Global existence and incompressible limit in critical spaces for ...

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT IN CRITICAL SPACES FOR COMPRESSIBLE FLOW OF LIQUID CRYSTALS

arXiv:1507.08804v1 [math.AP] 31 Jul 2015

QUNYI BIE, HAIBO CUI, QIRU WANG∗ , AND ZHENG-AN YAO Abstract. The Cauchy problem for the compressible flow of nematic liquid crystals in the framework of critical spaces is considered. We first establish the existence and uniqueness of global solutions provided that the initial data are close to some equilibrium states. This result improves the work by Hu and Wu [SIAM J. Math. Anal., 45 (2013), pp. 2678-2699] through relaxing the regularity requirement of the initial data in terms of the director field. We then consider the incompressible limit problem for ill prepared initial data. We prove that as the Mach number tends to zero, the global solution to the compressible flow of liquid crystals converges to the solution to the corresponding incompressible model in some function spaces. Moreover, the accurate converge rates are obtained.

Contents 1. Introduction and main results

2

2. Homogeneous and hybrid Besov spaces

8

3. Global existence for initial data near equilibrium

11

4. Uniqueness

15

5. Incompressible limit

21

References

35

2010 Mathematics Subject Classification. 35Q35, 76N10, 35B40. Key words and phrases. Liquid crystal flow; global well-posedness; critical space; incompressible limit. Research Supported by the NNSF of China (nos. 11271379, 11271381), the National Basic Research Program of China (973 Program) (Grant No. 2010CB808002) and the Science Foundation of China Three Gorges University (no. KJ2013B030). ∗

Corresponding author, email: [email protected], tel.: 86-20-84037100, fax: 86-20-84037978. 1

2

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

1. Introduction and main results In this paper we consider the global well-posedness and incompressible limit to the following compressible flow of nematic liquid crystals in critical spaces:   ∂t ρ + div(ρu) = 0,        ∂t (ρu) + div(ρu ⊗ u) − µ∆u − (µ + λ)∇divu + ∇P (ρ)     = −ξdiv ∇d ⊙ ∇d − 12 |∇d|2 I ,       ∂t d + u · ∇d = θ(∆d + |∇d|2 d),

(1.1)

where ρ ∈ R is the density function of the fluid, u ∈ RN (N ≥ 2) is the velocity, and d ∈ SN −1

represents the director field for the averaged macroscopic molecular orientations. The scalar function P ∈ R is the pressure, which is an increasing and convex function in ρ. We denote

by λ and µ the two Lam´e coefficients of the fluid, which are constant and satisfy µ > 0 and ν := λ + 2µ > 0. Such a condition ensures ellipticity for the operator µ∆ + (λ + µ)∇div and is satisfied in the physical cases. The constants ξ > 0, θ > 0 stand for the competition between the kinetic energy and the potential energy, and the microscopic elastic relaxation time (or the Debroah number) for the molecular orientation field, respectively. The symbol ⊗ denotes the

Kronecker tensor product such that u ⊗ u = (ui uj )1≤i,j≤N and the term ∇d ⊙ ∇d denotes a

matrix whose (i, j)−th entry is ∂xi d · ∂xj d (1 ≤ i, j ≤ N ). I is the N × N identity matrix. To complete the system (1.1), the initial data are given by

ρ|t=0 = ρ0 (x), u|t=0 = u0 (x), d|t=0 = d0 (x), with d0 ∈ SN −1 .

(1.2)

The hydrodynamic theory of liquid crystals was first proposed by Ericksen [11, 12] and Leslie [24] in 1960s. In 1989, Lin [25] first derived a simplified Ericksen-Leslie equation modeling liquid crystal flows when the fluid is an incompressible and viscous fluid. Subsequently, Lin and Liu [27] showed the global existence of weak solutions and smooth solutions for the approximation system. Recently, Hong [16] and Lin et al. [26] showed independently the global existence of a weak solution of an incompressible model of system (1.1) in two-dimensional space. Furthemore, in [26], the regularity of solutions except for a countable set of singularities whose projection on the time axis is a finite set had been obtained. Very recently, in dimension three, Lin and Wang [28] have proved the existence of global weak solutions under the assumption that d0 ∈ S2+ by

developing some new compactness arguments, here S2+ is the upper hemisphere.

As for the compressible case, Huang et al. [19] proved the local existence of unique strong solution of (1.1) provided that the initial data ρ0 , u0 , d0 are sufficiently regular and satisfy a natural compatibility condition. And a criterion for possible breakdown of such a local strong ∞ and k∇dk 3 ∞ . solution at finite time was given in terms of blow up of the quantities kρkL∞ Lt Lx t Lx

In [18], an alternative blow-up criterion was derived in terms of the temporal integral of both the L∞ -norm of the deformation tensor Du and the square of the L∞ -norm of ∇d. In terms of

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

3

the global well-posedness, results in one dimensional space have been obtained in [9, 10]. In two dimensions, Jiang et al. [21] established the global existence of weak solutions under the small initial energy. In two or three dimensions, if some component of initial direction field is small, Jiang et al. [20] established the global existence of weak solutions to the initial-boundary problem with large initial energy and without any smallness condition on the initial density and velocity. Recently, Lin et al. [29] established the existence of global weak solutions in three-dimensional space, provided the initial orientational director field d0 lies in the hemisphere S2+ . The low Mach number limit of the system (1.1)-(1.2) has also been studied recently. Hao and Liu [15] investigated the so-called incompressible limit (i.e., the low Mach number limit) for solutions in the whole space RN (N = 2, 3) and a bounded domain of RN (N = 2, 3) with Dirichlet boundary conditions. Ding et al. [8] studied the incompressible limit with periodic boundary conditions in RN (N = 2, 3). Wang and Yu [35] proved the incompressible limit for weak solutions in a bounded domain. For more about the incompressible limit problem, one can refer to [22, 31, 32] and the references therein. Let us mention that all of the above results were performed in the framework of Sobolev spaces. Inspired by [5] for the compressible Navier-Stokes equations, it is natural to study the system (1.1)-(1.2) in critical Besov spaces. We observe that the system (1.1) is invariant by the transformation ˜ = d(l2 t, lx) ˜ = lu(l2 t, lx), d ρ˜ = ρ(l2 t, lx), u

(1.3)

up to a change of the pressure law P˜ = l2 P . A critical space is a space in which the norm is invariant under the scaling ˜ )(x) = (e(lx), lf (lx), g(lx)). (˜ e, ˜f , g Very recently, in the case N = 3, Hu and Wu [17] studied the global strong solution to (1.1)-(1.2) ˆ in critical Besov spaces provided that the initial datum is close to an equilibrium state (1, 0, d) ˆ ∈ S2 . More precisely, there exist two positive constants η0 and Γ0 such with a constant vector d 1 1 1 ˆ∈B ˜ν2 ,∞ (R3 ) satisfy ˜ν2 ,∞ (R3 ), u0 ∈ B˙ 2 (R3 ), and d0 − d that if ρ0 − 1 ∈ B 2,1

kρ0 − 1k

1

˜ν2 ,∞ B

+ ku0 k

1

2 B˙ 2,1

ˆ + kd0 − dk

1

˜ν2 ,∞ B

≤ η0 ,

(1.4)

then system (1.1)-(1.2) has a unique global strong solution (ρ, u, d) with 3

3

˜ν2 ,∞ ), ˜ν2 ,1 ) ∩ C(R+ ; B ρ − 1 ∈ L1 (R+ ; B 3  1 5 + ˙ 2 1 + ˙ 2 u ∈ L (R ; B2,1 ) ∩ C(R ; B2,1 ) , ˆ∈ d−d



3 7 3 ,∞ + ˜ 2 ,∞ 2 ˜ L (R ; Bν ) ∩ C(R ; Bν ) 1

+

(1.5)

4

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

satifying kρ − 1k

3

˜ν2 ,∞ ) L∞ (R+ ;B

+ kρ − 1k

+ kuk

3 ˜ν2 ,1 ) L1 (R+ ;B

1

2 ) L∞ (R+ ;B˙ 2,1

+ kuk

ˆ + kd − dk

5 2 ) L∞ (R+ ;B˙ 2,1

3

˜ν2 ,∞ ) L∞ (R+ ;B

ˆ + kd − dk

7 ˜ν2 ,∞ ) L∞ (R+ ;B

(1.6) ≤ Γ0 η0 .

s and B ˜νs1 ,r denote the homogeneous Besov space and hybrid Besov space, respectively. Here B˙ p,r

We are going to explain these notations in Section 2. The purpose of this paper includes the following two aspects: On one hand, we establish global strong solutions to the Cauchy problem of (1.1)-(1.2) in critical Besov spaces with initial data close to a stable equilibrium. From [17], when N = 3 ˆ and (1.4) holds true, the system (1.1)-(1.2) has a unique global strong solution (ρ − 1, u, d − d) satisfying (1.6). Concerning the global well-posedness with respect to d, we carry out in the 3 3 ˜ν2 ,∞ in [17]. framework of critical Besov space B˙ 2 (if N = 3) but not the hybrid Besov space B 2,1

3 ,∞ 2

1 2

3 2

3 2

≈ B˙ 2,1 ∩ B˙ 2,1 ⊂ B˙ 2,1 , the regularity requirement of the initial data in terms of the ˆ is relaxed. The key point is that, different from the estimate of d in hybrid director field d0 − d ˜ν Since B

Besov space in [17, Proposition 4.1], we make use of the estimate of d in the homogeneous Besov 3 space B˙ 2 (see Proposition 3.2 below when N = 3). In addition, the global estimates of a linear 2,1

hyperbolic-parabolic system given by Danchin [5] (see also [1, Chapter 10]) play an important role. On the other hand, we give the rigorous justification of the convergence of the incompressible limit for global strong solutions to the compressible equations of liquid crystals when the initial data are ill prepared and small in a critical space. Meanwhile, the accurate converge rates are obtained. Our proof follows the ideas of Danchin [6] and the key point is to use some dispersive inequalities for the wave equation: the so-called Strichartz estimates (see e.g., [14, 23, 34] and the references therein). We would like to point out that [7] is the first paper devoted to the incompressible limit problem where Strichartz estimates have been used. In the spirit of [7], Danchin [6] studied the zero Mach number limit in critical spaces for barotropic compressible Navier-Stokes equations. Fang and Zi [13] investigated the incompressible limit of Oldroyd-B fluids in the whole space. Before presenting the main statements of this paper, we introduce the following function space: n   ˜νs,1 ) ∩ C([0, T ]; B ˜νs,∞ ) Bsν (T ) = (e, f , g) ∈ L1 (0, T ; B  N s+1 s−1 × L1 (0, T ; B˙ 2,1 ) ∩ C([0, T ]; B˙ 2,1 )  N o s+2 1 s ˙ ˙ × L (0, T ; B2,1 ) ∩ C([0, T ]; B2,1 )

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

5

and k(e, f , g)kBsν (T ) = kekL∞ (B˜νs,∞ ) + kf kL∞ (B˙ s−1 ) + kgkL∞ (B˙ s T

2,1

T

T

2,1 )

+ νkekL1 (B˜νs,1 ) + νkf kL1 (B˙ s+1 ) + θkgkL1 (B˙ s+2 ) . T

T

2,1

T

2,1

Here T > 0, s ∈ R, ν := λ + 2µ and ν := min(µ, λ + 2µ). We use the notation Bsν if T = +∞

by changing the interval [0, T ] into [0, ∞) in the definition above. Our first result of this paper reads as follows.

ˆ ∈ RN be an arbitrary constant unit vector, and assume that P ′ (1) = 1. Theorem 1.1. Let d N N ˆ∈ ˜ν2 ,∞ , u0 ∈ B˙ 2 −1 and d0 − d There exist two positive constants η and Γ such that if ρ0 − 1 ∈ B 2,1

N 2

B˙ 2,1 satisfy

kρ0 − 1k

N ,∞

˜ν2 B

+ ku0 k

N

2 −1 B˙ 2,1

ˆ + kd0 − dk

≤ η,

N

2 B˙ 2,1

(1.7)

then the following results hold true: N

ˆ in Bν2 (i) System (1.1)-(1.2) has a global strong solution (ρ, u, d) with (ρ − 1, u, d − d)

satisfying

ˆ k(ρ − 1, u, d − d)k

N Bν2

 ≤ Γ kρ0 − 1k

N ˜ν2 ,∞ B

+ ku0 k

N −1 2 B˙ 2,1

ˆ + kd0 − dk

N 2 B˙ 2,1



.

(1.8)

N

(ii) Uniqueness holds in Bν2 if N ≥ 3 and in B1ν ∩ Bsν (1 < s < 2) if N = 2. Remark 1.1. Since P (ρ) is an increasing convex function of ρ, we assume P ′ (1) = 1 for simplicity. The general barotropic case P ′ (1) > 0 can be verified by a slight modification of the argument below. Recall that the Mach number for the compressible flow (1.1) is defined as: |u| . M=p P ′ (ρ) Thus, letting M approach zero, we hope that ρ, d keep a typical size 1, and u is of order ǫ, where ǫ ∈ (0, 1) is a small parameter. As in [30], we scale ρ, u and d in the following way: ρ = ρǫ (ǫt, x), u = ǫuǫ (ǫt, x), d = dǫ (ǫt, x), and we take the viscosity coefficients as: µ = ǫµǫ , λ = ǫλǫ , ξ = ǫ2 ξ ǫ , θ = ǫθ ǫ .

6

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Under this scaling, system (1.1)-(1.2) becomes   ∂t ρǫ + div(ρǫ uǫ ) = 0,       ∇P (ρǫ )  ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ  ∂ (ρ u ) + div(ρ u ⊗ u ) − µ ∆u − (µ + λ )∇divu + t   ǫ2      = −ξ ǫ div ∇dǫ ⊙ ∇dǫ − 12 |∇dǫ |2 I ,        ∂t dǫ + uǫ · ∇dǫ = θ ǫ (∆dǫ + |∇dǫ |2 dǫ ),        ǫ ǫ ǫ (ρ , u , d )|t=0 = (ρǫ0 , uǫ0 , dǫ0 ).

(1.9)

For the simplicity of notations and presentation, we shall assume that µǫ , λǫ , ξ ǫ and θ ǫ are

constants, independent of ǫ, and still denote them as µ, λ, ξ and θ with an abuse of notations. Formally, we get by letting ǫ → 0 the following incompressible model   ∂t u + u · ∇u + ∇π = µ∆u − ξdiv(∇d ⊙ ∇d),        ∂ d + u · ∇d = θ(∆d + |∇d|2 d), t

(1.10)

  divu = 0,       (u, d)| t=0 = (u0 , d0 ).

