Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force

J. Math. Anal. Appl. 427 (2015) 600–617

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Large time behavior of the compressible magnetohydrodynamic equations with Coulomb force ✩ Zhong Tan, Leilei Tong ∗ , Yong Wang School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China

a r t i c l e

i n f o

Article history: Received 1 January 2015 Available online 27 February 2015 Submitted by D. Wang Keywords: MHD equations Optimal decay rates Energy method

a b s t r a c t In this paper, we consider the large time behavior of the solutions near a constant equilibrium state to the Cauchy problem for the compressible magnetohydrodynamic equations with Coulomb force in R3 . Under the assumptions that the H 3 norm of the initial data is small, but its higher order derivatives could be large, we obtain the optimal time decay rates of the solutions and their higher-order spatial derivatives by introducing the negative Sobolev and Besov spaces. As an immediate byproduct, the Lp –L2 (1  p  2) type of the decay rates follows without requiring the smallness for Lp norm of initial data. © 2015 Published by Elsevier Inc.

1. Introduction We consider the following compressible viscous magnetohydrodynamic equations with Coulomb force in the isentropic case: ⎧ ρt + div(ρv) = 0, ⎪ ⎪ ⎪ ⎨ (ρv) + div(ρv ⊗ v) + ∇P = curlB × B + μΔv + (λ + μ)∇divv + ρ∇Φ, t ⎪ B − curl(v × B) − νΔB = 0, divB = 0, t ⎪ ⎪ ⎩ ΔΦ = ρ − ρ¯,

(1.1)

for (x, t) ∈ R3 × [0, ∞). Here ρ = ρ(x, t), v = v(x, t), B = B(x, t) and Φ = Φ(x, t) represent the density, velocity, magnetic field and electric potential respectively. The pressure P = P (ρ) is a smooth function with P  (ρ) > 0 for ρ > 0. μ, λ are the viscosity coefficients of the flow satisfying μ > 0 and 2μ + 3λ  0. The ✩

This work was supported by the National Natural Science Foundation of China (No. 11271305).

* Corresponding author. E-mail addresses: [email protected] (Z. Tan), [email protected] (L.L. Tong), [email protected] (Y. Wang). http://dx.doi.org/10.1016/j.jmaa.2015.02.077 0022-247X/© 2015 Published by Elsevier Inc.

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constant ν > 0 is the magnetic diffusivity acting as a magnetic diffusion coefficient of the magnetic field. ρ¯ is the positive constant background ionic density. We complement (1.1) with the Cauchy data (ρ, v, B, ∇Φ)(x, 0) = (ρ0 (x), v0 (x), B0 (x), ∇Φ0 (x)) → (¯ ρ, 0, 0, 0) as |x| → ∞.

(1.2)

When we neglect the Coulomb force, (1.1) will become the compressible MHD equations, and then there are many results about it. First, we can refer to [2–4,10,18,19,34] for the 1-D or 2-D case. For the three-dimensional case, we can refer to [1,5,6,11,13,22,23,29,32,33] for the local or global existence and the convergence rates of weak, strong and smooth solutions and to [12,14,15,17] for the incompressible limit. When we neglect the effect of the magnetic field, (1.1) will become the compressible Navier–Stokes–Poisson equations, and then there are also many works about it. For instance, we can refer to [9,20,21,30,31,35,36] and the references cited therein. Considering the common effects of the Coulomb force and the magnetic field, Tan and Wang [28] showed the global existence of weak solutions to (1.1) in the framework of Lions–Feireisl for the compressible Navier–Stokes equations [7,24]. Motivated by [8,29,35], we will investigate the optimal time decay rates of smooth solutions to (1.1)–(1.2) in this paper. To obtain the optimal time decay rates of solutions, we −s additionally introduce the homogeneous Besov space B˙ 2,∞ (0 < s  3/2). Notations Throughout this paper, ∇l with an integer l > 0 stands for the usual any spatial derivatives of order l. When l  0, ∇l stands for Λl defined by Λl f := F −1 (|ξ| F f ), where F is the usual Fourier transform operator and F −1 is its inverse. The norm of the Lp (R3 ) spaces is denoted by  · Lp . We use H s (R3 ) to denote the usual Sobolev spaces with  · H s . And we use H˙ s (R3 ), s ∈ R, to denote the homogeneous Sobolev spaces on R3 with norm  · H˙ s defined by f H˙ s := ∇s f L2 . Now we review the homogeneous Besov spaces. Let φ ∈ C0∞ (R3ξ ) be such that φ(ξ) = 1 when |ξ|  1 and φ(ξ) = 0 when  |ξ|  2. Let ϕ(ξ) = φ(ξ) − φ(2ξ) and ϕj (ξ) = ϕ(2−j ξ) for j ∈ Z. Then by the construction, j∈Z ϕj (ξ) = 1 ˙ j f := F −1 (ϕj ) ∗ f , then for s ∈ R and 1  p, r  ∞, we define the homogeneous if ξ = 0. We define Δ  1/r s rjs ˙ r Besov spaces B˙ p,r (R3 ) with norm  · B˙ p,r by f B˙ p,r := 2  Δ f  . Particularly, if r = ∞, p s s j L j∈Z sj ˙ then f  ˙ s := sup 2 Δj f Lp . We employ the notation A  B to mean that A  CB for a constant Bp,∞

j∈Z

C > 0. We use C0 for a positive constant depending additionally on the initial data. For simplicity, we write (A, B)X := AX + BX . The main results of this paper can be stated as follows. Theorem 1.1. Assume that (ρ0 − ρ¯, v0 , B0 , ∇Φ0 ) ∈ H N with N  3, there exists a constant δ0 > 0 such that if (ρ0 − ρ¯, v0 , B0 , ∇Φ0 )H 3  δ0 ,

(1.3)

then the problem (1.1)–(1.2) admits a unique global solution (ρ(t), v(t), B(t), ∇Φ(t)) satisfying that for all t  0, t (ρ −

ρ¯, v, B, ∇Φ)(t)2H N

(ρ − ρ¯, ∇v, ∇B, ∇∇Φ)(τ )2H N dτ  C(ρ0 − ρ¯, v0 , B0 , ∇Φ0 )2H N .

