Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three
aa r X i v : . [ m a t h . A P ] A ug GLOBAL FINITE ENERGY WEAK SOLUTIONS TO THE COMPRESSIBLENEMATIC LIQUID CRYSTAL FLOW IN DIMENSION THREE
JUNYU LIN, BAISHUN LAI, AND CHANGYOU WANG
Abstract.
In this paper, we consider the initial and boundary value problem of a simplifiedcompressible nematic liquid crystal flow in Ω ⊂ R . We establish the existence of global weaksolutions, provided the initial orientational director field d lies in the hemisphere S . Introduction
The continuum theory of liquid crystals was developed by Ericksen [3] and Leslie [7] duringthe period of 1958 through 1968, see also the book by De Gennes [2]. Since then there have beenremarkable research developments in liquid crystals from both theoretical and applied aspects. Whenthe fluid containing nematic liquid crystal materials is at rest, we have the well-known Oseen-Franktheory for static nematic liquid crystals, see Hardt-Lin-Kinderlehrer [8] on the analysis of energyminimal configurations of nematic liquid crystals. In general, the motion of fluid always takes place.The so-called Ericksen-Leslie system is a macroscopic continuum description of the time evolutionof the material under influence of both the flow velocity field u and the macroscopic description ofthe microscopic orientation configurations d of rod-like liquid crystals.When the fluid is an incompressible, viscous fluid, Lin [10] first derived a simplified Ericksen-Lesliesystem (i.e. ρ = 1 and div u = 0 in the equation (1.1) below) modeling liquid crystal flows in 1989.Subsequently, Lin and Liu [11, 12] have made some important analytic studies, such as the globalexistence of weak and strong solutions and the partial regularity of suitable weak solutions, of thesimplified Ericksen-Leslie system, under the assumption that the liquid crystal director field is ofvarying length by Leslie’s terminology or variable degree of orientation by Ericksen’s terminology.When dealing with the system (1.1) with ρ = 1 and div u = 0, in dimension two Lin-Lin-Wang [13]and Lin-Wang [14] have established the existence of a unique global weak solution, that has at mostfinitely many possible singular time, for the initial-boundary value problem in bounded domains (seealso Hong [9], Xu-Zhang [36], and Lei-Li-Zhang [15] for some related works); and in dimension threeLin-Wang [18] have obtained the existence of global weak solutions very recently when the initialdirector field d maps to the hemisphere S .When the fluid is compressible, the simplified Ericksen-Leslie system (1.1) becomes more compli-cate, which is a strongly coupling system between the compressible Navier-Stokes equation and thetransported harmonic map heat flow to S . It seems worthwhile to be explored for the mathematicalanalysis of (1.1). We would like to mention that there have been both modeling study, see Morro [24],and numerical study, see Zakharov-Vakulenko [25], on the hydrodynamics of compressible nematicliquid crystals under the influence of temperature gradient or electromagnetic forces.Now let’s introduce the simplified Ericksen-Leslie system for compressible nematic liquid crystalflow. Let Ω ⊂ R be a bounded, smooth domain, S ⊂ R be the unit sphere, and 0 < T ≤ + ∞ . Wewill consider a simplified version of the three dimensional hydrodynamic flow of the compressible Date : August 7, 2018.