Thus, roughly speaking, it is also reasonable to expect from the mathematical point of view that the global strong solutions to (1.9) converge in suitable functional spaces to the global strong solutions of (1.10) as ǫ → 0, and the hydrostatic pressure π in the first equation of (1.10) is

the limit of

P (ρǫ ) ǫ2

−

ξǫ ǫ 2 2 |∇d | .

Our second goal is devoted to the rigorous justification of the

convergence of the above incompressible limit in the whole space. We remark that the existence of global strong solutions to the incompressible flow of liquid crystals (1.10) in critical Besov space was established in Xu et al. [36]. As in [6], we want to consider so-called ill prepared data of the form ρǫ0 = 1 + ǫbǫ0 , uǫ0 and dǫ0 , where (bǫ0 , uǫ0 , dǫ0 ) are bounded in a sense that will be specified later on. Setting ρǫ = 1 + ǫbǫ , it is easy to check that (bǫ , uǫ , dǫ ) satisfies  divuǫ ǫ  ∂ b + = −div(bǫ uǫ ),  t   ǫ      µ∆uǫ + (µ + λ)∇divuǫ P ′ (1 + ǫbǫ ) ∇bǫ  ǫ ǫ ǫ  + ∂ u + u · ∇u −  t   1 + ǫbǫ 1 + ǫbǫ ǫ    1 −ξ  div ∇dǫ ⊙ ∇dǫ − |∇dǫ |2 I , =  ǫ  1 + ǫb 2        ∂t dǫ + uǫ · ∇dǫ = θ(∆dǫ + |∇dǫ |2 dǫ ),      ǫ ǫ ǫ (b , u , d )|t=0 = (bǫ0 , uǫ0 , dǫ0 ). Our second result of the paper can be stated as follows.

(1.11)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

7

ˆ ∈ RN be an arbitrary constant Theorem 1.2. Denote Q := ∇∆−1 div and P := I − Q. Let d N N ˜ν2 ,∞ , uǫ ∈ B˙ 2 −1 unit vector. There exist two positive constants c and M such that if bǫ0 ∈ B 0 2,1 N

ˆ ∈ B˙ 2 satisfy (for all 0 < ǫ ≤ ǫ0 ) and dǫ0 − d 2,1 C0ǫν := kbǫ0 k

N

2 −1 B˙ 2,1

+ ǫνkbǫ0 k

N

2 B˙ 2,1

+ kuǫ0 k

N

2 −1 B˙ 2,1

ˆ + kdǫ0 − dk

N

2 B˙ 2,1

≤ c,

then the following results hold: 1. Existence: N

• System (1.11) has a solution in Bǫν2 such that for all 0 < ǫ ≤ ǫ0 , ˆ k(bǫ , uǫ , dǫ − d)k

N

2 Bǫν

≤ M C0ǫν .

• System (1.10) has a unique solution such that kuk

+ kuk

N

ˆ + kd − dk

N

2 +1 ) L1 (B˙ 2,1

˜ ∞ (B˙ 2 −1 ) L 2,1

 ≤ M ku0 k

N −1 2 B˙ 2,1

N

˜ ∞ (B˙ 2 ) L 2,1

ˆ + kd0 − dk

N 2 B˙ 2,1

ˆ + kd − dk

N

2 +2 ) L1 (B˙ 2,1

 .

2. Convergence: 2(N −1) N −3 ,

• If N ≥ 4: For all p ∈ [pN , ∞] with pN := kbǫ k

N −1 ˜ 2 (B˙ p 2 ) L p,1

we have 1

+ kQuǫ k

N −1 ˜ 2 (B˙ p 2 ) L p,1

≤ M C0ǫν ǫ 2 ,

and kPuǫ − uk

N −3 p 2 L∞ (B˙ p,1 )

+ kPuǫ − uk

+ kdǫ − dk

N +1 p 2 L1 (B˙ p,1 )

N +3 p 2 L1 (B˙ p,1 )

+ kdǫ − dk

N −1 2

p L∞ (B˙ p,1

 ≤ M kPuǫ0 − u0 k

N −3 p 2 B˙ p,1

+ kdǫ0 − d0 k

)

N −1 p 2 B˙ p,1

 1 + C0ǫν ǫ 2 .

• If N = 3: For all p ∈ [2, ∞), we have kbǫ k

2−1 2p ˜ p−2 (B˙ p 2 ) L p,1

1

+ kQuǫ k

4−1 ˜ 2 (B˙ p 2 ) L p,1

1

≤ M C0ǫν ǫ 2 − p ,

and kPuǫ − uk

4−3 2

p L∞ (B˙ p,1

+ kdǫ − dk

)

+ kPuǫ − uk

4 3 p+2 L1 (B˙ p,1 )

4+1 2

p L1 (B˙ p,1

)

 ≤ M kPuǫ0 − u0 k

+ kdǫ − dk

4 3 p−2 B˙ p,1

4−1 2

p L∞ (B˙ p,1

)

+ kdǫ0 − d0 k

4 1 p−2 B˙ p,1

• If N = 2: For all p ∈ [2, 6], we have kbǫ k

3 −3 4p ˜ p−2 (B˙ 2p 4 ) L p,1

+ kQuǫ k

5 −1 ˜ 2 (B˙ 2p 4 ) L p,1

1

1

≤ M C0ǫν ǫ 4 − 2p ,

 1 1 + C0ǫν ǫ 2 − p .

(1.12)

8

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

and kPuǫ − uk

5 5 2p − 4 ) L∞ (B˙ p,1

+ kPuǫ − uk

+ kdǫ − dk

5 7 2p + 4 L1 (B˙ p,1 )

3 5 2p + 4 L1 (B˙ p,1 )

+ kdǫ − dk

 ≤ M kPuǫ0 − u0 k

5 5 2p − 4 B˙ p,1

5 −1 4

2p L∞ (B˙ p,1

+ kdǫ0 − d0 k

)

5 1 2p − 4 B˙ p,1

 1 1 + C0ǫν ǫ 4 − 2p .

ˆ the compressible flow of nematic liquid crystals is reduced to the Remark 1.2. When d ≡ d, well-known compressible Navier-Stokes system. Our results coincide with the ones in [6] concerning the zero Mach number limit in critical spaces for compressible Navier-Stokes equations. Our paper is organized as follows. In the next section, we recall some basic facts about Littlewood-Paley decomposition and the homogeneous Besov spaces. In Section 3, we investigate the existence of global solutions for system (1.1)-(1.2). Section 4 is devoted to the proof of uniqueness. In Section 5, we will prove the convergence of the incompressible limit in the whole space RN . We end this section by introducing the notations used throughout this paper. C stands for a harmless constant which never depends on ǫ, and we sometimes use the notation A . B as an equivalent to A ≤ CB. The notation A ≈ B means that A . B and B . A. 2. Homogeneous and hybrid Besov spaces We first recall the definition and some basic properties of homogeneous Besov spaces. They could be defined through the use of a dyadic partition of unity in Fourier variables called homogeneous Littlewood-Paley decomposition. To this end, choose a radial function ϕ ∈ S(RN )

supported in C = {ξ ∈ RN , 34 ≤ |ξ| ≤ 83 } such that X ϕ(2−j ξ) = 1 if ξ 6= 0. j∈Z

˙ j and S˙ j are defined by The homogeneous frequency localization operator ∆ X ˙ j u = ϕ(2−j D)u, S˙ j u = ˙ k u for j ∈ Z. ∆ ∆ k≤j−1

With our choice of ϕ, one can easily verify that ˙ p∆ ˙ q u ≡ 0 if |p − q| ≥ 2 and ∆ ˙ p (S˙ q−1 u∆ ˙ q u) ≡ 0 if |p − q| ≥ 5. ∆

(2.1)

Let us denote the space Y ′ (RN ) by the quotient space of S ′ (RN )/P with the polynomials

space P. The formal equality

u=

X

˙ ku ∆

k∈Z

holds true for u ∈

Y ′ (RN )

and is called the homogeneous Littlewood-Paley decomposition.

We will repeatedly use the following Bernstein’s inequality:

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

9

Lemma 2.1. (see [3]) Let C be an annulus and B a ball, 1 ≤ p ≤ q ≤ +∞. Assume that f ∈ Lp (RN ), then for any nonnegative integer k, there exists constant C independent of f , k such that

1

1

suppfˆ ⊂ λB ⇒ kD k f kLq (RN ) := sup k∂ α f kLq (RN ) ≤ C k+1 λk+N ( p − q ) kf kLp (RN ) , |α|=k

suppfˆ ⊂ λC ⇒ C −k−1 λk kf kLp (RN ) ≤ kD k f kLp (RN ) ≤ C k+1 λk kf kLp (RN ) . Next, let us recall the definitions of the Besov spaces. s is defined by Definition 1. Let s ∈ R, 1 ≤ p, r ≤ +∞. The homogeneous Besov space B˙ p,r s B˙ p,r = {f ∈ Y ′ (RN ) : kf kB˙ s < +∞}, p,r

where ˙ k f kLp kℓr . kf kB˙ s := k2ks k∆ p,r

˜ ρ (B˙ s ), which is initiated in [4]. We next introduce the Besov-Chemin-Lerner space L p,r T s ) is defined by ˜ ρ (B˙ p,r Definition 2. Let s ∈ R, 1 ≤ p, ρ, r ≤ +∞, 0 < T ≤ +∞. The space L T s ˜ ρ (B˙ p,r ) = {f ∈ (0, +∞) × Y ′ (RN ) : kf kL˜ ρ (B˙ s L T T

p,r )

< +∞},

where kf kL˜ ρ (B˙ s T

p,r )

˙ k f (t)kLρ (0,T ;Lp ) kℓr . := k2ks k∆

A direct application of Minkowski’s inequality implies that s ˜ ρ (B˙ s ), if r ≥ ρ, LρT (B˙ p,r ) ֒→ L p,r T s s ˜ ρ (B˙ p,r ), if ρ ≥ r. ) ֒→ LρT (B˙ p,r L T

We also need the following hybrid Besov space introduced by Danchin in [5]: Definition 3. For ν > 0, r ∈ [1, +∞], s ∈ R, we define X 2 ˙ j f kL2 . kf kB˜νs,r := 2js max{ν, 2−j }1− r k∆ j∈Z

By the definition, it is easy to verify that kf kB˜νs,∞ ≈ kf kB˙ s−1 + νkf kB˙ s , 2,1

˜νs,∞ ≈ B˙ s−1 ∩ B˙ s . which means that B 2,1 2,1

2,1

Let us now state some classical properties for the Besov spaces.

10

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Proposition 2.1. The following properties hold true: 1) Derivation: There exists a universal constant C such that C −1 kf kB˙ s ≤ k∇f kB˙ p,r s−1 ≤ Ckf k ˙ s . B p,r

p,r

s− N + pN

p 2) Sobolev embedding: If 1 ≤ p1 ≤ p2 ≤ ∞ and 1 ≤ r1 ≤ r2 ≤ ∞, then B˙ ps1 ,r1 ֒→ B˙ p2 ,r21

2

.

3) Real interpolation: kf kB˙ θs1 +(1−θ)s2 ≤ kf kθB˙ s1 kf k1−θ s . B˙ 2 p,r

p,r

p,r

s ∩ L∞ is an algebra. 4) Algebraic properties: for s > 0, B˙ p,1

5) Scaling properties: s , we have (a) for all λ > 0 and f ∈ B˙ p,1 N

kf (λ·)kB˙ s ≈ λs− p kf kB˙ s , p,1

p,1

s ), we have (b) for f = f (t, x) in Lr (0, T ; B˙ p,1

kf (λa ·, λb ·)kLr (B˙ s T

N

p,1 )

a

≈ λb(s− p )− r kf kLr a

λ T

s ). (B˙ p,1

Next we recall a few nonlinear estimates in Besov spaces which may be obtained by means of paradifferential calculus. Firstly introduced by J.-M. Bony in [2], the paraproduct between f and g is defined by T˙f g =

X

˙ q g, S˙ q−1 f ∆

q∈Z

and the remainder is given by ˙ R(f, g) =

X

˜˙ g ˙ qf ∆ ∆ q

q∈Z

with ˜˙ g := (∆ ˙ q−1 + ∆ ˙ q +∆ ˙ q+1 )g. ∆ q We have the following so-called Bony’s decomposition: ˙ f g = T˙g f + T˙f g + R(f, g) . | {z }

(2.2)

T˙f′ g

The paraproduct T˙ and the remainder R˙ operators satisfy the following continuous properties. Proposition 2.2. Suppose that s ∈ R, σ > 0, and 1 ≤ p, p1 , p2 , r, r1 , r2 ≤ ∞. Then we have s to B s , and from ˙ p,r 1) The paraproduct T˙ is a bilinear, continuous operator from L∞ × B˙ p,r −σ × B ˙ s to B˙ s−σ with 1 = min{1, 1 + 1 }. B˙ ∞,r p,r2 p,r 1 r r1 r2 1 p

s1 +s2 with s + s > 0, 2) The remainder R˙ is bilinear continuous from B˙ ps11,r1 × B˙ ps22,r2 to B˙ p,r 1 2

=

1 p1

+

1 p2

≤ 1, and

1 r

=

1 r1

+

1 r2

≤ 1.