+

(1.4)

0 −s Moreover, if (ρ0 − ρ¯, v0 , B0 , ∇Φ0 ) ∈ H˙ −s for some s ∈ [0, 3/2), or (ρ0 − ρ¯, v0 , B0 , ∇Φ0 ) ∈ B˙ 2,∞ for some s ∈ (0, 3/2], then for all t  0

(ρ − ρ¯, v, B, ∇Φ)(t)2H˙ −s  C0 ,

(1.5)

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or (ρ − ρ¯, v, B, ∇Φ)(t)2B˙ −s  C0 ,

(1.6)

2,∞

and here are the following decay results: ∇l (ρ − ρ¯, v, B, ∇Φ)(t)H N −l  C0 (1 + t)−

l+s 2

for l = 0, 1, . . . , N − 1,

(1.7)

and ∇l (ρ − ρ¯)(t)L2  C0 (1 + t)−

l+1+s 2

for l = 0, 1, . . . , N − 2.

(1.8)



 −s Note that Lp ⊂ H˙ −s with s = 3 p1 − 12 ∈ [0, 3/2) (Lemma 2.4), and Lp ⊂ B˙ 2,∞ with s = 3 p1 − 12 ∈ (0, 3/2] (Lemma 2.5). Then by Theorem 1.1, we have the following Lp –L2 type of the optimal decay results: −s Corollary 1.2. Under the assumptions of Theorem 1.1 except that we replace the H˙ −s or B˙ 2,∞ assumption p by (ρ0 − ρ¯, v0 , B0 , ∇Φ0 ) ∈ L for some p ∈ [1, 2], then the following decay results hold:

∇l (ρ − ρ¯, v, B, ∇Φ)(t)H N −l  C0 (1 + t)−σp,l

for l = 0, 1, . . . , N − 1,

and ∇l (ρ − ρ¯)(t)L2  C0 (1 + t)−(σp,l + 2 ) for l = 0, 1, . . . , N − 2. 1

Here σp,l :=



3 1 2 p

−

1 2



+ 2l .

The following are several remarks for Theorem 1.1 and Corollary 1.2. Remark 1.3. We only need to assume that the H 3 norms of the initial density, velocity, magnetic field are small, while the higher order Sobolev norms can be arbitrarily large. We claim that the decay results of (ρ0 − ρ¯, v0 , B0 , ∇Φ0 ) in Theorem 1.1 are optimal in the sense that they are consistent with those in the linearized case. Remark 1.4. Note that the L2 decay rate of the higher order spatial derivatives of the solution is obtained. Then the general optimal Lq (2  q  ∞) decay rates of the solution follow by the Sobolev interpolation. Remark 1.5. From Corollary 1.2, we obtain the optimal Lp –L2 (1  p  2) type of the decay rates without requiring that the Lp norm of initial data is small. The rest of the paper is structured as follows. In Section 2, we will give some useful lemmas. In Section 3, we will establish the refined energy estimates and derive the negative Sobolev and Besov estimates. We will prove Theorem 1.1 in Section 4. 2. Preliminaries In this section, we will give some lemmas to be used in the next sections. Lemma 2.1. Let 0  m, α  l and 2  p  ∞, then we have 1−θ l θ ∇α f Lp  ∇m f L 2 ∇ f L2 ,

(2.1)

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where 0  θ  1 and α satisfy 1 1 − α+3 = m(1 − θ) + lθ. 2 p Here we require that 0 < θ < 1, m  α + 1 and l  α + 2, when p = ∞. Proof. See [25, p. 125, Theorem]. 2 Lemma 2.2. Let m  1 be an integer and define the commutator [∇m , f ]g = ∇m (f g) − f ∇m g.

(2.2)

[∇m , f ]gLp  ∇f Lp1 ∇m−1 gLp2 + ∇m f Lp3 gLp4 ,

(2.3)

∇m (f g)Lp  f Lp1 ∇m gLp2 + ∇m f Lp3 gLp4 ,

(2.4)

Then we have

and for m  0

where p, p2 , p3 ∈ (1, ∞) and

1 p

Proof. See Lemma 3.1 in [16].

=

1 p1

+

1 p2

1 p3

=

+

1 p4 .

2

Lemma 2.3. Assume that  L∞  1 and 2  p  ∞. Let g( ) be a smooth function of with bounded derivatives of any order, then for any integer m  1, we have ∇m (g( ))Lp  ∇m Lp .

(2.5)

Proof. Notice that for m  1, ∇m (g( )) = a sum of products g α1 ,...,αn ( )∇α1 ∇α2 · · · ∇αn , where the functions g α1 ,...,αn ( ) are some derivatives of g( ) and 1  αi  m, i = 1, . . . , n with α1 + · · · + αn = m. By Hölder’s inequality and Lemma A.1 of [35], we obtain ∇m (g( ))Lp  ∇α1 ∇α2 · · · ∇αn Lp  ∇α1 

mp

L α1

1−

α1

∇α2 

mp

L α2

α1

1−

· · · ∇αn 

mp

L αn

α2

α2

1− αn

αn

  L∞ m ∇m Lmp  L∞ m ∇m Lmp · · ·  L∞ m ∇m Lmp m   n−1 L∞ ∇ Lp .

Hence, we conclude our lemma since  L∞  1. 2 Lemma 2.4. Let 1 < p  2, and 1/2 + s/3 = 1/p, then 0  s < 3/2, and f H˙ −s  f Lp . Proof. It follows from the Hardy–Littlewood–Sobolev theorem, see [27, p. 119, Theorem 1]. 2

(2.6)

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−s Lemma 2.5. Suppose that s > 0 and 1  p < 2, then we have the embedding Lp ⊂ B˙ 2,∞ where s, p satisfy 1/2 + s/3 = 1/p. In particular −s  f Lp . f B˙ 2,∞

(2.7)

2

Proof. See Lemma 4.1 in [26].