Key words and phrases. renomalized solutions, compressible nematic liquid crystal flow, finite energy solutions. nematic liquid crystal flow in Ω × (0 , T ), i.e., ( ρ, u, d ) : Ω × (0 , T ) → R + × R × S solves ∂ t ρ + ∇ · ( ρu ) = 0 ,∂ t ( ρu ) + ∇ · ( ρu ⊗ u ) + a ∇ ρ γ = L u − ∇ · (cid:0) ∇ d ⊙ ∇ d − |∇ d | I (cid:1) ,∂ t d + u · ∇ d = △ d + |∇ d | d, (1.1)under the initial and boundary condition: ( ρ ( x,
0) = ρ ( x ) , ρu ( x,
0) = m ( x ) , d ( x,
0) = d ( x ) , x ∈ Ω ,u ( x, t ) = 0 , d ( x, t ) = d ( x ) , x ∈ ∂ Ω , t > , (1.2)where ρ : Ω × [0 , T ) → R + denotes density function of the fluid, u : Ω × [0 , T ) → R denotes velocityfield of the fluid, d : Ω × [0 , T ) → S denotes direction field of the averaged macroscopic molecularorientations, ∇· denotes the divergence operator in R , I is the 3 × P ( ρ ) = aρ γ ,with a > γ >
1, denotes the pressure function associated with an isentropic fluid, L is theLam´e operator defined by L u = µ △ u + ( µ + λ ) ∇ ( ∇ · u ) , where µ and λ represent the shear viscosity and the bulk viscosity coefficients of the fluid respectively,which satisfy the natural physical condition: µ > , e µ := µ + λ ≥ , (1.3) ∇ d ⊙ ∇ d denotes the 3 × i, j )-entry is h ∂ x i d, ∂ x j d i for 1 ≤ i, j ≤ u ⊗ u = ( u i u j ) ≤ i,j ≤ .Throughout this paper, we denote S = (cid:8) y = ( y , y , y ) ∈ S : y ≥ (cid:9) as the upper hemisphere, χ E denote the characteristic function of a set E ⊂ R , H (Ω , S ) = n d ∈ H (Ω , R ) : d ( x ) ∈ S a . e . x ∈ Ω o , and A : B = X i,j =1 A ij B ij denotes the scalar product of two 3 × < T ≤ + ∞ , denote Q T = Ω × (0 , T ) , ∂ p Q T = (Ω × { } ) ∪ ( ∂ Ω × (0 , T )) , D ′ ( Q T ) = ( C ∞ ( Q T )) ′ . We say ( ρ, u, d ) : Ω × [0 , T ) → R + × R × S is a finite energy weak solution of the initial-boundaryvalue problem (1.1)-(1.2) if the following properties hold:(i) ρ ≥ , ρ ∈ L ∞ ((0 , T ) , L γ (Ω)) , u ∈ L ((0 , T ) , H (Ω , R )), and d ∈ L ((0 , T ) , H (Ω , S )).(ii) the system (1.1) holds in D ′ ( Q T ), (1 . also holds in D ′ ( R × (0 , T )) provided ( ρ, u ) isprolonged by zero in R \ Ω, ( ρ, ρu, d )( x,
0) = ( ρ ( x ) , m ( x ) , d ( x )) for a.e. x ∈ Ω, and( u, d )( x, t ) = (0 , d ( x )) on ∂ Ω × (0 , T ) in the sense of traces.(iii) ( ρ, u ) satisfies (1 . in the sense of the renormalized solutions introduced by DiPerna-Lions[26], that is, ( ρ, u ) satisfies ∂ t (cid:0) b ( ρ ) (cid:1) + ∇ · ( b ( ρ ) u ) + (cid:0) b ′ ( ρ ) ρ − b ( ρ ) (cid:1) ∇ · u = 0 , (1.4)in the sense of distributions in R × (0 , + ∞ ) for any b ∈ C ((0 , + ∞ )) ∩ C ([0 , + ∞ )) suchthat b ′ ( z ) = 0 for all z ∈ (0 , + ∞ ) large enough, say z ≥ M, (1.5)where the constant M > b ’s. Here ( ρ, u ) is prolonged byzero outside Ω.(iv) ( ρ, u, d ) satisfies the following energy inequality E ( t ) + ˆ t ˆ Ω (cid:0) µ |∇ u | + e µ |∇ · u | + |△ d + |∇ d | d | (cid:1) ≤ E (0) , (1.6) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 3 for almost all 0 < t < T . Here E ( t ) := ˆ Ω (cid:16) ρ | u | + aρ γ γ − |∇ d | (cid:17) ( t ) (1.7)is the total energy of ( ρ, u, d ) at time t >
0, and E (0) = ˆ Ω (cid:16) | m | ρ χ { ρ ≥ } + aρ γ γ − |∇ d | (cid:17) (1.8)is the initial energy.There have been some earlier results on (1.1). In dimension one, the existence of global strongsolutions and weak solutions to (1.1) has been obtained by [28] and [29] respectively. In dimensiontwo, the existence of global weak solution of (1.1), under the condition that the image of d iscontained in S , was obtained by [33]. In dimension three, the local existence of strong solutions of(1.1) has been studied by [30] and [31]. The compressible limit of compressible nematic liquid crystalflow (1.1) has been studied by [32]. We also mention a related work [34]. When considering thecompressible nematic liquid crystal flow (1.1) under the assumption that the director d has variabledegree of orientations, the global existence of weak solutions in dimension three has been obtainedby [27] and [35] respectively.In this paper, we are mainly interested in the existence of finite energy weak solutions of (1.1)-(1.2)in dimension three. Our main states as follows. Theorem 1.1.