From (2.2) and Proposition 2.2, we have the following more accurate product estimate:

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

Corollary 2.1. If u ∈ B˙ ps11,1 and v ∈ B˙ ps22,1 with 1 ≤ p1 ≤ p2 ≤ ∞, s1 ≤ s1 + s2 > 0, then uv ∈ B˙

s1 +s2 − pN 1 p2 ,1

N p1 ,

s2 ≤

11 N p2

and

and there exists a constant C, depending only on N, s1 , s2 , p1

and p2 , such that

kuvk

N s +s2 − p 1

1 B˙ p ,1 2

≤ CkukB˙ s1 kvkB˙ s2 . p1 ,1

p2 ,1

We finally need the following two composition lemmas (see [5, 33]). s ∩ L∞ . Let F ∈ W [s]+2,∞ (RN ) such that Lemma 2.2. Let s > 0, p ∈ [1, ∞] and u ∈ B˙ p,1 loc s ˙ F (0) = 0. Then F (u) ∈ Bp,1 and there exists a constant C = C(s, p, N, F, kukL∞ ) such that

kF (u)kB˙ s ≤ CkukB˙ s . p,1

p,1

N

N

s for s ∈ (− N , N ] and G ∈ W [ 2 ]+3,∞ (RN ) 2 Lemma 2.3. If u and v belong to B˙ 2,1 , (v − u) ∈ B˙ 2,1 loc 2 2 ′ s ˙ satisfies G (0) = 0, then G(v) − G(u) belongs to B and there exists a function of two variables 2,1

C depending only on s, N and G, and such that

 kG(v) − G(u)kB˙ s ≤ C(kukL∞ , kvkL∞ ) kuk

N 2 B˙ 2,1

2,1

+ kvk

N 2 B˙ 2,1

 kv − ukB˙ s . 2,1

3. Global existence for initial data near equilibrium In this section, we will prove the part (i) of Theorem 1.1. Throughout this paper, we consider only viscous fluids, those for which µ > 0 and ν > 0. The following proposition plays an important role in obtaining the estimates of (u, b). Proposition 3.1. (see [1]) Let ∇v ∈ L1T (L∞ (RN )), s ∈ R, and (a, u) be a solution of the following system

  ∂t a + div(T˙v a) + divu = F,

(3.1)

 ∂ u + T˙ · ∇u − µ∆u − (µ + λ)∇divu + ∇a = G, t v

on [0, T ). Then there exists a constant C depending only on N and s, such that the following estimate holds on [0, T ): ka(t)kB˜νs,∞ +ku(t)kB˙ s−1 + 2,1

Z

0

t

Ck∇vkL1 (L∞ (RN ))

(νkakB˜νs,1 + νkukB˙ s+1 )dt′ ≤ Ce

t

2,1

Z t  (3.2)  −Ck∇vkL1 (L∞ (RN )) ′ ′ t × ka0 kB˜νs,∞ + ku0 kB˙ s−1 + e (kF kB˜νs,∞ + kGkB˙ s−1 )dt . 2,1

0

2,1

Next, we establish the estimate of the director field d in critical Besov sapce. Proposition 3.2. Let s ∈ (− N2 , 1 +

N 2 ),

and d be a solution of the following equation

∂t d + u · ∇d − θ∆d = M

(3.3)

12

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO N

+1

s ). Then there exists a constant C depending 2 ) and M ∈ L1T (B˙ 2,1 on [0, T ), where u ∈ L1T (B˙ 2,1

on N and s, such that the following estimate holds on [0, T ):    kdkL˜ ∞ (B˙ s ) + θkdkL1 (B˙ s+2 ) ≤ C exp Ckuk kd0 kB˙ s + kMkL1 (B˙ s ) . N +1 t

2,1

2,1

t

˙ q to (3.3) and taking the Proof. Applying ∆ ˙ q d, integrating by part, we deduce that ∆

2 L1t (B˙ 2,1 2 L inner

2,1

)

t

2,1

(3.4)

product of the resulting equation with

1 d ˙ ˙ q dk2 2 k∆q dk2L2 + θk∇∆ L 2 dt 1  ˙ q dkL2 + k[u, ∆ ˙ q ] · ∇dkL2 + k∆ ˙ q MkL2 k∆ ˙ q dkL2 . ≤ kdivukL∞ k∆ 2 Hence, according to Bernstein’s inequality, we get, for some universal constant κ, ˙ q dkL∞ (L2 ) + θκ22q k∆ ˙ q dk 1 2 k∆ Lt (L ) t Z t  1 ˙ q d0 kL2 + ˙ q dkL2 + k[u, ∆ ˙ q ] · ∇dkL2 + k∆ ˙ q MkL2 dt′ . ≤ k∆ kdivukL∞ k∆ 2 0 Now, multiplying both sides by 2qs and summing over q, we end up with + κθkdkL1 (B˙ s+2 ) t 2,1 Z t  X 1 ˙ q ] · ∇dkL2 + kMk ˙ s dt′ . kdivukL∞ kdkB˙ s + 2qs k[u, ∆ ≤ kd0 kB˙ s + B2,1 2,1 2,1 2 0

kdkL˜ ∞ (B˙ s t

2,1 )

(3.5)

q∈Z

According to Lemma 2.100 in [1], we get, for − N2 < s < X

N 2

˙ q ] · ∇dkL2 ≤ Ckuk 2qs k[u, ∆

+ 1, N

2 +1 B˙ 2,1

q∈Z

kdkB˙ s . 2,1

(3.6)

N

2 Substituting (3.6) into (3.5) and using the embedding B˙ 2,1 ֒→ L∞ , we get

kdkL˜ ∞ (B˙ s t

2,1 )

+ κθkdkL1 (B˙ s+2 ) 2,1

t

≤ kd0 kB˙ s + kMkL1 (B˙ s 2,1

t

2,1

)+C

Z

0

t

kuk

N

2 +1 B˙ 2,1

kdkB˙ s dt′ , 2,1

taking advantage of Gronwall’s inequality, we obtain (3.4) immediately. This competes the proof of Proposition 3.2.



Proof of the part (i) of Theorem 1.1. We are going to prove that if the initial data kρ0 − 1k

N ,∞

˜ν2 B

+ ku0 k

N

2 −1 B˙ 2,1

ˆ + kd0 − dk

N

2 B˙ 2,1

≤ η,

(3.7)

for some sufficiently small η, there exists a positive constant Γ such that ˆ k(ρ − 1, u, d − d)k

N

Bν2

≤ Γη.

(3.8)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

13

This uniform estimates will enable us to extend the local solution (ρ − 1, u, d) obtained by using

a Friedrichs method as in [1] to be a global one. To this end, we use a contradiction argument. Define

n ˆ T0 = sup T ∈ [0, ∞) : k(ρ − 1, u, d − d)k

N Bν2

(T )

≤ Γη

o

(3.9)

with Γ to be determined later. ˆ because d ˆ ∈ SN −1 is a constant vector. Letting b = ρ − 1, we We note that ∇d = ∇(d − d) rewrite the system (1.1) as   ∂ b + divu + div(T˙u b) = −div(T˙b′ u),   t  ∂t u + T˙u · ∇u − Au + ∇b = K(b)∇b − I(b)Au − T˙∂′ i u ui + H,     ˆ + u · ∇(d − d) ˆ − θ∆(d − d) ˆ = J, ∂t (d − d)

where

H := −

(3.10)

  ξ ˆ − 1 |∇(d − d)| ˆ 2I , ˆ ⊙ ∇(d − d) div ∇(d − d) 1+b 2

ˆ 2 (d − d) ˆ + θ|∇(d − d)| ˆ 2 d, ˆ J := θ|∇(d − d)| and A := µ∆ + (µ + λ)∇div, I(b) :=

b 1+b ,

K(b) := 1 −

P ′ (1+b) 1+b .

Suppose that T0 < ∞. We apply the linear estimates in Propositions 3.1 and 3.2 to the

solutions of reformulated system (3.10) such that for all t ∈ [0, T0 ], the following estimates hold: Z T0   νkbk N2 ,1 + νkuk N2 +1 dt′ kb(t)k N2 ,∞ + ku(t)k N2 −1 + B˙ 2,1

˜ν B

k∇ukL1

≤ Ce

T0

(L∞ (RN ))

n

˜ν B

0

kb0 k

N ,∞

˜ν2 B

+ kK(b)∇bk

N −1 2 L1T (B˙ 2,1 ) 0

B˙ 2,1

+ ku0 k

N

2 −1 B˙ 2,1

+ kdiv(T˙b′ u)k

N ,∞

˜ν2 L1T (B

+ kI(b)Auk

N −1 2 L1T (B˙ 2,1 ) 0

0

)

+ kT˙∂′ i u ui k

N −1 2 L1T (B˙ 2,1 ) 0

+ kHk

N −1 2 L1T (B˙ 2,1 ) 0

o ,

(3.11) and ˆ kd(t) − dk

N 2 B˙ 2,1

Z

+θ



≤ C exp Ckuk

T0

ˆ kd − dk

N

2 +2 B˙ 2,1

0

N

2 +1 ) L1T (B˙ 2,1 0

dt′

 ˆ kd0 − dk

N

2 B˙ 2,1

+ kJk

N

2 ) L1T (B˙ 2,1 0

 .

(3.12)

In what follows, we derive estimates for the nonlinear terms one by one. Similar to the case of isentropic Navier-Stokes equations [1], by Proposition 2.2, Corollary 2.1 and Lemma 2.2, we have the following inequalities: kdiv(T˙b′ u)k

N ,∞

˜ν2 L1T (B 0

)

. kbk

N ,∞

˜2 L∞ T (Bν 0

)

kuk

N +1

2 L1T (B˙ 2,1 0

)

,

(3.13)

14

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

kK(b)∇bk

N −1

2 L1T (B˙ 2,1 0

kI(b)Auk

)

. kbk2

N −1

2 L1T (B˙ 2,1

kT˙∂′ i u ui k

. kbk

N

0

N

2 −1 ) L1T (B˙ 2,1 0

0

. kbk

N ,∞

˜ 2 L∞ T (Bν

2 ) L2T (B˙ 2,1

0

N ,∞

˜ 2 L∞ T (Bν

)

0

. kuk

)

N

˙ 2 −1 ) L∞ T (B2,1 0

kuk

)

kbk

N +1

2 L1T (B˙ 2,1 0

kuk

N ,1

˜ν2 L1T (B

N

0

,

(3.14)

,

(3.15)

.

(3.16)

)

2 +1 ) L1T (B˙ 2,1

)

0

Compared with the isentropic Navier-Stokes equations, the new terms can be estimated as follows: kHk

N

2 −1 ) L1T (B˙ 2,1 0

ξ  

ˆ ⊙ ∇(d − d) ˆ − 1 |∇(d − d)| ˆ 2I = div ∇(d − d) N −1

1 2 1+b 2 ) LT (B˙ 2,1 0

 

ˆ 2I ˆ − 1 |∇(d − d)| ˆ ⊙ ∇(d − d) ≤ ξ div ∇(d − d) N −1

1 2 2 ) LT (B˙ 2,1 0

 

ˆ − 1 |∇(d − d)| ˆ 2I ˆ ⊙ ∇(d − d) + ξ I(b)div ∇(d − d) N −1

1 2 2 ) LT (B˙ 2,1 0



ˆ 2 I ˆ − 1 |∇(d − d)| ˆ ⊙ ∇(d − d) − d) N N

1

∇(d 2 ) ˙ 2 2 LT (B˙ 2,1 L∞ T0 (B2,1 ) 0

(3.17)

 . 1 + kbk  . 1 + kbk

 ˆ 2 k∇(d − d)k

 . 1 + kbk

 ˆ kd − dk

N ˜ 2 ,∞ ) L∞ T0 (Bν

N ,∞

˜2 L∞ T (Bν

kJk

N

2 ) L1T (B˙ 2,1 0

0

)

N

2 ) L2T (B˙ 2,1 0

N

˙ 2 L∞ T (B2,1 )

ˆ 2 (d − d)k ˆ = θk|∇(d − d)|

N

2 ) L1T (B˙ 2,1 0

0

ˆ kd − dk

N

2 +2 ) L1T (B˙ 2,1

,

0

ˆ 2 dk ˆ + θk|∇(d − d)|

N

2 ) L1T (B˙ 2,1

:= J1 + J2 .

(3.18)

0

For the estimate of J1 , it follows that ˆ 2 (d − d)k ˆ J1 = θk|∇(d − d)|

N

2 ) L1T (B˙ 2,1

ˆ 2k . k|∇(d − d)|

N

2 ) L1T (B˙ 2,1

ˆ 2 . kd − dk

0

N

2 +1 ) L2T (B˙ 2,1

ˆ . kd − dk

0

N +2

2 L1T (B˙ 2,1 0

)

0

ˆ kd − dk

N

˙ 2 L∞ T (B2,1 ) 0

(3.19)

ˆ kd − dk

N ˙ 2 L∞ T0 (B2,1 )

ˆ 2 kd − dk

N

˙ 2 L∞ T (B2,1 )

.

0

For J2 , we have by the definition of Besov’s spaces ˆ 2 dk ˆ J2 = θk|∇(d − d)|

N

2 ) L1T (B˙ 2,1

ˆ . kd − dk

N ˙ 2 L∞ T0 (B2,1 )

0

ˆ kd − dk

ˆ 2 . kd − dk

N +2 2 ) L1T (B˙ 2,1 0

N

2 +1 ) L2T (B˙ 2,1

.

0

(3.20)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

15

Plugging inequalities (3.13)-(3.20) in (3.11) and (3.12), we thus get Z T0   νkbk N2 ,1 + νkuk N2 +1 dt′ kbk ∞ N2 ,∞ + ku(t)k ∞ N2 −1 + ˜ν LT (B

LT (B˙ 2,1

)

0

k∇ukL1

.e

T0

+ kbk

)

0

(L∞ (RN ))

n kb0 k

+ ku0 k

N ,∞

N ,∞

0

)

kbk

N ,1

˜ν2 L1T (B

 + 1 + kbk

0

N ˜ 2 ,∞ ) L∞ T0 (Bν

N

2 −1 B˙ 2,1

˜ν2 B

˜ 2 L∞ T (Bν

˜ν B

0

)

+ kuk

B˙ 2,1

+ kbk

N −1

˙ 2 L∞ T (B2,1 0

 ˆ kd − dk

N ˙ 2 L∞ T0 (B2,1 )

N ,∞

˜ 2 L∞ T (Bν

)

0

kuk

)

kuk

N +1

2 L1T (B˙ 2,1 0

ˆ kd − dk

N +2 2 ) L1T (B˙ 2,1 0

N

2 +1 ) L1T (B˙ 2,1 0

(3.21)

)

o

,

and ˆ kd − dk

N ˙ 2 L∞ T0 B2,1

Z

+θ

0

T0

ˆ kd − dk

 . exp Ckuk

N +1 2 L1T (B˙ 2,1 ) 0

ˆ + kd − dk

N +2 2 L1T (B˙ 2,1 ) 0

N

2 +2 B˙ 2,1

n

dt′

ˆ kd0 − dk

N

2 B˙ 2,1

ˆ kd − dk

N ˙ 2 L∞ T0 (B2,1 )

ˆ + kd − dk

N +2 2 ) L1T (B˙ 2,1 0

ˆ 2 kd − dk

N

˙ 2 L∞ T (B2,1 )

(3.22)

0

o .

The above two inequalities combined with the definition of T0 yield that ˆ k(b, u, d − d)k

N

Bν2 (T0 )

≤ C1 eC1 Γη (η + Γ2 η 2 + Γ3 η 3 ).

(3.23)

We choose Γ = 4C1 and η > 0 satisfying eC1 Γη < 2, Γ2 η ≤

1 1 , Γ3 η 2 ≤ . 2 2

(3.24)

Consequently, it follows from (3.23) and the above choices of Γ and η that ˆ k(b, u, d − d)k

N

Bν2 (T0 )

< Γη,

which is a contradiction with the definition of T0 . Hence, we can deduce that T0 = ∞. Global

existence is thus proved.



