Asymptotic behavior of global smooth solutions for full compressible Navier–Stokes–Maxwell equations

Nonlinear Analysis: Real World Applications 19 (2014) 105–116

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

Asymptotic behavior of global smooth solutions for full compressible Navier–Stokes–Maxwell equations Yue-Hong Feng a,b,c,∗ , Yue-Jun Peng b,c , Shu Wang a a

College of Applied Sciences, Beijing University of Technology, Beijing 100022, China

b

Clermont Université, Université Blaise Pascal, 63000 Clermont-Ferrand, France CNRS, UMR 6620, Laboratoire de Mathématiques, 63171 Aubière Cedex, France

c

article

info

Article history: Received 6 May 2013 Accepted 25 March 2014 Keywords: Full compressible Navier–Stokes–Maxwell equations Global smooth solutions Energy estimates Large-time behavior

abstract In the present paper, the full compressible Navier–Stokes–Maxwell equations are investigated in R3 and the large-time behavior of global smooth solutions is established. It is shown that both the density and the temperature converge to the equilibrium states with the same norm ∥ · ∥H s−1 , while the other unknown variables converge to the equilibrium states with weaker norms than ∥ · ∥H s−1 . This phenomenon on the charge transport shows the essential difference of the equations with the non-isentropic Euler–Maxwell and the isentropic Navier–Stokes–Maxwell equations. © 2014 Elsevier Ltd. All rights reserved.

1. Introduction The Navier–Stokes–Maxwell equations are used to simulate the transport of viscosity charged particles in plasma [1–5]. When the external force term in the full Navier–Stokes equations is represented by the Lorentz force which is characterized by the self-consistent Maxwell equations, we obtain the full compressible Navier–Stokes–Maxwell equations. In this paper we consider the Cauchy problem for the full compressible Navier–Stokes–Maxwell equations:

 ∂t n + ∇ · (nu) = 0,   ′  ∂t (mnu) + ∇ · (mnu ⊗ u) + ∇ p = −n (E + γ u × B) + ν 1u + ν ∇(∇ · u),   E − el n ∂t E + ∇ · (E u + pu) = −nuE − + u1u, τ    2  λ2 ∇ · E = b − n,  γ λ ∂t E − ∇ × B = γ nu, γ ∂t B + ∇ × E = 0, ∇ · B = 0, (t , x) ∈ (0, +∞) × R3 ,

(1.1)

where the unknowns are the density n > u = (u1 , u2 , u3 ), the absolute temperature θ > 0, the internal  0, the velocity  energy e = 23 KB θ , the total energy E = n e + 12 m|u|2 , the pressure function p = 32 ne, the electric field E, and the magnetic 1

field B. The constants el = 23 KB θl , m > 0, θl > 0, b > 0, KB > 0, λ > 0, ν > 0, ν ′ , τ > 0, γ1 = c = (ε0 µ0 )− 2 , ε0 > 0, and µ0 > 0 are the background internal energy, mass, back ground temperature, back ground density, Boltzmann constant, scaled Debye length, shear viscosity coefficient, bulk viscosity coefficient, energy relaxation time, speed of light, vacuum permittivity, and permeability, respectively. Throughout this paper, we set m = KB = λ = ν = τ = γ = 1 and ν ′ = 0

∗ Corresponding author at: College of Applied Sciences, Beijing University of Technology, Beijing 100022, China. Tel.: +86 10 67392212, +86 10 67392182x202; fax: +86 10 67391738. E-mail addresses: [email protected] (Y.-H. Feng), [email protected] (Y.-J. Peng), [email protected] (S. Wang). http://dx.doi.org/10.1016/j.nonrwa.2014.03.004 1468-1218/© 2014 Elsevier Ltd. All rights reserved.

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without loss of generality. This is not an essential restriction in the investigation of the global existence of smooth solutions. Then, Eqs. (1.1) are equivalent to

 ∂t n + ∇ · (nu) = 0,    1 1   ∂t u + (u · ∇) u + ∇ (nθ ) = − (E + u × B) + 1u,   n n  1

2

∂t θ + θ∇ · u + u · ∇θ + θ − θl + |u|2 = 0,   3 3    ∂t E − ∇ × B = nu, ∇ · E = b − n,   ∇ · B = 0, (t , x) ∈ (0, +∞) × R3 . ∂t B + ∇ × E = 0,

(1.2)

Initial conditions are given as

(n, u, θ , E , B) |t =0 = (n0 , u0 , θ0 , E0 , B0 ) ,

x ∈ R3 ,

(1.3)

which satisfy the compatibility conditions:

∇ · E 0 = b − n0 ,

∇ · B0 = 0,

x ∈ R3 .

(1.4)

The full compressible Navier–Stokes–Maxwell equations (1.2) are a symmetrizable hyperbolic–parabolic system for n, θ > 0. For the full compressible Navier–Stokes equations, the local existence and uniqueness of classical solutions are known in [6,7] in the absence of vacuum, where vacuum is defined by the density n = 0. Then, according to the result of Kato [8], the Cauchy problem (1.2)–(1.3) has a unique local smooth solution when the initial data are smooth. Here we are concerned with stabilities of global smooth solutions to (1.2)–(1.3) around a constant state being a particular solution of (1.2). It is easy to see that this constant state is necessarily given by

  (n, u, θ , E , B) = b, 0, θl , 0, B¯ ∈ R11 . Proposition 1.1 (Local Existence of Smooth Solutions, See [6–9]). Assume (1.4) holds.  Let s ≥ 4 be an integer, b, θl > 0 and B¯ ∈ R3 be any given constant. Suppose (n0 − b, u0 , θ0 − θl , E0 , B0 − B¯ ) ∈ H s R3 with n0 , θ0 ≥ 2κ for some given constant κ > 0. Then there exists T > 0 such that problem (1.2)–(1.3) has a unique smooth solution satisfying n, θ ≥ κ in [0, T ] × R3 and u ∈ C 1 [0, T ] ; H s−2 R3

   ∩ C [0, T ] ; H s R3 ,         n − b, θ − θl , E , B − B¯ ∈ C 1 [0, T ] ; H s−1 R3 ∩ C [0, T ] ; H s R3 . 





