42)

θ pθ ∈ L 2 ((0, T ) × Ω ),

(1.43)

and ρ Q(θ )u ∈ L 1 ((0, T ) × Ω ).

In addition, the solutions constructed in Theorem 1.1 will satisfy the continuity equation (1.1) in the sense of the renormalized solutions introduced by Diperna and Lions [11], that is, the integral identity Z TZ b(ρ)∂t φ + b(ρ)u · ∇φ + (b(ρ) − b0 (ρ)ρ)div uφdxdt = 0 (1.44) 0

Ω

holds for any b ∈ C 1 [0, ∞),

b0 (ρ) = 0,

for all ρ large enough,

and any test function φ ∈ C ∞ ([0, T ] × Ω ),

φ(0) = φ(T ) = 0.

Furthermore, when µ, λ and σ are constants, the temperature θ will be a renormalized solution [11,6] especially θ will satisfy Z TZ ρ Q h (θ )∂t φ + ρ Q h (θ )u · ∇φ + Kh (θ )1φdxdt 0 Ω   Z TZ 1 2 ≤ h(θ ) θ pθ (ρ)div u − S : ∇u − (curl H ) φdxdt σ 0 Ω Z TZ Z Z TZ 0 2 h (θ )k(θ )|∇θ| φdxdt − φh(θ )ρgdxdt − ρ0 Q h (θ0 )φ(0)dx, (1.45) + 0

Ω

0

Ω

Ω

and holds for any h satisfying: h ∈ C 2 [0, ∞),

h(0) = 1,

h 00 (z)h(z) ≥ 3(h 0 (z))2

h non-increasing on [0, ∞),

lim h(z) = 0,

z→∞

for all z ≥ 0.

(1.46)

and any test function φ satisfying: φ ≥ 0, φ ∈ W 2,∞ ((0, T ) × Ω ),

∇φ · n|∂ Ω = 0,

support [φ] ⊂ [0, T ) × Ω .

(1.47)

The growth restrictions imposed on k, µ, λ, Q and σ are definitely not optimal, but on the other hand, it is known that γ > N /2 is a critical condition even in the isentropic case. The existence of local smooth solution to the problem (MHD) has been proved in [12] and [13]. When N = 1, the existence, the uniqueness and the long-time behavior of strong solutions to the problem (MHD) have been studied by several authors, see e.g., [14–19]. When H ≡ 0, a similar existence result as Theorem 1.1 was proved by Feireisl [6,9] and Lions [20]. The existence of time-periodic weak solutions was proved in [21] for the baratropic flow. However, it is open up to now whether there exists a time-periodic solutions under the time-periodic external forces and/or time-periodic heat sources. In this paper, we will prove the nonexistence of nontrivial time-periodic variational solutions to the problem (MHD) which contains the H ≡ 0 case. This interesting result follows from the entropy principle of Clausius–Duhem.

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

3643

The proofs in [6,9] are done by a three-level approximation scheme based on a regularized system, that is, simply adding a viscous term ε1ρ on the right-hand side of the continuity equation (1.1). The most difficult part is to prove that the strong convergence of the densities ρε → ρ in L 1 ((0, T ) × Ω ) by the same techniques used in [20,22–25] for the isentropic case. The proofs of the strong convergence of the temperature θε → θ in L 1 ((0, T ) × Ω ) are easier, based on the strong convergence of the densities and the concept of renormalized limit for sequences of non-negative functions which are merely bounded in L 1 (O), where O ⊂ R M is a bounded domain. To prove Theorem 1.1, our method is slightly different from that in [6,9]. Firstly, we regularize the initial data, thus there exists a local smooth solution (ρε , u ε , θε , Hε ) by [12,13]. Then we try to obtain the uniform a priori estimates independent of ε in a slightly different way in [6,9]. Based on this, we use the above mentioned weak convergence methods used in [6,9,20,22–25] to prove the strong convergence of the densities and the temperature. As a by-product, we prove the nonexistence of nontrivial time-periodic solutions. The paper is organized as follows. In Section 2, we get the a priori estimates. In Section 3, convergence of solutions is proved. In Section 4, we prove the nonexistence of nontrivial time-periodic solutions. 2. Proof of Theorem 1.1—Estimates Let the initial data (ρ0ε , u 0ε , θ0ε , H0ε ) ∈ C 2+ν (Ω ) satisfy ρ0ε ≥ ε > 0, u 0ε |∂ Ω = 0, θ0ε ≥ ε, θ0ε * θ0

strongly in L γ (Ω ),

ρ0ε → ρ0

θ0ε

(2.1)

m2 (ρ0ε u 0ε )2 → 0 strongly in L 1 (Ω ), ρ0ε ρ0 → θ0 strongly in L p (Ω ) (∀ p > 1),

weakly in L ∞ (Ω ),

ρ0ε Q(θ0ε ) → ρ0 Q(θ0 )

H0ε · n|∂ Ω = curl H0ε × n|∂ Ω = 0,

H0ε → H0

(2.2)

strongly in L 1 (Ω ),

strongly in L (Ω ), 2

(2.3) (2.4)

Similarly, we take f ε (x, t), gε (x, t) such that f ε ∈ C ∞ ([0, T ] × Ω ), fε * f gε * g

gε ∈ C ∞ ([0, T ] × Ω ),

weakly in L ((0, T ) × Ω ), weakly∗ in L ∞ ((0, T ) × Ω ), ∗

∞

gε ≥ 0,

strongly in L p ((0, T ) × Ω ) for any p > 1, strongly in L p ((0, T ) × Ω ) for any p > 1.

(2.5) (2.6)

Based on the results in [12,13], there exists a unique smooth solution (ρε , u ε , θε , Hε ) such that ρε ≥

ε , 2

θε ≥

ε 2

on [0, T ∗ ] × Ω

(2.7)

for some small time T ∗ := T ∗ (ε). In the following calculations, we will prove some uniform estimates of solutions (ρε , u ε , θε , Hε ) independent of ε and T ∗ . Since the solutions are smooth, we firstly give the well-known equations which can be derived from (1.1)–(1.5).        1 2 1 1 2 1 2 ρe + ρu + p u − Su + div curl H × H − (u × H ) × H ∂t ρe + ρu + H + div 2 2 2 σ = div(k∇θ ) + ρ f u + ρg.

(2.8)

Using (1.1), (1.11), (1.12), (1.29)–(1.32) and (2.8), we see that Lemma 2.1. (2.9)

kρε k L ∞ (0,T ;L γ (Ω )) ≤ C, √ k ρε u ε k L ∞ (0,T ;L 2 (Ω )) ≤ C,

(2.10)

kρε Q(θε )k L ∞ (0,T ;L 1 (Ω )) ≤ C,

(2.11)

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

kHε k L ∞ (0,T ;L 2 (Ω )) ≤ C,

(2.12)

k∂t ρε k L ∞ (0,T ;W −1,2γ /(γ +1) (Ω )) ≤ C.