Lemma 2.6. Let s  0 and l  0, then we have 1−θ −s ∇l f L2  ∇l+1 f L f θL2 , where θ = 2 Λ

1 . l+s+1

(2.8)

2

Proof. See Lemma A.4 in [8].

The following lemma is known: Lemma 2.7. If 1  r1  r2  ∞, then s s B˙ p,r ⊂ B˙ p,r . 1 2

(2.9)

Lemma 2.8. Let s > 0 and l  0, then we have 1−ϑ ϑ ∇l f L2  ∇l+1 f L 2 f  ˙ −s , where ϑ = B 2,∞

1 . l+s+1

−s −s Proof. We refer to Lemma 4.2 in [26] by noting that B˙ 2,p ⊂ B˙ 2,q for p  q.

(2.10)

2

We will give the special Besov interpolation estimates: Lemma 2.9. Fix m > l  k, and 1  p  q  r  ∞, then we have gB˙ 2,q  gϑB˙ k g1−ϑ , l B˙ m 2,r

where l = kϑ + m(1 − ϑ) and

1 q

Proof. See Lemma 4.3 in [26].

=

ϑ r

+

(2.11)

2,p

1−ϑ p .

2

3. Nonlinear energy estimates 3.1. Energy estimates Using some vector analysis formula, (1.1) can be expressed as ⎧ ρt + ρdivv + v · ∇ρ = 0, ⎪ ⎪ ⎪ ⎨ ρvt + ρv · ∇v − μΔv − (λ + μ)∇divv + ∇P = B · ∇B − 12 ∇(|B|2 ) + ρ∇Φ, ⎪ ⎪ Bt − νΔB = B · ∇v − v · ∇B − Bdivv, divB = 0, ⎪ ⎩ ΔΦ = ρ − ρ¯. Define γ=



P  (¯ ρ), μ1 =

μ λ+μ , μ2 = , ρ¯ ρ¯

(3.1)

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and introduce the unknowns

= ρ − ρ¯, u =

ρ¯ v. γ

Then Eqs. (3.1) are reformulated as ⎧ + γdivu = G , t 1 ⎪ ⎪ ⎪ ⎨ ut − μ1 Δu − μ2 ∇divu + γ∇ − γρ¯ ∇Φ = G2 , ⎪ ⎪ Bt − νΔB = G3 , divB = 0, ⎪ ⎩ ΔΦ = ,

(3.2)

with γ G1 = − ( divu + u · ∇ ), ρ¯ 



γ λ+μ μ G2 = − u · ∇u + − μ1 Δu + − μ2 ∇divu ρ¯

+ ρ¯

+ ρ¯

 ρ¯ P  (¯ ρ¯ ρ¯ ρ) P  ( + ρ¯) + − ∇ + B · ∇B − ∇(|B|2 ), γ ρ¯

+ ρ¯ γ( + ρ¯) 2γ( + ρ¯) γ G3 = (B · ∇u − u · ∇B − Bdivu). ρ¯ The initial data (1.2) becomes ( , u, B, ∇Φ)(x, 0) = ( 0 (x), u0 (x), B0 (x), ∇Φ0 (x)) ρ¯ = (ρ0 (x) − ρ¯, v0 (x), B0 (x), ∇Φ0 (x)). γ

(3.3)

In this section, to derive the a priori nonlinear energy estimates for (3.2)–(3.3), we assume a priori that for sufficiently small δ > 0 

E03 =  (t)H 3 + u(t)H 3 + B(t)H 3 + ∇Φ(t)H 3  δ.

(3.4)

This together with Sobolev’s inequality implies in particular that ρ¯  + ρ¯  2¯ ρ. 2

(3.5)

This will be kept in mind in the rest of this paper. Now, we first establish the energy estimates for (u, B). Lemma 3.1. For 0  k  N , we have d ∇k ( , u, B, ∇Φ)2L2 + C∇k+1 (u, B)2L2  δ(∇k 2L2 + ∇k+1 (u, B, ∇Φ)2L2 ). dt

(3.6)

Proof. Applying ∇k to (3.2)1 , (3.2)2 , (3.2)3 and multiplying by ∇k , ∇k u, ∇k B respectively, summing them up and then integrating over R3 by parts, we get

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ρ¯ 1 d ∇k ( , u, B)2L2 + μ1 ∇k+1 u2L2 + μ2 ∇k divu2L2 + ν∇k+1 B2L2 − 2 dt γ

∇k G1 ∇k dx +

= R3

∇k G2 ∇k udx +

R3

∇k ∇Φ · ∇k udx R3

∇k G3 ∇k Bdx.

(3.7)

R3

By the continuity equations (3.2)1 and (3.2)4 , and integrating by parts, we get −

∇k ∇Φ · ∇k udx =

R3

∇k Φ · ∇k divudx = − R3

1 = ρ¯ =

We now bound the term

1 ρ¯

then we get



1 ∇ ∇Φ∇ ( u)dx + γ k

R3

1 γ

R3



∇∇k Φ · ∇∇k Φt dx

k

1 d 1 ∇k ∇Φ2L2 + 2γ dt ρ¯



γ ∇k Φ · ∇k [ div( u) + t ]dx ρ¯

R3

∇k ∇Φ · ∇k ( u)dx.