There are some mathematical studies on the equations arising from plasma. For one-dimensional isentropic Euler–Maxwell equations, Chen–Jerome–Wang [10] proved the global existence of entropy solutions by using the compensated compactness method. For the three-dimensional isentropic Euler–Maxwell equations in which the energy equation is not contained, the existence of global smooth small solutions to the Cauchy problem in R3 is established by Ueda–Wang–Kawashima [11] when s ≥ 3 and the asymptotic behaviors of solutions when s ≥ 4. With the help of suitable choices of symmetrizers and energy estimates, Peng–Wang–Gu [12] and Peng [13] obtained the global existence and the long time behaviors of smooth solutions to the periodic problem in the torus and to the Cauchy problem in R3 when s ≥ 3. By using high- and low-frequency decomposition methods, Xu [14] construct uniform (global) classical solutions to the Cauchy problem in Chemin–Lerner’s spaces with critical regularity. When s ≥ 4, by using the tools of Fourier analysis, Duan [15] and Duan–Liu–Zhu [16] obtained the decay rates of global smooth solutions in Lq with 2 ≤ q ≤ ∞ when the time goes to infinity. And when s ≥ 6, Ueda–Kawashima [17] also obtained the large time decay rates of global smooth solutions in H s−2k with 0 ≤ k ≤ 2s . For asymptotic limits with parameters, see [18–20] and references therein. For the three-dimensional non-isentropic Euler–Maxwell equations, the diffusive relaxation limit was considered by Yang–Wang [21] and the existence of global smooth small solutions to the Cauchy problem in R3 is established by Feng–Wang–Kawashima [22] and Wang–Feng–Li [23]. For numerical analysis, see [24]. In the case without velocity damping term −u, an initial assumption B = ∇ × u was made in [25] to imply such a global existence result. For isentropic compressible Navier–Stokes–Maxwell equations in which the energy equation is not contained, by using Green’s function argument, Duan [2] proved the global existence and asymptotic behavior of smooth solutions around a steady state. For incompressible Navier–Stokes–Maxwell equations, with the help of Fujita–Kato’s method in l1 based (for the Fourier coefficients) functional spaces, Ibrahim–Yoneda [26] established the existence of local unique solution and loss of smoothness of the velocity and magnetic field for the periodic problem. Ibrahim–Keraani [27] also obtained the existence of global small mild solutions in three dimensions and the same results in a space ‘close’ to the energy space in two dimensions. By using an a priori L2t (L∞ x ) estimate for solutions of the forced Navier–Stokes equations, Germain–Ibrahim–Masmoudi [28] proved the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier–Stokes and refined the results in [27]. However, there is no analysis on the asymptotics and the global existence for the full compressible Navier–Stokes–Maxwell equations yet. The goal of the present paper is to establish such a result.

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The main result of this paper can be stated as follows. Theorem 1.1. Let s ≥ 4 be an integer. Assume (1.4) holds, b, θl > 0 and B¯ ∈ R3 be any given constant. Then there exist constants δ0 > 0 small enough and C > 0, independent of any given time t > 0, such that if

   n0 − b, u0 , θ0 − θl , E0 , B0 − B¯ 

≤ δ0 ,

Hs

then, the Cauchy problem (1.2)–(1.3) has a unique global solution u ∈ C 1 (0, +∞); H s−2 (R3 ) ∩ C (0, +∞); H s (R3 ) ,









    (n − b, θ − θl , E , B − B¯ ) ∈ C 1 (0, +∞); H s−1 (R3 ) ∩ C (0, +∞); H s (R3 ) ,

(1.5)

which satisfies, for all t > 0,

   n − b, u, θ − θl , E , B − B¯ 2 s + H

t

 0

 2   ∥(n − b, ∇ u, θ − θl ) (τ )∥2H s + ∥∇ E (τ )∥2H s−2 + ∇ 2 B (τ )H s−3 dτ

 2 ≤ C  n0 − b, u0 , θ0 − θl , E0 , B0 − B¯  s .

(1.6)

H

Moreover, lim ∥(n − b, θ − θl ) (t )∥H s−1 = 0,

t →+∞

lim ∥∇ u(t )∥H s−3 = 0,

t →+∞

(1.7)

lim ∥∇ E (t )∥H s−2 = 0,

(1.8)

lim ∇ 2 B(t )H s−4 = 0.

(1.9)

t →+∞

and





t →+∞

Remark 1.1. It should be emphasized that both the velocity viscosity term and the temperature relaxation term of the full Navier–Stokes–Maxwell equations (1.2) play a key role in the proof of global existence. We prove Theorem 1.1 by using careful energy estimates and the techniques of symmetrizer. It should be pointed out that the full Navier–Stokes–Maxwell system is much more complex than the isentropic Navier–Stokes–Maxwell system. For instance, Duan [2] introduced a new variable and reduced directly the isentropic Navier–Stokes–Maxwell system to a symmetric system by using a scaling technique. However, this technique does not work for the full Navier–Stokes–Maxwell equations due to the complexity of the coupled energy equations. To overcome this difficulty, we choose a new symmetrizer. Next, let us explain the main difference of proofs in the non-isentropic Euler–Maxwell and full Navier–Stokes–Maxwell equations. From (1.2), it is easy to see that both ∇ u and θ − θl  are dissipative. By using a classical H s energy estimate, we obtain an energy estimate for ∇ u and θ − θl in L2 [0, T ] ; H s R3 . In the non-isentropic Euler–Maxwell equations [22], this is achieved in estimate