(2.13)

Here and later on, the symbol C will denote various constants independent of ε and T ∗ . Proof. Integrating (1.1) over (0, t) × Ω immediately leads to Z Z ρ0ε dx ≤ C. ρε dx = Ω

(2.14)

Ω

Integrating (2.8) over Ω and using the boundary conditions (1.8)–(1.10), we get Z Z Z d 1 2 1 2 √ √ ρe + ρu + H dx = ρ · ρu · f dx + ρg dt Ω 2 2 Ω Ω √ √ ≤ k ρkk ρukk f k L ∞ (Q T ) + kρk L 1 (Ω ) kgk L ∞ (Q T ) which gives (2.9)–(2.12) by the Gronwall’s inequality. (2.13) follows from (1.1), (2.9) and (2.10).  Multiplying (1.15) by θ1 and integrating by parts, we obtain:   Z Z 1 0 Z tZ Z tZ k(θ ) 1 1 Q (z) 2 dzdx S : ∇u + (curl H )2 + ρg dxds + (∇θ ) dxds + ρ 2 σ z 0 Ω θ 0 Ω θ Ω ∩{θ ≤1} θ Z Z Z θ 0 Z θ0 0 Q (z) Q (z) dzdx − dzdx. = ρ ρ0 z z Ω Ω ∩{θ>1} 1 1

(2.15)

From (2.15) it follows that Lemma 2.2. kρε log θε k L ∞ (0,T ;L 1 (Ω )) ≤ C,

(2.16)

k∇ log θε k L 2 ((0,T )×Ω ) ≤ C,

(2.17)

k∇θεα/2 k L 2 ((0,T )×Ω ) ≤ C.

(2.18)

At this stage, we shall need the following elementary Poincar´e-type inequality which is slightly generalized from Lions [20], Feireisl [6] and Padula [32]. Lemma 2.3. Let ρ be a non-negative function such that Z Z 2N 0
when N ≥ 3,

and

kρkH ≤ k

when N = 2.

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

Proof. When N ≥ 3, by H¨older and Sobolev–Poincar´e’s inequality, it is easy to compute that



Z Z

1

1

vdx + vdx kvk ≤ v −

|Ω | Ω |Ω | Ω R   R R Ω ρ v − |Ω1 | Ω vdx dx − Ω ρvdx R ≤ C(Ω )k∇vk + Ω ρdx R C(Ω )kρk L γ k∇vk + Ω ρvdx R ≤ C(Ω )k∇vk + Ω ρdx R   ρvdx kρk L γ Ω R k∇vk. = + C(Ω ) 1 + kρk L 1 Ω ρdx When N = 2, we have



Z Z

1

1

+

v − kvk ≤ vdx vdx



|Ω | Ω |Ω | Ω R   R R Ω ρ v − |Ω1 | Ω vdx dx − Ω ρvdx R ≤ C(Ω )k∇vk + Ω ρdx R C(Ω )kρkH kvk B M O + Ω ρvdx R ≤ C(Ω )k∇vk + Ω ρdx R   ρvdx kρkH Ω R ≤ k∇vk.  + C(Ω ) 1 + kρk L 1 Ω ρdx Corollary 2.1. R

 

γ ρvdx

≤ C(Ω ) 1 + kρk L

v − RΩ k∇vk,

kρk L 1 Ω ρdx

N ≥3

and R

 

ρvdx

v − RΩ

≤ C(Ω ) 1 + kρkH k∇vk,

kρk L 1 Ω ρdx Proof. Setting w := v −

R ρvdx RΩ . Ω ρdx

N = 2.

Then applying Lemma 2.3 to w, and thus the conclusion follows.



By virtue of Lemma 2.3, the estimates (2.16)–(2.18) yield Lemma 2.4. k log θε k L 2 ((0,T )×Ω ) ≤ C, kθεα/2 k L 2 (0,T ;H 1 (Ω )) ≤ C.

(2.19) 

(2.20)

Moreover, the estimates (2.9) and (2.20), and the hypothesis (1.32) give Lemma 2.5. kθε pθ (ρε )k L 2 ((0,T )×Ω ) ≤ C.



(2.21)

Integrating (1.15) over (0, T ) × Ω and using (2.11) and the following inequalities [20,26]: kH k ≤ Ckcurl H k, we infer that

kH k H 1 ≤ Ckcurl H k,

(2.22)

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

Lemma 2.6. ku ε k L 2 (0,T ;H 1 (Ω )) ≤ C,

(2.23)

kHε k L 2 (0,T ;H 1 (Ω )) ≤ C,

(2.24)

0

0

k∂t Hε k L 2 (0,T ;W −1,3/2 (Ω )) ≤ C.



(2.25)

Multiplying (1.15) by θε−δ and integrating by parts for any δ ∈ (0, 1), we see that Z TZ k(θε ) (∇θε )2 dxdt ≤ C 1+δ 0 Ω θε which yields kθε(α+1−δ)/2 k L 2 (0,T ;H 1 (Ω )) ≤ C

for any δ ∈ (0, 1).

(2.26)

Thus one can use H¨older’s inequality in order to conclude that kθε1+α+1/N k L 1 (O) ≤ kθε1+α−1/N k L N /(N −2) (O) kθε2/N k L N /2 (O) 2/N

≤ kθε1+α−1/N k L N /(N −2) (Ω ) kθε k L 1 (O) , for any O ⊂ Ω which, together with (2.9) and (2.26), yields Z TZ θε1+α+1/N dxdt ≤ (c/d)2/N , 0

(2.27)

Ω ∩{ρε (t)≥d}

for any d > 0 and c is independent of d, ε and T ∗ . Now we have Z Z ρε (t)dx ≥ ρε (t)dx − d|Ω |. {ρε (t)≥d}

Ω

Here we can take Z Z Z 1 M ρε (t)dx = ρ0ε dx ≥ ρ0 dx =: . 2 2 Ω Ω Ω On the other hand, by virtue of H¨older’s inequality, Z ρε (t)dx ≤ (mes{ρε (t) ≥ d})(γ −1)/γ kρε k L ∞ (0,T ;L γ (Ω )) ≤ C (mes{ρε (t) ≥ d})(γ −1)/γ . {ρε (t)≥d}

Consequently, mes{ρε (t) ≥ d} ≥



M/4 C

γ /(γ −1) if d ≤

M . 4|Ω |

Similarly, mes{ρε (t) ≥ 2d} ≥



M/4 C

γ /(γ −1) if d ≤

M . 8|Ω |

(2.28)

In the following calculations, fix 0 < d ≤ 8|MΩ | , and take a function G ∈ C ∞ (R) such that G:R→R

non-increasing, G(z) = 0 for z ≤ d, G(z) = −1 for z ≥ 2d.

For each t ∈ (0, T ), one can solve the Neumann problem [6]: Z 1 1η(t) = G(ρε (t)) − G(ρε (t))dx in Ω , ∇η · n|∂ Ω = 0, |Ω | Ω

Z Ω

ηdx = 0.

(2.29)

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

3647

From [6], we know that kηk L ∞ ((0,T )×Ω ) ≤ C,

k∇ηk L ∞ ((0,T )×Ω ) ≤ C,

kηt k L 2 (0,T ;H 1 (Ω )) ≤ C.

(2.30)

and hence we can set η :=

inf [0,T ]×Ω

η.