(3.8)

R3

∇k ∇Φ · ∇k ( u)dx. If k = 0, by Eq. (3.2)4 , Hölder’s and Sobolev’s inequalities, R3



∇Φ · udx = R3

∇Φ · uΔΦdx  ∇ΦL3 uL6 ΔΦL2  δ(∇u2L2 + ∇∇Φ2L2 ). R3

If 1  k  N , by Eq. (3.2)4 , Hölder’s and Sobolev’s inequalities, Lemma 2.1 and product estimates (2.4) of Lemma 2.2, then we have

∇ ∇Φ · ∇ ( u)dx = − k

k

R3

∇k−1 (ΔΦu) · ∇k+1 ∇Φdx

R3

 ∇k−1 (ΔΦu)L2 ∇k+1 ∇ΦL2    ∇k−1 ΔΦL6 uL3 + ΔΦL3 ∇k−1 uL6 ∇k+1 ∇ΦL2  k 1 1 k  k+1 k+1 k+1 k+1 2k ∇Φ  δ∇k+1 ∇ΦL2 + ∇k+1 uLk+1 ∇ΦLk+1 ∇k+1 ∇ΦL2 2 uL2 ∇ 2 ∇ L2  δ(∇k+1 ∇Φ2L2 + ∇k+1 u2L2 ). Thus, we have ∇k ∇Φ · ∇k ( u)dx  δ(∇k+1 ∇Φ2L2 + ∇k+1 u2L2 ).

(3.9)

R3

We deduce from (3.8)–(3.9) that −

ρ¯ γ

∇k ∇Φ · ∇k udx  R3

We deduce from (3.7) and (3.10) that

1 d ∇k ∇Φ2L2 − Cδ(∇k+1 ∇Φ2L2 + ∇k+1 u2L2 ). 2γ dt

(3.10)

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1 d 1 d ∇k ( , u, B)2L2 + ∇k ∇Φ2L2 + μ1 ∇k+1 u2L2 + μ2 ∇k divu2L2 + ν∇k+1 B2L2 2 dt 2γ dt k k k k  ∇ G1 ∇ dx + ∇ G2 ∇ udx + ∇k G3 ∇k Bdx + δ(∇k+1 (u, ∇Φ)2L2 ). R3

R3

(3.11)

R3

∇k G1 ∇k dx. By Hölder’s, Sobolev’s inequalities and the product estimates (2.4) of

First, we estimate Lemma 2.2, we have

R3

−

∇k ∇k ( divu)dx  ∇k L2 ∇k ( divu)L2

R3

 ∇k L2 (∇k L2 ∇uL∞ + ∇k+1 uL2  L∞ )  δ(∇k+1 u2L2 + ∇k 2L2 ).

(3.12)

∇k G1 ∇k dx, if k = 0, we have

For the second term of R3

2

−

2

(u∇ )dx   L2 uL6 ∇ L3  δ( L2 + ∇uL2 ),

(3.13)

R3

and for k  1, by the commutator notation (2.2) of Lemma 2.2, we rewrite it as −

∇k ∇k (u∇ )dx = −

R3

∇k (u · ∇∇k + [∇k , u] · ∇ )dx.

R3

By the integration by parts, Hölder’s and Sobolev’s inequalities, we have −

∇k (u · ∇∇k )dx = −

R3

R3

u·∇

1 |∇k |2 dx = 2 2

divu|∇k |2 dx R3

 ∇uL∞ ∇k 2L2  δ∇k 2L2 .

(3.14)

By the commutator estimates (2.3) of Lemma 2.2, we have −

∇k ([∇k , u] · ∇ )dx  (∇uL∞ ∇k−1 ∇ L2 + ∇k uL6 ∇ L3 )∇k L2

R3

 δ(∇k 2L2 + ∇k+1 u2L2 ).

(3.15)

We deduce from (3.13)–(3.15) that −

∇k ∇k (u · ∇ )dx  δ(∇k 2L2 + ∇k+1 u2L2 ).

(3.16)

R3

(3.12) and (3.16) imply ∇k G1 ∇k dx  δ(∇k 2L2 + ∇k+1 u2L2 ). R3

(3.17)

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∇k G2 · ∇k udx.

Now, we estimate the term

∇

First we bound

k

R3

R3

ρ¯ B · ∇B

+ ρ¯

 · ∇k udx. If k = 0, by Hölder’s and Sobolev’s inequalities, we obtain

      ρ¯   ρ¯      + ρ¯ B · ∇B, u    + ρ¯ 

L∞

∇BL2 BL6 uL3  δ∇B2L2 .

If 1  k  N , by the integration by parts, Hölder’s and Sobolev’s inequalities, together with Lemma 2.1, the product estimates (2.4) of Lemma 2.2 and Lemma 2.3, then we have

∇k R3

ρ¯ B · ∇B

+ ρ¯



· ∇k udx = −

∇k−1

R3

ρ¯ B · ∇B

+ ρ¯

 · ∇k+1 udx



   k−1 ρ¯  BL∞ ∇BL3 ∇k+1 uL2   ∇

+ ρ¯ L6    ρ¯  k−1  + (B · ∇B)L2 ∇k+1 uL2  + ρ¯  ∞ ∇ L  ∇k L2 BL∞ ∇BL3 ∇k+1 uL2

 + BL3 ∇k−1 ∇BL6 + ∇k−1 BL6 ∇BL3 ∇k+1 uL2  BL∞ ∇BL3 ∇k L2 ∇k+1 uL2 + BL3 ∇k+1 BL2 ∇k+1 uL2 1

k

k+1

k

1

k+1 k+1 k+1 k+1 2k B + BLk+1 BLk+1 BLk+1 uL2 2 ∇ 2 ∇ 2 ∇ L2 ∇

k 2  k+1 2 k+1 2  δ ∇ L2 + ∇ BL2 + ∇ uL2 .

Thus we have

∇

k

R3

ρ¯ B · ∇B

+ ρ¯



 · ∇k udx  δ ∇k 2L2 + ∇k+1 B2L2 + ∇k+1 u2L2 .