∥w(t )∥

2 Hs

 +

t

Ds (w(τ )) dτ ≤ C ∥w (0)∥

0

provided that sup0≤t ≤T ∥w(t )∥H s

t



∥w (τ )∥H s Ds (w(τ )) dτ ,  ≤ λ, where w = n − b, u, θ − θl , E , B − B¯ , 2 Hs

+ 

(1.10)

0

Ds (w(t )) = ∥(n − b, u, θ − θl ) (t )∥2H s + ∥E (t )∥2H s−1 + ∥∇ B(t )∥2H s−2 ,

C > 0 and λ > 0 are constants independent of T . In the full Navier–Stokes–Maxwell equations, according to the coupling viscosity term, the proof of such an estimate is more technical. It is divided into two steps. In the first step, we show a similar estimate as (1.10) (see (2.20) of Lemma 2.4) which is sufficient to prove the  global existence   and large time behavior for (n − b, u, θ − θl ). In the second step, we establish estimates for ∇ E in L2 [0, T ] ; H s−2 R3 and for ∇ 2 B in    L2 [0, T ] ; H s−3 R3 , respectively. Thus, a classical argument yields the large-time behavior for (E , B). The rest of this paper is arranged as follows. In Section 2, we deal with the global existence for smooth solutions. The main goal is to prove the first part of Theorem 1.1 by establishing energy estimates. In Section 3, the large-time behavior of the solutions is presented, and we complete the second part of Theorem 1.1 by making further energy estimates. 2. Global existence of smooth solutions According to [29], the global existence of smooth solutions follows from the local existence and uniform estimates of solutions with respect to t. The main task of this section is devoted to the uniform estimates for proving the first part of Theorem 1.1. 2.1. Preliminary We first introduce some notations for later use. The expression f ∼ g means γ g ≤ f ≤ γ1 g for a constant 0 < γ < 1. We denote by ∥ · ∥s the norm of the usual Sobolev space H s (R3 ), and by ∥ · ∥ and ∥ · ∥L∞ the norms of L2 (R3 ) and L∞ (R3 ),

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respectively. We also denote by ⟨·, ·⟩ the inner product over L2 (R3 ). For a multi-index α = (α1 , α2 , α3 ) ∈ N3 , we denote α

α

α

∂ α = ∂xα11 ∂xα22 ∂xα33 = ∂1 1 ∂2 2 ∂3 3 and |α| = α1 + α2 + α3 . For α = (α1 , α2 , α3 ) and β = (β1 , β2 , β3 ) ∈ N3 , β ≤ α stands for βj ≤ αj for j = 1, 2, 3, and β < α stands for β ≤ α and β ̸= α . The Leibniz formulas

∂ α (fg ) =

 β≤α

Cαβ ∂ α−β f ∂ β g ,

∀ α ∈ N3 ,

where Cαβ > 0 for β ≤ α are constants. The following lemmas will be needed in the proof of Theorem 1.1. Lemma 2.1 (Moser-Type Calculus Inequalities, See Lemma A.1 in [30,9]). Let s ≥ 1 be an integer. Suppose u ∈ H s (R3 ), ∇ u ∈ L∞ (R3 ) and v ∈ H s−1 (R3 ) ∩ L∞ (R3 ). Then for all multi-index α with 1 ≤ |α| ≤ s, one has ∂ α (uv) − u∂ α v ∈ L2 (R3 ) and

  ∥∂ α (uv) − u∂ α v∥ ≤ Cs ∥∇ u∥L∞ ∥D|α|−1 v∥ + ∥D|α| u∥ ∥v∥L∞ , where ′

∥Ds u∥ =



∥∂ α u∥.

|α|=s′

Furthermore, if s ≥ 3, then the embedding H s−1 (R3 ) ↩→ L∞ (R3 ) is continuous and we have

∥uv∥s−1 ≤ Cs ∥u∥s−1 ∥v∥s−1 ,

∀u, v ∈ H s−1 (R3 ),

and for all u, v ∈ H s (R3 ),

∥∂ α (uv) − u∂ α v∥ ≤ Cs ∥u∥s ∥v∥s−1 ,

∀|α| ≤ s.

Remark 2.1. For all smooth functions f and u ∈ H s (R3 ) with s ≥ 3, we have

∥∂ α f (u)∥ ≤ C∞ (1 + ∥u∥s )s−1 ∥u∥s ,

∀|α| ≤ s.

Lemma 2.2. For ∇ u ∈ H 1 , there exists a constant C > 0 such that

∥u∥L∞ ≤ C ∥∇ u∥1 . Proof. From the Morrey theorem [31], the embedding W 1,p Rd ↩→ L∞ Rd is continuous if p > d. Then for p = 6 and d = 3, we have









  ∀ u ∈ W 1 ,6 R 3 .

∥u∥L∞ ≤ C ∥u∥W 1,6 (R3 ) ,

By the Sobolev inequality [31], we obtain

∥u∥L6 ≤ C ∥∇ u∥

and

∥∇ u∥L6 ≤ C ∥∇ u∥1 ,

which imply the result of Lemma 2.2.



Suppose (n, u, θ , E , B) is a local smooth solution of the Cauchy problem for the full Navier–Stokes–Maxwell equations (1.2) with initial value (1.3) which satisfies (1.4). Set n = b + N,

θ = θl + Θ ,

B = B¯ + G,

(2.1)

and N u

  U =

Θ

U E G

  ,

W =

.