4. Uniqueness In this section, we will prove the uniqueness of the solution to system (1.1), i.e., the part (ii) of Theorem 1.1. Proof of the part (ii) of Theorem 1.1. N

Case N ≥ 3: Assume that (bi , ui , di )i=1,2 in Bν2 solve (1.1) with the same initial data.

Denote

δb = b2 − b1 , δu = u2 − u1 , δd = d2 − d1 .

16

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO N

Then (δb, δu, δd) ∈ Bν2 (T ) for all T > 0. On the other hand, since (bi , ui , di )i=1,2 are solutions to system (1.1) with the same initial data, (δb, δu, δd) solves   ∂t δb + div(T˙u2 · δb) + divδu = δF,    ∂t δu + T˙u2 · ∇δu − Aδu + ∇δb = δG,     ∂t δd + u2 · ∇δd − θ∆δd = δM,

(4.1)

with

′ u2 ), δF = −δu · ∇b1 − b1 divδu − div(T˙δb

δG = −T˙∂′ i δu ui2 − δu · ∇u1 − (I(b2 ) − I(b1 ))Au2 − I(b1 )Aδu   ˆ ⊙ ∇(d2 − d) ˆ − 1 |∇(d2 − d)| ˆ 2I + ξ(I(b2 ) − I(b1 ))div ∇(d2 − d) 2  ξ ˆ ⊙ ∇δd + ∇(d2 − d) ˆ ⊙ ∇δd + div ∇(d1 − d) 1 + b1  1 ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd) + ∇(K(b2 ) − K(b1 )), − I(∇(d1 − d) 2 and   ˆ 2 δd + (∇(d1 − d) ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd)d2 , δM = −δu · ∇d1 + θ |∇(d1 − d)| where we used the notations A : B =

PN

i,j=1 Aij Bij

tions 3.1 and 3.2 to the system (4.1), we have k(δb, δu, δd)k

N −1

Bν2

and K(z) =

Rz 0

K(y)dy. Applying Proposi-

(T )

 ≤ C exp Cku2 k

N +1 2 L1T (B˙ 2,1 )



kδF k

N ˜ν2 −1,∞ ) L1T (B

+ kδGk

N −2 2 L1T (B˙ 2,1 )

+ kδMk

N −1 2 L1T (B˙ 2,1 )

N

N



(4.2) . N

−1 −1 −1 2 2 2 ) (i = 1, 2). Indeed, b0 ∈ B˙ 2,1 , bi ∈ L∞ (R+ ; B˙ 2,1 ) As in [5], we could get that bi ∈ C(R+ ; B˙ 2,1 N

N

N

−1 −1 2 2 ˙ 2 and ∂t bi = −ui · ∇bi − divui − bi divui ∈ L2 (R+ ; B˙ 2,1 ). And because bi ∈ C(B˙ 2,1 ) ∩ L∞ loc B2,1 ,

we have bi ∈ C([0, +∞) × RN ). On the other hand, if (1.7) is satisfied for some η suitably small,

we have

1 for all t ≥ 0 and x ∈ RN . (4.3) 4 In the following, we estimate the terms δF, δG and δM respectively. Firstly, according to |bi (t, x)| ≤

Proposition 2.2 and Corollary 2.1, one gets kδF k

N −1,∞

˜ν2 L1T (B

)

. kδuk

N

2 ) L1T (B˙ 2,1

kb1 k

N ,∞

˜2 L∞ T (Bν

)

+ kδbk

N −1,∞

˜ 2 L∞ T (Bν

)

ku2 k

N +1

2 L1T (B˙ 2,1

)

.

(4.4)

,

(4.5)

Similarly, the terms of δG could be estimated as follows: kT˙∂′ i δu ui2 k

N −2

2 L1T (B˙ 2,1

)

. k∂i δuk

N −2

2 L2T (B˙ 2,1

)

ku2 k

N

2 ) L2T (B˙ 2,1

. kδuk

N −1

2 L2T (B˙ 2,1

)

ku2 k

N

2 ) L2T (B˙ 2,1

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

kδu · ∇u1 k

N −2

2 L1T (B˙ 2,1

)

. kδuk

N −1

2 L2T (B˙ 2,1



(I(b ) − I(b ))Au

2 1 2

)

N −2

2 L1T (B˙ 2,1

k∇u1 k

N −1

2 L2T (B˙ 2,1

. kδuk

. kI(b2 ) − I(b1 )k

+ kb2 k

 k∇2 u2 k

 . 1 + kb1 k

+ kb2 k

 ku2 k

N ˙ 2 L∞ T (B2,1 )

kI(b1 )Aδuk

N −2

2 L1T (B˙ 2,1

)

N ˙ 2 L∞ T (B2,1 )

N ˙ 2 L∞ T (B2,1 )

. kb1 k

N

˙ 2 L∞ T (B2,1 )

N −1

˙ 2 L∞ T (B2,1

)

 . 1 + kb1 k

N ˙ 2 L∞ T (B2,1 )

N −1

2 L2T (B˙ 2,1

)

N −1

2 L1T (B˙ 2,1

N +1

2 L1T (B˙ 2,1

k∇2 δuk

N −2

2 L1T (B˙ 2,1

)

)

)

ku1 k

N

2 ) L2T (B˙ 2,1

kAu2 k

N −1

2 L1T (B˙ 2,1

kδbk

N −1

˙ 2 L∞ T (B2,1

kδbk

N −1

˙ 2 L∞ T (B2,1

. kb1 k

N

˙ 2 L∞ T (B2,1 )

)

)

)

,

17

(4.6)

)

(4.7)

)

,

kδuk

N

2 ) L1T (B˙ 2,1

,

(4.8)

 

ˆ ⊙ ∇(d2 − d) ˆ − 1 |∇(d2 − d)| ˆ 2I

ξ(I(b2 ) − I(b1 ))div ∇(d2 − d)

1 N2 −2 2 LT (B˙ 2,1 ) . kI(b2 ) − I(b1 )k

N

˙ 2 −1 ) L∞ T (B2,1

 . 1 + kb1 k

N ˙ 2 L∞ T (B2,1 )



ˆ 2k k|∇(d2 − d)|

(4.9)

N

2 ) L1T (B˙ 2,1

+ kb2 k

N ˙ 2 L∞ T (B2,1 )



ˆ 2 kd2 − dk

N

2 +1 ) L2T (B˙ 2,1

kδbk

N

˙ 2 −1 ) L∞ T (B2,1

,

  ξ ˆ ⊙ ∇δd + ∇(d2 − d) ˆ ⊙ ∇δd div ∇(d1 − d)

1 N2 −2 1 + b1 LT (B˙ 2,1 )    ˆ ˆ . 1 + kb1 k k∇δdk k∇(d1 − d)k + k∇(d2 − d)k N N −1 N ˙ 2 L∞ T (B2,1 )

 . 1 + kb1 k

N

˙ 2 L∞ T (B2,1 )

2 L2T (B˙ 2,1



kδdk

N

2 ) L2T (B˙ 2,1

N 2 ) L2T (B˙ 2,1

2 ) L2T (B˙ 2,1

)

 ˆ kd1 − dk

N

2 +1 ) L2T (B˙ 2,1

ˆ + kd2 − dk

N

2 +1 ) L2T (B˙ 2,1

Similar to (4.10), there holds that

ξ  1

ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd) div I(∇(d1 − d)

1 N2 −2 1 + b1 2 LT (B˙ 2,1 )    ˆ ˆ . 1 + kb1 k ∞ N2 kδdk 2 N2 kd1 − dk N +1 + kd2 − dk 2 2 LT (B˙ 2,1 )

LT (B˙ 2,1 )

LT (B˙ 2,1



(4.10)

.

N +1

2 L2T (B˙ 2,1

)



)

 .

(4.11)

Moreover, applying Corollary 2.1 and Lemma 2.3, one easily gets k∇(K(b2 ) − K(b1 ))k

N

2 −2 ) L1T (B˙ 2,1

 . T kb1 k

N ˙ 2 L∞ T (B2,1 )

+ kb2 k

. kK(b2 ) − K(b1 )k

N ˙ 2 L∞ T (B2,1 )

N

2 −1 ) L1T (B˙ 2,1

 kδbk

N ˙ 2 −1 ) L∞ T (B2,1

(4.12) .

18

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Note that in the above inequalities, we have used that N > 2. Collecting the above estimates (4.5)-(4.12), it follows that kδGk

N −2

2 L1T (B˙ 2,1

)

 . ku1 k

N 2 ) L2T (B˙ 2,1

+ ku2 k

N 2 ) L2T (B˙ 2,1

 + 1 + kb1 k + 1 + kb1 k

N

˙ 2 L∞ T (B2,1 )

 + 1 + kb1 k

N ˙ 2 L∞ T (B2,1 )

 + T kb1 k

N ˙ 2 L∞ T (B2,1 )

kδuk

N −1

2 L2T (B˙ 2,1

)

+ kb1 k



ku2 k

+ kb2 k



ˆ 2 kd2 − dk

N

˙ 2 L∞ T (B2,1 )

kδbk

kδuk N

(4.13) N +1

)

kδbk

N

˙ 2 −1 ) L∞ T (B2,1

ˆ + kd2 − dk

N +1 2 L2T (B˙ 2,1 )

 kδbk

N

2 ) L1T (B˙ 2,1

˙ 2 −1 ) L∞ T (B2,1

2 L2T (B˙ 2,1

N +1 2 L2T (B˙ 2,1 )

N ˙ 2 L∞ T (B2,1 )

N

2 +1 ) L1T (B˙ 2,1

 ˆ kd1 − dk

+ kb2 k

N

˙ 2 L∞ T (B2,1 )

+ kb2 k

N ˙ 2 L∞ T (B2,1 )

N ˙ 2 L∞ T (B2,1 )





N

˙ 2 −1 ) L∞ T (B2,1



kδdk

N

2 ) L2T (B˙ 2,1

.

For the estimate of δM, note that kδu · ∇d1 k

N

2 −1 ) L1T (B˙ 2,1

. kδuk

N

2 −1 ) L2T (B˙ 2,1

ˆ = kδu · ∇(d1 − d)k

N

2 −1 ) L1T (B˙ 2,1

ˆ k∇(d1 − d)k

. kδuk

N

ˆ 2 δdk kθ|∇(d1 − d)|

N −1

2 L1T (B˙ 2,1

N

2 −1 ) L2T (B˙ 2,1

2 ) L2T (B˙ 2,1

. kδdk

N −1

˙ 2 L∞ T (B2,1

)

. kδdk

N

)

˙ 2 −1 ) L∞ T (B2,1

ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd)d2 k kθ(∇(d1 − d)

N −1

2 L1T (B˙ 2,1

(4.14)

ˆ kd1 − dk

N

2 +1 ) L2T (B˙ 2,1

ˆ 2k k|∇(d1 − d)|

,

N

2 ) L1T (B˙ 2,1

(4.15)

ˆ 2 kd1 − dk

N

2 +1 ) L2T (B˙ 2,1

,

)

ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd)(d2 − d)k ˆ . k(∇(d1 − d)

N

2 −1 ) L1T (B˙ 2,1

ˆ : ∇δd + ∇(d2 − d) ˆ : ∇δd)dk ˆ + k(∇(d1 − d)

N

2 −1 ) L1T (B˙ 2,1

ˆ . kd2 − dk

N ˙ 2 L∞ T (B2,1 )

 ˆ + kd1 − dk

 ˆ kd1 − dk

N +1 2 L2T (B˙ 2,1 )

N

2 +1 ) L2T (B˙ 2,1

 ˆ . 1 + kd2 − dk

N ˙ 2 L∞ T (B2,1 )

ˆ + kd2 − dk

ˆ + kd2 − dk

N +1 2 L2T (B˙ 2,1 )

N

2 +1 ) L2T (B˙ 2,1

 ˆ kd1 − dk

 kδdk

N +1 2 L2T (B˙ 2,1 )



(4.16) kδdk

N 2 ) L2T (B˙ 2,1

N

2 ) L2T (B˙ 2,1

ˆ + kd2 − dk

N +1 2 L2T (B˙ 2,1 )



kδdk

N

2 ) L2T (B˙ 2,1

.

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

19

Therefore, kδMk

N −1

2 L1T (B˙ 2,1

. kδuk

) N −1

2 L2T (B˙ 2,1

)

ˆ kd1 − dk

N +1

2 L2T (B˙ 2,1

 ˆ + 1 + kd2 − dk

N ˙ 2 L∞ T (B2,1 )



)

+ kδdk

N −1

˙ 2 L∞ T (B2,1

ˆ kd1 − dk

N +1 2 L2T (B˙ 2,1 )

)

ˆ 2 kd1 − dk

(4.17)

N

2 +1 ) L2T (B˙ 2,1

ˆ + kd2 − dk

N +1 2 L2T (B˙ 2,1 )

 kδdk

N

2 ) L2T (B˙ 2,1

.

Consequently, it follows that k(δb, δu, δd)k

N −1

Bν2

(T )

≤ Z(T )k(δb, δu, δd)k

N −1

Bν2

(T )

,

with  Z(T ) = C exp Cku2 k

N +1 2 L1T (B˙ 2,1 )

+ ku2 k

N 2 ) L2T (B˙ 2,1

N ,∞

˜ 2 L∞ T (Bν

N ˙ 2 L∞ T (B2,1 )

N +1 2 L1T (B˙ 2,1 )

N ˙ 2 L∞ T (B2,1 )

 ˆ × kd1 − dk

kb1 k

 + 1 + kb1 k

 × ku2 k

 + T kb1 k

n

N +1 2 ) L2T (B˙ 2,1



N +1

2 L1T (B˙ 2,1

N ˙ 2 L∞ T (B2,1 )

)

+ ku1 k

N +1 2 ) L2T (B˙ 2,1

 ˆ + 1 + kd2 − dk

N ˙ 2 L∞ T (B2,1 )

ˆ + kd2 − dk

N +1 2 L2T (B˙ 2,1 )

N

2 ) L2T (B˙ 2,1



ˆ 2 + kd2 − dk

N +1 2 L2T (B˙ 2,1 )

N ˙ 2 L∞ T (B2,1 )

+ ku2 k

+ kb2 k

ˆ 2 + kd1 − dk + kb2 k

)



+ kb1 k

N ˙ 2 L∞ T (B2,1 )



o .

It is now clear that lim sup Z(T ) ≤ Ckb1 k

N ,∞

˜ 2 L∞ T (Bν

T →0+

)

.