(3.18)

∇k u · ∇k (u · ∇u)dx  δ∇k+1 u2L2

(3.19)

Similarly, we have − R3

and

− R3

∇

k



 ρ¯ 2 ∇(|B| ) · ∇k udx  δ ∇k 2L2 + ∇k+1 u2L2 + ∇k+1 B2L2 . 2( + ρ¯)

(3.20)

 μ   − μ1 Δu · ∇k udx. By Hölder’s, Sobolev’s inequalities, Then we bound the term − ∇k

+ ρ¯ R3

Lemma 2.3 and the integration by parts, we get for k = 0 − R3

        μ  μ μ    − μ1 Δu · udx  ∇uL2 ∇ − μ1  uL6 + ∇uL2  − μ1   ∞ ∇uL2

+ ρ¯

+ ρ¯

+ ρ¯ L3 L  ∇ L3 ∇u2L2 +  L∞ ∇u2L2  δ∇u2L2 ,

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and for k = 1 

−

∇

R3

   μ μ − μ1 Δu · ∇udx = − μ1 Δu · ∇2 udx

+ ρ¯

+ ρ¯ R3

    μ  ∇2 uL2 ∇ − μ  ∇2 uL2  1  

+ ρ¯ L∞  δ∇2 u2L2 .

Integrating by parts and by (2.2)–(2.3) of Lemma 2.2 and Lemma 2.3, let f ( ) = for 2  k  N , −

− μ1 , then we have

 μ   − μ1 Δu dx = ∇k+1 u · ∇k−1 (f ( )Δu)dx ∇ u·∇

+ ρ¯ k

R3

μ

+ρ¯

k

R3

 ∇k+1 uL2 (f ( )∇k−1 ΔuL2 + [∇k−1 , f ( )]ΔuL2 )  ∇k+1 uL2 (f ( )L∞ ∇k−1 ΔuL2 + ∇f ( )L3 ∇k−2 ΔuL6 + ∇k−1 f ( )L6 ΔuL3 )   L∞ ∇k+1 u2L2 + ∇ L3 ∇k+1 u2L2 + ∇k L2 ∇2 uL3 ∇k+1 uL2  δ(∇k+1 u2L2 + ∇k 2L2 ). Thus we have −

∇k u · ∇k

R3

 μ   − μ1 Δu dx  δ(∇k+1 u2L2 + ∇k 2L2 ).

+ ρ¯

(3.21)

Like (3.21), we have 

−

∇k u · ∇k

R3

  λ+μ − μ2 ∇divu dx  δ(∇k+1 u2L2 + ∇k 2L2 )

+ ρ¯

and −

   P  ( + ρ¯) ∇ dx  δ(∇k+1 u2L2 + ∇k 2L2 ). ρ¯ − ∇ u·∇

+ ρ¯ k

R3

k

Thus ∇k G2 · ∇k udx  δ(∇k 2L2 + ∇k+1 u2L2 + ∇k+1 B2L2 ).

(3.22)

R3

Like (3.18), we obtain ∇k G3 · ∇k Bdx  δ(∇k+1 B2L2 + ∇k+1 u2L2 ). R3

Plugging the estimates (3.17), (3.22), (3.23) into (3.11), we prove (3.6) for 0  k  N . 2 The following lemma provides the dissipation estimates for and ∇Φ.

(3.23)

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Lemma 3.2. For 0  k  N − 1, we have d dt

∇k u · ∇∇k dx + C(∇k ( , ∇∇Φ)2L2 + ∇k+1 ( , ∇∇Φ)2L2 ) R3

 ∇k+1 (u, B)2L2 + ∇k+2 (u, B)2L2 .

(3.24)

Proof. Applying ∇k to (3.2)2 and then multiplying by ∇∇k , we get |∇k+1 |2 dx −

γ R3



ρ¯ γ

∇k ∇Φ · ∇∇k dx R3

∇k ut · ∇∇k dx + C∇k+1 L2 ∇k+2 uL2 + ∇k G2 L2 ∇k+1 L2 .

−

(3.25)

R3

Integrating by parts and (3.2)4 imply −

∇k ∇Φ · ∇∇k dx =

R3

∇k ΔΦ · ∇k dx =

R3

|∇k |2 dx. R3

The delicate first term in the right-hand side of (3.25) involves ∇k ut , by using the continuity equation (3.2)1 and integrating by parts for both the t and x variables, we obtain −

∇k ut · ∇∇k dx = − R3

=− =−

d dt d dt

d dt





∇k u · ∇∇k dx −

R3

R3



∇k u · ∇∇k dx − R3

∇k divu · ∇k (G1 − γdivu)dx R3



∇k divu · ∇k t dx

∇k u · ∇∇k dx + γ∇k+1 u2L2 + R3

γ ρ¯

∇k divu · ∇k (div( u))dx.

(3.26)

R3

Using Hölder’s, Sobolev’s inequalities and the product estimates (2.4) of Lemma 2.2, we obtain ∇k divu · ∇k (div( u))dx  ∇k+1 uL2 ∇k+1 ( u)L2 R3

 ∇k+1 uL2 (∇k+1 L2 uL∞ +  L∞ ∇k+1 uL2 )  δ(∇k+1 2L2 + ∇k+1 u2L2 ).

(3.27)

Thus, we obtain − R3

∇k ut · ∇∇k dx  −

d dt

∇k u · ∇∇k dx + C∇k+1 u2L2 + δ∇k+1 2L2 .

(3.28)

R3

On the other hand, note that it has been proved along the proof of Lemma 3.1 that ∇k G2 L2  δ(∇k+1 L2 + ∇k+2 uL2 + ∇k+2 BL2 ).