(2.2)

Then, we can rewrite Eqs. (1.2)–(1.4) as

 ∂t N + u · ∇ N + (b + N ) ∇ · u = 0,      θl + Θ 1  ¯   ∂t u + b + N ∇ N + (u · ∇) u + ∇ Θ + E + u × B + G − b + N 1u = 0,  2

1

∂t Θ + (θl + Θ ) ∇ · u + u · ∇ Θ + Θ + |u|2 = 0,    3 3    ∂ E − ∇ × G = b + N u , ∇ · E = − N , ( )  t  ∂t G + ∇ × E = 0, ∇ · G = 0, (t , x) ∈ (0, +∞) × R3 ,

(2.3)

Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

109

with an initial condition: W |t =0 = W0 := (N0 , u0 , Θ0 , E0 , G0 ) ,

x ∈ R3 ,

(2.4)

which satisfies the compatibility conditions:

∇ · E0 = −N0 ,

∇ · G0 = 0,

x ∈ R3 .

(2.5)

Here, N0 = n0 − b,

Θ0 = θ0 − θl ,

G0 = B0 − B¯ .

Furthermore, the first three equations of (2.3) can be rewritten as

∂t U +

3 

Aj (n, u, θ) ∂j U = KI (W ) + KII (U ),

(2.6)

j =1

with



uj

θ  

Aj (n, u, θ) =  n ej

neTj

0

uj I 3

ej  ,

2

0

3



 

θ eTj

uj



0

E + u × B¯ + G  , 1 |u|2

KI (W ) = − 

j = 1, 2, 3,

(2.7)







0



1  KII (U ) =  1u , n

(2.8)

−Θ

3

where (e1 , e2 , e3 ) is the canonical basis of R3 , I3 is the 3 × 3 unit matrix. It is clear that (2.6) for U is symmetrizable hyperbolic–parabolic when b + N , θl + Θ > 0. More precisely, since b, θl ≥ const. > 0 and we consider small solutions for which N , Θ are close to zero, we have b + N , θl + Θ ≥ const. > 0. Let



θ

0

n  A0 (n, θ) =  0 

0

nI3

0

0

  

0 . 3 n 2θ

Then



θ

uj

n  A˜ j (n, u, θ) = A0 (n, θ) Aj (n, u, θ) =  θ ej 

0

θ eTj nuj I3 neTj

0

  

nej  . 3n  uj 2θ

Since A0 is symmetric positively definite and A˜ j is symmetric for all 1 ≤ j ≤ 3, system (2.6) is symmetrizable hyperbolic–parabolic. Let T > 0 and W be a smooth solution of (2.6) defined on time interval [0, T ] with initial value W0 . This solution is given by Proposition 1.1. From now on, we define

ωT = sup ∥W (t )∥s ,

(2.9)

0 ≤t ≤T

and by C > 0 various constants independent of any time t and T . From the continuous embedding H s R3 ↩→ L∞ R3 for s ≥ 2, there exists a constant Cm > 0 such that



∥f ∥L∞ ≤ Cm ∥f ∥s , If ωT ≤

min{b,θl } , 2Cm

  ∀f ∈ H s R3 .

from (2.9) it is easy to get

min{b, θl } and ≤ n = b + N, 2 2 Furthermore, by Remark 2.1, for any smooth function g one has

∥(N , Θ )∥L∞ ≤

min{b, θl }

θ = θl + Θ ≤

3 max{b, θl } 2

.

sup ∥g (W (t ))∥s ≤ C . 0≤t ≤T

Note that in the proof of Lemmas 2.2–2.4, we always suppose ωT ≤ δ ≤ δ0 with δ0 sufficiently small. Now, let us establish the classical energy estimate for W .







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Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

Lemma 2.3. Under the assumptions of Theorem 1.1, if ωT ≤ δ ≤ δ0 with δ0 sufficiently small, it holds

∥W (t )∥2s +

t



∥(∇ u, Θ ) (τ )∥2s dτ ≤ C ∥W0 ∥2s + C 0

t



∥W (τ )∥s ∥(N , ∇ u, Θ ) (τ )∥2s dτ ,

∀t ∈ [0, T ].

(2.10)

0

Proof. For α ∈ N3 with |α| ≤ s. Applying ∂ α to (2.6) and multiplying the resulting equations by the symmetrizer matrix A0 (n, θ), we have A0 (n, θ) ∂t ∂ α U +

3 

A˜ j (n, u, θ) ∂j ∂ α U = A0 (n, θ) ∂ α [KI (W ) + KII (U )] + Jα ,

(2.11)

j =1

where Jα = −

3 

A0 (n, θ) ∂ α Aj (n, u, θ) ∂j U − Aj (n, u, θ) ∂ α ∂j U .









j =1

It is easy to see that Jα will vanish when |α| = 0. Taking the inner product of (2.11) with 2∂ α U in L2 (R3 ), we have d dt

⟨A0 (n, θ) ∂ α U , ∂ α U ⟩ = 2 ⟨Jα , ∂ α U ⟩ + ⟨div A (n, u, θ) ∂ α U , ∂ α U ⟩ + 2 ⟨A0 (n, θ) ∂ α U , ∂ α [KI (W ) + KII (U )]⟩ , (2.12)

where div A (n, u, θ) = ∂t A0 (n, θ) +

3 

∂j A˜ j (n, u, θ) .

(2.13)

j =1

Since

1 n  ∂t A0 (n, θ) =  

∂t Θ −

θ n

∂N 2 t

0

∂t NI3

0 0

0



0 0 3



1

θ

2

∂t N −

n

θ2

∂t Θ

  ,  

using the first and third equations of (2.3), Lemma 2.2 and the smallness of ωT , we get

∥∂t N ∥ ≤ C ∥∇ u∥1 ,

∥∂t N ∥L∞ ≤ C ∥∇ u∥2 ,

∥∂t Θ ∥L∞ ≤ C (∥∇ u∥2 + ∥Θ ∥2 ) .