Thus, if η > 0 is sufficiently small, we have k(δb, δu, δd)k

N −1

Bν2

(T )

=0

for certain T > 0 small enough. So (b1 , u1 , d1 ) ≡ (b2 , u2 , d2 ) on [0, T ].

Then we could argue as in [5] for the isentropic compressible Navier-Stokes equations. Let

Tm be the largest time such that the two solutions coincide on [0, Tm ]. Taking Tm as the initial time, we define ˜ i (t)) := (bi (t + Tm ), ui (t + Tm ), di (t + Tm )). ˜ i (t), d (˜bi (t), u Repeating the above arguments and using the fact that k˜bi (0)kL∞ ≤

that

˜ 1 (t)) = (˜b2 (t), u ˜ 2 (t)) ˜ 1 (t), d ˜ 2 (t), d (˜b1 (t), u

1 4

(see (4.3)), we can prove

20

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

on a suitably small interval [0, ǫ] (ǫ > 0). This contradicts the assumption that Tm is the largest time such that the two solutions coincide. As a result, Tm = ∞, which means that the uniqueness

result holds in R+ .

Case N = 2: Using again Propositions 3.1 and 3.2, it follows that k(δb, δu, δd)kBs−1 (T ) ν    ≤ C exp Cku2 kL1 (B˙ 2 ) kδF kL1 (B˜νs−1,∞ ) + kδGkL1 (B˙ s−2 ) + kδMkL1 (B˙ s−1 ) . 2,1

T

2,1

T

T

(4.18)

2,1

T

Let us estimate δF, δG and δM, respectively. kδF kL1 (B˜νs−1,∞ ) . T

s−1 2

kδuk

2

1 ) LT3−s (B˙ 2,1

T

kb1 kL∞ (B˜νs,∞ ) + kδukL1 (B˙ s ) kb1 kL∞ (B˜ν1,∞ ) T

2,1

T

T

+ kδbkL∞ (B˜νs−1,∞ ) ku2 kL1 (B˙ 2 .T

s−1 2

2,1 )

T

T

3−s

s−1

2 kb k ∞ ˜νs,∞ ) + kδukL1 (B˙ s ) kb1 kL∞ (B˜ν1,∞ ) kδukL12 (B˙ s ) kδukL∞ (B˙ s−2 ) 1 L (B 2,1

T

T

2,1

T

T

2,1

T

+ kδbkL∞ (B˜νs−1,∞ ) ku2 kL1 (B˙ 2 .T

s−1 2

2,1 )

T

T

 kδukL1 (B˙ s

2,1

T

 + kδuk s−2 ∞ ˜νs,∞ ) ) kb1 kL∞ (B ) L (B˙ 2,1

T

T

+ kδukL1 (B˙ s ) kb1 kL∞ (B˜ν1,∞ ) + kδbkL∞ (B˜νs−1,∞ ) ku2 kL1 (B˙ 2 ) , 2,1

T

T

T

T

2,1

where we have used the following inequality: 3−s

kδuk

2 1 ) LT3−s (B˙ 2,1

s−1

2 ≤ kδukL1 (B˙ s ≤ kδukL12 (B˙ s ) kδukL∞ (B˙ s−2 ) 2,1

T

2,1 )

T

2,1

T

+ kδukL∞ (B˙ s−2 ) . 2,1

T

Moreover, kδGkL1 (B˙ s−2 ) T

2,1

 . ku1 kL2 (B˙ 1 T

2,1

) + ku2 kL2 (B˙ 1 T

 + 1 + kb1 kL∞ (B˙ 1

2,1

T

 + 1 + kb1 kL∞ (B˙ 1

T

) + kb2 kL∞ (B˙ 1

2,1 )

T

2,1

 ) kδukL2 (B˙ s−1 ) + kb1 kL∞ (B˙ 1 ) kδukL1 (B˙ s 2,1

T

+ kb2 kL∞ (B˙ 1

2,1

T

2,1

T

T

2,1 )

T

+ kb2 kL∞ (B˙ 1 T

2,1 )



2,1

T

2,1 )

T

 ) ku2 kL1 (B˙ 2 ) kδbkL∞ (B˙ s−1 ) 2,1

T

2,1

T

 ˆ 2 ) kd2 − dkL2 (B˙ 2 ) kδbkL∞ (B˙ s−1 ) T

  ˆ 2 ˙2 + 1 + kb1 kL∞ (B˙ 1 ) kd1 − dk L (B  + T kb1 kL∞ (B˙ 1

2,1

2,1

T

2,1

ˆ ) + kd2 − dkL2 (B˙ 2 T

kδbkL∞ (B˙ s−1 ) , T

2,1

2,1

2,1

 ) kδdkL2 (B˙ s T

2,1 )

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

21

kδMkL1 (B˙ s−1 ) T

2,1

ˆ 2 ˙2 . kδukL2 (B˙ s−1 ) kd1 − dk L (B T

2,1

2,1 )

T

ˆ 22 2 + kδdkL∞ (B˙ s−1 ) kd1 − dk L (B˙ 2,1

T

  ˆ 2 ˙2 ˆ ∞ ˙1 kd1 − dk + 1 + kd2 − dk L (B L (B ) 2,1

T

2,1

T

2,1 )

T

ˆ ) + kd2 − dkL2 (B˙ 2

2,1 )

T



kδdkL2 (B˙ s ) . 2,1

T

Consequently, it follows that k(δb, δu, δd)kBs−1 (T ) ≤ Z(T )k(δb, δu, δd)kBs−1 (T ) ν ν with  n Z(T ) = C exp Cku2 kL1 (B˙ 2 ) kb1 kL∞ (B˜ν1,∞ ) + ku2 kL1 (B˙ 2 2,1

T

+ ku2 kL2 (B˙ 1

2,1

T



 + 1 + kb1 kL∞ (B˙ 1 )

× ku2 kL1 (B˙ 2

2,1

T

) + kb2 kL∞ (B˙ 1

2,1

T

) + kb2 kL∞ (B˙ 1

 ˆ 2 ˙2 × kd1 − dk L (B T

2,1

T





 ˆ ∞ ˙1 + 1 + kd2 − dk L (B

T

T

2,1 )

2,1

T

ˆ ) + kd2 − dkL2 (B˙ 2

2,1 )



2,1 )



2,1

2,1 )

T

2,1 )

T

+ ku1 kL2 (B˙ 1

ˆ 22 ˙2 + kd2 − dk ) L (B

ˆ 2 ) + kd1 − dkL2 (B˙ 2

 + T kb1 kL∞ (B˙ 1 T

2,1

T

2,1 )

T

T

) + kb1 kL∞ (B˙ 1

2,1 )

T



o 1 + T 1− r kb1 kL∞ (B˜νs,∞ ) . T

It is clear that lim sup Z(T ) ≤ Ckb1 kL∞ (B˜ν1,∞ ) . T

T →0+

Thus, if η > 0 is sufficiently small, we have k(δb, δu, δd)kBs−1 (T ) = 0 ν for certain T > 0 small enough. So (b1 , u1 , d1 ) ≡ (b2 , u2 , d2 ) on [0, T ]. We can now achieve the

proof as in the case N ≥ 3.



5. Incompressible limit This part is devoted to the proof of Theorem 1.2. Firstly we list the global well-posedness of incompressible model (1.10). Proposition 5.1. (see [36]) Let N ≥ 2. Suppose that the initial data (u0 , d0 ) belong to N N −1 2 2 ¯ B˙ 2,1 (SN −1 ) with divu0 = 0. Then there exist two positive constants c¯ and C, (RN ) × B˙ 2,1 depending on N, µ, ξ and θ, such that if ku0 k

N

2 −1 B˙ 2,1

ˆ + kd0 − dk

N

2 B˙ 2,1

≤ c¯,

then there exists a unique global solution (u, d) to (1.10) in the class N

N

N

N

−1 +1 +2 2 2 2 2 ) ∩ L1 ([0, ∞); B˙ 2,1 ) ∩ L1 ([0, ∞); B˙ 2,1 ) × L∞ ([0, ∞); B˙ 2,1 ). L∞ ([0, ∞); B˙ 2,1

22

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Moreover, the following estimate holds true kuk

N

˜ ∞ (B˙ 2 −1 ) L 2,1 T

 ≤ C¯ ku0 k

+ kuk

N +1

2 L1T (B˙ 2,1

N

2 −1 B˙ 2,1

)

ˆ + kd − dk

N

˜ ∞ (B˙ 2 ) L 2,1 T

ˆ + kd0 − dk

N

2 B˙ 2,1



ˆ + kd − dk

N +2

2 L1T (B˙ 2,1

)

(5.1) .

The study of incompressible limit problem of (1.9) relies on Strichartz estimates for the following system of acoustics: Proposition 5.2. (see [1]) Let (b, v) be a solution of the following system of acoustics:   ∂t b + ǫ−1 Λv = F,     ∂t v − ǫ−1 Λb = G,      (b, v)t=0 = (b0 , v0 ),

where Λ =

√

k(b, v)k

(5.2)

−∆. Then, for any s ∈ R and T ∈ (0, ∞], the following estimate holds: 1

1

1 − 1 )+ 1 s+N( p r 2

˜ r (B˙ L p,1 T

)

1

≤ Cǫ r k(b0 , v0 )kB˙ s + ǫ1+ r − r¯′ k(F, G)k 2,1

1 −1 s+N( 1′ − 1 2 )+ r p ¯ ¯′

˜ r¯′ (B˙ L T p ¯′ ,1

,

(5.3)

)

with 2 ≤ min(1, γ(p)), (r, p, N ) 6= (2, ∞, 3), r 2 ≤ min(1, γ(¯ p)), (¯ r , p¯, N ) 6= (2, ∞, 3), p¯ ≥ 2, r¯ p ≥ 2,

where γ(q) := (N − 1)( 12 − 1q ),

1 p¯

+

1 p¯′

= 1, and

1 r¯

+

1 r¯′

= 1.

Let us make the following change of functions: c(t, x) := ǫbǫ (ǫ2 t, ǫx), v(t, x) := ǫuǫ (ǫ2 t, ǫx), h(t, x) := dǫ (ǫ2 t, ǫx). Then (bǫ , uǫ , dǫ ) solves (1.11) if and only if (c, v, h) solves   ∂t c + divv = −div(cv),         µ∆v + (µ + λ)∇divv P ′ (1 + c)   ∂ v + v · ∇v − + ∇c  t  1+c 1+c    −ξ 1 = div(∇h ⊙ ∇h − |∇h|2 I),   1+c 2       ∂t h + v · ∇h = θ(∆h + |∇h|2 h),         (c, v, h)| = (c , v , h ), t=0

0

0

with (c0 , v0 , h0 ) := (ǫbǫ0 (ǫx), ǫuǫ0 (ǫx), dǫ0 (ǫx)).

0

(5.4)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

23

According to Theorem 1.1, there exist two positive constants η = η(N, λ, µ, ξ, θ, P ) and M = N

M (N, λ, µ, ξ, θ, P ) such that (5.4) has a solution (c, v, h) in Bν2 as soon as kc0 k

N ,∞

˜ν2 B

+ kv0 k

N

2 −1 B˙ 2,1

ˆ + kh0 − dk

N

2 B˙ 2,1

≤ η.

In addition, we have the estimate ˆ k(c, v, h − d)k

N Bν2

 ≤ M kc0 k

N ˜ν2 ,∞ B

ˆ + kh0 − dk

+ kv0 k

N −1 2 B˙ 2,1

N 2 B˙ 2,1

 .

According to the scaling properties of Besov space in Proposition 2.1, it is easy to verify that kc0 k

N ,∞

˜ν2 B

+ kv0 k

N

2 −1 B˙ 2,1

ˆ + kh0 − dk

N

≈ kbǫ0 k

k(c, v, h)k

N

≈ k(b, u, d)k

2 B˙ 2,1

+ kuǫ0 k

N ,∞

N

2 −1 B˙ 2,1

˜ν2 B

ˆ + kdǫ0 − dk

N

2 B˙ 2,1

,

and Bν2

N

Bǫν2

,

which combined with Proposition 5.1 conclude the part (i) of Theorem 1.2. Now we turn to the proof of part (ii) of Theorem 1.2. Case 1: N ≥ 4. Let us first focus on the convergence of (bǫ , Quǫ ). Applying Q := ∇∆−1 div

to the second equation of (1.11), we conclude that (bǫ , Quǫ ) satisfies  divQuǫ ǫ   ∂ b + = −div(bǫ uǫ ),   t ǫ      ǫ ǫbǫ ∇bǫ ǫ ǫ ∇b = Q − uǫ · ∇uǫ − Au + K(ǫb ) ∂t Quǫ − ν∆Quǫ +  ǫ 1 + ǫbǫ ǫ         − ξ div(∇dǫ ⊙ ∇dǫ − 1 |∇dǫ |2 I) . 1 + ǫbǫ 2

Setting lǫ := Λ−1 divQuǫ , then system (5.5) becomes  ǫ −1 ǫ ǫ   ∂t b + ǫ Λl = F ,

(5.5)

(5.6)

  ∂ lǫ − ǫ−1 Λbǫ = Gǫ , t

with

F ǫ := −div(bǫ uǫ ),

ǫbǫ ∇bǫ Auǫ − K(ǫbǫ ) ǫ 1 + ǫb ǫ  1 ξ div(∇dǫ ⊙ ∇dǫ − |∇dǫ |2 I) . + 1 + ǫbǫ 2 ǫ −1 ǫ Remark that Qu = −∇Λ l so that estimating Quǫ or lǫ is equivalent (up to an irrelevant  Gǫ := ν∆lǫ − Λ−1 div uǫ · ∇uǫ +

constant). Taking p¯ = 2, r¯ = ∞, s = N −1 ˜ 2 (B˙ p 2 ) L p,1

2(N −1) N −3 .