Assume γ > and the condition (1.3) holds. If the initial data ( ρ , m , d ) satisfiesthe following condition: ≤ ρ ∈ L γ (Ω) , (1.9) m ∈ L γγ +1 (Ω) , m χ { ρ =0 } = 0 , | m | ρ χ { ρ > } ∈ L (Ω) , (1.10) and d ∈ H (Ω , S ) , with d ( x ) ∈ S a.e. x ∈ Ω . (1.11) Then there exists a global finite energy weak solution ( ρ, u, d ) : Ω × [0 , + ∞ ) → R + × R × S to theinitial and boundary value problem (1.1)-(1.2) such that (i) d = ( d , d , d ) ∈ L ∞ ((0 , + ∞ ) , H (Ω , S )) and d ( x, t ) ≥ a.e. ( x, t ) ∈ Ω × (0 , + ∞ ) . (ii) it holds ˆ ∞ η ( t ) ˆ Ω (cid:16) ∇ d ⊙ ∇ d − |∇ d | I (cid:17) : ∇ X + ˆ ∞ η ( t ) ˆ Ω (cid:10) ∂ t d + u · ∇ d, X · ∇ d (cid:11) = 0 , (1.12) for any X ∈ C (Ω , R ) and η ∈ C ((0 , + ∞ )) . The main ideas of proof of Theorem 1.1 rely on (i) the precompactness results, due to Lin-Wang[18], on approximated Ginzburg-Landau equations { d ǫ } with bounded energies, bounded L -tensionfields, and the condition | d ǫ | ≤ d ǫ ≥ − δ for δ >
0, and (ii) suitable adaption of compactnessproperties of renormalized solutions of compressible Navier-Stokes equations established by Lions[26] and Feireisl and his collaborators [4], [5], and [6].For any global finite energy weak solutions of (1.1) and (1.2) that satisfies the properties stated inTheorem 1.1, we are able to establish the following preliminary result on its large time asymptoticbehavior.
Corollary 1.2.
Under the same assumptions of Theorem 1.1, let ( ρ, u, d ) : Ω × [0 , + ∞ ) → R + × R × S be any global finite energy weak solution of (1.1) and (1.2) that satisfies the properties ofTheorem 1.1. Then there exist t n → ∞ and a harmonic map d ∞ ∈ H ∩ C ∞ (Ω , S ) , with d ∞ = d on ∂ Ω , such that (cid:16) ρ ( · , t n ) , u ( · , t n ) , d ( · , t n ) (cid:17) → (cid:16) ρ , ∞ , , d ∞ (cid:17) in L γ (Ω) × L p (Ω) × H (Ω) , (1.13) J. LIN, B. LAI, AND C. WANG for any < p < , where ρ , ∞ := 1 | Ω | ˆ Ω ρ > is the average of the initial mass. Remark 1.3.
