Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field

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ScienceDirect J. Differential Equations 260 (2016) 5450–5480 www.elsevier.com/locate/jde

Global solutions to the 2D viscous, non-resistive MHD system with large background magnetic field Ting Zhang Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China Received 14 May 2015; revised 20 October 2015 Available online 30 December 2015

Abstract In this paper, we consider the 2D viscous, non-resistive MHD system. If the background magnetic field is sufficiently large, then we can obtain the global existence of strong solutions. © 2015 Elsevier Inc. All rights reserved. Keywords: MHD system; Existence; Uniqueness

1. Introduction In this paper, we consider the global existence of strong solutions to the following 2D incompressible viscous and non-resistive magnetohydrodynamics (MHD) system, ⎧ ∂t b + v · ∇b = b · ∇v, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎨ ∂ v + v · ∇v − νv + ∇p = − 1 ∇(|b|2 ) + b · ∇b, t 2 ⎪ divv = divb = 0, ⎪ ⎪ ⎩ (b, v)|t=0 = (b0 , v0 ), and the boundary condition E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2015.12.005 0022-0396/© 2015 Elsevier Inc. All rights reserved.

(1.1)

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1 v(t, x) → 0, b(t, x) → ( , 0) , when |x| → ∞, ε i.e. the background magnetic field is (ε −1 , 0) , ε > 0. Here, b = (b1 , b2 ) , v = (v1 , v2 ) and p denote the magnetic field, velocity and scalar pressure of the fluid respectively. Magnetohydrodynamics (MHD) is the study of the dynamics of electrically conducting fluids, such as plasmas, liquid metals, and salt water or electrolytes. The MHD system describes many phenomena such as the geomagnetic dynamo in geophysics and solar winds and solar flares in astrophysics [1,4, 6,16]. There are many interesting results on the global regularity problem for the 2D MHD system with or without dissipation [3,5,7,18]. Physicists point out that, in the nonlinear MHD system, a strong enough magnetic field will reduce the nonlinear interaction [10] and inhibit formation of strong gradients. This effect was also observed in direct numerical simulations of the ideal MHD system (i.e. inviscid and non-resistive), with periodic boundary conditions [8]. Considering the ideal MHD system, Bardos, Sulem and Sulem [3] gave a rigorous proof of the global well-posedness when the initial data (b0 , v0 ) close to the equilibrium state (B0 , 0). They showed that the ideal MHD system is global well-posed with large initial data, when the background magnetic field |B0 | is sufficiently large. The physical background is that the Alfvén wave generated by the strong enough magnetic field will guarantee the stability of the MHD system [3, 10]. Since the ideal MHD system is hyperbolic–hyperbolic, one cannot use the method of characteristics in [3] to study the hyperbolic–parabolic system (1.1). There are some interesting results on the global regularity problem for the 2D MHD system (1.1) with ε = 1. Under some special conditions on the initial data, F.H. Lin, L. Xu and P. Zhang [12] can transfer the system (1.2) to the 2D viscoelastic fluid system. Using the Lagrangian transformation method and the anisotropic Littlewood–Paley analysis techniques, they [12] obtained the global well-posedness result when the initial data (b0 , v0 ) is close to the equilibrium state ((1, 0) , (0, 0) ). We [20] also obtained the global well-posedness result when (b0 − (1, 0) , v0 )H 2 ∩B˙ 0 1, by a new 2,1 and simple proof, which involves only the energy estimate method, interpolating inequalities and couple elementary observations. Recently, using the anisotropic Littlewood–Paley analysis techniques, X.X. Ren, J.H. Wu, Z.Y. Xiang and Z.F. Zhang [17] obtained the global existence and decay estimates of global strong solution when (b0 − (1, 0) , v0 )H 8 ∩H −s,−s ∩H −s,8 1, s = 12 − . Under some special conditions on the initial data, X.P. Hu and F.H. Lin [9] obtained the global existence of the strong solution when the small initial data are in the critical space. The aim of this paper is to get the global well-posedness result with large initial data, when the parameter ε is sufficiently small, and to improve the previous result for the ideal MHD system to the hyperbolic–parabolic system (1.1). In (1.1), the condition divb = 0 implies the existence of a scalar function φ such that b = (∂2 φ, −∂1 φ) , and the corresponding system becomes ⎧ ∂t φ + v · ∇φ = 0, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎨ ∂t v + v · ∇v − v + ∇p = −div[∇φ ⊗ ∇φ], ⎪ divv = 0, ⎪ ⎪ ⎩ (φ, v)|t=0 = (φ0 , v0 ), and the boundary condition

(1.2)

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1 v(t, x) → 0, φ(t, x) − x2 → 0, when |x| → ∞, ε where φ, v = (v1 , v2 ) and p denote the magnetic potential, velocity and scalar pressure of the fluid respectively. The system (1.2) is a coupled system between the Navier–Stokes equations and a free transport equation with a universal nonlinear coupling structure. Nonlinear Hyperbolic–Parabolic system as (1.2) arises in many problems, such as the incompressible inviscid MHD equations, the viscoelastic fluid system, the evolution of nematic liquid crystals, the diffusive free interface motion and the immersed boundaries in flow fields [11–13]. After substituting (φ, v) = ( 1ε x2 + ψ, v) into (1.2), one obtains the following system for (ψ, v): ⎧ ∂t ψ + v · ∇ψ + ε −1 v2 = 0, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎪ −1 −1 ⊥ ⎪ ⎪ ⎨ ∂t v + v · ∇v − v + ∇p + 2ε ∇∂2 ψ + ε ∇ ∂1 ψ = −div[∇ψ ⊗ ∇ψ], divv = 0, ⎪ ⎪ ⎪ v(t, x) → 0, ψ(t, x) → 0, when |x| → ∞, ⎪ ⎪ ⎪ ⎩ (ψ, v)|t=0 = (ψ0 , v0 ).

(1.3)

Here and in what follows, we denote ∇ ⊥ = (−∂2 , ∂1 ) , ∂i = ∂xi , i ∈ {1, 2}. By taking divergence of the v equation of (1.3), we can solve the pressure function p via p = −2ε −1 ∂2 ψ +

2 

(−)−1 [∂i vj ∂j vi + ∂i ∂j (∂i ψ∂j ψ)].

(1.4)

i,j =1

Substituting (1.4) into (1.3), we have ⎧ ∂t ψ + v · ∇ψ + ε −1 v2 = 0, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎨ ∂t v + v · ∇v − v + ε−1 ∇ ⊥ ∂1 ψ = g, ⎪ divv = 0, ⎪ ⎪ ⎩ (ψ, v)|t=0 = (ψ0 , v0 ),

(1.5)

where g := −

2 

∇(−)−1 [∂i vj ∂j vi + ∂i ∂j (∂i ψ∂j ψ)] −

i,j =1

2  ∂j [∇ψ∂j ψ]. j =1

Let (ψ L , v L ) be the solution for the following linear system: ⎧ L −1 L + 2 ⎪ ⎨ ∂t ψ + ε v2 = 0, (t, x) ∈ R × R , ∂t v L − v L + ε −1 ∇ ⊥ ∂1 ψ L = 0, ⎪ ⎩ L L (ψ , v )|t=0 = (ψ0 , v0 ).

(1.6)

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When ε is small enough, we will show that the solution (ψ ε , v ε ) for the system (1.3) is close to the linear solution (ψ L , v L ). Using the classical method in [15], one can easily obtain the local existence and uniqueness of the solution for the system (1.3). In this paper, we will prove the following global existence and uniqueness of the solution of the system (1.3) when ε is sufficient small. Assume that the initial data satisfy 0 (ξ ), |ξ |−1 ∇ψ0 , v0 ∈ H 2 , divv0 = 0, and ψ v0 (ξ ) ∈ L1ξ .

(1.7)

Let 0  1 + |ξ |−1 C0 := (∇ψ0 , v0 )X = ∇ψ0 H 2 + v0 H 2 + ψ v0 L1 , L ξ

ξ

(∇ψ, v)YT = (∇ψ, v)Y1,T + (∇ψ, v)Y2,T , with (∇ψ, v)Y1,T = (∇ψ, v)L∞ ([0,T ];H 2 ) + ∇vL2 ([0,T ];H 2 ) + ∂1 ∇ψL2 ([0,T ];H 1 ) , (∇ψ, v)Y2,T =  v (ξ )L2 ([0,T ];L1 ) + ∂ 1 ψ(ξ )L2 ([0,T ];L1 ) . ξ

ξ

For simplify, we denote Y , Y1 and Y2 respectively, when T = ∞. Denote



∇ψ, v, ∈ C([0, T ]; H n ), ∇p ∈ C([0, T ]; H n−1 ),

. = (ψ, v, ∇p)

1 2

(∂1 ∇ψ, ∇v) ∈ L2 ([0, T ]; H n−1 × H n ),  v (ξ ), ∂ 1 ψ(ξ ) ∈ L ([0, T ]; Lξ ),

ETn

We use the notation E n if T = ∞ by changing the time interval [0, T ] into [0, ∞) in the above definition. Theorem 1.1. Assume that the initial data (ψ0 , v0 ) satisfy (1.7), then there exists a positive constant ε0 such that if ε ∈ (0, ε0 ),

(1.8)

then the system (1.3) has a unique global solution (ψ ε , v ε , ∇p ε ) ∈ E 2 . Furthermore, we have lim (∇ψ ε − ∇ψ L , v ε − v L )Y = 0.