In fact,

∥∂t N ∥ 6 ∥u∥L∞ ∥∇ N ∥ + ∥(1 + N )∥L∞ ∥∇ · u∥ 6 C ∥∇ u∥1 , and similarly for the rest two estimates. Now, let us estimate each term on the right hand side of (2.12). For the first term, we obtain



     α   α    α   α θ  α θ  α α α α A0 (n, θ) ∂ Aj ∂j U − Aj ∂ ∂j U , ∂ U = ∂ uj ∂j N − uj ∂ ∂j N , ∂ N + ∂ n∂j uj − n∂ ∂j uj , ∂ N n n             θ θ + n ∂α ∂j N − ∂ α ∂j N , ∂ α uj + n ∂ α uj ∂j u − uj ∂ α ∂j u , ∂ α u n n n    n        + ∂ α θ ∂ j uj − θ ∂ α ∂ j uj , ∂ α Θ + ∂ α uj ∂j θ − uj ∂ α ∂j θ , ∂ α Θ θ θ ≤ C ∥(N , u, Θ )∥s ∥(N , ∇ u, Θ )∥2s ,

and then 2 ⟨Jα , ∂ α U ⟩ ≤ C ∥(N , u, Θ )∥s ∥(N , ∇ u, Θ )∥2s .

(2.14)

For the second term on the right hand side of (2.12), we have

⟨∂t A0 (n, θ) ∂ α U , ∂ α U ⟩ =



|∂ α N |2 n

     θ , ∂t Θ − 2 |∂ α N |2 , ∂t N + |∂ α u|2 , ∂t N n

3 |∂ α Θ |2



+

2

θ



, ∂t N −

3n 2 θ2

 |∂ α Θ |2 , ∂t Θ .

Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

111

When |α| = 0, it follows that

        3 |Θ |2 θ 3n 2 | | , ∂t Θ − 2 |N |2 , ∂t N + |u|2 , ∂t N + , ∂t N − Θ , ∂ Θ t n n 2 θ 2 θ2   2 2 ≤ C (∥∂t N ∥∞ + ∥∂t Θ ∥∞ ) ∥N ∥ + ∥Θ ∥ + C ∥u∥∞ ∥u∥ ∥∂t N ∥   ≤ C (∥∇ u∥2 + ∥Θ ∥2 ) ∥N ∥2 + ∥Θ ∥2 + C ∥u∥ ∥∇ u∥21   ≤ C ∥(N , u, Θ )∥ ∥N ∥2 + ∥∇ u∥22 + ∥Θ ∥22 ,

⟨∂t A0 (n, θ) U , U ⟩ =



|N |2

where the Cauchy–Schwarz inequality is used. And when |α| ≥ 1, it can be controlled as follows

  ⟨∂t A0 (n, θ) ∂ α U , ∂ α U ⟩ ≤ C (∥∂t N ∥∞ + ∥∂t Θ ∥∞ ) ∥∂ α N ∥2 + ∥∂ α u∥2 + ∥∂ α Θ ∥2 ≤ C ∥(u, Θ )∥s ∥∇ (N , u, Θ )∥2s−1 . Hence,

⟨∂t A0 (n, θ) ∂ α U , ∂ α U ⟩ ≤ C ∥(u, Θ )∥s ∥∇ (N , u, Θ )∥2s−1 ,

∀α, |α| ≤ s.

(2.15)

Next, from the definition of A˜ j (n, u, θ), we obtain

   θ u ∂  j n j  ∂j A˜ j (n, u, θ) =   ∂j Θ ej 

∂j Θ eTj  

∂j nuj I3 ∂j NeTj

0



0

  ∂j Nej  .    3 n ∂j uj 2 θ

Then,

      u  θ   2 θ α 2 j |∂ α N |2 , ∂j Θ − 2 uj |∂ α N |2 , ∂j N + |∂ N | , ∂j uj + ∂ α uj  , ∂j Θ ∂j A˜ j (n, u, θ) ∂ α U , ∂ α U = n n n         + |∂ α N |2 , ∂j Θ + uj |∂ α u|2 , ∂j N + n |∂ α u|2 , ∂j uj + |∂ α Θ |2 , ∂j N  3  n  u  n   2 j 2 |∂ α Θ |2 , ∂j N − |∂ α Θ |2 , ∂j uj uj |∂ α Θ | , ∂j Θ − + ∂ α uj  , ∂j N − 2 2 θ θ θ ≤ C ∥(N , u, Θ )∥s ∥∇ (N , u, Θ )∥2s−1 . This together with (2.15) gives

 α

α

⟨div A (n, u, θ) ∂ U , ∂ U ⟩ =

∂t A0 (n, θ) +

3 

  α α ˜ ∂j Aj (n, u, θ) ∂ U , ∂ U

j=1

≤ C ∥(N , u, Θ )∥s ∥∇ (N , u, Θ )∥2s−1 .

(2.16)

Now, for the last term on the right hand side of (2.12), from (2.8) it holds 2 ⟨A0 (n, θ) ∂ α U , ∂ α KI (W )⟩ = −2 ⟨∂ α (nu) , ∂ α E ⟩ −

−2

 β<α

 β

α

n

β

θ

    Cαβ ∂ α−β N ∂ β u, ∂ α E ∂ α Θ , u∂ α u + 2 β<α

Cα n∂ u, ∂ u × ∂

α−β



G −

 β<α

Cαβ

n θ

∂ α Θ , ∂ α−β u∂ β u



≤ −2 ⟨∂ α (nu) , ∂ α E ⟩ + C ∥(u, Θ , E , G)∥s ∥(N , ∇ u, Θ )∥2s , and 2 ⟨A0 (n, θ) ∂ α U , ∂ α KII (U )⟩ = 2 ⟨∂ α u, ∂ α 1u⟩ + 2

 β<α

α

2

≤ −2 ∥∂ ∇ u∥ − 3

n θ



Cαβ n∂ α u, ∂ α−β

 

 n  ∂ β 1u − 3 , |∂ α Θ |2 n θ

1

 , |∂ α Θ |2 + C ∥N ∥s ∥∇ u∥2s .