It is a very interesting question to ask whether the convergence in (1.13) holds for t → + ∞ . We plan to address it in a future work. We would like to point out that such a propertyhas been established by [5] for the compressible Navier-Stokes equation. For the compressible flowof nematic liquid crystals with variable degree of orientations, see Wang-Yu [35] for the large timeasymptotic behavior of global weak solutions.The paper is written as follows. In section 2, we provide some preliminary estimates of (1.1).In section 3, we briefly review a compactness theorem due to Lin and Wang [18]. In section 4, wereview the main results by Wang-Yu [35] on nematic liquid crystal flows with variable lengths ofdirectors. In section 5, we prove Theorem 1.1. In section 6, we prove Corollary 1.2.2. Global energy inequality and estimates based on the maximum principle
In this section, we will provide several basic properties of the hydrodynamic flow of compressiblenematic liquid crystals (1.1) and (1.2). First, we will derive an energy equality for sufficiently smoothsolutions of (1.1) and (1.2).
Lemma 2.1.
Assume the conditions (1.3), (1.9), (1.10), and (1.11) hold. For < T ≤ + ∞ , if ( ρ, u, d ) ∈ C ( Q T , R + ) × C ( Q T , R ) × C ( Q T , S ) is a solution of (1.1) and (1.2), then the followingenergy equality E ( t ) + ˆ t ˆ Ω (cid:0) µ |∇ u | + e µ |∇ · u | + |△ d + |∇ d | d | (cid:1) = E (0) , (2.1) holds for any ≤ t < T , where E ( t ) and E (0) are given by (1.7) and (1.8) respectively.Proof. Multiplying (1 . by u , integrating the resulting equation over Ω, applying integration byparts, and using (1 . , we obtain ddt ˆ Ω (cid:18) ρ | u | + aρ γ γ − (cid:19) + ˆ Ω (cid:0) µ |∇ u | + e µ |∇ · u | (cid:1) = − ˆ Ω ∇ · ( ∇ d ⊙ ∇ d − |∇ d | I ) u, (2.2)where we have used the fact ˆ Ω ρ γ ∇ · u = ˆ Ω ρ γ − ρ ∇ · u = − ˆ Ω ( ∂ t ρ + u · ∇ ρ ) ρ γ − = − ddt ˆ Ω ρ γ γ + ˆ Ω ρ γ γ ∇ · u, so that − a ˆ Ω ρ γ ∇ · u = ddt ˆ Ω aρ γ γ − . Direct calculations show ∇ · ( ∇ d ⊙ ∇ d − |∇ d | I ) = h△ d, ∇ d i . Note also, since | d | = 1, that we have h ∂ t d, d i = h∇ d, d i = 0 , and hence − ˆ Ω ∇ · ( ∇ d · ∇ d − |∇ d | I ) u = − ˆ Ω u · h ∆ d + |∇ d | d, ∇ d i . (2.3)Multiplying (1 . by − ( △ d + |∇ d | d ) and integrating over Ω yields that ddt ˆ Ω (cid:12)(cid:12) ∇ d (cid:12)(cid:12) + ˆ Ω (cid:12)(cid:12) △ d + |∇ d | d (cid:12)(cid:12) = ˆ Ω u · h△ d + |∇ d | d, ∇ d i . (2.4) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 5
Putting (2.2), (2.3), and (2.4) together implies ddt E ( t ) + ˆ Ω (cid:0) µ |∇ u | + e µ |∇ · u | + |△ d + |∇ d | d | (cid:1) = 0 . (2.5)This, after integrating over t , implies (2.1). (cid:3) In order to construct global finite energy weak solutions to (1.1)-(1.2), we need some importantestimates of transported Ginzburg-Landau equations based on the maximum principle.
Lemma 2.2.