ε→0

(1.9)

0 ∩ H 2 , then (1.7) holds. Remark 1.1. One can see that if ∇ψ0 , v0 ∈ B˙ 2,1

Remark 1.2. Under the assumptions of Theorem 1.1, if (∇ψ0 , v0 ) ∈ H n (R2 ) × H n (R2 ), n ≥ 3, we can easily obtain that (ψ, v, ∇p) ∈ E n and omit the details. Remark 1.3. Considering new variables and new functions, x˜ x˜ x˜ ˜ x) ˜ x) ˜ = εv(t, ), p(t, ˜ x) ˜ = εp(t, ), b(t, ˜ = εb(t, ), v(t, ε ε ε

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˜ v), from (1.1), we have new system for (b, ˜ ⎧ ˜ ˜ (t, x) ˜ ∈ R + × R2 , ∂t b + v˜ · ∇x˜ b˜ = b˜ · ∇x˜ v, ⎪ ⎪ ⎪ ⎪ 1 2 ˜2 ˜ ˜ ⎪ ⎪ ⎨ ∂t v˜ + v˜ · ∇x˜ v˜ − νε x˜ v˜ + ε∇x˜ p˜ = − 2 ∇x˜ (|b| ) + b · ∇x˜ b, divx˜ v˜ = divx˜ b˜ = 0, ⎪ ⎪ ⎪ v(t, ˜ x) ⎪ ˜ x) ˜ → 0, b(t, ˜ → (1, 0) , when |x| → ∞, ⎪ ⎪ ⎩ ˜ v)| (b, ˜ t=0 = (b˜0 , v˜0 ).

(1.10)

From this viewpoint, the system (1.1) equals the system with small viscosity coefficient and the background magnetic field (1, 0) . Since 1 (b˜0 − (1, 0) , v˜0 )H˙ 1 ≈ ε(b0 − ( , 0) , v0 )H˙ 1 , ε 1 (b˜0 − (1, 0) , v˜0 )L2 ∩B˙ 0 ≈ ε 2 (b0 − ( , 0) , v0 )L2 ∩B˙ 0 , 2,1 2,1 ε then the initial data are very close to the stationary solution. When ε = 0, the system (1.10) is the ideal MHD system in [3], i.e., ⎧ ∂t b + v · ∇b = b · ∇v, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t v + v · ∇v + ∇p = b · ∇b, divv = divb = 0, ⎪ ⎪ ⎪ v(t, x) → 0, b(t, x) → B0 , when |x| → ∞, ⎪ ⎪ ⎪ ⎩ (b, v)|t=0 = (b0 , v0 ).

(1.11)

In [3], Bardos, Sulem and Sulem obtained the global existence of the classical solution (b, v) for the system (1.11), v ± (b − B0 ) ∈ C(R+ ; C 1,α (R2 )), when (1 + |x|2 )(v ± (b − B0 ))C 1,α (R2 ) < β|B0 | for some β > 0. Following the ideas in [3], we could give some new methods to obtain the global well-posedness result for the system (1.1) with large initial data, while the global well-posedness result for the system (1.10) with suitable initial data. It is pointed out in [3, 10] that the strong background magnetic field will generate Alfvén wave, and the fluctuations v + b − B0 and v − b + B0 are propagated as Alfvén wave in opposite directions along the lines of force of the B0 field. For the details, let z± = v ± (b − B0 ), we could rewrite the system (1.11) as follows ⎧ ± ∂t z ∓ B0 · ∇z± + z∓ · ∇z± + ∇p = 0, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎨ divz± = 0, ⎪ z± (t, x) → 0, when |x| → ∞, ⎪ ⎪ ⎩ ± z |t=0 = z0± := v0 ± (b0 − B0 ).

(1.12)

Considering the linear solutions (zL )± for the linearized system ∂t z± ∓ B0 · ∇z± = 0, we have (zL )± (x) = z0± (x ± B0 t). Then the linear solutions have the decay (zL )± (x)  1 + (x ± B0 t)2 . If the initial data have the compact supported, supp z0± (x) ⊂ BR , we have supp(zL )± (t, x) ⊂ BR ∓ B0 t. Then the bilinear terms of the linear solutions (zL )− · ∇(zL )+ = (zL )+ · ∇(zL )− = 0 when

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t ≥ |BR0 | , which means that a strong enough magnetic field will reduce the nonlinear interaction and inhibit formation of strong gradients [3,8,10]. This idea implies that the small viscosity is not important to obtain the global well-posed result, and nondegenerate background magnetic field will help us to obtain the global well-posed result. This idea will play a very important role in our proof. Remark 1.4. To prove Theorem 1.1, we set L , v ε = vN ψ ε = ψNL + ψ + v, L ) is the linear solution with the modified initial data. We will show that the linear where (ψNL , vN L L ) is bounded and the error part (ψ L ) is the , solution (ψN , vN v ) is very small. Here, (ψNL , vN solution of the linear system,

⎧ L −1 L + 2 ⎪ ⎨ ∂t ψN + ε vN,2 = 0, (t, x) ∈ R × R , L − v L + ε −1 ∇ ⊥ ∂ ψ L = 0, ∂t vN 1 N N ⎪ ⎩ L L (ψN , vN )|t=0 = (ψ0,N M , v0,N M ),

(1.13)

where ψ0,N M = F −1 (hM ∗ (1|ξ1 |≥ 1 ,|ξ |≤N ψˆ 0 )), v0,N M = F −1 (hM ∗ (1|ξ1 |≥ 1 ,|ξ |≤N vˆ0 )), N

N

M ≥ 2N , hM (ξ ) = M 2 h(Mξ ), the mollifier h ∈ Cc∞ (R2 ) satisfies supp h ⊂ {ξ ∈ R2 ; |ξ | ≤ 1} , and hdξ = 1. Then (ψ v ) satisfies the following system, ⎧ L ) · ∇ψ + ε −1 v2 = fN , (t, x) ∈ R+ × R2 , v + vN ∂t ψ + ( ⎪ ⎪ ⎪ ⎨ L + ε −1 ∇ ⊥ ∂1 ψ = −div[∇ ψ ⊗ ∇ψ ] + gN , v + ( v + vN ) · ∇ v −  v + ∇p + 2ε −1 ∇∂2 ψ ∂t ⎪ div v = 0, ⎪ ⎪ ⎩ 0 , v0 ), (ψ , v )|t=0 = (ψ (1.14) 0 = ψ0 − ψ0,N M and where ψ v0 = v0 − v0,N M , L fN := − v · ∇ψNL − vN · ∇ψNL , L L L ⊗ ∇ψNL + ∇ψNL ⊗ ∇ ψ ] − div[vN gN = − v · ∇vN − div[∇ ψ ⊗ vN + ∇ψNL ⊗ ∇ψNL ].

0 , It is easy to see that the initial error (ψ v0 ) is small when N, M  1. So, the key point is using the classical method for the oscillatory integrals to obtain the smallness estimates for the bilinear L ), such as, terms of the linear solutions (ψNL , vN   L L L vN · ∇ψNL , P div[vN ⊗ vN + ∇ψNL ⊗ ∇ψNL ] , where P = I + ∇(−)−1 div. The simple computation implies

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 L P(div[vN

L ⊗ vN

+ ∇ψNL

⊗ ∇ψNL ]) = Pdiv

L )2 −(∂ ψ L )2 (vN 2 N 1

L vL + ∂ ψ L ∂ ψ L vN1 1 N 2 N N2

L vL + ∂ ψ L ∂ ψ L vN 1 N 2 N 1 N2

L )2 −(∂ ψ L )2 (vN2 1 N

 .

From the ideas in [3], we will prove that the above bilinear terms are small when ε 1, see Propositions 2.1–2.2. Scheme of the proof and organization of the paper. In Section 2, we consider the linear system (1.6), obtain some important estimates for the linear solution (ψ L , v L ). Section 3 is devoted to , give some a priori estimates for the error part (ψ v ). Finally, we prove Theorem 1.1 in Section 4.  Notation. We shall denote by (a|b) the L2 inner product of a and b, (a|b)H˙ s = |α|=s (∂ α a|∂ α b),  p and (a|b)H s = sk=0 (a|b)H˙ k , CT (X) = C([0, T ]; X) and LT (X) = Lp ([0, T ]; X). 2. The linearized system In this section, we will investigate carefully the linear system (1.6). From (1.6) and divv0 = 0, we have ∂t divv L − divv L = 0 and divv L = 0. Using the classical energy method, we can obtain the following lemma and give the proof in Appendix. Lemma 2.1. Let (ψ, v) be sufficiently smooth solutions for the following system, ⎧ ∂t ψ + ε −1 v2 = f, (t, x) ∈ R+ × R2 , ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎨ ∂t v1 − v1 − ε ∂1 ∂2 ψ = g1 , ∂t v2 − v2 + ε −1 ∂12 ψ = g2 , ⎪ ⎪ ⎪ ⎪ divv = 0, ⎪ ⎪ ⎩ (ψ, v)|t=0 = (ψ0 , v0 ),

(2.1)

then there holds     d 1 ε2 ε v2H 2 + ∇ψ2H 2 + ψ2H 1 + (v2 |ψ)H 1 dt 2 4 4 1 1 + ∇v2H 2 − ∇v2 2H 1 + ∇∂1 ψ2H 1 4 4 ε ε ε2 = (g|v)H 2 − (ψ|f )H 2 + (g2 |ψ)H 1 + (v2 |f )H 1 + (f |ψ)H 1 . 4 4 4

(2.2)

Remark 2.1. In the proof of Lemma 2.1, we have  1 d  v2H 2 + ∇ψ2H 2 + ∇v2H 2 = (g|v)H 2 − (ψ|f )H 2 , 2 dt and

(2.3)

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 d ε ψ2H 1 + (v2 |ψ)H 1 + ε −1 ∂1 ∇ψ2H 1 − ε −1 ∇v2 2H 1 dt 2 = (g2 |ψ)H 1 + (v2 |f )H 1 + ε(f |ψ)H 1 .