(3.29)

Plugging the estimates (3.28)–(3.29) into (3.25), by Cauchy’s inequality, since δ is small, and the fact that

Z. Tan et al. / J. Math. Anal. Appl. 427 (2015) 600–617

∇k+1 ∇Φ2L2 = ∇k ΔΦ2L2 = ∇k 2L2 ,

611

∇k+2 ∇Φ2L2 = ∇k+1 2L2 ,

then we obtain (3.24). The proof of Lemma 3.2 is completed. 2 3.2. Negative Sobolev estimates In this subsection, we will derive the evolution of the negative Sobolev norms of the solution. Lemma 3.3. For s ∈ (0, 1/2], we have d −s Λ ( , u, B, ∇Φ)2L2 + C∇Λ−s (u, B)2L2 dt  ( 2H 2 + ∇(u, B)2H 1 )Λ−s ( , u, B, ∇Φ)L2 ,

(3.30)

and for s ∈ (1/2, 3/2), we have d −s Λ ( , u, B, ∇Φ)2L2 + C∇Λ−s (u, B)2L2 dt s− 1

 ( , u, B)(t)L2 2 ( H 2 + ∇(u, B)H 1 ) 2 −s Λ−s ( , u, B, ∇Φ)L2 . 5

(3.31)

Proof. Applying Λ−s to (3.2)1 , (3.2)2 , (3.2)3 and multiplying the resulting identities by Λ−s , Λ−s u, Λ−s B respectively, summing them up and then integrating the result over R3 by parts, we get ρ¯ 1 d −s Λ ( , u, B)2L2 + μ1 ∇Λ−s u2L2 + μ2 Λ−s divu2L2 + ν∇Λ−s B2L2 − 2 dt γ



Λ−s ∇Φ · Λ−s udx

R3

 Λ−s G1 L2 Λ−s L2 + Λ−s G2 L2 Λ−s uL2 + Λ−s G3 L2 Λ−s BL2 .

(3.32)

If s ∈ (0, 1/2], then 1/2 + s/3  1 and 3/s  6. Using the estimate (2.6) of Riesz potential in Lemma 2.4 and the Sobolev interpolation of Lemma 2.1, together with Hölder’s and Young’s inequalities, we get Λ−s ( divu)L2   divu 1

  L3/s ∇uL2

1

L 1/2+s/3

+s

1

−s

 ∇ L2 2 ∇2 L2 2 ∇uL2  ∇ 2H 1 + ∇u2L2 .

(3.33)

Similarly to (3.33), we have Λ−s G1 L2 + Λ−s G2 L2 + Λ−s G3 L2   2H 2 + ∇(u, B)2H 1 .

(3.34)

If s ∈ (1/2, 3/2), then 1/2 + s/3 < 1 and 2 < 3/s < 6. Using the Sobolev interpolation, we obtain     −s Λ (B · ∇B)  

s− 1

 B · ∇B L2

1 L 1/2+s/3

3

−s

 BL3/s ∇BL2  BL2 2 ∇BL2 2 ∇BL2 .

(3.35)

Similarly to (3.35), we have s− 1

Λ−s G1 L2 + Λ−s G2 L2 + Λ−s G3 L2  ( , u, B)(t)L2 2 ( H 2 + ∇(u, B)H 1 ) 2 −s . 5

(3.36)

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For the remaining Poisson term, we use (3.2)1 , (3.2)4 and the integration by parts to obtain −

Λ−s ∇Φ · Λ−s udx =

R3

1 =− γ 1 = γ =

Λ

−s

Φ·Λ

−s

R3

Λ

Λ−s Φ · Λ−s divudx

R3







−s

∇Φ · Λ

R3

−s

γ ∂t dx + ρ¯

Λ

1 1 d −s Λ ∇Φ2L2 + 2γ dt ρ¯

Φ·Λ

−s

 div( u)dx

R3

1 ∂t ∇Φdx + ρ¯

−s



Λ−s ∇Φ · Λ−s ( u)dx

R3

Λ−s ∇Φ · Λ−s ( u)dx.

(3.37)

R3

About the second term on the right-hand side of (3.37), we have for s ∈ (0, 1/2], 1

1

+s

−s

Λ−s ( u)L2   L2 uL3/s   L2 ∇uL2 2 ∇2 uL2 2 ,

(3.38)

and for s ∈ (1/2, 3/2), s− 1

3

−s

Λ−s ( u)L2   L2 uL3/s   L2 uL2 2 ∇uL2 2 .

(3.39)

Consequently, plugging (3.34) and (3.37)–(3.38) with s ∈ (0, 1/2], and (3.36)–(3.37), (3.39) with s ∈ (1/2, 3/2) respectively into (3.32), we get (3.30)–(3.31). 2 3.3. Negative Besov estimates In this subsection, we will derive the evolution of the negative Besov norms of the solution. The argument is similar to the previous subsection. Lemma 3.4. For s ∈ (0, 1/2], we have d ( , u, B, ∇Φ)2B˙ −s + C∇(u, B)2B˙ −s  ( 2H 2 + ∇(u, B)2H 1 )( , u, B, ∇Φ)B˙ −s , 2,∞ 2,∞ 2,∞ dt

(3.40)

and for s ∈ (1/2, 3/2], we have d ( , u, B, ∇Φ)2B˙ −s + C∇(u, B)2B˙ −s 2,∞ 2,∞ dt  5 −s s− 12

−s .  ( , u, B)L2  H 2 + ∇(u, B)H 1 2 ( , u, B, ∇Φ)B˙ 2,∞

(3.41)

˙ j to (3.2)1 , (3.2)2 , (3.2)3 and multiplying the resulting identities by Δ ˙ j , Δ ˙ j u, Δ ˙ jB Proof. Applying Δ respectively, summing them up and then integrating the result over R3 by parts, we get 1 d ˙ ˙ j ∇u2 2 + μ2 Δ ˙ j divu2 2 + νΔ ˙ j ∇B2 2 − ρ¯ Δj ( , u, B)2L2 + μ1 Δ L L L 2 dt γ ˙ j G1 · Δ ˙ j + Δ ˙ j G2 · Δ ˙ ju + Δ ˙ j G3 · Δ ˙ j Bdx. Δ

= R3

˙ j ∇Φ · Δ ˙ j udx Δ R3

(3.42)

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Like (3.37), we easily obtain

˙ j ∇Φ · Δ ˙ j ∇Φ2 2 + 1 ˙ j udx = 1 d Δ Δ L 2γ dt ρ¯

− R3

˙ j ∇Φ · Δ ˙ j ( u)dx. Δ

(3.43)