Now multiplying (1.15) by η − η and integrating by parts, we find that Z

T



 Z 1 K(θε ) G(ρε ) − G(ρε )dx dxdt |Ω | Ω 0 Ω Z TZ Z TZ ρε Q(θε )∂t ηdxdt − ρε Q(θε )u ε · ∇ηdxdt =− Z

Ω

0

T

Z

Z

+ Ω

0 T

Z − 0

T

Z ≤−

T

Ω

ρε Q(θε )(η − η)dx

 Z 1 2 ρ0ε Q(θ0ε )(η0 − η)dx Sε : ∇u ε + (curl Hε ) + ρε g (η − η)dxdt − σ Ω Ω Z Z TZ ρε Q(θε )∂t ηdxdt − ρε Q(θε )u ε · ∇ηdxdt

Z Ω

0 4 X

Z

Z 

+ =:

Ω

θε pθ (ρε )div u ε (η − η)dxdt +

Ω

0

Z

0

0

Ω

θε pθ (ρε )div u ε (η − η)dxdt +

Z Ω

ρε Q(θε )(η − η)dx

Ii .

i=1

Each term Ii can be bounded as follows I1 ≤ kρε k L ∞ (0,T ;L γ (Ω )) kQ(θε )k L 2 (0,T ;L 2N /(N −2) (Ω )) k∂t ηk L 2 (0,T ;L 2N /(N −2) (Ω )) ≤ C, I2 ≤ kρε k L ∞ (0,T ;L γ (Ω )) kQ(θε )k L 2 (0,T ;L 2N /(N −2) (Ω )) ku ε k L 2 (0,T ;L 2N /(N −2) (Ω )) k∇ηk L ∞ ((0,T )×Ω ) ≤ C, I3 ≤ kθε pθ (ρε )k L 2 ((0,T )×Ω ) kdiv u ε k L 2 ((0,T )×Ω ) kη − ηk L ∞ ((0,T )×Ω ) ≤ C, I4 ≤ kρε Q(θε )k L ∞ (0,T ;L 1 (Ω )) kη − ηk L ∞ ((0,T )×Ω ) ≤ C. This proves that the right-hand side of (2.31) is bounded above by C. The left-hand side of (2.31) can be written as   Z TZ Z 1 K(θε ) G(ρε ) − G(ρε )dx dxdt |Ω | Ω 0 Ω   Z TZ Z 1 = K(θε ) G(ρε ) − G(ρε )dx dxdt |Ω | Ω 0 Ω ∩{ρε (t)0 − G(ρε )dx ≥ − G(ρε )dx = |Ω | Ω |Ω | {ρε (t)≥2d} |Ω | |Ω | 4C by (2.28). Consequently,

(2.31)

3648

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

Z

T

Z

T

Z

  Z 1 G(ρε )dx dxdt K(θε ) G(ρε ) − |Ω | Ω 0 Ω ∩{ρε (t)
whence Z

Ω ∩{ρε (t)
0

K(θε )dxdt ≤ C.

(2.32)

Combining (2.27) and (2.32) gives the key estimates on the temperature. Lemma 2.7. (2.33)

kθε k L α+1 ((0,T )×Ω ) ≤ C. Now we are in a position to show the key estimates on the density. *

To this end, as in [6,21,22,25], we introduce the operator B := [B1 , B2 , B3 ] enjoying the properties: • *

Z  h i3 1, p B : g ∈ L (Ω ) gdx = 0 7→ W0 (Ω ) 

p

Ω

is a bounded linear operator, i.e., *

k B [g]kW 1, p (Ω ) ≤ C( p)kgk L p (Ω ) 0

for any 1 < p < ∞;

*

*

• the function v := B [g] solves the problem *

div v = g

*

v |∂ Ω = 0;

in Ω ,

 3 * * * • if, moreover, g can be written in the form g = div h for a certain h ∈ L p (Ω ) , h ·n|∂ Ω = 0, then *

*

k B [g]k L p (Ω ) ≤ C( p)k h k L p (Ω ) for arbitrary 1 < p < ∞. *

The operator B was introduced by Bogovskii [27]. A complete proof of the above mentioned properties may be found in Galdi [28, Theorem 3.3] or Borchers and Sohr [29, Proof of Theorem 2.4]. Let ω be a positive constant which will be determined later, then multiplying the momentum equation (1.2) by R * B [ρεω − |Ω1 | Ω ρεω dx] and integrating by parts and using the simple identity:   1 curl H × H = div H ⊗ H − H 2 I . 2 We obtain Z TZ 0

Ω

( pe (ρε ) + θε pθ (ρε ))ρεω dxdt = T

Z + 0

T

Z + 0

*

T

Z 0

Z Ω

( pe (ρε ) + θε pθ (ρε ))dx ·

 Z 1 ω ω Sε : ∇ B ρε − ρ dx dxdt |Ω | Ω ε Ω     Z Z * 1 1 2 ω ω Hε ⊗ Hε − Hε I : ∇ B ρε − ρ dx dxdt 2 |Ω | Ω ε Ω Z



1 |Ω |

Z Ω

 ρεω dx dt

3649

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660 T

  Z * 1 ρεω dx dxdt ρε u ε ⊗ u ε : ∇ B ρεω − |Ω | Ω 0 Ω     Z Z Z Z * * 1 1 ω ω ρ0ε u 0ε · B ρ0ε − ρε u ε · B ρεω − ρεω dx dx − ρ0ε dx dx + |Ω | Ω |Ω | Ω Ω Ω   Z TZ Z Z TZ * * 1 ρε u ε B [div(ρεω u ε )]dxdt ρεω dx dxdt + ρε f ε B ρεω − − |Ω | 0 Ω Ω 0 Ω   Z TZ Z * 1 ω ω ρε u ε B ρε div u ε − + (ω − 1) ρ div u ε dx dxdt |Ω | Ω ε 0 Ω Z

Z

−

=:

9 X

Ji .

(2.34)

i=1 *

By the H¨older, Young and Sobolev inequalities and the properties of the operator B , each term Ji can be bounded as follows |J1 | ≤ C

if ω ≤ γ ,

  Z

* ω 1 ω

∇ ρ − dx |J2 | ≤ kSε k L 2 ((0,T )×Ω ) ρ B ε ε

2

|Ω | Ω L ((0,T )×Ω )

Z

ω 1 ω

≤ C ρε − ρ dx |Ω | Ω ε L 2 ((0,T )×Ω ) Z TZ ≤ ε0 ρεγ +ω dxdt + C(ε0 ) for any positive constant ε0 if ω ≤ γ , 0 Ω

  Z

* ω

1 3 2 ω

|J3 | ≤ kHε k L 2 (0,T ;L 2N /(N −2) (Ω )) ∇ B ρε − ρ dx

∞ 2 |Ω | Ω ε L (0,T ;L N /2 (Ω )) ω ω ≤ Ckρε k L ∞ (0,T ;L N /2 (Ω )) ≤ Ckρε k L ∞ (0,T ;L N ω/2 (Ω )) ≤ C if ω ≤ 2γ /N and N ≥ 3,

  Z

* ω 1 3 ω 2

ρε dx |J3 | ≤ kHε k L 2 (0,T ;L 2γ (Ω )) ∇ B ρε −

∞ 2 |Ω | γ /(γ −1) Ω

L (0,T ;L

(Ω ))