(2.4)

To control the term −ε −1 ∇v2 2H 1 in (2.4), we need use (2.3) + 4ε ×(2.4) to obtain (2.2), where the coefficient of the dissipation ∂1 ∇ψ2H 1 will not depend on the small parameter ε. On the other hand, considering one term in the expression of ψˆ L(ξ ) (2.7), we have   |ξ |2 |ξ |2 β β ˆ M1 (t, ξ )ψ0 = ε sin( ) + cos( ) e− 2 t ψˆ 0 , 2β ε ε

(2.5)

when supp(ψˆ 0 ) ⊂ N (ξ ) and ε ∈ (0, ε1 ), where N (ξ ), β and ε1 are given in (2.11) and (2.12). Then, we cannot get that the dissipation ε−1 ∂1 ∇ψ2 2 1 is uniform bounded with respect to LT H

the small parameter ε. Remark 2.2. Since ε ∈ (0, 1), we have

  1 ε2 ε v2H 2 + ∇ψ2H 2 + ψ2H 1 + (v2 |ψ)H 1 ≈ v2H 2 + ∇ψ2H 2 . 2 4 4 Thus, we let (∇ψ, v)Y1,T = (∇ψ, v)L∞ ([0,T ];H 2 ) + ∇vL2 ([0,T ];H 2 ) + ∂1 ∇ψL2 ([0,T ];H 1 ) , which is independent of the small parameter ε. From equations (1.6) and the condition divv L = 0, we have that (ψ L , v L ) satisfies the following system, ⎧ ⎪ ∂t2 ψ L − ∂t ψ L − ε −2 ∂12 ψ L = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ∂ 2 v L − ∂t v L − ε −2 ∂12 v L = 0, ⎪ ⎨ t ψ L |t=0 = ψ0 , ∂t ψ L |t=0 = −ε −1 v0,2 , ⎪ ⎪ ⎪ v1L |t=0 = v0,1 , ∂t v1L |t=0 = v0,1 + ε −1 ∂1 ∂2 ψ0 , ⎪ ⎪ ⎪ ⎪ ⎩ vL| = v , ∂ vL| = v − ε −1 ∂ 2 ψ . 2 t=0

0,2

t 2 t=0

0,2

1

(2.6)

0

Using the classical Fourier analysis method, one can easily obtain that

where

ψˆ L (ξ ) = M1 (t, ξ )ψˆ 0 + M2 (t, ξ )vˆ0,2 ,

(2.7)

vˆ1L (ξ ) = M2 (t, ξ )ξ1 ξ2 ψˆ 0 + M3 (t, ξ )vˆ0,1 ,

(2.8)

vˆ2L (ξ ) = −M2 (t, ξ )|ξ1 |2 ψˆ 0 + M3 (t, ξ )vˆ0,2 ,

(2.9)

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M1 (t, ξ ) =

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λ+ eλ− t − λ− eλ+ t eλ− t − eλ+ t λ+ eλ+ t − λ− eλ− t , M2 (t, ξ ) = , , M3 (t, ξ ) = λ + − λ− ε(λ+ − λ− ) λ + − λ− (2.10)

and λ± = −

|ξ |2 ±

 |ξ |4 − 4ε −2 |ξ1 |2 . 2

Let N (ξ ) = {ξ ∈ R2 , |ξ1 | ≥

1 , |ξ | ≤ 2N }. 2N

(2.11)

If supp(ψˆ 0 , vˆ0 ) ⊂ N (ξ ), then we have λ± = −

|ξ |2 ± ε −1 βi, when ε ∈ (0, ε1 ), 2

where  β=

4|ξ1 |2 − ε 2 |ξ |4 and ε1 = (2N )−3 . 2

(2.12)

Lemma 2.2. When supp(ψˆ 0 , vˆ0 ) ⊂ N (ξ ) and ε ∈ (0, ε1 ), we have (∇ψ L , v L )Y ≤ C(∇ψ0 , v0 )X = CC0 .

(2.13)

In what follows, we always use C to denote a generic positive constant independent of the initial data. Proof. From Lemmas 2.1, we have (∇ψ L , v L )Y1 ≤ C(∇ψ0 , v0 )X .

(2.14)

When ξ ⊂ N(ξ ) and ε ∈ (0, ε1 ), we have β ≈ |ξ1 |, |ξ1 | ≥ ε|ξ |2 ,



|ξ |2 t ξ 2 sin(ε −1 βt) + 2ε −1 β cos(ε −1 βt)

|ξ |2 t

≤ Ce− 2 , |M1 | =

e− 2

2ε −1 β



|ξ |2 t 2 sin(ε −1 βt)

|ξ |2 t

≤ C|ξ1 |−1 e− 2 . |M2 | =

e− 2

2β Combining (2.7)–(2.9), we have

(2.15) (2.16)

T. Zhang / J. Differential Equations 260 (2016) 5450–5480 − L ∂ 1 ψ L2 ([0,∞);L1 ) ≤ C(e

|ξ |2 t 2

ξ

− ∂ 1ψ 0, e

|ξ |2 t 2

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vˆ0,2 )L2 ([0,∞);L1 ) ≤ C(∇ψ0 , v0 )X = CC0 . ξ

(2.17) L Similarly, we can estimate the term v L2 ([0,∞);L1ξ ) and omit the details, which finishes the proof of (2.13). 2

From Remark 1.4, we need to estimate the following bilinear terms of the linear solutions, v L · ∇ψ L , (v1L )2 − (∂2 ψ L )2 , (v2L )2 − (∂1 ψ L )2 , v1L v2L + ∂1 ψ L ∂2 ψ L , where v L = (v1L , v2L ) . But the complicated expressions of the linear solutions (2.7)–(2.9) will imply the complicated computations. Noting that β ≈ |ξ1 |, we consider the following approximate expressions, ψ L = L + εL , v L = V L + VεL ,

(2.18)

where F L = e−

F εL

|ξ |2 2 t

 −

   sin(ξ1 εt ) t vˆ0,2 + cos ξ1 ψˆ 0 , ξ1 ε

(2.19)

 sin(|ξ1 | εt ) sin(β εt ) =e − vˆ0,2 |ξ1 | β        |ξ |2 sin(β εt ) t t + cos β − cos |ξ1 | +ε ψˆ 0 , ε ε 2β 2

− |ξ2| t

FV = e L



2

− |ξ2| t



cos(ξ1 εt ) t vˆ0,2 + sin(ξ1 )ψˆ 0 ξ1 ε



−ξ2 ξ1

 ,

     |ξ |2 sin(β εt ) vˆ0,2 t t − cos |ξ1 | −ε cos β ε ε 2β ξ1      sin(β εt ) sin(|ξ1 | εt ) −ξ2 + − ξ1 ψˆ 0 . β |ξ1 | ξ1

FVεL = e−

|ξ |2 2 t

(2.20)

(2.21)



(2.22)

Then, one can easily get the following bounded estimates for the approximate linear solution ( L , V L ), as well as the smallness estimates for the error terms ( εL , VεL ). Proposition 2.1. Suppose ψ0 (x) and v0 (x) are two smooth functions satisfying 0 , supp suppψ v0 ∈ N (ξ ).

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When ε ∈ (0, ε1 ), we get the following estimates L , ∇ L ) 2 (∇ L , V L )Y + (∇ εL , VεL )Y + (∇ ε L ([0,∞);L1 ) ≤ C(∇ψ0 , v0 )X , ξ

(2.23) ∇V L1 ([0,∞);L∞ ) + ∇ L2 ([0,+∞);H 2 ) ≤ C(∇ψ0 , v0 )X ,

(2.24)

(∇ψ L , v L )L1 ([0,+∞);L2 ) ≤ CN ,

(2.25)

(∇ εL , VεL )L1 ([0,+∞);L2 ) ≤ CN ε,

(2.26)

2

L

L

where CN = C(N, (∇ψ0 , v0 )L2 ) denotes a uniform constant independent of ε. Proof. From (2.19)–(2.22), we can easily obtain (2.23)–(2.25) and omit the details. Since |ξ1 | − β =

ε2 4 4 |ξ |

|ξ1 | + β

,

when ξ ∈ N (ξ ) and ε ∈ (0, ε1 ), we have |F( εL )| + |F(VεL )| ≤ CN εe and finish the proof of (2.26) easily.

−

1 t 8N 2

(|vˆ0 | + |ψˆ 0 |),

2

Using the simple expressions of the approximate linear solutions, we have F(V L · ∇ L )(ξ )  |η|2 t |ξ −η|2 t t vˆ0,2 (η) 1 e(2η1 −ξ1 ) ε i e− 2 − 2 [−η2 (ψˆ 0 (η) + i )] = 2 η1 R2

× [(ξ1 − η1 )(ψˆ 0 (ξ − η) − i 1 − 2



t

e−(2η1 −ξ1 ) ε i e−

vˆ0,2 (ξ − η) )]dη ξ 1 − η1

|η|2 t |ξ −η|2 t 2 − 2

R2

× [(ξ1 − η1 )(ψˆ 0 (ξ − η) + i

vˆ0,2 (η) [−η2 (ψˆ 0 (η) − i )] η1

vˆ0,2 (ξ − η) )]dη. ξ 1 − η1

(2.27)

There are just the oscillatory integrals in the right hand side. Using the classical method for the oscillatory integrals [2], we can obtain the following lemma. Lemma 2.3. Suppose f (ξ ) and g(ξ ) are two smooth functions. Let  t (ξ ) = f (η)g(ξ − η)e±i ε (2η1 −ξ1 ) dη, R2

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5461

then we get the following estimate |(ξ )| ≤

C f H 2 gH 2 . ξ ξ 1 + 4( εt )2

(2.28)

Proof. Since t

(1 − ∂η21 )e±i ε (2η1 −ξ1 ) 1 + 4( εt )2

t

= e±i ε (2η1 −ξ1 ) ,

we have  (ξ ) =

t

f (η)g(ξ − η) R2

=

1 1 + 4( εt )2



(1 − ∂η21 )e±i ε (2η1 −ξ1 ) 1 + 4( εt )2

dη

t

(1 − ∂η21 )[f (η)g(ξ − η)]e±i ε (2η1 −ξ1 ) dη.