− 1 and r = 2 in Proposition 5.2 yields 1

1

k(bǫ , Quǫ )k for all p ≥ pN :=

N 2

≤ Cǫ 2 k(bǫ0 , Quǫ0 )k

N

2 −1 B˙ 2,1

+ Cǫ 2 k(F ǫ , Gǫ )k

Note that

kdiv(bǫ uǫ )k

N −1

2 L1 (B˙ 2,1

)

. kbǫ k

N

2 ) L2 (B˙ 2,1

N

2 −1 ) L1 (B˙ 2,1

kuǫ k

N

2 ) L2 (B˙ 2,1

. C0ǫν ,

24

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

kuǫ · ∇uǫ k

N −1

2 L1 (B˙ 2,1

)

. kuǫ k2

. kuǫ k

N

N −1

2 L∞ (B˙ 2,1

2 ) L2 (B˙ 2,1

)

kuǫ k

N +1

2 L1 (B˙ 2,1

ǫbǫ

ǫ Au

1 N2 −1 . ǫkbǫ k ∞ N2 kAuǫ k 1 N2 −1 1 + ǫbǫ L (B˙ 2,1 ) L (B˙ 2,1 ) L (B˙ 2,1 ) . ν −1 kbǫ k

N ˜ǫν2 ,∞ ) L∞ (B

kuǫ k

N +1 2 L1 (B˙ 2,1 )

)

. C0ǫν ,

. C0ǫν ,

ǫ

ǫ ∇b ǫ 2 . C0ǫν . N −1 . kb k

K(ǫb )

N 2 2 2 ˙ ǫ L1 (B˙ 2,1 ) L (B2,1 )

Moreover,



1 ξ ǫ 2 ǫ ǫ |∇d | I) div(∇d ⊙ ∇d −

1 N2 −1 1 + ǫbǫ 2 L (B˙ 2,1 )

=

  ξ ǫ ˆ ⊙ ∇(dǫ − d) ˆ − 1 |∇(dǫ − d)| ˆ 2I div ∇(d − d)

1 N2 −1 1 + ǫbǫ 2 L (B˙ 2,1 )   ǫν ˆ 2 . 1 + ǫkbǫ k kdǫ − dk N N +1 . C0 . 2 ) L∞ (B˙ 2,1

2 L2 (B˙ 2,1

)

Collecting all the estimates above, we conclude that 1

k(bǫ , Quǫ )k

N −1 ˜ 2 (B˙ p 2 ) L p,1

. C0ǫν ǫ 2 , p ≥

2(N − 1) . N −3

(5.7)

On the other hand, define P := Id − Q, then the rest part (Puǫ , dǫ ) of system (1.11) satisfies     ∂t Puǫ − µ∆Puǫ = P − uǫ · ∇uǫ −  

ǫ

ǫ

ǫ

ǫ

 ξ ǫbǫ ǫ ǫ ǫ Au − div(∇d ⊙ ∇d ) , 1 + ǫbǫ 1 + ǫbǫ

(5.8)

ǫ 2 ǫ

∂t d + u · ∇d = θ(∆d + |∇d | d ).

¯ ǫ = dǫ − d, then it follows from (1.10) and (5.8) that (wǫ , d ¯ ǫ ) solves Letting wǫ = Puǫ − u and d    ǫbǫ ǫ   Au ∂t wǫ − µ∆wǫ = −P(uǫ · ∇uǫ − u · ∇u) − P   1 + ǫbǫ      ξ ǫ ǫ div(∇d ⊙ ∇d ) − ξdiv(∇d ⊙ ∇d) , −P  1 + ǫbǫ       ¯ǫ ¯ ǫ = −uǫ · ∇d ¯ ǫ − wǫ · ∇d − Quǫ · ∇d + θ|∇d|2 d ¯ ǫ + θ|∇dǫ + ∇d||∇d ¯ ǫ |dǫ . ∂t d − θ∆d (5.9)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

25

We infer from the estimates for heat equation (see [1, 6]) that kwǫ (t)k

+ µkwǫ k

N −3 2

p B˙ p,1

. kw0ǫ k

N −3 p 2 B˙ p,1



+ P

N +1 2

p L1 (B˙ p,1

)

+ kP(uǫ · ∇uǫ − u · ∇u)k

N −3 p 2 L1 (B˙ p,1 )

 ǫbǫ 

ǫ + P Au N −3

p 2 1 + ǫbǫ ) L1 (B˙ p,1

(5.10)

 ξ

ǫ ǫ div(∇d ⊙ ∇d ) − ξdiv(∇d ⊙ ∇d) N −3 .

ǫ p 2 1 + ǫb L1 (B˙ p,1 )

By virtue of Corollary 2.1 and interpolation, there holds

 ǫbǫ

ǫbǫ 

ǫ ǫ Au Au N −3 .

1 N2 − 32 . kǫbǫ k 4 N2 kAuǫ k 4 N2 − 32

P p 2 1 + ǫbǫ 1 + ǫbǫ L (B˙ 2,1 ) L (B˙ 2,1 ) L 3 (B˙ 2,1 ) L1 (B˙ p,1 ) 1 2

ǫ

. ǫ kb k

N ,4

˜ǫν2 L4 (B

ǫ

)

ku k

1

N

4 2 +2 ) L 3 (B˙ 2,1

.

(5.11)

1 (C0ǫν )2 ǫ 2 ,

where we have used that N

N

N

˜ǫν2 ,1 ) ∩ L∞ (B ˜ǫν2 ,∞ ) ⇒ b ∈ Lm (B ˜ǫν2 ,m ), for 1 ≤ m ≤ ∞, b ∈ L1 (B N

N

2

N

−1 +1 −1+ m 2 2 2 u ∈ L∞ (B˙ 2,1 ), for 1 ≤ m ≤ ∞, ) ∩ L1 (B˙ 2,1 ) ⇒ u ∈ Lm (B˙ 2,1

and 1

+ (ǫν) 2 kbǫ k

≈ kbǫ k

kbǫ k

N −1 2 2 B˙ 2,1

N ˜ǫν2 ,4 B

N 2 B˙ 2,1

⇒ kbǫ k

1

N 2 B˙ 2,1

. ǫ− 2 kbǫ k

N ,4

˜ǫν2 B

.

Noting that uǫ · ∇uǫ − u · ∇u = uǫ · ∇Quǫ + uǫ · ∇wǫ + Quǫ · ∇u + wǫ · ∇u, which, together with Corollary 2.1, (5.7), interpolation and Young inequality yields, for any δ>0 kP(uǫ · ∇uǫ − u · ∇u)k

N −3 2

p L1 (B˙ p,1

≤ Ckuǫ k

N

2 ) L2 (B˙ 2,1

N −3 2

p L2 (B˙ p,1

Z t kuǫ k +C

k∇wǫ k

1 C(C0ǫν )2 ǫ 2

ǫ

0

≤

k∇Quǫ k

)

N 2 B˙ 2,1

N −3 p 2 B˙ p,1

+ δkw k

N +1 p 2 L1 (B˙ p,1 )

+ Ck∇uk

N

2 −1 ) L2 (B˙ 2,1

)

+ k∇uk

N −1 2 B˙ 2,1

N −1 p 2 B˙ p,1

N 2 B˙ 2,1

N −1 2

p L2 (B˙ p,1

kwǫ k

Z t kuǫ k2 + Cδ 0

kQuǫ k

+ kuk2

)

(5.12)

 dt′

N 2 B˙ 2,1

 kwǫ k

N −3 2

p B˙ p,1

dt′ .

26

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Moreover,



P

 ξ

ǫ ǫ div(∇d ⊙ ∇d ) − ξdiv(∇d ⊙ ∇d) N −3

p 2 1 + ǫbǫ ) L1 (B˙ p,1



. ξdiv(∇dǫ ⊙ ∇dǫ − ∇d ⊙ ∇d) N −3 p 2 L1 (B˙ p,1

)

(5.13)

ǫbǫ

ǫ ǫ + ξdiv(∇d ⊙ ∇d ) N −3

p 2 1 + ǫbǫ ) L1 (B˙ p,1

:= I1 + I2 . For the estimate of I1 , we have

I1 . k∇dǫ + ∇dk

N

2 ) L2 (B˙ 2,1



ǫ

. kd k

N

2 +1 ) L2 (B˙ 2,1

¯ ǫk k∇d

N −1 2

p L2 (B˙ p,1

+ kdk

N

2 +1 ) L2 (B˙ 2,1

)

(5.14)

 ¯ ǫk kd

N +1 2

p L2 (B˙ p,1

. )

For I2 , it follows that I2 . ǫkbǫ k

N −1 2 2) L∞ (B˙ 2,1 1

1



ˆ ⊙ ∇(dǫ − d)) ˆ

div(∇(dǫ − d)

. ǫ 2 kbǫ k 2

N −1 2 L∞ (B˙ 2,1 )

1

. ǫ 2 kbǫ k

N ˜ǫν2 ,∞ ) L∞ (B

 ǫνkbǫ k

1 2

N 2 ) L∞ (B˙ 2,1

N −1

p L1 (B˙ p,1

ˆ 2 kdǫ − dk

N +1

p L2 (B˙ p,1

)

(5.15) )

1

ˆ 2 kdǫ − dk

N +1 2 L2 (B˙ 2,1 )

. (C0ǫν )3 ǫ 2 .

Thus, substituting (5.11)-(5.15) into (5.10), and choosing δ in (5.12) sufficiently small, it is not difficult to obtain kwǫ k

N −3 p 2 ) L∞ (B˙ p,1

3 + µkwǫ k N +1 p 2 4 ) L1 (B˙ p,1

kw0ǫ k Np − 3 2 B˙

.

+

1 C0ǫν ǫ 2

p,1

 + kdǫ k

N +1 2 ) L2 (B˙ 2,1

Z t kuǫ k2 + 0

+ kdk

N 2 B˙ 2,1

N +1 2 L2 (B˙ 2,1 )

+ kuk2

N 2 B˙ 2,1

 ¯ǫk kd



N +1 2

p L2 (B˙ p,1

kwǫ k

dt′

N −3 2

p B˙ p,1

(5.16)

. )

¯ǫk In order to close the estimates of wǫ , we now aim to bound the term kd

N +1 2

p L2 (B˙ p,1

of (5.16). )

For this, we take advantage of the estimates of heat equation (5.9)2 and obtain that ¯ ǫk kd

N −1 2

p L∞ (B˙ p,1

¯ǫ k . kd 0

)

¯ ǫk + θkd

N −1 2

p B˙ p,1

N +3 2

p L1 (B˙ p,1

¯ ǫk + kuǫ · ∇d

)

N −1 p 2 ) L1 (B˙ p,1

¯ ǫk + k|∇d|2 d

N −1 2

p L1 (B˙ p,1

)

+ kwǫ · ∇dk

N −1 p 2 L1 (B˙ p,1 )

¯ ǫ |dǫ k + k|∇dǫ + ∇d||∇d

N −1 2

p L1 (B˙ p,1

+ kQuǫ · ∇dk .

)

N −1 2

p L1 (B˙ p,1

)

(5.17)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

27

Next, we estimate the terms of the right hand of (5.17) one by one. ¯ ǫk kuǫ · ∇d

N −1 2

p L1 (B˙ p,1

. kuǫ k

N

2 ) L2 (B˙ 2,1

)

N 2 ) L2 (B˙ 2,1

N −1 2

p L1 (B˙ p,1

. kwǫ k

)

k∇dk

N −1 2

. kQuǫ k

¯ ǫk k|∇d|2 d

. k|∇d|2 k

N −1 p 2 L1 (B˙ p,1 )

N −1 2

p L1 (B˙ p,1

¯ ǫ |dǫ k . k|∇dǫ ||∇d

N −1 2

p L1 (B˙ p,1

N

2 ) L2 (B˙ 2,1

. k∇dǫ k

N

2 ) L2 (B˙ 2,1

 . kdǫ k

)

N +3 2

p L1 (B˙ p,1

)

N

(5.19)

N −3 2

)

k∇dk

kwǫ k 2

N +1 2

p L1 (B˙ p,1

)

kdk

. kQuǫ k

N

2 ) L2 (B˙ 2,1

N +1 2 L2 (B˙ 2,1 )

N −1 2

p L2 (B˙ p,1

¯ ǫk kd

. kdk2

N −1 p 2 L∞ (B˙ p,1 )

N +1 2 ) L2 (B˙ 2,1

)

.

kdk

N

.

(5.20)

.

(5.21)

2 +1 ) L2 (B˙ 2,1

¯ ǫk kd

N −1 2

p L∞ (B˙ p,1

)

)

)

¯ ǫ |dǫ k + k|∇d||∇d N −1 2

p L2 (B˙ p,1

N

N −1 2

p L1 (B˙ p,1

¯ ǫk k∇d

kdǫ k

2 ) L∞ (B˙ 2,1

N +1 2 ) L2 (B˙ 2,1

N

2 ) L1 (B˙ 2,1

¯ ǫ |dǫ k k|∇dǫ + ∇d||∇d

. k|∇dǫ |dǫ k

N −1 2

p L2 (B˙ p,1

)

)

.

1

p L∞ (B˙ p,1

p L1 (B˙ p,1

(5.18)

¯ ǫk 2 kd

2 ) L2 (B˙ 2,1

1

. kwǫ k 2 kQuǫ · ∇dk

N −1 2

p L∞ (B˙ p,1

N −1 2

) 1

¯ ǫk 2 kd

p L2 (B˙ p,1

)

N −1 2

p L2 (B˙ p,1

1

. kuǫ k kwǫ · ∇dk

¯ ǫk k∇d

)

+ k|∇d|dǫ k

2 ) L2 (B˙ 2,1

¯ ǫk kd

N +1 2

p L2 (B˙ p,1

+ kdk

N +1 2 L2 (B˙ 2,1 )

N

)

)

+ k∇dk

¯ ǫk k∇d N

2 ) L2 (B˙ 2,1

 kdǫ k

kdǫ k

N

(5.22)

)

2 ) L∞ (B˙ 2,1

1

N 2 ) L∞ (B˙ 2,1

N −1 2

p L2 (B˙ p,1

¯ ǫk kd

N +1 2

p L2 (B˙ p,1

)

1

¯ ǫk 2 kd

N −1 p 2 L∞ (B˙ p,1 )

¯ ǫk 2 kd

N +3 2

p L1 (B˙ p,1

. )

It follows from the above estimates (5.17)-(5.22) that ¯ ǫk kd

N −1 2

p L∞ (B˙ p,1

)

¯ǫ k . kd 0

¯ ǫk + θkd

N −1 p 2 B˙ p,1

) 1

+ kuǫ k

N 2 ) L2 (B˙ 2,1

1

¯ ǫk 2 kd

N −1 p 2 L∞ (B˙ p,1 )

¯ ǫk 2 kd

N +3 2

p L1 (B˙ p,1

)

1

1

+ kwǫ k 2

N −3 p 2 L∞ (B˙ p,1 )

+ kQuǫ k

N −1 2

p L2 (B˙ p,1

 + kdǫ k

N +3 2

p L1 (B˙ p,1

N +1 2 ) L2 (B˙ 2,1

kwǫ k 2

N +1 2

p L1 (B˙ p,1

)

kdk

N +1

2 L2 (B˙ 2,1

+ kdk

)

)

kdk

N +1

2 L2 (B˙ 2,1

+ kdk2

N +1 2 L2 (B˙ 2,1 )

N +1 2 ) L2 (B˙ 2,1

 kdǫ k

(5.23)

)

N 2 ) L∞ (B˙ 2,1

¯ ǫk kd

N −1 2

p L∞ (B˙ p,1

)

1

¯ ǫk 2 kd

N −1 p 2 L∞ (B˙ p,1 )

1

¯ ǫk 2 kd

N +3 2

p L1 (B˙ p,1

. )

28

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

Now combining (5.16) with (5.23) and using Young inequality, we get kwǫ k

N −3 p 2 ) L∞ (B˙ p,1

µ ǫ θ ¯ǫ ¯ǫ kw k kd k N + 1 + kd k N +3 N −1 + p p p 2 2 2 1 ∞ ˙ ˙ 2 2 L (Bp,1 ) L1 (B˙ p,1 L (Bp,1 ) )

+

 . kw0ǫ k

¯ǫ k + kd 0

N −3 p 2 B˙ p,1

+

Z t

ku (t

Z t

kdǫ (t′ )k2

0

+

N −1 p 2 B˙ p,1

0

ǫ

′

)k2 N B˙ 2

2,1

+ ku(t

N +1 2 B˙ 2,1

′



1

+ C0ǫν ǫ 2

)k2 N B˙ 2

2,1

+ kd(t

+ kd(t′ )k2

N +1 2 B˙ 2,1

′

)k2 N +1 B˙ 2 2,1

(5.24)

 kwǫ (t′ )k

N −3 2

p B˙ p,1

+ kuǫ (t′ )k2

N 2 B˙ 2,1

 ¯ ǫ (t′ )k kd

dt

N −1 2

p B˙ p,1

′

dt′ .