≤ Ckρεω k L ∞ (0,T ;L γ /(γ −1) (Ω )) ≤ Ckρε kωL ∞ (0,T ;L γ ω/(γ −1) (Ω )) ≤ C if ω ≤ γ − 1 and N = 2,

  Z

* ω 1 ω

ρ dx |J4 | ≤ kρε k L ∞ (0,T ;L γ (Ω )) ku ε k2L 2 (0,T ;L 2N /(N −2) (Ω )) ∇ ρ − ε

∞

B ε |Ω | Ω L (0,T ;L N γ /(2γ −N ) (Ω )) 2 N ≤ Ckρεω k L ∞ (0,T ;L N γ /(2γ −N ) (Ω )) ≤ C if ω ≤ γ − 1, γ > and N ≥ 3, N 2   Z

* ω

1 ω

∇ ρ − |J4 | ≤ kρε k L ∞ (0,T ;L γ (Ω )) ku ε k L 2 (0,T ;L 2γ /(γ −1−ω) (Ω )) ρ dx B ε ε

∞ |Ω | Ω L (0,T ;L γ /ω (Ω )) ω ≤ Ckρε k L ∞ (0,T ;L γ /ω (Ω )) ≤ C if ω < γ − 1 and N = 2,

  Z

* ω 1 √ √ ω

|J5 | ≤ k ρε k L ∞ (0,T ;L 2γ (Ω )) k ρε u ε k L ∞ (0,T ;L 2 (Ω )) B ρε − ρε dx

∞ |Ω | Ω L (0,T ;L 2γ /(γ −1) (Ω ))

  Z

* ω

1 ≤ C ρεω dx

B ρε − |Ω |

∞ 1,2N γ /((N +2)γ −N ) Ω

≤

L (0,T ;W

Ckρεω k L ∞ (0,T ;L 2N γ /((N +2)γ −N ) (Ω ))

Similarly, |J6 | ≤ C

if ω ≤

2 γ − 1 and N ≥ 2, N

≤C

(Ω ))

2 if ω ≤ γ − 1 and N ≥ 2. N

3650

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

  Z

* ω 1 ω

|J7 | ≤ kρε k L ∞ (0,T ;L γ (Ω )) k f ε k L ∞ ((0,T )×Ω ) B ρε − ρε dx

1 |Ω | Ω L (0,T ;L γ /(γ −1) (Ω )) 2 ω ≤ Ckρε k L 1 (0,T ;L N γ /(N (γ −1)+γ ) (Ω )) ≤ C if ω ≤ γ − 1 and N ≥ 2, N *

|J8 | ≤ kρε k L ∞ (0,T ;L γ (Ω )) ku ε k L 2 (0,T ;L 2N /(N −2) (Ω )) k B [div(ρεω u ε )]k L 2 (0,T ;L 2N γ /((N +2)γ −2N ) (Ω )) ≤ Cku ε k L 2 (0,T ;L 2N /(N −2) (Ω )) kρεω k L ∞ (0,T ;L N γ /(2γ −N ) (Ω )) 2 ≤ C if ω ≤ γ − 1 and N ≥ 3, N *

|J8 | ≤ kρε k L ∞ (0,T ;L γ (Ω )) ku ε k L 2 (0,T ;L 2γ /(γ −1−ω) (Ω )) k B [div(ρεω u ε )]k L 2 (0,T ;L 2γ /(γ −1+ω) (Ω )) ≤ Ckρεω k L ∞ (0,T ;L γ /ω (Ω )) ku ε k L 2 (0,T ;L 2γ /(γ −1−ω) (Ω )) ≤ C if ω < γ − 1 and N = 2, |J9 | ≤ |ω − 1|kρε k L ∞ (0,T ;L γ (Ω )) ku ε k L 2 (0,T ;L 2N /(N −2) (Ω ))   Z

* ω

1 ω

× B ρε div u ε − ρ div u ε dx

|Ω | Ω ε

  L 2 (0,T ;L 2N γ /((N +2)γ −2N ) (Ω )) Z

* ω 1 ρεω div u ε dx ≤ C

2

B ρε div u ε − |Ω | Ω L (0,T ;W 1,2N γ /((N +4)γ −2N ) (Ω )) ω ≤ Ckρε div u ε k L 2 (0,T ;L 2N γ /((N +4)γ −2N ) (Ω )) ≤ Ckdiv u ε k L 2 ((0,T )×Ω ) kρεω k L ∞ (0,T ;L N γ /(2γ −N ) (Ω )) 2 ≤ Ckρε kωL ∞ (0,T ;L ωN γ /(2γ −N ) (Ω )) ≤ C if ω ≤ γ − 1 and N ≥ 3, N |J9 | ≤ |ω − 1|kρε k L ∞ (0,T ;L γ (Ω )) ku ε k L 2 (0,T ;L γ /(γ −1−ω) (Ω ))   Z

* ω

1 ω

× ρ div u − ρ div u dx B ε ε ε ε

2 |Ω | Ω L (0,T ;L γ /ω (Ω ))

  Z

* ω 1 ρεω div u ε dx ≤ C

2

B ρε div u ε − |Ω | 1,2γ /(γ +2ω) L (0,T ;W

Ω

(Ω ))

≤ Ckρεω k L ∞ (0,T ;L γ /ω (Ω )) kdiv u ε k L 2 ((0,T )×Ω ) ≤ C if ω < γ − 1 and N = 2.

On the other hand, we know that Z T Z Z ω θ p (ρ )ρ dxdt ≤ ε ε θ ε ε 0 Ω

0

≤ ε0

T

Z

T

Z

Ω

0

Z 0

Ω

ρε2ω dxdt +

1 4ε0

ρεγ +ω dxdt + C

T

Z 0

Z Ω

θε2 pθ2 (ρε )dxdt

if ω ≤ γ .

(2.35)

Now inserting the above estimates into (2.34) and taking ε0 small enough, and using the hypothesis (1.29), we arrive at Lemma 2.8. Z TZ 0

for ω :=

Ω 2 Nγ

ρεγ +ω dxdt ≤ C,

(2.36)

− 1 if N ≥ 3 and ω < γ − 1 if N = 2.

Finally we give the estimates of ∂t (ρε u ε ) and ∂t (ρε Q(θε )) whose proofs are very simple and so are omitted. Lemma 2.9. k∂t (ρε u ε )k L 2 (0,T ;W −m,l (Ω )) ≤ C, for a certain m ≥ 1, l > 1.