R2

Then, by Young’s inequality, one can easily obtain (2.28). 2 Then, we can get the smallness estimates for the bilinear terms of the approximate linear solutions. Proposition 2.2. Under the conditions in Proposition 2.1, we get the following estimates V L · ∇ L L1 ([0,+∞);L2 ) ≤ CN ε(ψˆ 0 2H 2 + vˆ0 2H 2 ), ξ

(2.29)

ξ

(V1L )2 − (∂2 L )2 L1 ([0,+∞);L2 ) + (V2L )2 − (∂1 L )2 L1 ([0,+∞);L2 ) ≤ CN ε(ψˆ 0 2H 2 + vˆ0 2H 2 ), ξ

(2.30)

ξ

V1L V2L + ∂1 L ∂2 L L1 ([0,+∞);L2 ) ≤ CN ε(ψˆ 0 2H 2 + vˆ0 2H 2 ), ξ

(2.31)

ξ

where CN denotes a uniform constant independent of ε. Proof. From (2.27) and Lemma 2.3, using the conditions that supp ψˆ 0 , supp vˆ0 ⊂ N (ξ ), we have V L · ∇ L L2x = CF(V L · ∇ L )(ξ )L2 ≤ ξ

CN (ψˆ 0 2H 2 + vˆ0 2H 2 ) ξ ξ 1 + 4( εt )2

and finish the proof of (2.29). Similarly, we can obtain (2.30)–(2.31) and omit the details.

2

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

3. A priori estimates , In this section, we will give the smallness estimates for the error part (ψ v ). Let AT = A1,T + A2,T , , , A1,T = (∇ ψ v )Y1,T , A2,T = (∇ ψ v )Y2,T . It is easy to obtain that lim

lim (∇ψ0 − ∇ψ0,N M , v0 − v0,N M )X = 0,

(3.1)

L suppψˆ NL , suppvˆN ∈ N (ξ ), if M ≥ 2N,

(3.2)

N→∞ M→∞

L lim (∇ψ L − ∇ψNL , v L − vN )Y = 0.

lim

(3.3)

N→∞ M→∞

Then, for any δ ∈ (0, δ0 ), there exist N > 0 and M > 2N such that 0 , (∇ ψ v0 )X = (∇ψ0 − ∇ψ0,N M , v0 − v0,N M )X ≤ δ,

(3.4)

where δ0 is a positive constant given in (4.3). At first, using the estimates in Section 2, we can obtain some useful estimates of the linear L ). From (2.13), Propositions 2.1 and 2.2, and (3.2), we obtain that there exists a solution (ψNL , vN positive constant ε1 ≤ 1 depending on N such that for all ε ∈ (0, ε1 ), L (∇ψNL , vN )Y ≤ C1 C0 .

(3.5)

L  2 L L L1 ([0,∞);L∞ ) + (∇ 2 ψNL , ∇vN )L2 ([0,∞);H 2 ) + ∇ψ ∇vN N L ([0,∞);L1 ) T

T

ξ

≤ C1 C0 + CN ε,

(3.6)

L ψNL L1 ([0,∞);L2 ) + (∇ψNL , vN )L1 ∩L2 ([0,∞);H 5 ) ≤ CN ,

(3.7)

L vN · ∇ψNL L1 ([0,+∞);L2 ) ≤ CN M ε,

(3.8)

L 2 L 2 (vN,1 ) − (∂2 ψNL )2 L1 ([0,+∞);L2 ) + (vN,2 ) − (∂1 ψNL )2 L1 ([0,+∞);L2 ) ≤ CN M ε,

(3.9)

L L vN,1 vN,2 + ∂1 ψNL ∂2 ψNL L1 ([0,+∞);L2 ) ≤ CN M ε.

(3.10)

Here CN M = C(N, M, (∇ψ0 , v0 )L2 ) denotes a uniform constant independent of ε. For simplify, we just denote CN in what follows. Then, using the similar argument as that in [20], we can obtain the following a priori estimates. 3.1. Estimates of A1,T Using Sobolev embedding theorem and Minkowski’s inequality, we can obtain the following lemma.

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5463

satisfies that ∇ ψ ∈ L∞ (H 2 ) and ∂1 ∇ ψ ∈ Lemma 3.1. (See Lemma 2.1, [20].) If the function ψ T 2 1 LT (H ), then there holds 1

 4 ∞ ≤ C∂1 ∇ ψ  2 2 ∇ ψ L (L ) T

LT

1

(H 1 )

 2 ∞ 1 ≤ CAT , ∇ ψ L (H )

(3.11)

T

where C is a positive constant independent of T . From Lemma 2.1, we can obtain the following lemma, and omit the details. , Lemma 3.2. Let (ψ v ) be sufficiently smooth functions which solve (1.14), then there holds d dt

    ε2 1 ε 2 2 + ψ 2 1 + ( | ψ )  v 2H 2 + ∇ ψ v 1 2 H H H 2 4 4

1 1 2 1 v2 2H 1 + ∇∂1 ψ + ∇ v 2H 2 − ∇ H 4 4 L L )H 2 + (( |ψ )H 2 = −(( v + vN ) · ∇ v | v )H 2 + (gN | v )H 2 − (fN |ψ v + vN ) · ∇ψ ε ε L ⊗ ∇ψ )| )H 1 + (g2 |ψ )H 1 − (div(∇ ψ v )H 2 − (( ) · ∇ v2 |ψ v + vN 4 4 ε ε ε L L L )H 1 + (∇ ))H 1 − (∇ − (( ) · ∇vN,2 |ψ v + vN ) · ∇ψ v + vN v2 |∇(( v2 |∇fN )H 1 4 4 4

− :=

ε2 ε2 L )|ψ )H 1 + (fN |ψ )H 1 ) · ∇ψ ((( v + vN 4 4

12 

(3.12)

Ij ,

j =1

where g2 := −

2 

∂2 (−)−1 [∂i vjε ∂j viε + ∂i ∂j (∂i ψ ε ∂j ψ ε )] −

i,j =1

2  ∂j [∂2 ψ ε ∂j ψ ε ]. j =1

, Lemma 3.3. Let ε ∈ (0, ε1 ), (ψ v ) be sufficiently smooth functions which solve (1.14) and satisfy ∞ 2 , v , ∇ ψ ∈ LT (H ), ∂1 ∇ ψ ∈ L2T (H 1 ), ∇ v ∈ L2T (H 2 ), and ∂1 ψ v ∈ L2T (L∞ ), then there holds A21,T

≤ C( v0 2H 2

0 2 2 ) + CA3T (1 + A2T ) + CN M ε(1 + A5T ) + C + ∇ ψ H

T L A2t ∇vN L∞ dt 0

+ CAT

⎧ T ⎨ ⎩

L 2 A2t ((∇ 2 ψNL , ∇vN )H 2 + ∇ψNL 2L∞ )dt

⎫ ⎬ ⎭

0

where C and CN M are two positive constants independent of T .

1 2

,

(3.13)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

Proof. From (3.9)–(3.10), using Hölder’s inequality and Sobolev embedding theorem, we obtain T

T I2 dt =

(gN | v )H 2 dt

0

0

T L ∇vN H 2 (∇ v H 2 +  v L∞ ) v H 2

≤C 0

T +C

H 2 (∇ 2 ψNL H 2 + ∇ψNL L∞ )dt ∇ v H 2 ∇ ψ

0

T L L (Pdiv(vN ⊗ vN + ∇ψNL ⊗ ∇ψNL )| v )H 2 dt

+C 0

≤ CAT

⎧ T ⎨ ⎩

L 2 2 )((∇ 2 ψNL , ∇vN ( v 2H 2 + ∇ ψ )H 2 + ∇ψNL L∞ )2 dt H

⎫ 12 ⎬ ⎭

0

 L 2 L 2 + CN (vN,1 ) − (∂2 ψNL )2 L1 (L2 ) + (vN,2 ) − (∂1 ψNL )2 L1 (L2 ) T T  L L L L + vN,1 vN,2 + ∂1 ψN ∂2 ψN L1 (L2 )  v L∞ (H 2 ) T

T

≤ CAT

⎧ T ⎨

L A2t ((∇ 2 ψNL , ∇vN )H 2 + ∇ψNL L∞ )2 dt

⎩

⎫ 12 ⎬ ⎭

+ CN M εAT ,

(3.14)

0 L , ψ L ) ⊂ {|ξ | ≤ 2N }. From (3.8), using Hölder’s inequality where we use the fact that suppF(vN N and Sobolev embedding theorem, we obtain

T

T I3 dt = −

0

)H 2 dt (fN |ψ

0

T ≤C

H 2 ((∇ v H 2 +  v L∞ )(∇ 2 ψNL H 2 + ∇ψNL L∞ )∇ ψ

0 L L∞ (H 2 ) + CN vN · ∇ψNL L1 (L2 ) ∇ ψ T

≤ CAT

⎧ T ⎨ ⎩ 0

T

A2t (∇ 2 ψNL H 2 + ∇ψNL L∞ )2 dt

⎫ 12 ⎬ ⎭

+ CN M εAT .