Then, 2 ⟨A0 (n, θ) ∂ α U , ∂ α [KI (W ) + KII (U )]⟩ ≤ −2 ⟨∂ α (nu) , ∂ α E ⟩ − 2 ∥∂ α ∇ u∥ − 3 2

+ C ∥(N , Θ , E , G)∥s ∥(N , ∇ u)∥ . 2 s

n θ

, |∂ α Θ |2

 (2.17)

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Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

On the other hand, an easy high order energy estimate for the Maxwell equations of (2.3) gives d  dt

 ∥∂ α E ∥2 + ∥∂ α G∥2 = 2 ⟨∂ α (nu) , ∂ α E ⟩ .

(2.18)

It follows from (2.12)–(2.18) that d 

n   ⟨A0 (n, θ) ∂ α U , ∂ α U ⟩ + ∥∂ α E ∥2 + ∥∂ α G∥2 + 2 ∥∂ α ∇ u∥2 + 3 , |∂ α Θ |2 dt θ ≤ C ∥W ∥s ∥(N , ∇ u, Θ )∥2s .

(2.19)

Noting the fact that

⟨A0 (n, θ) ∂ α U , ∂ α U ⟩ + ∥∂ α E ∥2 + ∥∂ α G∥2 ∼ ∥∂ α W ∥2

and

n θ

 , |∂ α Θ |2 ∼ ∥∂ α Θ ∥2 ,

summing (2.19) for all α with |α| ≤ s, and then integrating over [0, t ], we obtain (2.10).



Estimate (2.10) stands for the dissipation of ∇ u and Θ . It is clear that this estimate is not sufficient to control the higher order term on the right hand side of (2.10) and the dissipation estimates of N are necessary. Lemma 2.4. Under the assumptions of Lemma 2.3, there exist positive constants C1 and C2 , independent of t and T , such that

∥W (t )∥2s +

t



∥(N , ∇ u, Θ ) (τ )∥2s dτ ≤ C1 ∥W0 ∥2s + C2

t



0

∥W (τ )∥s ∥(N , ∇ u, Θ ) (τ )∥2s dτ ,

∀t ∈ [0, T ].

(2.20)

0

Proof. For α ∈ N3 with |α| ≤ s − 1, applying ∂ α to the second equation of (2.3), and then taking the inner product of the  resulting equation with ∇∂ α N in L2 R3 yields



θ n

α

, |∇∂ N |

2



+ ⟨∇∂ α N , ∂ α E ⟩ = −

d dt

⟨∇∂ α N , ∂ α u⟩ + ⟨∇∂ α ∂t N , ∂ α u⟩ − I1 (t ) −

 β<α

Cαβ I2 (t ),

(2.21)

where

  α     ∂ 1u α α α α α ⟨ ⟩ ⟨∇∂ ⟩ , ∇∂ N + ∂ α u × B¯ + G , ∇∂ α N , I1 (t ) = u∇∂ u, ∇∂ N + Θ , ∇∂ N − n       θ I2 (t ) = ∂ α−β ∇∂ β N , ∇∂ α N + ∂ α−β u∇∂ β u, ∇∂ α N n      β  1 + ∂ u × ∂ α−β G, ∇∂ α N − ∂ α−β ∂ β 1u, ∇∂ α N . n

First, noting that n = b + N , θ = θl + Θ ≥

min{b,θl } 2

≥ const. > 0, one has     θ θ , |∇∂ α N |2 + ⟨∇∂ α N , ∂ α E ⟩ = , |∇∂ α N |2 − ⟨∂ α N , ∂ α ∇ · E ⟩ n n   θ = , |∇∂ α N |2 + ∥∂ α N ∥2

θ n

≥ C −1 . Hence,

n

  ≥ C −1 ∥∂ α N ∥2 + ∥∂ α ∇ N ∥2 .

(2.22)

By the first equation of (2.3) and an integration by parts, it follows that

⟨∇∂ α ∂t N , ∂ α u⟩ = − ⟨∂ α ∂t N , ∂ α ∇ · u⟩    β  α−β β  = ⟨u · ∂ α ∇ N , ∂ α ∇ · u⟩ + n, |∂ α ∇ · u|2 + Cα ∂ u∂ ∇ N + ∂ α−β N ∂ β ∇ · u, ∂ α ∇ · u β<α

2 s −1

≤ C ∥∇ u∥

+ C ∥(N , u)∥s ∥(N , ∇ u)∥ . 2 s

(2.23)

When |α| = 0, using an integration by parts, we get

         1u |I1 (t )| = ⟨u∇ u, ∇ N ⟩ + ⟨∇ Θ , ∇ N ⟩ − , ∇ N + u × B¯ + G , ∇ N  n ≤ ∥u∥∞ ∥∇ N ∥ (∥∇ u∥ + ∥G∥) + ε ∥∇ N ∥2 + C ∥∇ Θ ∥2 + ε ∥N ∥2 + C ∥∇ u∥21 ≤ ε ∥N ∥21 + C ∥(∇ u, Θ )∥21 + C ∥(N , G)∥2 ∥(N , ∇ u)∥21 .

(2.24)

Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

113

When |α| ≥ 1, we obtain similarly

|I1 (t ) + I2 (t )| ≤ ε ∥N ∥2s + C ∥(∇ u, Θ )∥2s + C ∥(N , u, Θ , G)∥s ∥(N , ∇ u)∥2s .