R3

Plugging (3.43) into (3.42), we get 1 d ˙ 1 d ˙ ˙ j ∇u2 2 + μ2 Δ ˙ j divu2 2 + νΔ ˙ j ∇B2 2 Δj ( , u, B)2L2 + Δj ∇Φ2L2 + μ1 Δ L L L 2 dt 2γ dt 1 ˙ j ∇Φ · Δ ˙ j G1 · Δ ˙ j ( u)dx + Δ ˙ j + Δ ˙ j G2 · Δ ˙ ju + Δ ˙ j G3 · Δ ˙ j Bdx. Δ =− ρ¯ R3

(3.44)

R3

Then multiplying (3.44) by 2−2sj and then taking the supremum over j ∈ Z, we obtain 1 d 1 d ( , u, B)2B˙ −s + ∇Φ2B˙ −s + μ1 ∇u2B˙ −s + μ2 divu2B˙ −s + ν∇B2B˙ −s 2,∞ 2,∞ 2,∞ 2,∞ 2,∞ 2 dt 2γ dt −s  u ˙ −s + G1  ˙ −s   ˙ −s + G2  ˙ −s u ˙ −s + G3  ˙ −s B ˙ −s .  ∇ΦB˙ 2,∞ B2,∞ B2,∞ B2,∞ B2,∞ B2,∞ B2,∞ B2,∞

−s Then as in the proof of Lemma 3.3, applying Lemma 2.5 instead to estimate the B˙ 2,∞ norm, we have completed the proof of Lemma 3.4. 2

4. Proof of Theorem 1.1 In this section, we shall combine all the energy estimates that we have derived in the previous sections and the Sobolev interpolation to prove Theorem 1.1. We first close the energy estimates at each lth level in our weak sense. Let N  3 and 0  l  m − 1 with 1  m  N . Summing up the estimates (3.6) of Lemma 3.1 from k = l to m, by changing the index and since δ is small, we obtain  d  ∇k ( , u, B, ∇Φ)2L2 + C1 ∇k (u, B)2L2 dt lkm l+1km+1      C2 δ ∇k 2L2 + ∇k ∇Φ2L2 . lkm

(4.1)

l+1km+1

Summing up the estimates (3.24) of Lemma 3.2 from k = l to m − 1, we obtain d dt



∇k u · ∇∇k dx + C3

lkm−1 R3

 C4

  lkm





∇k 2L2 +

∇k ∇Φ2L2



l+1km+1

∇k (u, B)2L2 .

(4.2)

l+1km+1 2δ Multiplying (4.2) by 2C C3 , adding it to (4.1), since δ is small, we deduce that there exists a constant C5 > 0 such that for 0  l  m − 1,

2C2 δ d  ∇k ( , u, B, ∇Φ)2L2 + dt C3 lkm

+ C5

  lkm

∇k 2L2 +





∇k u · ∇∇k dx



lkm−1 R3

l+1km+1

 ∇k (u, B, ∇Φ)2L2  0.

(4.3)

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Next, we define Elm (t) to be C5−1 times the expression under the time derivative in (4.3). Observe that since δ is small, then Elm (t) is equivalent to ∇l (t)2H m−l + ∇l u(t)2H m−l + ∇l B(t)2H m−l + ∇l ∇Φ(t)2H m−l . Then we can rewrite (4.3) as that for 0  l  m − 1, d m E (t) + ∇l (t)2H m−l + ∇l+1 u(t)2H m−l + ∇l+1 B(t)2H m−l + ∇l+1 ∇Φ(t)2H m−l  0. dt l

(4.4)

Taking l = 0 and m = 3 in (4.4) and integrating in time, we get ( , u, B, ∇Φ)(t)2H 3  E03 (t)  E03 (0)  ( 0 , u0 , B0 , ∇Φ0 )2H 3 .

(4.5)

By a standard continuity argument, this closes the a priori estimates (3.4) if we assume ( 0 , u0 , B0 , ∇Φ0 )2H 3  δ is small enough. This in turn allows us to take l = 0 and m = N in (4.4), integrating it in time, we obtain t ( , u, B, ∇Φ)(t)2H N

( , ∇u, ∇B, ∇∇Φ)(τ )2H N dτ  ( 0 , u0 , B0 , ∇Φ0 )2H N .

+

(4.6)

0

This proves (1.4). In the following, we will prove the optimal time decay rates of the unique global solution to system (1.1) obtained in Theorem 1.1. First we prove (1.7)–(1.8). Assume for the moment that we have proved (1.5)–(1.6). If l = 0, 1, . . . , N − 1, then Lemma 2.6 gives −

1

1+

1

∇l+1 f L2  CΛ−s f L2l+s ∇l f L2 l+s ,

(4.7)

and Lemma 2.8 implies −

1

1+

1

l+s ∇l+1 f L2  Cf B˙ −s ∇l f L2 l+s .

(4.8)

2,∞

Combining (1.5) and (4.7) or (1.6) and (4.8) yields 1

1+ l+s ∇l+1 (u, B, ∇Φ)2L2  C0 ∇l (u, B, ∇Φ)2L2 .

(4.9)

This together with (1.4) implies in particular that for l = 0, 1, . . . , N − 1, 1

1+ l+s (∇l , ∇l+1 u, ∇l+1 B, ∇l+1 ∇Φ)2H N −l  C0 ∇l ( , u, B, ∇Φ)2H N −l .

(4.10)

Thus, we deduce from (4.4) with m = N the following time differential inequality 1 d N E (t) + C0 (ElN (t))1+ l+s  0 for l = 0, 1, . . . , N − 1. dt l

(4.11)

Solving this inequality directly and together with (4.6), we get ElN (t)  C0 (1 + t)−(l+s) for l = 0, 1, . . . , N − 1.

(4.12)

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Thus, we obtain from (4.12) that ∇l ( , u, B, ∇Φ)(t)2H N −l  C0 (1 + t)−(l+s) for l = 0, 1, . . . , N − 1.