(3.15)

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5465

Using the similar argument as that in [20] (also see [14], for studying a 3D complex fluid v = 0, Hölder’s model), we can estimate the terms I4 + I5 as follows. From Lemma 3.1, using div inequality and Sobolev embedding theorem, we have T

|ψ )H˙ 1 dt + ( v · ∇ψ

T

0

|ψ )H˙ 2 dt ( v · ∇ψ

0

L∞ (L∞ ) + ∇  2 2 (∇ 2 L∞ (L4 ) ) ≤ C∇∂1 ψ v L2 (L2 ) ∇ ψ v L2 (L4 ) ∇ 2 ψ L (L ) T T

T

T

T

2

 2 ∞ ∇ + C∂1 ψ v1 L2 (L2 ) ∇ ψ L∞ (L2 ) L (L ) 2

T

T  −

T

T

+ 2∂2 )∂22 ψ dxdt (∂22 v2 ∂2 ψ v2 ∂22 ψ

0

L∞ (L∞ ) + ∇ 2  2 2 (∇ 3 L∞ (L4 ) + C∂1 ∇ 2 ψ v L2 (L2 ) ∇ ψ v L2 (L4 ) ∇ 2 ψ L (L ) T T

T

3

T

T

+ ∇ v L2 (L∞ ) ∇ ψ L∞ (L2 ) ) T

T

3

 2 ∞ + ∇ 2  2 4 + C∇ ψ L∞ (L2 ) (∇ v1 L2 (L2 ) ∂1 ψ v1 L2 (L4 ) ∇∂1 ψ L (L ) L (L ) 3

T

T

T

T

T

 2 2 ) + ∇ v1 L2 (L∞ ) ∇ ∂1 ψ L (L ) 2

T

T  −

T

(∂23 + 3∂22 + 3∂2 )dxdt. ∂23 ψ v2 ∂2 ψ v2 ∂22 ψ v2 ∂23 ψ

(3.16)

0

Then we need the following two lemmas to estimate the other terms in (3.16). Using the condition v1 to control ∂2 v2 and get the following lemma. div v = 0, we can use ∂1 Lemma 3.4. (See Lemma 2.4, [20].) Under the conditions in Lemma 3.3, there holds

T

T

T

 

 

 







3 3 2 3 2 2 2







∂2 ψ ∂2 ψ ∂2 ∂2 ψ ∂2 ψ ∂2 ∂2 ψ ∂2 ψ ∂2 v2 dxdt +

v2 dxdt +

v2 dxdt









0 0 0

T

T

 

 





2 2 2 2 3



(∂2 ψ ) ∂2 (∂2 ψ ) ∂2 v2 dxdt +

v2 dxdt

+





0

0

≤ CA3T ,

(3.17)

where C is a positive constant independent of T . Lemma 3.5. Under the conditions in Lemma 3.3, there holds

T

 



3 2

≤ A2 (1 + A2 )(CAT + CN ε),

∂ v (∂ dxdt ψ ) 2 2 2 T T





0

where C and CN are two positive constants independent of T .

(3.18)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

The proof of Lemma 3.5 is similar to the proof of Lemma 2.5 in [20]. For the convenience of the reader, we also give the detail proof in Appendix. From (3.16), Lemmas 3.4 and 3.5, we get T

|ψ )H˙ 1 + ( |ψ )H˙ 2 dt ≤ A2T (1 + A2T )(CAT + CN ε). ( v · ∇ψ v · ∇ψ

(3.19)

0

Similar to (3.17) and (3.19), we can easily obtain the following estimates and omit the details, T L (vN

|ψ )H 2 dt ≤ C · ∇ψ

0

T

L L H 2 (∇ ψ H 2 ∇vN H 1 ∇vN ∇ ψ L∞ + ∂1 ∇ ψ H 2 )dt

0

T L A2t ∇vN L∞ dt + CAT

≤C

⎧ T ⎨ ⎩

0

L 2 A2t ∇vN H 2 dt

⎫ 12 ⎬ ⎭

.

(3.20)

0

From Lemma 3.4, Hölder’s inequality and Sobolev embedding theorem, we have T −

⊗ ∇ψ )| ⊗ ∇ψ )| (div(∇ ψ v )H˙ 1 + (div(∇ ψ v )H˙ 2 dt

0

L∞ (L∞ ) ∇∂1 ψ  2 ∞ ∇ 2 ψ L∞ (L2 ) + ∇ ψ  2 2 ) v L2 (L2 ) (∂1 ψ ≤ C∇ 2 L (L ) L (L ) T T

T +

T

T

T

∂22 ψ ∂22 2∂2 ψ v2 dxdt

0

L∞ (L4 ) ∇∂1 ψ  2 4 + ∂1 ψ  2 ∞ ∇ 3 ψ L∞ (L2 ) v L2 (L2 ) (∇ 2 ψ + C∇ 3 L (L ) L (L ) T

T

L∞ (L∞ ) ∇ 2 ∂1 ψ  2 2 ) + 2 + ∇ ψ L (L ) T

T

T

T



T

T

)2 + ∂2 ψ ∂23 ψ )dxdt ∂23 v2 ((∂22 ψ

0

≤ CA3T .

(3.21)

Combing (3.19)–(3.21), we obtain T

T (I4 + I5 )dt ≤ C

0

L A2t ∇vN L∞ dt + CAT

⎧ T ⎨ ⎩

0

L 2 A2t ∇vN H 2 dt

⎫ 12 ⎬ ⎭

0

+ A2T (1 + A2T )(CAT + CN ε). Using Hölder’s inequality and Sobolev embedding theorem, from (3.5), we get

(3.22)

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

−

2 T 

5467

)H 1 dt (∂2 (−)−1 (∂i vjε ∂j viε )|ψ

i,j =1 0

T ≤C

H 1 dt (∇v ε )2 H 1 ∇ ψ

0

L∞ (H 2 ) ≤ CAT (C0 + AT )2 , ≤ C∇v ε 2L2 (H 2 ) ∇ ψ

(3.23)

T

T

and

⎞



⎝− ⎠ ∂2 (−)−1 (∂i ∂j (∂i ψ ε ∂j ψ ε )) − ∂j (∂2 ψ ε ∂j ψ ε )

ψ

i,j =1 j =1 ⎛

T 0

=

T 

2 

2 



−∂2 (−)−1 (∂12 (∂1 ψ ε )2 ) ψ

H1

dt H1

dt

0

+

T 



2∂2 (−)−1 (∂2 (∂2 ψ ε ∂1 ψ ε )) + (∂2 ψ ε ∂1 ψ ε ) ∂1 ψ

H1

dt

0

T +

)H 1 (−(−)−1 ∂23 (∂2 ψ ε )2 − ∂2 (∂2 ψ ε )2 |ψ

0

T ≤C

H 1 dt + ∇(∇ψ ∂1 ψ )H 1 ∇∂1 ψ ε

T

ε

0

)H 1 ((−)−1 ∂2 ∂12 (∂2 ψ ε )2 |ψ

0

 2 1 (∇∂1 ψ  2 1 + ∂1 ψ ε  2 ∞ )∇ψ ε L∞ (H 2 ) ≤ C∇∂1 ψ L (H ) L (H ) L (L ) ε

T

T +C

T

T

T

H 1 dt ∂1 (∇ψ ε )2 H 1 ∇∂1 ψ

0

≤ CAT (C0 + AT )2 .

(3.24)

From (3.23)–(3.24), we have T

T I7 dt =

0

ε )H 1 dt ≤ CεAT (C02 + A2T ). (g2 |ψ 4

0

Using the similar argument as that in the proofs of (3.19)–(3.20) we have

(3.25)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

T

T I11 dt = −

0

ε2 L )|ψ )H 1 dt ≤ CN ε 2 A2t (1 + A3T ). ) · ∇ψ ((( v + vN 4

(3.26)

0

Similarly, we can easily obtain the following estimates and omit the details, T

T I1 dt = −

0

T L (( v + vN ) · ∇ v | v )H 2 dt

0

L ∇(vN + v )H 2 ∇ v H 2  v H 2 dt, (3.27) 0

T

T (I6 + I8 )dt = −

0

≤

≤C

ε L L )H 1 dt ) · ∇( v2 + vN,2 )|ψ (( v + vN 4

0

L )L2 (H 2 ) Cε(∇( v + vN T

L L L∞ (H 2 ) , (3.28) +  v + vN L2 (L∞ ) )∇( v + vN )L2 (H 2 ) ∇ ψ T

T

T I9 dt =

0

T

T

ε L ))H 1 dt v + vN ) · ∇ψ (∇ v2 |∇(( 4

0

L L∞ (H 1 ) , v + vN L2 (W 1,∞ ) ∇ ψ ≤ Cε∇ v L2 (H 2 )  T

T

T I10 dt = −

0

T

(3.29)

T

ε v L∞ (H 2 ) ∇fN L1 (H 1 ) , (∇ v2 |∇fN )H 1 dt ≤ Cε T T 4

(3.30)

0

T

T I12 dt =

0

ε2 )H 1 dt ≤ Cε 2 ∇ ψ L∞ (H 2 ) ∇fN  1 2 . (fN |ψ LT (H ) T 4

(3.31)

0

From (3.2), (3.5)–(3.6) and (3.8), using Hölder’s inequality and Sobolev embedding theorem, we obtain fN L1 (H 3 ) ≤ CN (AT + 1).