(2.25)

Then, combining (2.21)–(2.25) yields C −1 ∥∂ α N ∥ + ∥∂ α ∇ N ∥



2

2



+

d dt

⟨∇∂ α N , ∂ α u⟩ ≤ ε ∥N ∥2s + C ∥(∇ u, Θ )∥2s + C ∥(N , u, Θ , G)∥s ∥(N , ∇ u)∥2s .

Summing up this inequality for all |α| ≤ s − 1 and choosing ε > 0 small enough, the term ε∥N ∥2s can be controlled by the left hand side. Hence, integrating the resulting equation over [0, t ], we have t





t



t

∥(N , u, Θ , G) (τ )∥s ∥(N , ∇ u) (τ )∥2s dτ , ∥(∇ u, Θ ) (τ )∥ dτ + C 0 0  [⟨∇∂ α N0 , ∂ α u0 ⟩ − ⟨∇∂ α N (t ), ∂ α u(t )⟩] . +

∥N (τ )∥ dτ ≤ C 2 s

0

2 s

|α|≤s−1

Finally,

⟨∇∂ α N0 , ∂ α u0 ⟩ ≤ C ∥u0 ∥s−1 ∥N0 ∥s ≤ C ∥W0 ∥2s , and

⟨∇∂ α N (t ), ∂ α u(t )⟩ ≤ C ∥u (t )∥s−1 ∥N (t )∥s ≤ C ∥W (t )∥2s . Thus, together with (2.10), we obtain (2.20).



Proof of the global existence of solutions in Theorem 1.1. By Lemma 2.4, we deduce that if C2 ωT < 1, the integral term on the right hand side of (2.20) can be controlled by that of the left hand side. It follows that 1

∥W (t )∥s ≤ C12 ∥W0 ∥s ,

∀t ∈ [0, T ] .

Then, it is sufficient to choose initial data ∥W 0 ∥s ≤ δ0 with the constant δ0 satisfying 1

C12 δ0 < min



min{b, θl } 2Cm

,

1



C2

,

min{b,θ }

l which ensures both ωT ≤ and C2 ωT < 1. Thus, the global existence of smooth solutions follows from the 2Cm local existence result given in Proposition 1.1 and a standard argument on the continuous extension of local solutions. See [29]. 

3. Large time behavior of smooth solutions 3.1. Dissipation of the electromagnetic fields The large timebehavior of smooth solutions follows from uniform energy estimates of N , ∇ u, Θ , ∇ E and ∇ 2 G with  ′

respect to T in L2 [0, T ] ; H s R3





for appropriate integers s′ ≥ 1. We will establish these estimates in the following two

lemmas. Lemma 3.1. Under the assumptions of Lemma 2.3, there exists a small constant ε > 0, such that for all t ∈ [0, T ], t



∥∇ E (τ )∥2s−2 dτ ≤ C ∥W0 ∥2s + ε 0

 0

t

 2  ∇ G (τ )2

s−3

dτ

t



  ∥W (τ )∥s ∥(N , ∇ u, Θ ) (τ )∥2s + ∥∇ E (τ )∥2s−2 dτ .

+C

(3.1)

0

Proof. For α ∈ N3 with 1 ≤ |α| ≤ s − 1, applying ∂ α to the second equation of (2.3) and taking the inner product of the resulting equations with ∂ α E in L2 (R3 ), we have

∥∂ α E ∥2 = −

d dt

⟨∂ α u, ∂ α E ⟩ + ⟨∂ α u, ∂ α ∂t E ⟩ − R1 (t ) −

 β<α

Cαβ R2 (t ) ,

(3.2)

where R1 (t ) = ⟨∂ α ∇ Θ , ∂ α E ⟩ +



θ n

  α      ∂ 1u α ∂ α ∇ N , ∂ α E + ⟨u∇∂ α u, ∂ α E ⟩ + ∂ α u × B¯ + G , ∂ α E − ,∂ E , n

114

Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

and



R2 (t ) = ∂ α−β

  θ n

         1 ∇∂ β N , ∂ α E + ∂ α−β u∇∂ β u, ∂ α E + ∂ β u × ∂ α−β G, ∂ α E − ∂ α−β ∂ β 1u, ∂ α E . n

From the fourth equation of (2.3), we deduce that

   β α  ⟨∂ α u, ∂ α ∂t E ⟩ = ⟨∂ α u, ∂ α ∇ × G⟩ + n, |∂ α u|2 + Cα ∂ u, ∂ α−β N ∂ β u β<α

 2 ≤ ε ∇ 2 Gs−3 + C ∥∇ u∥2s−1 + C ∥N ∥s ∥∇ u∥2s .

(3.3)

Similarly as before, we obtain

|R1 (t )| +

 β<α

Cαβ |R2 (t )| ≤ ε ∥∂ α E ∥ + C ∥(N , ∇ u, Θ )∥2s + C ∥(E , G)∥s ∥(N , ∇ u, Θ )∥2s + ∥∇ E ∥2s−2 . 2





(3.4)

It follows from (3.2)–(3.4) that

∥∂ α E ∥2 ≤ −

d dt

 2   ⟨∂ α u, ∂ α E ⟩ + ε ∇ 2 Gs−3 + C ∥(N , ∇ u, Θ )∥2s + C ∥(N , E , G)∥s ∥(N , ∇ u, Θ )∥2s + ∥∇ E ∥2s−2 . (3.5)

Note that for all t ∈ [0, T ],

|⟨∂ α u, ∂ α E ⟩| ≤ C ∥W (t )∥2s ,

∀α, 1 ≤ |α| ≤ s − 1.