For ǫ > , T > , and u ǫ ∈ L ([0 , T ] , L ∞ (Ω , R )) , assume d ǫ ∈ L ([0 , T ] , H (Ω , R )) ,with (1 − | d ǫ | ) ∈ L ( Q T ) , solves the transported Ginzburg-Landau equation: ( ∂ t d ǫ + u ǫ · ∇ d ǫ = △ d ǫ + ǫ (1 − | d ǫ | ) d ǫ , in Q T ,d ǫ = g ǫ , on ∂ p Q T . (2.6) If g ǫ ∈ H (Ω , R ) satisfies | g ǫ ( x ) | ≤ for a.e. x ∈ Ω , then | d ǫ ( x, t ) | ≤ . e . ( x, t ) ∈ Q T . Proof.
We will follow the proof of Lemma 2.1 of Lin-Wang [18] with some modifications. For any k > , define f kǫ : Q T → R + by f kǫ = k − , if | d ǫ ( x, t ) | > k, | d ǫ ( x, t ) | − , if 1 < | d ǫ ( x, t ) | ≤ k, , if | d ǫ ( x, t ) | ≤ . By direct calculations, we have that f kǫ satisfies, in the sense of distributions, ( ∂ t f kǫ + u ǫ · ∇ f kǫ = △ f kǫ − χ { < | d ǫ |≤ k } (cid:16) |∇ d ǫ | + ǫ ( | d ǫ | − | d ǫ | (cid:17) ≤ △ f kǫ in Q T ,f kǫ = 0 on ∂ p Q T . (2.7)Multiplying (2.7) by f kǫ and integrating over Ω, we obtain ddt ˆ Ω | f kǫ | + 2 ˆ Ω |∇ f kǫ | ≤ ˆ Ω u ǫ · ∇ f kǫ f kǫ ≤ ˆ Ω |∇ f kǫ | + k u ǫ ( t ) k L ∞ (Ω) ˆ Ω | f kǫ | . Hence we have ddt ˆ Ω | f kǫ | dx ≤ k u ǫ ( · ) k L ∞ (Ω) ˆ Ω | f kǫ | dx. (2.8)Since u ǫ ∈ L ([0 , T ] , L ∞ (Ω)) and f kǫ ( x,
0) = 0 for a.e. x ∈ Ω, applying Gronwall’s inequality to (2.8)yields that f kǫ = 0 a.e. in Q T . By the definition of f kǫ , this implies that d ǫ ≤ Q T . (cid:3) We also have the following lemma.
Lemma 2.3.
For ǫ > , T > , and u ǫ ∈ L ([0 , T ] , L ∞ (Ω , R )) , assume d ǫ ∈ L ([0 , T ] , H (Ω , R )) ,with (1 − | d ǫ | ) ∈ L ( Q T ) , solves the transported Ginzburg-Landau equation (2.6). If g ǫ ∈ H (Ω , R ) satisfies | g ǫ ( x ) | ≤ and g ǫ ( x ) ≥ . e . x ∈ Ω , then | d ǫ ( x, t ) | ≤ d ǫ ( x, t ) ≥ . e . ( x, t ) ∈ Q T . J. LIN, B. LAI, AND C. WANG
Proof.