(2.37)

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

kρε Q(θε )k L 2 (0,T ;L 2N γ /(2N +γ (N −2)) (Ω )) ≤ C, here

2N γ 2N +γ (N −2)

>

2N N +2

is equivalent to γ >

3651

(2.38)

N 2.

k∂t (ρε Q(θε ))k L 1 (0,T ;W −m,l (Ω )) ≤ C,

(2.39)

for a certain m ≥ 1, l > 1. 3. Proof of Theorem 1.1—Convergence 3.1. Convergence of geometric nonlinearities The first part of the convergence proof entirely holds. We may assume that γ

ρε → ρ

in C([0, T ]; L weak (Ω )) whence in C([0, T ]; H −1 (Ω )),

uε → u

weakly in L (0, T ;

H01 (Ω )),

2

(3.1) (3.2)

and consequently, ρε u ε → ρu

weakly∗ in L ∞ (0, T ; L 2γ /(γ +1) (Ω )) 2γ /(γ +1)

and in C([0, T ]; L weak

(Ω )) whence in C([0, T ]; H −1 (Ω )),

(3.3)

and weakly in L 2 (0, T ; L 2N γ /(N +2γ (N −1)) (Ω )),

ρε u ε ⊗ u ε * ρu ⊗ u

(3.4)

where the limit functions ρ, u satisfy the continuity equation (1.1) in D0 ((0, T ) × R N ), provided they were extended to be zero outside of Ω . Moreover, ρ ≥ 0 and ρu satisfy the initial conditions (1.17) and (1.18), respectively. Furthermore, pθ (ρε ) → pθ (ρ)

N in C([0, T ]; L weak (Ω )),

(3.5)

in C([0, T ]; H −1 (Ω )).

(3.6)

whence pθ (ρε ) → pθ (ρ)

Now we may assume that θε * θ

weakly in L 2 (0, T ; H 1 (Ω )),

(3.7)

and consequently, θε pθ (ρε ) * θ pθ (ρ) Q(θε ) * Q(θ )

weakly in L 2 ((0, T ) × Ω ),

(3.8)

weakly in L (0, T ; H (Ω )),

ρε Q(θε ) * ρ Q(θ )

2

1

weakly in L (0, T ; L 2

2N γ /(2N +γ (N −2))

(3.9) (Ω )).

(3.10)

Moreover, by virtue of (2.39) and Simon’s compactness principle [30], one has ρε Q(θε ) → ρ Q(θ )

in L 2 (0, T ; H −1 (Ω ))

(3.11)

which yields the desired conclusion. ρ Q(θ )θ = ρ Q(θ)θ .

(3.12)

Thus, in accordance with the hypothesis (1.35), Z TZ Z TZ CV ρε |θε − θ|2 dxdt ≤ ρε (Q(θε ) − Q(θ ))(θε − θ )dxdt 0

Ω

0

Ω

3652

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660 T

Z

Z

− Ω

0

Z

T

Z

Z

Ω T Z

= 0

ρε Q(θε )θε − ρε θε Q(θ ) − ρε Q(θε )θ + ρε Q(θ )θdxdt

− Ω

0 T

Z

Z

→ Ω

0

= 0,

ρε (Q(θ ) − Q(θ ))(θε − θ )dxdt

ρε (Q(θ ) − Q(θ ))(θε − θ )dxdt

ρ Q(θ )θ − ρθ Q(θ ) − ρ Q(θ )θ + ρ Q(θ )θdxdt

as ε → 0.

(3.13)

This proves that θε → θ

strongly in L 2 ({(x, t)|ρ(x, t) > 0}).

(3.14)

In particular, we get Sε → S weakly in L 2 ((0, T ) × Ω ),

(3.15)

where ρS = ρ[µ(θ )(∇u + ∇u T ) + λ(θ )div uI].

(3.16)

Due to the convergence of Hε , by the standard Lions–Aubin compactness principle, we know,  Hε * H weakly in L 2 (0, T ; H 1 (Ω )), weakly∗ in L ∞ (0, T ; L 2 (Ω )),   Hε → H in C([0, T ]; L 2weak (Ω )) whence in C([0, T ]; H −1 (Ω )),  2(N + 2) q  Hε → H strongly in L ((0, T ) × Ω ) for q < ,   N  2 −1,3/2 ∂t Hε * ∂t H weakly in L (0, T ; W (Ω )). 3.2. Strong convergence of the densities Prolonging (ρε , u ε , θε , Hε ) to be zero outside of Ω , we test the momentum equation (1.2) by ψ(t)φ(x)1−1 ∂xi [Tk (ρε )],

ψ ∈ D(0, T ), φ ∈ D(Ω ),

and

Tk (z) := min{z, k}, k ≥ 1, z ≥ 0.

After a lengthy but straightforward computation, we obtain Z TZ ∂u i ψφ( p(ρε , θε ) − λ(θε )div u ε )Tk (ρε ) − 2ψφµ(θε ) ε Ri, j [Tk (ρε )]dxdt ∂x j 0 Ω ! Z TZ j ∂u ε ∂u iε ∂x j φ1−1 ∂xi [Tk (ρε )]dxdt + = ψµ(θε ) ∂ x ∂ x j i 0 Ω Z TZ + ψ[λ(θε )div u ε − p(ρε , θε )]∂x j φ1−1 ∂xi [Tk (ρε )]dxdt Ω

0

Z

T

Z

T

Z

− Ω

0

Z −

Ω

0

Z

T

Z

T

Z

+ Ω

0

Z + 0

Ω

n o φρε u iε ∂t ψ1−1 ∂xi [Tk (ρε )] + ψ1−1 ∂xi [(Tk (ρε ) − Tk0 (ρε )ρε )div u ε ] dxdt ψρε u iε u εj ∂x j φ1−1 ∂xi [Tk (ρε )]dxdt

T

Z

Z

− 0

Ω

ψφρε f εi 1−1 ∂xi [Tk (ρε )]dxdt

n o ψu iε Tk (ρε )Ri, j [φρε u εj ] − φρε u εj Ri, j [Tk (ρε )] dxdt n o ψ Hεi Hεj φRi, j [Tk (ρε )] + ∂x j φ1−1 ∂xi [Tk (ρε )] dxdt

(3.17)

−

T

Z

1 2

Z Ω

0

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

3653

8 n o X lεi , ψ Hε2 φTk (ρε ) + ∂xi φ1−1 ∂xi [Tk (ρε )] dxdt =:

(3.18)

i=1

where Ri, j := ∂x j 1−1 ∂xi . On the other hand, from Section 3.1. we know that the limit equation of the momentum equation (1.2) is:   1 2 ∂t (ρu) + div(ρu ⊗ u) + ∇ p(ρ, θ ) − div H ⊗ H − H I = div(S) + ρ f, 2

(3.19)

with S := µ(θ )(∇u + ∇u T ) + λ(θ )div uI. Test Eq. (3.19) by ψ(t)φ(x)1−1 ∂xi [Tk (ρ)], we find that Z TZ 0

ψφ( p(ρ, θ ) − λ(θ )div u)Tk (ρ) − 2ψφµ(θ )

Ω T

∂u i Ri, j [Tk (ρ)]dxdt ∂x j

 ∂u i ∂u j = ψµ(θ ) + ∂x j φ1−1 ∂xi [Tk (ρ)]dxdt ∂x j ∂ xi 0 Ω Z TZ + ψ[λ(θ )div u − p(ρ, θ )]∂xi φ1−1 ∂xi [Tk (ρ)]dxdt Z



Z

Ω

0

Z

T

Z

T

Z

n o φρu i ∂t ψ1−1 ∂xi [Tk (ρ)] + ψ1−1 ∂xi [(Tk (ρ) − Tk0 (ρ)ρ)div u] dxdt

− Ω

0

Z

ψρu u ∂x j φ1 i

− Ω

0 T

Z

T

Z

Z

T

Z

Ω

0

Ω

0

=:

8 X

0

∂xi [Tk (ρ)]dxdt −

T

Z

Z

0

Ω

ψφρ f i 1−1 ∂xi [Tk (ρ)]dxdt

n o ψ H i H j φRi, j [Tk (ρ)] + ∂x j φ1−1 ∂xi [Tk (ρ)] dxdt

+ 1 − 2

−1

n o ψu i Tk (ρ)Ri, j [φρu j ] − φρu j Ri, j [Tk (ρ)] dxdt

+ Z

j

Z Ω

n o ψ H 2 φTk (ρ) + ∂xi φ1−1 ∂xi [Tk (ρ)] dxdt

li .