Thus, Gronwall’s inequality guarantees that kwǫ k

N −3 p 2 ) L∞ (B˙ p,1

 . kw0ǫ k

+

N −3 2

p B˙ p,1

µ ǫ θ ¯ǫ ¯ǫ kw k kd k N −1 + N + 1 + kd k N +3 p p p 2 2 2 2 2 L∞ (B˙ p,1 ) ) L1 (B˙ p,1 ) L1 (B˙ p,1 ¯ǫ k + kd 0

N −1 2

p B˙ p,1



+

(5.25)

1 C0ǫν ǫ 2 .

Case 2: N = 3. Applying Proposition 5.2 to (5.5) with p¯ = 2, r¯ = ∞, s =

1 2

and r =

2p p−2 ,

similar to (5.7), we obtain k(bǫ , Quǫ )k

2 1 2p p−2 ) L p−2 (B˙ p,1

. k(bǫ , Quǫ )k

1

2−1 2

2p

˜ p−2 (B˙ p L p,1

)

. C0ǫν ǫ 2

− p1

, p ≥ 2.

Use the following interpolation for 2 ≤ q < +∞,   h 2 1 i 5 2q −2 (14−q)/(2q+4) q 2 = L1 (R+ ; B˙ 2,1 L2 R+ ; B˙ q+2 ); L q−2 (R+ ; B˙ q,1 ) ,1

2

Make the change of parameter p =

q+2 2 .

2 q+2

(5.26)

.

Due to (5.26), there holds 1

kQuǫ k

4 1 p−2 L2 (B˙ p,1 )

1

. C0ǫν ǫ 2 − p , 2 ≤ p < +∞.

(5.27)

In the following, we want to prove that kwǫ k

4−3 2

p L∞ (B˙ p,1

)

+ kwǫ k

4+1 2

p L1 (B˙ p,1

 . kw0ǫ k

¯ǫ k + kd 0

4 3 p−2 B˙ p,1

4 1 p−2 B˙ p,1

)



¯ǫk + kd

4−1 2

p L∞ (B˙ p,1 1

+ C0ǫν ǫ 2

− p1

)

¯ ǫk + kd

4+3 2

p L1 (B˙ p,1

)

(5.28) .

To this end, similar to the estimates of (5.11)-(5.15), we have

 ǫbǫ 

ǫ ǫ kAuǫ k 3 4 3 . ǫkb k Au

P 1+ 1 p−2 p p −1 1 ∞ ˙ ˙ 1 + ǫbǫ L1 (B˙ p,1 ) L (Bp,1 ) L (B2,1 ) − 12 − p1

. ǫ(ǫν) 1

. ǫ2

− p1

kbǫ k

1

kbǫ k 2

− p1

1 2 ) L∞ (B˙ 2,1

3 2 ,∞ ) ˜ǫν L∞ (B

1 +1 2 p 3 2 L∞ (B˙ 2,1 )

(ǫνkbǫ k

kuǫ k

5 2 ) L1 (B˙ 2,1

)

1

. (C0ǫν )2 ǫ 2

kuǫ k

− p1

5

2 ) L1 (B˙ 2,1

,

(5.29)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

kP(uǫ · ∇uǫ − u · ∇u)k

4−3 2

p L1 (B˙ p,1

≤ Ckuǫ k

N

2 ) L2 (B˙ 2,1

≤

1

1

C(C0ǫν )2 ǫ 2 − p

4−3 2

p L2 (B˙ p,1

3 2 B˙ 2,1

0

)

k∇Quǫ k

Z t kuǫ k +C

k∇wǫ k

4 3 p−2 B˙ p,1

ǫ

29

+ Ck∇uk

N

2 −1 ) L2 (B˙ 2,1

)

+ k∇uk

1 2 B˙ 2,1

+ δkw k

4 1 p+2 L1 (B˙ p,1 )

+ Cδ

Z t 0

kQuǫ k

4−1 2

p L2 (B˙ p,1

(5.30)

kwǫ k

 dt′

kuǫ k2

+ kuk2

4 1 p−2 B˙ p,1

3 2 B˙ 2,1

)

3 2 B˙ 2,1

 kwǫ k

4−3 2

p B˙ p,1

dt′ .

Moreover,



P

 ξ

ǫ ǫ div(∇d ⊙ ∇d ) − ξdiv(∇d ⊙ ∇d) 4−3

p 2 1 + ǫbǫ L1 (B˙ p,1 )



. ξdiv(∇dǫ ⊙ ∇dǫ − ∇d ⊙ ∇d) 4−3 p 2 L1 (B˙ p,1

(5.31)

)

ǫbǫ

ǫ ǫ ξdiv(∇d ⊙ ∇d ) 4 3 := H1 + H2 . +

p−2 1 + ǫbǫ L1 (B˙ p,1 )

For the estimate of H1 , we have

H1 . k∇dǫ + ∇dk

3 2 ) L2 (B˙ 2,1

 . kdǫ k

5

2 ) L2 (B˙ 2,1

¯ ǫk k∇d

4−1 2

p L2 (B˙ p,1

+ kdk

5

2 ) L2 (B˙ 2,1

)

(5.32)

 ¯ ǫk kd

4+1 2

p L2 (B˙ p,1

. )

For H2 , it follows that H2 . ǫkbǫ k

1+ 1 L∞ (B˙ 2,1 p )

− 12 − p1

. ǫ(ǫν) 1

. ǫ2

− p1



ˆ ⊙ ∇(dǫ − d)) ˆ

div(∇(dǫ − d)

3 −1

p L1 (B˙ p,1 )

1

kbǫ k 2

− p1

1 2 ) L∞ (B˙ 2,1

kbǫ k

3 2 ,∞ ) ˜ǫν L∞ (B

1 +1 2 p 3 2 ) L∞ (B˙ 2,1

(ǫνkbǫ k

)

ˆ 2 kdǫ − dk 1

ˆ 2 kdǫ − dk

5 2 ) L2 (B˙ 2,1

3 +1

p L2 (B˙ p,1 )

. (C0ǫν )3 ǫ 2

− p1

(5.33)

.

Therefore, in view of the estimates of heat equation, similar to (5.16), we have kwǫ k

4 3 p−2 ) L∞ (B˙ p,1

.

3 + µkwǫ k 4 1 p+2 4 L1 (B˙ p,1 )

kw0ǫ k p4 − 32 B˙

+

1

1

C0ǫν ǫ 2 − p

p,1

 + kdǫ k

5 2 ) L2 (B˙ 2,1

+ kdk

Z t kuǫ k2 +

3 2 B˙ 2,1

0

5 2 ) L2 (B˙ 2,1



¯ ǫk kd

+ kuk2

3 2 B˙ 2,1

4+1 2

p L2 (B˙ p,1

. )

 kwǫ k

4−3 2

p B˙ p,1

dt′

(5.34)

30

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

¯ ǫk In order to close the estimates of wǫ , we must bound the term kd

4+1 2

p L2 (B˙ p,1

of (5.34). For this )

purpose, similar to (5.17), it follows that ¯ ǫk kd

4−1 2

p L∞ (B˙ p,1

¯ǫ k . kd 0

)

¯ ǫk + θkd

4+3 2

p L1 (B˙ p,1

4−1 2

p B˙ p,1

)

¯ ǫk + kuǫ · ∇d

4−1 2

p L1 (B˙ p,1

¯ ǫk + k|∇d|2 d

4−1 2

p L1 (B˙ p,1

)

)

+ kwǫ · ∇dk

4−1 2

p L1 (B˙ p,1

¯ ǫ |dǫ k + k|∇dǫ + ∇d||∇d

+ kQuǫ · ∇dk

4−1 2

p L1 (B˙ p,1

4−1 2

p L1 (B˙ p,1

)

(5.35)

)

. )

Next, we estimate the terms of the right hand of (5.35) as follows. ¯ ǫk kuǫ · ∇d

4 1 p−2 ) L1 (B˙ p,1

. kuǫ k

3 2 ) L2 (B˙ 2,1

. ku k

3

2 ) L2 (B˙ 2,1

4−1 2

p L1 (B˙ p,1

)

. kwǫ k

)

k∇dk

¯ ǫk k|∇d|2 d

4−1 2

p L1 (B˙ p,1

)

4 1 p−2 L2 (B˙ p,1 )

. k|∇d|2 k

¯ ǫ |dǫ k k|∇dǫ + ∇d||∇d

4−1 2

p L1 (B˙ p,1

¯ ǫ |dǫ k . k|∇dǫ ||∇d

4 1 p−2 ) L1 (B˙ p,1

. k|∇dǫ |dǫ k

3

2 ) L2 (B˙ 2,1

. k∇dǫ k

3

2 ) L2 (B˙ 2,1

 . kdǫ k

5 2 ) L2 (B˙ 2,1

3

2 ) L1 (B˙ 2,1

4−3 2

2 ) L∞ (B˙ 2,1

+ kdk

k∇dk

)

, )

3

(5.37)

kwǫ k 2

3

¯ǫk kd

4+1 2

p L1 (B˙ p,1

2 ) L2 (B˙ 2,1

4−1 2

p L∞ (B˙ p,1

¯ ǫ |dǫ k + k|∇d||∇d

)

)

kdk

5 2 ) L2 (B˙ 2,1

. kQuǫ k

4−1 2

p L2 (B˙ p,1

. kdk2

5

2 ) L2 (B˙ 2,1

4−1 2

p L1 (B˙ p,1

4−1 2

p L2 (B˙ p,1

3

4+3 2

p L1 (B˙ p,1

)

,

kdk

5

,

(5.38)

,

(5.39)

2 ) L2 (B˙ 2,1

¯ ǫk kd

4−1 2

p L∞ (B˙ p,1

)

)

¯ ǫk k∇d

kdǫ k

)

1

p L∞ (B˙ p,1

. kQuǫ k

(5.36)

¯ ǫk 2 kd

2 ) L2 (B˙ 2,1

1

4 1 p−2 L1 (B˙ p,1 )

4−1 2

p L∞ (B˙ p,1

4−1 2

) 1

¯ ǫk 2 kd

p L2 (B˙ p,1

. kwǫ k 2

kQuǫ · ∇dk

4−1 2

p L2 (B˙ p,1

1

ǫ

kwǫ · ∇dk

¯ ǫk k∇d

+ k|∇d|dǫ k

2 ) L2 (B˙ 2,1

)

¯ ǫk kd

5 2 ) L2 (B˙ 2,1

4 1 p+2 ) L2 (B˙ p,1



3

)

kdǫ k

+ k∇dk

3 2 ) L∞ (B˙ 2,1

¯ ǫk k∇d 3

2 ) L2 (B˙ 2,1

1

¯ ǫk 2 kd

4−1 2

p L2 (B˙ p,1

4 1 p−2 L∞ (B˙ p,1 )

kdǫ k

(5.40)

)

3

2 ) L∞ (B˙ 2,1

¯ ǫk kd

1

¯ ǫk 2 kd

4+3 2

p L1 (B˙ p,1

4+1 2

p L2 (B˙ p,1

. )

)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

31

It follows from the above estimates (5.35)-(5.40) that ¯ ǫk kd

4−1 2

p L∞ (B˙ p,1

)

¯ǫ k . kd 0

¯ ǫk + θkd

4−1 p 2 B˙ p,1

4+3 2

p L1 (B˙ p,1

1

1

+ kuǫ k

3 2 ) L2 (B˙ 2,1

1

)

¯ ǫk 2 kd

4 1 p−2 L∞ (B˙ p,1 )

¯ǫk 2 kd

4+3 2

p L1 (B˙ p,1

)

1

+ kwǫ k 2

4 3 p−2 ) L∞ (B˙ p,1

+ kQuǫ k

4−1 2

p L2 (B˙ p,1

 + kdǫ k

5 2 ) L2 (B˙ 2,1

kwǫ k 2

4+1 2

p L1 (B˙ p,1

)

kdk

5

2 ) L2 (B˙ 2,1

+ kdk

5 2 ) L2 (B˙ 2,1

)

kdk

(5.41)

5

2 ) L2 (B˙ 2,1

+ kdk2

5

2 ) L2 (B˙ 2,1

 kdǫ k

¯ ǫk kd

4−1 2

p L∞ (B˙ p,1

) 1

1

3 2 ) L∞ (B˙ 2,1

¯ǫk 2 kd

4 1 p−2 ) L∞ (B˙ p,1

¯ ǫk 2 kd

4+3 2

p L1 (B˙ p,1

. )

Now combining (5.34) with (5.41) and using Young inequality, we get kwǫ k

+

4 3 p−2 L∞ (B˙ p,1 )

µ ǫ θ ¯ǫ ¯ǫ kw k kd k 4 + 1 + kd k 4 3 4−1 + p p p+2 2 2 2 2 L1 (B˙ p,1 ) L1 (B˙ p,1 L∞ (B˙ p,1 ) )

 . kw0ǫ k

¯ǫ k + kd 0

4 3 p−2 B˙ p,1

+

Z t

ku (t

Z t

kdǫ (t′ )k2

0

+

4 1 p−2 B˙ p,1

0

ǫ

′

)k2 3 B˙ 2

2,1

5 2 B˙ 2,1



+ ku(t

′

1

+ C0ǫν ǫ 2

)k2 3 B˙ 2

− p1

+ kd(t

2,1

+ kd(t′ )k2

′

)k2 5 B˙ 2

2,1

+ kuǫ (t′ )k2

5 2 B˙ 2,1

(5.42)

 kwǫ (t′ )k

3 2 B˙ 2,1

4−3 2

p B˙ p,1

 ¯ ǫ (t′ )k kd

4−1 2

p B˙ p,1

dt

′

dt′ .