We will modify the proof of Lemma 2.2 by Lin-Wang [18]. First it follows from Lemma 2.2that 0 ≤ ǫ (1 − | d ǫ | ) ≤ ǫ . Set e d ǫ := e − tǫ d ǫ . Then we have ∂ t e d ǫ + u ǫ · ∇ e d ǫ − △ e d ǫ = h ǫ e d ǫ , where h ǫ ( x, t ) = (cid:16) ǫ (1 − | d ǫ | ) − ǫ (cid:17) ≤ . e . ( x, t ) ∈ Q T . Since e d ǫ ≥ ∂ p Q T , we have that ( e d ǫ ) − := − min (cid:8) e d ǫ , (cid:9) satisfies ( ∂ t ( e d ǫ ) − + u ǫ · ∇ ( e d ǫ ) − − △ ( e d ǫ ) − = h ǫ ( e d ǫ ) − , in Q T , ( e d ǫ ) − = 0 , on ∂ p Q T . (2.9)Multiplying (2 . by ( e d ǫ ) − and integrating the resulting equation over Ω, we have ddt ˆ Ω | ( e d ǫ ) − | + 2 ˆ Ω |∇ ( e d ǫ ) − | = − ˆ Ω u ǫ · ∇ ( e d ǫ ) − ( e d ǫ ) − + 2 ˆ Ω h ǫ | ( e d ǫ ) − | ≤ − ˆ Ω u ǫ · ∇ ( e d ǫ ) − ( e d ǫ ) − ≤ ˆ Ω |∇ ( e d ǫ ) − | + k u ǫ ( t ) k L ∞ (Ω) ˆ Ω | ( e d ǫ ) − | , where we have used the fact that h ǫ ( x, t ) ≤ x, t ) ∈ Q T . Thus we have ddt ˆ Ω | ( e d ǫ ) − | ≤ k u ǫ ( t ) k L ∞ (Ω) ˆ Ω | ( e d ǫ ) − | . Applying Gronwall’s inequality and using the initial condition ( e d ǫ ) − ( x,
0) = 0 a.e. x ∈ Ω, we obtainthat ( e d ǫ ) − = 0 a.e. in Q T . Therefore d ǫ ≥ . e . Q T . This completes the proof of Lemma 2.3. (cid:3) Review of Lin-Wang’s compactness results
In order to show that a family of global finite weak solutions ( ρ ǫ , u ǫ , d ǫ ) to the Ginzburg-Landauapproximation of compressible nematic liquid crystal flow converges to a global finite weak solution( ρ, u, d ) of the compressible nematic liquid crystal flow (1.1) and (1.2), we need to establish the com-pactness of d ǫ in L ([0 , T ] , H (Ω , R )). Under suitable conditions, this has recently been achievedby Lin-Wang [18] in their studies of the existence of global weak solutions to the incompressiblenematic liquid crystal flow.Since such a compactness property also plays a crucial role in this paper, we will state it and referthe interested readers to the paper [18] for more detail. For a ∈ (0 , , denote S − a = n y = ( y , y , y ) ∈ S (cid:12)(cid:12) y ≥ − a o . For any a ∈ (0 , , L > L > , let X ( L , L , a ; Ω) denote the set consisting of all maps d ǫ ∈ H (Ω , R ), with ǫ ∈ (0 , , that are solutions of △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = τ ǫ in Ω , with τ ǫ ∈ L (Ω , R ) , (3.1)such that for all 0 < ǫ ≤
1, the following properties hold:(i) | d ǫ | ≤ d ǫ ≥ − a for a.e. x ∈ Ω.(ii) E ǫ ( d ǫ ) := ˆ Ω (cid:0) |∇ d ǫ | + 34 ǫ (1 − | d ǫ | ) (cid:1) ≤ L . OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 7 (iii) (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) ≤ L . We have
Theorem 3.1. ([18])