Let ε > 0 be small enough. Integrating (3.5) over [0, t ] and summing for all 1 ≤ |α| ≤ s − 1, together with (2.20), we obtain (3.1).  Lemma 3.2. Under the assumptions of Lemma 2.3, for all t ∈ [0, T ], it holds

 t  t  2    ∥∇ E (τ )∥2s−2 + ∇ 2 G (τ )s−3 dτ ≤ C ∥W0 ∥2s + C ∥W (τ )∥s ∥(N , ∇ u, Θ ) (τ )∥2s + ∥∇ E (τ )∥2s−2 dτ . (3.6) 0

0

Proof. For α ∈ N3 with 1 ≤ |α| ≤ s − 2, applying ∂ α to the fourth equation of (2.3) and taking the inner product of the resulting equation with −∇ × ∂ α G, one has

∥∇ × ∂ α G∥2 = = ≤ ≤

d dt d dt d dt d dt

⟨∇ × ∂ α G, ∂ α E ⟩ − ⟨∇ × ∂ α ∂t G, ∂ α E ⟩ − ⟨∇ × ∂ α G, ∂ α (nu)⟩ ⟨∇ × ∂ α G, ∂ α E ⟩ + ∥∇ × ∂ α E ∥2 − ⟨∇ × ∂ α G, n∂ α u⟩ −

 β<α

Cαβ ∇ × ∂ α G, ∂ α−β N ∂ β u





⟨∇ × ∂ α G, ∂ α E ⟩ + ∥∇ × ∂ α E ∥2 + ε ∥∇ × ∂ α G∥2 + C ∥∂ α u∥2 + C ∥G∥s ∥(N , ∇ u)∥2s  2 ⟨∇ × ∂ α G, ∂ α E ⟩ + ∇ 2 E s−3 + ε ∥∇ × ∂ α G∥2 + C ∥∇ u∥2s−3 + C ∥G∥s ∥(N , ∇ u)∥2s .

(3.7)

Note that for all t ∈ [0, T ],

|⟨∇ × ∂ α G, ∂ α E ⟩| ≤ C ∥W (t )∥2s ,

∀α, 1 ≤ |α| ≤ s − 2.

Let ε > 0 be small enough, integrating (3.7) over [0, T ] and summing for all 1 ≤ |α| ≤ s − 2, together with (3.1), we get (3.6). In this estimate we have used

  ∥∂ α ∂i G∥ = ∂i ∆−1 ∇ × (∇ × ∂ α G) ≤ C ∥∇ × ∂ α G∥ for 1 ≤ i ≤ 3, due to the fact that ∇ · G = 0 and ∂i ∆−1 ∇ is bounded from Lp to Lp with 1 < p < ∞, see [32].



3.2. Proof of the large time behavior of solutions in Theorem 1.1 Recall the following lemma that is used in the following proof. Lemma 3.3. Let f : (0, +∞) → R be a uniformly continuous function such that f ∈ L1 (0, +∞). Then limt →+∞ f (t ) = 0. In particular, the conclusion holds when f ∈ L1 (0, +∞) ∩ W 1,+∞ (0, ∞).

Y.-H. Feng et al. / Nonlinear Analysis: Real World Applications 19 (2014) 105–116

115

Now, we combine all the proceeding estimates to establish the large time behavior of solutions to complete the proof of Theorem 1.1. From Lemma 2.4, there exists a constant δ0 such that if ωT ≤ δ0 , it holds

∥W (t )∥2s +

t



∥(N , ∇ u, Θ ) (τ )∥2s dτ ≤ C ∥W0 ∥2 ,

(3.8)

0

which implies

   ∩ L∞ (0, +∞) ; H s R3 ,       ∇ u ∈ L2 (0, +∞) ; H s R3 ∩ L∞ (0, +∞) ; H s−1 R3 . n − b, θ − θl ∈ L2 (0, +∞) ; H s R3







Using the first three equations of (2.3), we obtain

   ∂t (n − b) , ∂t (θ − θl ) ∈ L∞ (0, +∞) ; H s−1 R3 ,    ∂t ∇ u ∈ L∞ (0, +∞) ; H s−3 R3 . Therefore,

   ∩ W 1,∞ (0, +∞) ; H s−1 R3 ,       ∇ u ∈ L2 (0, +∞) ; H s−3 R3 ∩ W 1,∞ (0, +∞) ; H s−3 R3 ,

n − b, θ − θl ∈ L2 (0, +∞) ; H s−1 R3







which imply (1.7). Similarly as before, from (3.6) and (3.8), we get

      ∇ E ∈ L2 (0, +∞) ; H s−2 R3 ∩ L∞ (0, +∞) ; H s−1 R3 ,       ∇ 2 G ∈ L2 (0, +∞) ; H s−3 R3 ∩ L∞ (0, +∞) ; H s−2 R3 . It follows from the Maxwell equations in (2.3) that

      ∂t ∇ E ∈ L2 (0, +∞) ; H s−3 R3 ∩ L∞ (0, +∞) ; H s−2 R3 . Therefore,

      ∇ E ∈ L2 (0, +∞) ; H s−2 R3 ∩ W 1,∞ (0, +∞) ; H s−2 R3 , which implies (1.8). We further deduce that

      ∂t G = −∇ × E ∈ L2 (0, +∞) ; H s−2 R3 ∩ W 1,∞ (0, +∞) ; H s−2 R3 . Then

      ∂t ∇ 2 G ∈ L2 (0, +∞) ; H s−4 R3 ∩ W 1,∞ (0, +∞) ; H s−4 R3 , which implies (1.9). Acknowledgments The first and third authors were partially supported by the National Basic Research Program of China (973 Program, 2011CB808002), NSFC (No. 11371042), BNSF (No. 1132006), the fund of the Beijing education committee of China, the state scholarship fund (No. 201206540015) of CSC and the Foundation Project of Doctor Graduate Student Innovation of Beijing University of Technology of China (No. ykj-2012-6724). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

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