(3.32)

T

From Lemma 3.2, (3.14)–(3.15), (3.22) and (3.25)–(3.32), one can obtain (3.13).

2

3.2. Estimates of A2,T Then, we estimate A2,T as follows. At first, we estimate the easy one , (∂1 ψ v )1{|ξ |≥1}∪{|ξ |<1,|ξ1 |≤ε|ξ |2 } . Using the similar argument as in the proof of Lemma 3.1 in [20], we can obtain the following lemma and omit the details.

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5469

Lemma 3.6. Under the conditions in Lemma 3.3, there holds % % % % % % % ψ 1 v 1{|ξ |≥1} %L2 (L1 ) + %∂ 1 {|ξ |≥1} % T

L2T (L1ξ )

ξ

≤ CA1,T ,

% % % % % % % v 1{|ξ |<1,|ξ1 |≤ε|ξ |2 } %L2 (L1 ) + %∂ 1 ψ 1{|ξ |<1,|ξ1 |≤ε|ξ |2 } % T

L2T (L1ξ )

ξ

≤ CA1,T ,

(3.33) (3.34)

where C is a positive constant independent of T . , To estimate the difficult part (∂1 ψ v )1{|ξ |<1,|ξ1 |>ε|ξ |2 } , we need obtain the following expressions of solutions. From (1.14), we have ⎧ + ε −1 v2 = F, (t, x) ∈ R+ × R2 , ∂t ψ ⎪ ⎪ ⎪ ⎪ ⎪ = G1 , v1 −  v1 − ε −1 ∂1 ∂2 ψ ⎪ ⎨ ∂t = G2 , (3.35) v2 −  v2 + ε −1 ∂12 ψ ∂t ⎪ ⎪ ⎪ v = 0, ⎪ div ⎪ ⎪ ⎩ 0 , v0 ), (ψ , v )|t=0 = (ψ where F = −v ε · ∇ψ ε , G = (G1 , G2 ) = −v ε · ∇v ε −

2 

∇(−)−1 [∂i vjε ∂j viε + ∂i ∂j (∂i ψ ε ∂j ψ ε )]

i,j =1

−

2  ∂j [∇ψ ε ∂j ψ ε ]. j =1

From (2.7)–(2.9), we obtain that   0 (ξ ) + M2 (t, ξ ) (t, ξ ) = M1 (t, ξ )ψ v 0,2 (ξ ) + ψ t +

t

(ξ, s)ds M1 (ξ, t − s)F

0

2 (ξ, s)ds, M2 (ξ, t − s)G

(3.36)

0

  0 (ξ ) + M3 (t, ξ ) v 0,1 (ξ ) + v 1 (t, ξ ) = M2 (t, ξ )ξ1 ξ2 ψ t + 0

t

(ξ, s)ds M2 (ξ, t − s)ξ1 ξ2 F

0

1 (ξ, s)ds, M3 (ξ, t − s)G

(3.37)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

 0 (ξ ) + M3 (t, ξ )  v 0,2 (ξ ) − v 2 (t, ξ ) = −|ξ1 |2 M2 (t, ξ )ψ t +

t

(ξ, s)ds |ξ1 |2 M2 (ξ, t − s)F

0

2 (ξ, s)ds, M3 (ξ, t − s)G

(3.38)

0

where Mi , i = 1, 2, 3, are given in (2.10) and & |ξ |2 λ± = − ± iε −1 β, β = 2

4ξ12 − ε 2 ξ 4 2

, when |ξ1 | > ε|ξ |2 .

(3.39)

Lemma 3.7. Under the conditions in Lemma 3.3, there holds % % % % % % % v 1{|ξ |<1,|ξ1 |>|ξ |2 } %L2 (L1 ) + %∂ 1 ψ 1{|ξ |<1,|ξ1 |>|ξ |2 } % T

L2T (L1ξ )

ξ

 0 , |ξ |−1 ≤ C(ψ v 0 )L1 + CA2T ,

(3.40)

ξ

where C is a positive constant independent of T . Proof. When ξ ∈ A = {|ξ | < 1, |ξ1 | > ε|ξ |2 }, from (2.10), (3.39), we get



− |ξ |2 t ξ 2 sin(ε −1 βt) + 2ε −1 β cos(ε −1 βt)

|ξ |2 t

≤ Ce− 2 , 2 |M1 | = e

2ε −1 β



|ξ |2 t 2 sin(ε −1 βt)

|ξ |2 t

≤ C|ξ1 |−1 e− 2 , |M2 | =

e− 2

2β

(3.41) (3.42)

then we can estimate the linear part as follows − 0  2 1 M1 ∂ 1ψ L (L (A)) ≤ Cξ1 e ξ

T

ξ1 M2 v 0,2 L2 (L1 (A)) ≤ Ce− T

ξ

|ξ |2 t 2

|ξ |2 t 2

  0  1 , 0  2 1 ≤ Cψ ψ L (L ) L ξ

T

(3.43)

ξ

 v 0,2 L2 (L1 ) ≤ C|ξ |−1 v 0 L1 . T

ξ

ξ

Let F = F 1 + F 2 + F3 , G = E 1 + E 2 + E 3 , where L L , F2 = − , F3 = −vN F1 = − v · ∇ψ v · ∇ψNL − vN · ∇ψ · ∇ψNL ,

and E1 = − v · ∇ v−

2  i,j =1

∂j ψ )] − ∇(−)−1 [∂i vj ∂j vi + ∂i ∂j (∂i ψ

2  ∂j ψ ], ∂j [∇ ψ j =1

(3.44)

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

L L E2 = − v · ∇vN − vN · ∇ v−

5471

2  + ∇ψ ∂j ψNL ] ∂j [∇ψNL ∂j ψ j =1

−

2 

L L ∂j ψNL + ∂i ψNL ∂j ψ )], ∇(−)−1 [∂i vj ∂j vN,i + ∂i vN,j ∂j vi + ∂i ∂j (∂i ψ

i,j =1 L E3 = −vN

L · ∇vN

2 

−

∇(−)

−1

L L [∂i vN,j ∂j vN,i

+ ∂i ∂j (∂i ψNL ∂j ψNL )] −

i,j =1

2  ∂j [∇ψNL ∂j ψNL ]. j =1

From (3.41)–(3.42), we obtain ξ1 M1 (t)L2

∞ ξ2 (Lξ1 (A))

ξ1 M2 (t)L1

ξ2

(L2ξ (A)) 1

≤ Cξ1 e−

≤ Ce−

|ξ |2 t 2

|ξ |2 t 2

L1

ξ2

1

L2

∞ ξ2 (Lξ1 )

(L2ξ ) 1

≤ Ct − 2 e−

≤ Ce−

ξ12 t 2

ξ22 t 2

3

L2 ≤ Ct − 4 ,

(3.45)

ξ2

L2 e− ξ1

ξ22 t 2

3

L1 ≤ Ct − 4 . ξ2

(3.46)

Using Young’s inequality and the interpolation inequality, we get F1 

4

LT3 (L2ξ (L1ξ )) 2

  4 2 ≤ C v L2 (L1 ) ∇ ψ L (L T

ξ

T

1

1 ξ2 (Lξ1 ))

1

1

  2 ∞ ≤ C v L2 (L1 ) ∇ ψ T

LT (L2ξ )

ξ

  2 2 |ξ1 |∇ ψ

LT (L2ξ )

≤ CA1,T A2,T ,

(3.47)

and combining the idea in (3.24),  E 1,2 

4

2 LT3 (L∞ ξ (Lξ )) 2

≤ C v L4 (L2 T

1

1 ξ2 (Lξ1 ))

 v  2 2 + C∂  2 2 ∇  4 2 ∇ ψ 1∇ ψ L (L ) L (L ) L (L T

ξ

T

ξ

T

≤ CA21,T ,

1 ξ2 (Lξ1 ))

(3.48)

where E1 = (E1,1 , E1,2 ) . From (3.45)–(3.48), using the Riesz potential inequality ([19], Theorem 1, p. 119), we obtain % % % % t % % %ξ1 M1 (ξ, t − s)F1 (ξ, s)ds % % % % 2 1 % 0

LT (Lξ (A))

% t % % % % % %  ≤ % ξ1 M1 (ξ, t − s)L2 (L∞ (A)) F1 (ξ, s)L2 (L1 ) ds % % ξ2 ξ1 ξ2 ξ1 % % 0

% t % % % % % − 34  % ≤ C % (t − s) F1 (ξ, s)L2 (L1 ) ds % % ξ2 ξ1 % % 0

% % ≤ C %F1 (ξ, s)%

4

LT3 (L2ξ (L1ξ )) 2

1

L2T

L2T

≤ CA1,T A2,T ,

(3.49)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

and % % % % t % % % %ξ1 M2 (ξ, t − s)E  1,2 (ξ, s)ds % % % % 0

L2T (L1ξ (A))

% t % % % % % %  ≤ % ξ1 M2 (ξ, t − s)L1 (L2 (A)) E1,2 (ξ, s)L∞ (L2 ) ds % % ξ2 ξ1 ξ2 ξ1 % % 0

% t % % % % % − 34  % ≤C% (t − s)  E (ξ, s) ds 2 ) 1,2 L∞ (L % % ξ2 ξ1 % % 0

% % %  ≤ C %E 1,2 (ξ, s)

4

2 LT3 (L∞ ξ (Lξ )) 2

L2T

L2T

≤ CA21,T .