(4.13)

Note that = div∇Φ, and then we have ∇l (t)2L2  ∇l+1 ∇Φ(t)2L2  C0 (1 + t)−(l+1+s) for l = 0, 1, . . . , N − 2.

(4.14)

Consequently, by (4.13)–(4.14) and the interpolation, we deduce (1.7)–(1.8). Now, we turn back to prove (1.5)–(1.6). First, we prove (1.5) by Lemma 3.3. However, we are not able to prove it for all s ∈ [0, 3/2) at this moment. We shall first prove it for s ∈ [0, 1/2]. We denote E−s (t) to be the expression under the time derivative in the estimates (3.30)–(3.31), which is equivalent to Λ−s ( , u, B, ∇Φ)(t)2L2 . Then, integrating (3.30) in time, by the bound (1.4), we obtain t E−s (t)  E−s (0) + C



( (τ )2H 2 + ∇(u, B)(τ )2H 1 )

  E−s (τ )dτ  C0 1 + sup E−s (τ ) . 0τ t

0

This implies (1.5) for s ∈ [0, 1/2], and this verifies (1.7)–(1.8) for s ∈ [0, 1/2]. For s ∈ (1/2, 3/2), we notice that the arguments for the case s ∈ [0, 1/2] cannot be applied to this case.  Observe that we have 0 , u0 , B0 , ∇Φ0 ∈ H˙ −1/2 since H˙ −s ∩ L2 ⊂ H˙ −s for any s ∈ [0, s). We then deduce from what we have proved for (1.5) and (1.7) with s = 1/2 that the following decay results hold: ∇l ( , u, B, ∇Φ)(t)2H N −l  C0 (1 + t)−(l+ 2 ) 1

for l = 0, 1, . . . , N − 1

(4.15)

and ∇l (t)2L2  C0 (1 + t)−(l+ 2 ) for l = 0, 1, . . . , N − 2. 3

(4.16)

Therefore, by (4.15)–(4.16), we deduce from (3.31) that for s ∈ (1/2, 3/2), t E−s (t)  E−s (0) + C

s− 1

( , u, B)(τ )L2 2 ( (τ )H 2 + ∇(u, B)(τ )H 1 ) 2 −s 5



E−s (τ )dτ

0

t  C0 + C0

(1 + τ )−( 4 − 2 ) dτ sup 7

s

0τ t

0

  E−s (τ )  C0 1 + sup E−s (τ ) . 0τ t

(4.17)

This implies (1.5) for s ∈ (1/2, 3/2), and this verifies (1.7)–(1.8) for s ∈ (1/2, 3/2). Now we will prove (1.6). As in the proof of (1.5), integrating (3.40) of Lemma 3.4 in time, by (1.4), we obtain that for s ∈ (0, 1/2], ( , u, B, ∇Φ)(t)2B˙ −s

2,∞

t 

( 0 , u0 , B0 , ∇Φ0 )2B˙ −s

−s dτ ( (τ )2H 2 + ∇(u, B)(τ )2H 1 )( , u, B, ∇Φ)(τ )B˙ 2,∞

+

2,∞

0 −s ).  C0 (1 + sup ( , u, B, ∇Φ)(τ )B˙ 2,∞

0τ t

(4.18)

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This proves (1.6) for s ∈ (0, 1/2], and this verifies (1.7)–(1.8) for s ∈ (0, 1/2]. Next, let s ∈ (1/2, 3/2), observing that we have −1/2 −s −s ( 0 , u0 , B0 , ∇Φ0 ) ∈ B˙ 2,∞ since B˙ 2,∞ ∩ L2 ⊂ B˙ 2,∞ for any s ∈ [0, s),

(4.19)

where we have used Lemma 2.7 and Lemma 2.9. Then as in the proof of (4.17), we deduce from (3.41) that for s ∈ (1/2, 3/2), ( , u, B, ∇Φ)(t)2B˙ −s  ( 0 , u0 , B0 , ∇Φ0 )2B˙ −s 2,∞

t +

2,∞

 5 −s s− 1

−s dτ ( , u, B)(τ )L2 2  (τ )H 2 + ∇(u, B)(τ )H 1 2 ( , u, B, ∇Φ)(τ )B˙ 2,∞

0

t

 C0 1 +

−( 74 − s2 )

(1 + τ )

 dτ sup ( , u, B, ∇Φ)(τ )B˙ −s

2,∞

0τ t

0

 −s  C0 1 + sup ( , u, B, ∇Φ)(τ )B˙ 2,∞ .

(4.20)

0τ t

This gives (1.6) for s ∈ (1/2, 3/2), and this verifies (1.7)–(1.8) for s ∈ (1/2, 3/2). It remains to prove the −1 case s = 3/2. For s = 3/2, we still have ( 0 , u0 , B0 , ∇Φ0 ) ∈ B˙ 2,∞ since (4.19). We then deduce from what we have proved for (1.6) and (1.7)–(1.8) with s = 1 that the following decay results hold: ∇l ( , u, B, ∇Φ)(t)L2  C0 (1 + t)−

l+1 2

for l = 0, 1, 2

(4.21)

and ∇l (t)L2  C0 (1 + t)−( 2 +1) for l = 0, 1, 2. l

(4.22)

Thus, by (4.21)–(4.22), we deduce from (3.41) that for s = 3/2, ( , u, B, ∇Φ)(t)2B˙ −3/2  ( 0 , u0 , B0 , ∇Φ0 )2B˙ −3/2 2,∞

2,∞

t ( (τ )L2 + (u, B)(τ )H 1 ) ( (τ )H 2 + ∇(u, B)(τ )H 1 ) ( , u, B, ∇Φ)(τ )B˙ −3/2 dτ

+C

2,∞

0

t

 C0 1 +

(1 + τ )

−3/2

0

 dτ sup ( , u, B, ∇Φ)(τ )B˙ −3/2 0τ t

2,∞



 C0 1 + sup ( , u, B, ∇Φ)(τ )B˙ −3/2 0τ t

.

(4.23)

2,∞

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