Gronwall’s inequality then yields that kwǫ k

4 3 p−2 L∞ (B˙ p,1 )

 . kw0ǫ k

+

4−3 2

p B˙ p,1

µ ǫ θ ¯ǫ ¯ ǫk 4−1 + kw k 4 + 1 + kd kd k 4 3 p p p+2 2 2 2 2 L∞ (B˙ p,1 ) ) L1 (B˙ p,1 ) L1 (B˙ p,1 ¯ǫ k + kd 0

4−1 2

p B˙ p,1



+

1

(5.43)

1

C0ǫν ǫ 2 − p .

Case 3: N = 2. Applying Proposition 5.2 to (5.5) with p¯ = 2, r¯ = ∞, s = 0 and r =

4p p−2

yields

k(bǫ , Quǫ )k

3 3 4p 2p − 4 L p−2 (B˙ p,1 )

1

. k(bǫ , Quǫ )k

3 −3 4p ˜ p−2 (B˙ 2p 4 ) L p,1

1

. C0ǫν ǫ 4 − 2p , p ≥ 2.

(5.44)

Use the following interpolation for 2 ≤ q < +∞,   h 3 4q − 34 i (14+q)/(6q+4) 2q 2 L2 R+ ; B˙ 6q+4 = L1 (R+ ; B˙ 2,1 ); L q−2 (R+ ; B˙ q,1 ) q+2 . q+6

,1

3q+2

Make the change of parameter p = kQuǫ k

6q+4 q+6 .

Thanks to estimate (5.44), we conclude that

5 1 2p − 4 L2 (B˙ p,1 )

1

1

. C0ǫν ǫ 4 − 2p , 2 ≤ p ≤ 6.

(5.45)

32

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

¯ ǫ . For this purpose, it follows from Next, we are going to prove the convergence of wǫ and d Corollary 2.1 that

 ǫbǫ 

ǫ ǫ ǫ Au 5 5 . ǫkb k 3 + 1 kAu k 2

P p −1 2p − 4 4 2p 1 + ǫbǫ ) L1 (B˙ p,1 L1 (B˙ p,1 ) L∞ (B˙ 2,1 ) 1

1

3

−

1

1

3

(5.46)

. ǫ(ǫν)− 4 − 2p kbǫ kL4 ∞ (2pB˙ 0 ) (ǫνkbǫ kL∞ (B˙ 1 ) ) 4 + 2p kuǫ kL1 (B˙ 2 1

1

2,1 )

5 −5 4

2p L1 (B˙ p,1

+C

Z t 0

5 5 2p − 4 ) L2 (B˙ p,1

kuǫ kB˙ 1 k∇wǫ k 1

≤ C(C0ǫν )2 ǫ 4

1 − 2p

+ Ck∇ukL2 (B˙ 0 ) kQuǫ k 1 5 2p − 4 B˙ p,1

2,1

+ δkwǫ k

3 5 2p + 4 ) L1 (B˙ p,1

+ Cδ

5 −5 4

2p L2 (B˙ p,1

2,1

+ k∇ukB˙ 0 kwǫ k

5 5 2p − 4 B˙ p,1

2,1

. (C0ǫν )2 ǫ 4 − 2p ,

)

≤ Ckuǫ kL2 (B˙ 1 ) k∇Quǫ k 2,1

1

1

ǫ . ǫ 4 − 2p kbǫ kL∞ (B˜ǫν 1,∞ ku k 1 ˙ 2 L (B )

kP(uǫ · ∇uǫ − u · ∇u)k

2,1 )

2,1

2,1

)

(5.47)

 dt′

Z t  kuǫ k2B˙ 1 + kuk2B˙ 1 kwǫ k 0

5 −5 4

2p B˙ p,1

2,1

2,1

dt′ ,



P

 ξ

ǫ ǫ div(∇d ⊙ ∇d ) − ξdiv(∇d ⊙ ∇d) 5 5

ǫ 2p − 4 1 + ǫb L1 (B˙ p,1 )



. ξdiv(∇dǫ ⊙ ∇dǫ − ∇d ⊙ ∇d) 5 5 2p − 4 L1 (B˙ p,1

(5.48)

)

ǫbǫ

ǫ ǫ + ξdiv(∇d ⊙ ∇d ) 5 5 := K1 + K2 . 2p − 4 1 + ǫbǫ L1 (B˙ p,1 )

Attention is now focused on bounding K1 and K2 . For the estimate of K1 , we have ¯ ǫk K1 . k∇dǫ + ∇dkL2 (B˙ 1 ) k∇d  . kdǫ kL2 (B˙ 2

2,1

5 −1 4

2p L2 (B˙ p,1

2,1

) + kdkL2 (B˙ 2

2,1 )



)

(5.49) ¯ ǫk kd

5 3 2p + 4 L2 (B˙ p,1 )

.

For K2 , it follows that K2 . ǫkbǫ k

3+ 1 4 2p L∞ (B˙ 2,1

)



ˆ ⊙ ∇(dǫ − d)) ˆ

div(∇(dǫ − d)

1

1

2 −1

p ) L1 (B˙ p,1

− ˆ 2 . ǫ(ǫν)− 4 − 2p kbǫ kL4 ∞ (2pB˙ 0 ) (ǫνkbǫ kL∞ (B˙ 1 ) ) 4 + 2p kdǫ − dk 3

1

3

2,1

1

1

2,1

1

ǫ ˆ 2 . ǫ 4 − 2p kbǫ kL∞ (B˜ǫν 1,∞ kd − dk 2 ˙ 2 ) L (B

2,1 )

2 +1

p L2 (B˙ p,1 )

1

1

. (C0ǫν )3 ǫ 4 − 2p .

(5.50)

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

33

Therefore, in view of the estimates of heat equation, similar to (5.16), we have kwǫ k

5 5 2p − 4 ) L∞ (B˙ p,1

3 + µkwǫ k 5 3 2p + 4 4 L1 (B˙ p,1 )

kw0ǫ k 2p 5 −5 4 B˙

.

+

1

1

− C0ǫν ǫ 4 2p

+

0

p,1

 + kdǫ kL2 (B˙ 2

2,1

¯ ǫk In order to bound kd

5 +3 4

2p L2 (B˙ p,1

¯ ǫk kd

1 5 2p − 4 ) L∞ (B˙ p,1

¯ǫ k . kd 0

¯ ǫk + θkd

) + kdkL2 (B˙ 2

2,1 )

5 +7 4

1 5 2p − 4 ) L1 (B˙ p,1

 ¯ ǫk kd

5 +3 4

2p L2 (B˙ p,1

5 −5 4

2p B˙ p,1

dt′

(5.51)

. )

)

+ kwǫ · ∇dk

5 −1 4

¯ ǫ |dǫ k + k|∇dǫ + ∇d||∇d

5 −1 4

5 −1 4

2p L1 (B˙ p,1

¯ ǫk + k|∇d|2 d

2,1

2,1

 kwǫ k

, similar to (5.17), we have

¯ ǫk + kuǫ · ∇d

5 −1 4

kuǫ k2B˙ 1 + kuk2B˙ 1

)

2p L1 (B˙ p,1

2p B˙ p,1

Z t

)

2p L1 (B˙ p,1

2p L1 (B˙ p,1

+ kQuǫ · ∇dk

5 −1 4

2p L1 (B˙ p,1

)

)

(5.52)

. )

Next, we estimate the terms of the right hand of (5.52) by Corollary 2.1 and Lemma 2.2 as follows. ¯ ǫk kuǫ · ∇d

1 5 2p − 4 L1 (B˙ p,1 )

¯ ǫk . kuǫ kL2 (B˙ 1 ) k∇d 1

ǫ

. ku kL2 (B˙ 1

2,1

kwǫ · ∇dk

5 −1 4

2p L1 (B˙ p,1

)

. kwǫ k . kw k

1 2

)

5 −5 4

2p L∞ (B˙ p,1

kQuǫ · ∇dk

5 1 2p − 4 L1 (B˙ p,1 )

¯ǫk k|∇d|2 d

5 −1 4

2p L1 (B˙ p,1

)

. kQuǫ k

1 5 2p − 4 L2 (B˙ p,1 )

¯ ǫk . k|∇d|2 kL1 (B˙ 1 ) kd 2,1

5 −1 4

2p L∞ (B˙ p,1

5 −1 4

)

¯ǫk 2 kd

(5.53) 5 +7 4

2p L1 (B˙ p,1

, )

k∇dkL2 (B˙ 1

2,1 )

ǫ

)

kw k

1 2

2,1 )

5 −1 4

(5.54) 5 +3 4

2p L1 (B˙ p,1

k∇dkL2 (B˙ 1

2p L∞ (B˙ p,1

) 1

¯ǫ 2 ) kd k

2p L2 (B˙ p,1

ǫ

5 −1 4

2p L2 (B˙ p,1

2,1

)

)

1

. C0ǫν ǫ 4

kdkL2 (B˙ 2 ) ,

1 − 2p

2,1

kdkL2 (B˙ 2 ) ,

¯ ǫk . kdk2L2 (B˙ 2 ) kd 2,1

(5.55)

2,1

5 −1 4

2p L∞ (B˙ p,1

, )

(5.56)

34

Q. BIE, H. CUI, Q. WANG, AND Z.-A. YAO

¯ ǫ |dǫ k k|∇dǫ + ∇d||∇d

5 −1 4

2p L1 (B˙ p,1

¯ ǫ |dǫ k . k|∇dǫ ||∇d

5 −1 4

2p L1 (B˙ p,1

)

¯ ǫ |dǫ k + k|∇d||∇d

¯ ǫk . k|∇dǫ |dǫ kL2 (B˙ 1 ) k∇d

1 5 2p − 4 ) L2 (B˙ p,1

2,1

 . kdǫ kL2 (B˙ 2

2,1

5 +3 4

) + kdkL2 (B˙ 2

2,1

)

5 −1 4

2p L2 (B˙ p,1

2,1

2p L2 (B˙ p,1

2,1

)

¯ ǫk + k|∇d|dǫ kL2 (B˙ 1 ) k∇d

¯ ǫk . k∇dǫ kL2 (B˙ 1 ) kdǫ kL∞ (B˙ 1 ) kd 2,1

5 −1 4

2p L1 (B˙ p,1

)

)

¯ǫk + k∇dkL2 (B˙ 1 ) kdǫ kL∞ (B˙ 1 ) kd 2,1

 1 ǫ ¯ǫ 2 ) kd kL∞ (B˙ 1 ) kd k

)

1

1 5 2p − 4 ) L∞ (B˙ p,1

2,1

5 +3 4

2p L2 (B˙ p,1

2,1

¯ǫk 2 kd

5 +7 4

2p L1 (B˙ p,1

. )

(5.57) It follows from the above estimates (5.52)-(5.57) that ¯ ǫk kd

5 −1 4

2p L∞ (B˙ p,1

)

¯ ǫk + θkd

5 +7 4

2p L1 (B˙ p,1

) 1

1

¯ǫ k . kd 0

5 1 2p − 4 B˙ p,1

¯ ǫk 2 + kuǫ kL2 (B˙ 1 ) kd 2,1

¯ ǫk 2 kd

5 +7 4

2p L1 (B˙ p,1

)

1

1

+ kwǫ k 2

5 5 2p − 4 L∞ (B˙ p,1 )

1

5 1 2p − 4 L∞ (B˙ p,1 )

kwǫ k 2

1

+ C0ǫν ǫ 4 − 2p kdkL2 (B˙ 2

2,1 )

 + kdǫ kL2 (B˙ 2

2,1

5 +3 4

2p L1 (B˙ p,1

)

kdkL2 (B˙ 2

(5.58)

2,1 )

¯ ǫk + kdk2L2 (B˙ 2 ) kd

) + kdkL2 (B˙ 2

2,1

5 −1 4

2p L∞ (B˙ p,1

2,1

)

 1 ǫ ¯ǫ 2 ) kd kL∞ (B˙ 1 ) kd k

1

5 1 2p − 4 L∞ (B˙ p,1 )

2,1

¯ ǫk 2 kd

5 +7 4

2p L1 (B˙ p,1

. )

Now combining (5.51) with (5.58) and using Young inequality, we get kwǫ k

5 5 2p − 4 ) L∞ (B˙ p,1

+

 . kw0ǫ k

5 5 2p − 4 B˙ p,1

+

µ ǫ θ ¯ǫ ¯ǫ kw k kd k 5 + 3 + kd k 5 7 5 −1 + 2p 4 2p 4 2p + 4 2 2 L1 (B˙ p,1 L1 (B˙ p,1 ) L∞ (B˙ p,1 ) ) ¯ǫ k + kd 0

5 1 2p − 4 B˙ p,1



1

1

+ C0ǫν ǫ 4 − 2p

Z t  kuǫ (t′ )k2B˙ 1 + ku(t′ )k2B˙ 1 + kd(t′ )k2B˙ 2 kwǫ (t′ )k 2,1

0

2,1

2,1

5 −5 4

2p B˙ p,1

Z t  ǫ ′ 2 ′ 2 ǫ ′ 2 ¯ ǫ (t′ )k kd (t )kB˙ 2 + kd(t )kB˙ 2 + ku (t )kB˙ 1 kd + 2,1

0

2,1

2,1

5 −1 4

2p B˙ p,1

dt′ dt′ ,

which together with Gronwall’s lemma yields that kwǫ k

5 5 2p − 4 ) L∞ (B˙ p,1

 . kw0ǫ k

+

5 5 2p − 4 B˙ p,1

µ ǫ θ ¯ǫ ¯ǫ kw k kd k 7 5 −1 + 5 + 3 + kd k 5 2p 4 2p 4 2p + 4 2 2 ) L∞ (B˙ p,1 ) ) L1 (B˙ p,1 L1 (B˙ p,1 ¯ǫ k + kd 0

5 1 2p − 4 B˙ p,1



1

1

+ C0ǫν ǫ 4 − 2p .

GLOBAL EXISTENCE AND INCOMPRESSIBLE LIMIT FOR FLOW OF LIQUID CRYSTALS

Thus the proof of Theorem 1.2 is completed.

35



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