(3.50)

1

Similarly, we get F2 

 L 2 ≤ Cv N L2 (L1 ) ∇ ψ  ∞ 1

4 LT3 (L2ξ (L1ξ )) 2 1

T

1

LT (L2ξ )

ξ

1

  2 2 |ξ1 |∇ ψ

1

+ C v 2 ∞

LT (L2ξ )

|ξ1 | v 2 2

LT (L2ξ )

LT (L2ξ )

L  2 1 ∇ψ N L (L ) ξ

T

≤ (CC0 + CN ε)A1,T , % % % t % % % %ξ1 M1 (ξ, t − s)F2 (ξ, s)ds % % % % % 0

 E 2,2 

4

2 LT3 (L∞ ξ (Lξ )) 2

1

≤ C v L4 (L2 T

1 ξ2 (Lξ1 ))

L + Cv N L4 (L2 T

% % ≤ C %F2 (ξ, s)%

L  2 2 ∇v N L (L )

1 ξ2 (Lξ1 ))

T

ξ

4

LT3 (L2ξ (L1ξ )) 2

L2T (L1ξ (A))

1

≤ (CC0 + CN ε)A1,T , L   2 2 ∇ψ + C∂ ∇ψ 1

(3.51)

N L4T (L2ξ (L1ξ ))

LT (Lξ )

2

1

 L v  2 2 + C∂  4 2 ∇ 1 ∇ψN L2 (L2 ) ∇ ψ L (L ) L (L T

ξ

ξ

T

T

1 ξ2 (Lξ1 ))

≤ CC0 A1,T , and % % % % t % % % %ξ1 M2 (ξ, t − s)E  (ξ, s)ds 2,2 % % % % 0

% % %  ≤ C %E 2,2 (ξ, s) L2T (L1ξ (A))

4

2 LT3 (L∞ ξ (Lξ )) 2

≤ CC0 A1,T .

1

(3.52) Similarly, from (3.9)–(3.10), we have

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

% % % % t % % %ξ1 M1 (ξ, t − s)F3 (ξ, s)ds % % % % % 0

L2T (L1ξ (A))

5473

% t % % % % % %  ≤ % ξ1 M1 (ξ, t − s)L2 (A) F3 (ξ, s)L2 ds % % ξ ξ % % 0

% t % % % % % 1 −2 L L % ≤ C % (t − s) vN · ∇ψN L2 ds % % % %

L2T

L2T

0

L ≤ CvN · ∇ψNL L1 (L2 ) ≤ CN M ε, T

(3.53)

and % % % % t % % % %ξ1 M2 (ξ, t − s)E  3,2 (ξ, s)ds % % % % 0

L2T (L1ξ (A))

% t % % % % % −1  % ≤% ξ |ξ |M (ξ, t − s) |ξ | (ξ, s) ds E 2 2 1 2 3,2 Lξ (A) Lξ % % % % 0

% t % % % % % − 12 −1  % ≤C% (t − s) |ξ | (ξ, s) ds E 2 3,2 Lξ % % % % 0

L2T

L2T

L 2 L 2 ≤ C(vN,1 ) − (∂2 ψNL )2 L1 (L2 ) + C(vN,2 ) − (∂1 ψNL )2 L1 (L2 ) T

L L + CvN,1 vN,2

T

+ ∂1 ψNL ∂2 ψNL L1 (L2 ) T

≤ CN M ε.

(3.54)

Combining (3.43)–(3.44) and (3.49)–(3.54), we get % % % 1A % %∂1 ψ %

L2T (L1ξ )

 0 , |ξ |−1 ≤ C(ψ v 0 )L1 + CC0 A1,T + CA2T + CN M ε(1 + AT ). ξ

% % Similarly, we can estimate % v 1A %L2 (L1 ) and omit the details. ξ

T

(3.55)

2

From Lemmas 3.6 and 3.7, we can immediately obtain the following lemma. Lemma 3.8. Under the conditions in Lemma 3.3, there holds  0 , |ξ |−1 A2,T ≤ C(ψ v 0 )L1 + C(C0 + 1)A1,T + CA2T + CN M ε(1 + AT ), ξ

where C is a positive constant independent of T .

(3.56)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

4. Proof of Theorem 1.1 Using the classical method in [15], we can get the following local existence result and omit the details. Theorem 4.1. Assume that the initial data (ψ0 , v0 ) satisfy (1.7), then there exists T0 > 0 such that system (1.3) has a unique local solution (ψ, v, ∇p) ∈ ET20 on [0, T0 ]. Proof of the global existence. Theorem 4.1 implies that the system (1.3) has a unique local strong solution (ψ, v, ∇p) on [0, T ∗ ), where [0, T ∗ ) is the maximal existence interval of the above solution. The goal of this section is to prove that T ∗ = ∞ provided that the initial data (ψ0 , v0 ) satisfy (1.8). From (3.13) and (3.56), we have T A2T ≤ C2 (1 + C02 )δ 2 + C2 (1 + C04 )

L L 2 A2t (∇vN L∞ + (∇ 2 ψNL , ∇vN )H 2 + ∇ψNL 2L∞ )dt 0

+ C2 (1 + C02 )A3T (1 + A2T ) + CN M ε(1 + A5T ),

(4.1)

for all T ∈ (0, T ∗ ). Let T1 = sup{t ∈ [0, T ∗ ); At ≤ min{1,

1 }}. 10C2 (1 + C02 )

(4.2)

Then, we have T A2T ≤ 2C2 (1 + C02 )δ 2 + 4CN M ε + 2C2 (1 + C04 )

L A2t (∇vN L∞ 0

+ (∇

2

L 2 ψNL , ∇vN )H 2

+ ∇ψNL 2L∞ )dt

for all T ∈ (0, T1 ). Using Gronwall’s inequality and (3.6), we get A2T ≤ (2C2 (1 + C02 )δ 2 + 4CN M ε) exp{2C2 (1 + C04 )(1 + C1 C0 + CN ε)2 }, for all T ∈ (0, T1 ). Choosing a positive constant δ0 such that 16C2 (1 + C02 )δ02 exp{8C2 (1 + C04 )(1 + C1 C0 )2 } ≤ min{1,

1 }, 100C22 (1 + C02 )2

(4.3)

for any fixed δ ∈ (0, δ0 ), choosing a positive constant ε2 ≤ ε1 such that 4CN M ε2 ≤ C2 δ 2 , CN ε2 ≤ 1, we can easily obtain that, for all T ∈ (0, T1 ), ε ∈ (0, ε2 ),

(4.4)

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5475

& 1 1 }. (4.5) AT ≤ δ 4C2 (1 + C02 ) exp{4C2 (1 + C04 )(1 + C1 C0 )2 } ≤ min{1, 2 10C2 (1 + C02 ) Thus, one can easily obtain that T1 = T ∗ = ∞. From (1.4), we have that ∇p ∈ L∞ ([0, ∞); H 1 ). From (2.13), (3.4) and (4.5), we have that for all ε ∈ (0, ε2 ), L (∇ψ ε − ∇ψ L , v ε − v L )Y ≤ C(C0 )δ + (∇ψNL − ∇ψ L , vN − v L )Y ≤ C(C0 )δ, (4.6)

which finishes the proof of (1.9). This finishes proof of Theorem 1.1.

2

Acknowledgments Part of this work was done when the author was visiting Department of Mathematics, Princeton University (2012), Courant Institute of Mathematical Sciences, New York University (2012). The author appreciate the hospitality from Professor Fanghua Lin, and Yakov Sinai. This work is partially supported by National Natural Science Foundation of China under Grants 11271322, 11331005 and 11271017, National Program for Special Support of Top-Notch Young Professionals. The author thanks the referees for their useful comments and suggestions. Appendix Proof of Lemma 2.1. Taking the standard H 2 inner product of (2.1)2,3 and v, using the integration by parts, we have 1 d v2H 2 + ∇v2H 2 = ε −1 (∂1 ∂2 ψ|v1 )H 2 − ε −1 (∂12 ψ|v2 )H 2 + (g|v)H 2 . 2 dt

(4.7)

Using the fact that divv = 0 and the integration by parts, we get (∂1 ∂2 ψ|v1 )H 2 = −(∂2 ψ|∂1 v1 )H 2 = (∂2 ψ|∂2 v2 )H 2 = −(∂22 ψ|v2 )H 2 .

(4.8)

From (2.1)1 , we obtain −ε −1 (ψ|v2 )H 2 = (ψ|(∂t ψ − f ))H 2 = −

1 d ∇ψ2H 2 − (ψ|f )H 2 . 2 dt

(4.9)

From (4.7)–(4.9), we have  1 d  v2H 2 + ∇ψ2H 2 + ∇v2H 2 = (g|v)H 2 − (ψ|f )H 2 . 2 dt

(4.10)

Taking the standard H 1 inner product of (2.1)3 and ψ, using the integration by parts, we have (∂t v2 |ψ)H 1 − (v2 |ψ)H 1 + ε −1 ∂1 ∇ψ2H 1 = (g2 |ψ)H 1 . From (2.1)1 , using the integration by parts, we get

(4.11)

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T. Zhang / J. Differential Equations 260 (2016) 5450–5480

(∂t v2 |ψ)H 1 d (v2 |ψ)H 1 − (v2 |∂t ψ)H 1 dt d = (v2 |ψ)H 1 + ε −1 (v2 |v2 )H 1 − (v2 |f )H 1 dt d = (v2 |ψ)H 1 − ε −1 ∇v2 2H 1 − (v2 |f )H 1 . dt =

(4.12)

From (2.1)1 , we obtain −(v2 |ψ)H 1 = ε((∂t ψ − f )|ψ)H 1 =

ε d ψ2H 1 − ε(f |ψ)H 1 . 2 dt

(4.13)

From (4.11)–(4.13), we have  d ε ψ2H 1 + (v2 |ψ)H 1 + ε −1 ∂1 ∇ψ2H 1 − ε −1 ∇v2 2H 1 dt 2 = (g2 |ψ)H 1 + (v2 |f )H 1 + ε(f |ψ)H 1 .