(3.20)

i=1

It is easy to show that p

Tk (ρε ) → Tk (ρ)

in C([0, T ]; L weak (Ω ))

Tk (ρε ) → Tk (ρ)

in C([0, T ]; H −1 (Ω )).

for all p > 1,

and Consequently, by virtue of the properties of the operator 1−1 ∂xi and Ri, j , we have [22], 1−1 ∂xi [Tk (ρε )] → 1−1 ∂xi [Tk (ρ)] Ri, j [Tk (ρε )] → Ri, j [Tk (ρ)]

in

in C([0, T ] × Ω ),

(3.21)

p C([0, T ]; L weak (Ω )).

(3.22)

Using (3.21) and (3.22) and the results of Section 3.1, it is standard to show that lεi → l i

except i = 6

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

and as ε → 0

lε6 → l 6

can be proved by the div–curl lemma [22,25]. This means, Z TZ ∂u i η( p(ρε , θε ) − λ(θε )div u ε )Tk (ρε ) − 2ηµ(θε ) ε Ri, j [Tk (ρε )]dxdt lim ε→0 0 ∂x j Ω Z TZ ∂u i Ri, j [Tk (ρ)]dxdt = η( p(ρ, θ ) − λ(θ )div u)Tk (ρ) − 2ηµ(θ ) ∂x j 0 Ω

(3.23)

for any η ∈ D((0, T ) × Ω ). We can write    i  Z TZ Z TZ  ∂u iε ∂u ε ∂u iε ηµ(θε ) Ri, j [Tk (ρε )]dxdt = Ri, j ηµ(θε ) − ηµ(θε )Ri, j Tk (ρε )dxdt ∂x j ∂x j ∂x j 0 Ω 0 Ω Z TZ + ηµ(θε )div u ε Tk (ρε )dxdt. 0

Ω

Thus, one can use Lemma 4.2 in [9] to obtain " ) # Z TZ Z TZ ( ∂u iε ∂u i Ri, j ηµ(θ ) − ηµ(θ )div u Tk (ρ)dxdt lim ηµ(θε ) Ri, j [Tk (ρε )]dxdt = ε→0 0 ∂x j ∂x j 0 Ω Ω Z TZ ηµ(θε )div u ε Tk (ρε )dxdt, + lim ε→0 0

Ω

which, in combination with (3.23) yields, Z TZ lim η( p(ρε , θε ) − (λ(θε ) + 2µ(θε ))div u ε )Tk (ρε )dxdt ε→0 0

Ω

T

Z

Z

= Ω

0

η( p(ρ, θ ) − (λ(θ ) + 2µ(θ ))div u)Tk (ρ)dxdt.

Finally, as ρε are non-negative, we have ρε → 0

strongly in L 1 ({(x, t)|ρ(x, t) = 0}),

which, together with (3.14), implies (λ(θ ) + 2µ(θ ))div uTk (ρ) = (λ(θ ) + 2µ(θ ))div uTk (ρ), that is, T

Z

Z

lim

ε→0 0

Ω T

Z

Z

= 0

Ω

η( p(ρε , θε ) − (λ(θε ) + 2µ(θε ))div u ε )Tk (ρε )dxdt η( p(ρ, θ ) − (λ(θ ) + 2µ(θ ))div u)Tk (ρ)dxdt.

From which, it follows that [22,25,31]   Z TZ sup lim sup |Tk (ρε ) − Tk (ρ) |γ +1 dxdt < ∞. k≥1

ε→0

0

(3.24)

(3.25)

Ω

This implies, in particular, that the limit functions ρ, u satisfy the continuity equation (1.1) also in the sense of renormalized solutions (Lemma 4.4 in [22]).

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

Thus we obtain Z tZ Z Z tZ Tk (ρε )div u ε dxdτ, Tk (ρ)div udxdτ − lim (L k (ρ) − L k (ρ))(t)dx = Ω

ε→0 0

Ω

0

3655

(3.26)

Ω

for any t ∈ [0, T ], where Z ρ Tk (z) L k (ρ) = ρ dz. z2 1 Finally, utilizing (3.24) again, one can let k → ∞ in (3.26) in order to conclude (ρ log ρ)(t) = (ρ log ρ)(t)

for any t ∈ [0, T ],

which yields ρε → ρ

strongly in L 1 ((0, T ) × Ω ).

(3.27)

3.3. Strong convergence of the temperature To begin with, observe that one can pass to the limit in the renormalized inequality (1.45). Indeed, by virtue of (3.14) and (3.27), the functions θε converge to θ a.e. on the set where ρ is positive and, moreover,  ρε Q h (θε ) → ρ Q h (θ )  weakly in L 1 ((0, T ) × Ω ) ρε Q h (θε )u ε → ρ Q h (θ )u  h(θε )θε pθ (ρε )div u ε → h(θ )θ pθ (ρ)div u for any h satisfying (1.46). On the other hand, by our assumptions on λ(θ ), µ(θ ), σ (ρ, θ ) and h(θ ), it is easy to infer that Z TZ Z TZ h(θ )S : ∇uφdxdt ≤ lim inf h(θε )Sε : ∇u ε φdxdt, ε→0

Ω

0

T

h(θ )

0

Ω

h(θε ) (curl Hε )2 φdxdt, ε→0 σ (ρ , θ ) σ (ρ, θ ) ε ε 0 Ω 0 Ω  Z TZ  Z TZ 0 2 0 2 − h (θ )k(θ )|∇θ| φdxdt ≤ lim inf − h (θε )k(θε )|∇θε | φ dxdt, Z

Z

0

(curl H ) φdxdt ≤ lim inf 2

Z

T

Z

ε→0

Ω

0

Ω

for any non-negative test function φ. Furthermore, as the function h tend to zero for large arguments, we see that Kh (θε ) → Kh (θ )

weakly in L 1 ((0, T ) × Ω )

for any function h as in (1.46), where ρKh (θ ) = ρKh (θ )

on (0, T ) × Ω .

Finally, in accordance with our hypotheses, we get the following limit that Z Z ρε (0+)Q h (θ0,ε )φ(0)dx → ρ0 Q h (θ0 )φ(0)dx for ε → 0 Ω

Ω

in order to conclude that ρ, u, and θ satisfy Z TZ ρ Q h (θ )∂t φ + ρ Q h (θ )u · ∇φ + Kh (θ )1φdxdt 0 Ω   Z TZ 1 ≤ h(θ ) θ pθ (ρ)div u − S : ∇u − (curl H )2 φdxdt σ (ρ, θ ) 0 Ω Z TZ Z TZ Z + h 0 (θ )k(θ )(∇θ )2 φdxdt − φh(θ )ρgdxdt − ρ0 Q h (θ0 )φ(0)dx 0

Ω

0

Ω

Ω

(3.28)

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

for any test function φ as in (1.47) and ρKh (θ ) = ρKh (θ ), and kKh (θ )k L 1 ((0,T )×Ω ) ≤ C,

C independent of h.