(4.14)

From (4.10) and (4.14), one can finish the proof of (2.2). 2 To prove Lemma 3.5, we give the following lemma, where we use equation (1.14)1 to bounded

T ∂2 )2 dxdt. v2 (∂23 ψ the term 0 ∂2 ψ Lemma 4.1. Under the conditions of Lemma 3.3, there holds

T

 



3 2

∂2 ψ ∂2 v2 (∂2 ψ ) dxdt

≤ CA4T + CN εA3T (1 + A2T ),





(A.9)

0

where C and CN are two positive constants independent of T . Proof. From (1.14)1 , we have L ). − ( v2 = ε(fN − ∂t ψ ) · ∇ψ v + vN

(A.10)

From Lemma 3.1, (A.10), the integration by parts, Hölder’s inequality and Sobolev embedding theorem, we get

T 



' 3 (2

∂2 dxdt

∂2 ψ v2 ∂2 ψ



0



T 



' ( (' 3 (2 '

L − fN ∂2 ψ ∂2 ∂t ψ + dxdt

∂2 ψ v + vN · ∇ ψ = ε



0

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5477



T

1

' ( 2 3 2

= ε

(∂2 ψ ) ∂2 ψ dx

2 0 T  +

)

'' ( (' ( * L − fN ∂23 ψ )2 ∂23 ψ ∂23 ∂t ψ + ∂2 ψ ∂2 2 dxdt| −(∂2 ψ v + vN · ∇ψ

0

3 ∞ 2 ∂2 fN  1 ∞ 2 ∞ ∞ ∂23 ψ 2 ∞ 2 + Cε∇ ψ ≤ Cε∂2 ψ L (L ) L (L ) L (L ) L (H ) T

T

T

T

T 

) '' ( (

L + ε −1 )2 ∂23 ψ ∂23 (∂2 ψ v + vN · ∇ψ + ε

v2 − fN

0



'' ( (' * ( 2 L ∂23 ψ ∂2 dxdt

+ ∂2 ψ v + vN · ∇ψ

4 ∞ 2 + CN εA3T (1 + AT ) ≤ Cε∇ ψ L (H ) T

'' ( ( ' ( L L − 2 4 ∞ ∂23 ψ L∞ (L2 ) ∂23  2 2 v + vN · ∇ψ v + vN · ∇∂23 ψ + Cε∂2 ψ L (L ) L (L ) T

T

T

2 4 ∞ ∂23 ψ L∞ (L2 ) ∂23 + C∂2 ψ v2 L2 (L2 ) L (L ) T

T

T



T   



' ( ' 3 (2 ' ( ' 3 (2 * 1)



2 L L 2 ) + ) ∂2 ψ v + vN · ∇ ∂2 ψ dxdt

+ ε

v + vN · ∇(∂2 ψ (∂2 ψ



2 0



T 



+ ' ( ' , ( 2

L ∂23 ψ ∂2 dxdt

∂2 ψ v + vN · ∇ψ + ε



0

' ( ≤ CA4T + CN εA3T 1 + A2T , where we use the estimates L L ) − (  2 2 ∂23 (( v + vN ) · ∇ψ v + vN ) · ∇∂23 ψ L (L ) T

≤ C∇

3

L L∞ (L∞ ) ( v + vN )L2 (L2 ) ∇ ψ T T

L L∞ (L4 ) + C∇ 2 ( v + vN )L2 (L4 ) ∇ 2 ψ T

T

L L∞ (L2 ) + C∇( v + vN )L2 (L∞ ) ∇ 3 ψ T T L L∞ (H 2 ) , ≤ C∇( v + vN )L2 (H 2 ) ∇ ψ T

T

and



T

   



L 3 2



∂ ( v + v ) · ∇ ψ (∂ dxdt ψ ∂ ψ ) 2 2 N 2





0

L 2 ∞ 2 . 2 4 ∞ ∇( v + vN )L2 (L∞ ) ∇ 3 ψ ≤ C∇ ψ L (L ) L (L ) T

T

T

2

To prove Lemma 3.5, we also need the following lemma, whose proof is similar to Lemma 3.4.

5478

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

Lemma 1.2. Under the conditions in Lemma 3.3, there holds

T

 



3 2 L 2 L

≤ C∇ ψ

 2 1 ∇( 2 ∞ 2 ∂1 ∇ ψ ∂ ( v + v )∂ v + vN )L2 (H 2 ) , ψ ∂ ψ ∂ ψ dxdt 2 2 2 N,2 2 2 LT (H )



LT (H ) T



0

(A.11) where C is a positive constant independent of T . From the above lemma, we can obtain the proof of the key lemma in this paper. Proof of Lemma 3.5. From Lemmas 3.1, 3.4, 4.1 and 1.2, (A.10), the integration by parts, Hölder’s inequality and Sobolev embedding theorem, we get

T

 



3 2



∂ v (∂ dxdt ψ ) 2 2 2





0

T

 



L 3 2

∂2 (∂t ψ + ( v + vN ) · ∇ ψ − fN )(∂2 ψ ) dxdt

=ε



0



T T  

 



3 2 3 3 L 3 2



v + vN ) · ∇ ψ − fN )(∂2 ψ ) dxdt

−2∂2 ψ ∂2 ψ ∂2 ∂t ψ + ∂2 (( = ε ∂2 ψ (∂2 ψ ) dx +



0 0

2 ∞ 2 ∂2 fN  1 ∞ L∞ (L∞ ) ∂23 ψ 2 ∞ 2 + Cε∇ ψ ≤ Cε∂2 ψ LT (L ) LT (L ) LT (H ) T

T

  





L L + ε −1 )(∂23 ψ ∂23 ψ ∂23 (( )2 dxdt

+ ε

v + vN ) · ∇ψ v2 − fN ) + ∂2 (( v + vN ) · ∇ψ 2∂2 ψ



0

3 ∞ 2 + CN εA2T (1 + AT ) ≤ Cε∇ ψ L (H ) T

L∞ (H 2 ) ∇∂1 ψ  2 1 ∇ + C∇ ψ v L2 (H 2 ) LT (H ) T T

T

  

L L L ) − ( ∂23 ψ [∂23 (( ] + ∂2 ψ ( )2 + ε

v + vN ) · ∇ψ v + vN ) · ∇∂23 ψ ) · ∇(∂23 ψ v + vN 2∂2 ψ

0

 L L (∂23 ψ (∂23 ψ )2 + ∂2 ( )2 dxdt

+ ( v + vN ) · ∇∂2 ψ v + vN ) · ∇ψ ≤ CA3T + CN εA2T (1 + AT )3 , where we use the estimate

T

   



3 3 L L 3 L 3 2



2∂ (( v + v ) · ∇ ψ ) − ( v + v ) · ∇∂ ( v + v ) · ∇ ψ (∂ dxdt ψ ∂ ψ [∂ ψ ] + ∂ ψ ) 2 2 N N N 2 2 2 2





0

T. Zhang / J. Differential Equations 260 (2016) 5450–5480

5479

T

 



3 3 L 2 L 2 L 2

2∂2 ψ ∂2 ψ (∂2 ( ≤

v + vN ) · ∇ ψ + 3∂2 ( v1 + vN,1 )∂1 ∂2 ψ + 3∂2 ( v2 + vN,2 )∂2 ψ )dxdt



0

T

T

 

 





3 L 2 L 3 2



∂2 ψ ∂2 ( dxdt + ∂2 ( ) dxdt

6∂2 ψ 7∂2 ψ v1 + vN,1 )∂1 ∂2 ψ v2 + vN,2 )(∂2 ψ +









0 0

T

 



L (∂23 ψ )2 dxdt

∂2 ( v1 + vN,1 )∂1 ψ +





0

L L∞ (L2 ) ∇ 3 ( 2 4 ∞ ∂23 ψ v + vN )L2 (L2 ) ≤ C∇ ψ L (L ) T

T

T

L L∞ (L∞ ) ∂33 ψ  2 4 L∞ (L2 ) ∇ 2 ( + C∇ ψ v + vN )L2 (L4 ) ∂1 ∇ ψ LT (L ) T T T L 2 ∞ 2 ∂1 ∇ ψ  2 2 ∇( + C∇ ψ v + vN )L2 (H 2 ) L (H ) L (H ) T

T

T

L L∞ (L∞ ) ∂23 ψ L∞ (L2 ) ∂2 (  2 2 + C∂2 ψ v + vN )L2 (L∞ ) ∂1 ∂22 ψ L (L ) T T

+ CA4T

T

T

L L∞ (L∞ ) ∇vN 2 ∞ 2 + CN εA3T (1 + A2T ) + C∂2 ψ L1 (L∞ ) ∂23 ψ LT (L ) T T

L  2 ∞ ∂33 ψ 2 ∞ 2 + C∂2 ( v + vN )L2 (L∞ ) ∂1 ψ L (L ) L (L ) T

≤ CA4T

+ CN A3T

T

+ CN εA3T (1 + A2T ).

T

2

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