(3.29)

Next, we can use (2.19) together with convexity of the function − log to deduce log Kh (θ ) ∈ L 2 ((0, T ) × Ω ).

(3.30)

The idea now is the same as in [6], that is, we use the concept of a renormalized limit introduced in [6, Section 6.8.]. Take h(θ ) =

1 , (1 + θ )ω

0 < ω < 1/2

in (3.28), and let ω → 0 to obtain (1.25). 1 One has (1+θ) ω % 1 for ω → 0, and the monotone convergence theorem for the Lebesgue integral can be used. Note that, in accordance with (3.29), Kh (θ ) % K(θ ), with K(θ ) ∈ L 1 ((0, T ) × Ω ), where ρK(θ ) = ρK(θ )

on (0, T ) × Ω .

Finally,we set θ := K−1 (K(θ )). Obviously, the new function θ is non-negative, specifically, in accordance with (3.30), log θ ∈ L 2 ((0, T ) × Ω ). Moreover, θ satisfies ρθ = ρθ

a.a. on (0, T ) × Ω ,

and the proof of (1.22) is immediate. Theorem 1.1 has been proved. Remark 3.1. If θ = const. > 0, we have proved the existence of globally defined weak solutions to the (MHD) problem for baratropic fluid which are similar to that in [20,22,25]. Remark 3.2. The classical theory postulates the existence of another state variable – the specific entropy s := s(ρ, θ ) – related to e through a thermodynamic equation   ∂s 1 ∂e p ∂s 1 ∂e = − 2 , = . ∂ρ θ ∂ρ ∂θ θ ∂θ ρ Accordingly, Z θ 0 Z ρ Q (z) pθ (z) s= dz − dz z z2 1 1 and the variational solutions obtained in Theorem 1.1 must satisfy the entropy equation:  q  1  1 k(θ ) 2 ≥ S : ∇u + (curl H ) + ρg + 2 (∇θ )2 ∂t (ρs) + div(ρsu) + div (3.31) θ θ σ (ρ, θ ) θ

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

at least when the following conditions are satisfied:    2  µ(θ ) µ(θ ) 0 µ(θ ) 00 ≥2 , · θ θ θ " !00 !0 # 2 λ(θ ) + N2 µ(θ ) λ(θ ) + N2 µ(θ ) λ(θ ) + N2 µ(θ ) ≥2 · , θ θ θ and the following matrix is semi-positive definite, i.e.,   ηρρ ηρθ ηρ ηρθ ηθθ ηθ  ≥0  1  η ηρ ηθ 4 for η := η(ρ, θ ) =

3657

(3.32)

(3.33)

(3.34)

1 θσ (ρ,θ) .

The property (3.31) can be used to prove the nonexistence of time-periodic (nontrivial) weak solutions to the (MHD) problem even when H ≡ 0. This result is new and will be proved in the next section. 4. Nonexistence of time-periodic weak solutions Firstly, we give the definition of variational time-periodic solutions to the (MHD) problem on the basis of Theorem 1.1. We shall say that (ρ, u, θ, H ) is a variational time-periodic solutions of (MHD) problem if the following conditions hold: • Positivity: the functions ρ and θ are non-negative on the set (0, T ) × Ω . Moreover, the set {θ = 0} := {(t, x)|θ (t, x) = 0} is of zero (Lebesgue) measure. • Time-periodicity: Z Z ρ(t, x)φ(x)dx = ρ(t + T, x)φ(x)dx, N RN ZR Z (ρu)(t, x)φ(x)dx = (ρu)(t + T, x)φ(x)dx, Ω Ω Z Z (ρ Q(θ ))(t, x)φ(x)dx = (ρ Q(θ ))(t + T, x)φ(x)dx, Ω ZΩ Z (ρs(ρ, θ ))(t, x)φ(x)dx = (ρs(ρ, θ))(t + T, x)φ(x)dx, Ω ZΩ Z H (t, x)φ(x)dx = H (t + T, x)φ(x)dx Ω

Ω

for any t ∈ R and any φ(x) ∈ C ∞ (Ω ). • Mass conservation: The equation of continuity (1.1) is satisfied in the sense of distributions provided the functions ρ, u were extended to be zero outside Ω . More specifically, the extended quantities ρ and ρu are locally integrable on (0, T ) × R N , and the integral identities Z Z TZ ρ∂t φ + ρu · ∇φdxdt = 0, (4.1) ρdx = M > 0, Ω

0 RN ∞ C ([0, T ] × R N ),

hold for any function φ ∈ φ(0, ·) = φ(T, ·) and some positive constant M. • Balance of momentum: The quantities ρu, ρu ⊗ u, H ⊗ H, p, ρ f and S are locally integrable on (0, T ) × Ω , u ∈ L 2 (0, T ; H01 (Ω )) and the momentum equation Z TZ Z TZ 1 S : ∇φ − ρ f φdxdt ρu∂t φ + ρu ⊗ u : ∇φ + p div φ − H ⊗ H : ∇φ + H 2 div φdxdt = 2 0 Ω 0 Ω (4.2) holds for any φ ∈ C ∞ (R; C0∞ (Ω )) and φ(t, x) = φ(t + T, x).

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J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

(curl H ) q∇θ • The second law of thermodynamic-entropy production: The quantities ρs, ρsu, qθ , S:∇u , θ 2 and ρg θ , σθ θ are locally integrable on the set (0, T ) × Ω ; the integral inequality  Z TZ Z TZ  q S : ∇u q∇θ (curl H )2 ρg ρs∂t φ + ρsu∇φ + ∇φdxdt ≤ − φdxdt (4.3) − − θ θ σθ θ θ2 0 Ω 0 Ω holds for any non-negative test function 2

φ ∈ C 1.1 (R × Ω )

and

φ(t, x) = φ(t + T, x).

(4.4)

• The equations for the magnetic field H : The quantities H, u × H , and (0, T ) × Ω and the equations Z TZ H ∇φdxdt = 0

1 σ curl

H are locally integrable on the set

(4.5)

Ω

0

holds for any φ ∈ C ∞ ([0, T ] × Ω ) while Z TZ Z H ∂t φ − (u × H ) · curl φdxdt =

T

Z

1 curl H curl φdxdt, σ 0 Ω 0 Ω holds for any φ ∈ C ∞ ([0, T ] × Ω ), φ · n|∂ Ω = curl φ × n|∂ Ω = 0, and φ(0) = φ(T ). • Renormalization: The density ρ and the velocity u satisfy the renormalized continuity equation: ∂t b(ρ) + div(b(ρ)u) + (b0 (ρ)ρ − b(ρ))div u = 0 in

D0 ((0, T ) × R N )

for any b ∈

C 1 [0, ∞]

such that

b0 (z)

(4.6)

(4.7) = 0 for all z ≥ z b .

Now we are ready to state our MAIN RESULT in this section. Theorem 4.1. Assume that f and g are bounded measurable functions on (0, T ) × Ω , g ≥ 0, f (0) = f (T ), g(0) = g(T ) for any given positive number T and γ ∈ (1, 3]. Moreover, if (ρ, u, θ, H ) is any variational time-periodic solutions satisfying ρ ∈ L ∞ (0, T ; L γ (Ω )), θ α/2 ∈ L 2 (0, T ; H 1 (Ω )),

u ∈ L 2 (0, T ; H01 (Ω )), H ∈ V2 ((0, T ) × Ω ),

θ ∈ L α+1 ((0, T ) × Ω ), div H = 0

log θ ∈ L 2 (0, T ; H 1 (Ω )),

in (0, T ) × Ω , H · n|∂ Ω = 0,

then there holds u ≡ 0,

∇θ ≡ 0,

H ≡ 0,

ρg ≡ 0,

∇p = ρf θ ≡ const. > 0.

(4.8) (4.9) (4.10)

Proof. (4.8) follows from (4.3) by taking φ = 1 and thus (4.2) gives (4.9). From (4.3) and Corollary 2.1, we infer that ! R Z TZ Z TZ e )dx 2 ρ Q(θ 2 ΩR e e Q(θ ) − dxdt ≤ C |∇ Q(θ)| dxdt = 0 ρdx 0 Ω 0 Ω Ω which implies R e )dx ρ Q(θ e Q(θ ) = ΩR , Ω ρdx R e ) := θ Q 0 (z) dz. where Q(θ 1 z On the other hand, from (4.3) and (4.8), we obtain Z TZ ρs∂t φdxdt ≤ 0, 0

Ω

for any φ given in (4.4).

(4.11)

(4.12)

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

By an approximation procedure, we can take b(ρ) = ρ T

Z

Z Ω

0

Rρ 1

pθ (z) dz z2

3659

in (4.7) and using (4.8) we get

b(ρ)∂t φdxdt = 0

(4.12) and (4.13) gives Z TZ e )∂t φdxdt ≤ 0. ρ Q(θ

(4.13)

(4.14)

Ω

0

Now testing (4.14) by φ = max η(t, x) ± η(t, x)

(4.15)

[0,T ]×Ω

respectively, for any η ∈ C ∞ ([0, T ] × Ω ), η(0) = η(T ). We arrive at Z TZ e )∂t ηdxdt = 0. ρ Q(θ 0

(4.16)

Ω

e ) = const. (4.11) and (4.16) implies Q(θ This proves (4.10).  Remark 4.1. For the radiation model considered in [10], we can prove (4.8) and (4.9), however, we still cannot prove (4.10). Acknowledgement The authors would like to thank the referee for the careful reading of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

H. Cabannes, Theoretical Magnetofluiddynamics, Academic Press, New York, London, 1970. L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Butterworth-Heinemann, 1999. Ta-tsien Li, Tiehu Qin, Physics and Partial Differential Equations, 2nd ed., vol. I, Higher Education Press, Beijing, PR China, 2005. R.V. Polovin, V.P. Demutskii, Fundamentals of Magnetohydrodynamics, Consultants, Bureau, New York, 1990. R. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 1990. E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, 2004. R.J. Diperna, P.-L. Lions, On the Fokker–Planck–Boltzmann equation, Comm. Math. Phys. 120 (1988) 1–23. R. Alexandre, C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math. 55 (2002) 30–70. E. Feireisl, On the motion of a viscous, compressible, and heat conducting fluid, Indiana Univ. Math. J. 53 (2004) 1705–1738. B. Ducomet, E. Feireisl, A regularizing effect of radiation in the equations of fluid dynamics, Math. Methods Appl. Sci. 28 (2005) 661–685. R.J. Diperna, P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989) 511–547. A.I. Vol’pert, S.I. Hudjaev, On the Cauchy problem for composite systems of nonlinear differential equations, Math. USSR-Sb. 16 (1972) 517–544. G. Str¨ohmer, About compressible viscous fluid flow in a bounded region, Pacific J. Math. 143 (1990) 359–375. S. Kawashima, M. Okada, Smooth global solutions for the one-dimensional equations in magnetohydrodynamics, Proc. Japan Acad. Ser. A, Math. Sci. 58 (1982) 384–387. T.-P. Liu, Y. Zeng, Large-time behavior of solutions for general quasilinear hyperbolic–parabolic systems of conservation laws, Memoirs Amer. Math. Soc. 599 (1997). A.V. Kazhikhov, Sh.S. Smagulov, Well-posedness and approximation methods for a model of magnetogasdynamics, Izv. Akad. Nauk Kazakh. SSR Ser. Fiz.-Mat. 5 (1986) 17–19. Sh.S. Smagulov, A.A. Durmagambetov, D.A. Iskenderova, The Cauchy problem for the equations of magnetogasdynamics, Differential Equations 29 (1993) 278–288. G.Q. Chen, D. Wang, Existence and continuous dependence of large solutions for the magnetohydrodynamics equations, Z. Angew. Math. Phys. 54 (2003) 608–632. G.Q. Chen, D. Wang, Global solutions of nonlinear magnetohydrodynamics with large initial data, J. Differential Equations 182 (2002) 344–376. P.-L. Lions, Mathematical Topics in Fluid Dynamics, in: Compressible Models, vol. 2, Oxford Science Publication, Oxford, 1998.

3660

J. Fan, W. Yu / Nonlinear Analysis 69 (2008) 3637–3660

[21] E. Feireisl, S. Matuˇsu´ -Neˇcasov´a, H. Petzeltov´a, I. Straˇskraba, On the motion of a viscous compressible flow driven by a time-periodic external flow, Arch. Ration. Mech. Anal. 149 (1999) 69–96. [22] E. Feireisl, A. Novotn´y, H. Petzeltov´a, On the existence of globally defined weak solutions to the Navier–Stokes equation, J. Math. Fluid Mech. 3 (2001) 358–392. [23] S. Jiang, P. Zhang, Global spherically symmetry solutions of the compressible isentropic Navier–Stokes equations, Comm. Math. Phys. 215 (2001) 559–581. [24] S. Jiang, P. Zhang, Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids, J. Math. Pures Appl. 82 (2003) 949–973. [25] E. Feireisl, On compactness of solutions to the compressible isentropic Navier–Stokes equations when the density is not square integrable, Comment. Math. Univ. Carolin. 42 (2001) 83–98. [26] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland, Amsterdam, 1979. [27] M.E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad, Trudy. Sem. S. L. Sobolev 80 (1980) 5–40 (in Russian). [28] G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, I, Springer-Verlag, New York, 1994. [29] W. Borchers, H. Sohr, On the equation rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990) 67–87. [30] J. Simon, Compact sets and the space L p (0, T ; B), Ann. Mat. Pura Appl. 146 (1987) 65–96. [31] E. Feireisl, A. Novotn´y, H. Petzeltov´a, On the domain dependence of solutions to the compressible Navier–Stokes equations of a baratropic fluid, Math. Methods Appl. Sci. 25 (2002) 1045–1073. [32] M. Padula, Mathematical properties of motions of viscous compressible fluids, in: G.P. Galdi, J. M´alek and J. Neˇcas (Eds.), Progress in Theoretical and Computational Mechanics, Winter school, Paseky, 1993.