For any a ∈ (0 , , L > , and L > , the set X ( L , L , a ; Ω) is pre-compact in H (Ω , R ) . In particular, if for ǫ → , { d ǫ } ⊂ H (Ω , R ) is a sequence of maps in X ( L , L , a ; Ω) , then there exists a map d ∈ H (Ω , S − a ) ∩ Y ( L , L , a ; Ω) such that after passingto possible subsequences, d ǫ → d in H (Ω , R ) and e ǫ ( d ǫ ) dx := (cid:0) |∇ d ǫ | + (1 − | d ǫ | ) ǫ (cid:1) dx ⇀ |∇ d | dx as convergence of Radon measures. The idea of proof of Theorem 3.1 is based on: (1) almost energy monotonicity inequality of d ǫ ∈ X ( L , L , a ; Ω); (2) an δ -regularity and compactness property of d ǫ ∈ X ( L , L , a ; Ω); (3) theblowing-up analysis of d ǫ ∈ X ( L , L , a ; Ω) as ǫ → ν , motivated by that of harmonic maps by Lin [16] and approximated harmonicmaps [20, 21, 22]; and (4) the ruling out of possible harmonic S ’s generated at Σ.In order to study the large time behavior of global finite energy weak solutions to the compress-ible nematic liquid crystal flow (1.1) and (1.2), we also need the following compactness result onapproximated harmonic maps to S − a for 0 < a ≤ < a ≤ L >
0, and L >
0, let Y ( L , L , a ; Ω) be the set consisting of maps d ∈ H (Ω , S )that are approximated harmonic maps, i.e.,∆ d + |∇ d | d = τ in Ω , with τ ∈ L (Ω , R ) , (3.2)that satisfy the following properties:(i) d ( x ) ≥ − a for a.e. x ∈ Ω.(ii) F ( d ) := 12 ˆ Ω |∇ d | ≤ L . (iii) (cid:13)(cid:13) τ (cid:13)(cid:13) L (Ω) ≤ L .(iv) (almost energy monotonicity inequality) for any x ∈ Ω and 0 < r ≤ R < d( x , ∂ Ω),Ψ R ( d, x ) ≥ Ψ r ( d, x ) + 12 ˆ B R ( x ) \ B r ( x ) | x − x | − (cid:12)(cid:12) ∂d∂ | x − x | (cid:12)(cid:12) , (3.3)whereΨ r ( d, x ) := 1 r ˆ B r ( x ) (cid:0) |∇ d | − h ( x − x ) · ∇ d, τ i (cid:1) + 12 ˆ B r ( x ) | x − x || τ | . Theorem 3.2. ([18])
For any a ∈ (0 , , L > , and L > , the set Y ( L , L , a ; Ω) is precompactin H (Ω , S ) . In particular, if { d i } ⊂ H (Ω , R ) is a sequence of approximated harmonic mapsin Y ( L , L , a ; Ω) with tension fields { τ i } , then there exist τ ∈ L (Ω , R ) and an approximatedharmonic map d ∈ Y ( L , L , a ; Ω) with tension field τ such that after passing to possible subse-quences, d i → d in H (Ω , S ) and τ i ⇀ τ in L (Ω , R ) . In fact, { d i } is bounded in H (Ω , S ) .In particular, d ∈ H (Ω , S ) . Ginzburg-Landau approximation of compressible nematic liquid crystal flow
In this section, we will consider the Ginzburg-Landau approximation of compressible nematicliquid crystal flow and state the existence of global weak solutions, which is an improved version ofan earlier result obtained by Wang-Yu [35] (see also [27]).
J. LIN, B. LAI, AND C. WANG
For ǫ > < T ≤ + ∞ , the Ginzburg-Landau approximation equation of (1.1) and (1.2)seeks ( ρ ǫ , u ǫ , d ǫ ) : Q T → R + × R × R that satisfies: ∂ t ρ ǫ + ∇ · ( ρu ǫ ) = 0 ,∂ t ( ρ ǫ u ǫ ) + ∇ · ( ρ ǫ u ǫ ⊗ u ǫ ) + a ∇ ρ γǫ = L u ǫ − ∇ · (cid:0) ∇ d ǫ ⊙ ∇ d ǫ − ( |∇ d ǫ | + ǫ (1 − | d ǫ | ) ) I (cid:1) ,∂ t d ǫ + u ǫ · ∇ d ǫ = △ d ǫ + ǫ (1 − | d ǫ | ) d ǫ , (4.1)along with the initial and boundary condition (1.2). We would like to point out that the notion offinite energy weak solutions of (4.1) and (1.2) can be defined in the same way as that of (1.1) and(1.2) given in § Theorem 4.1.
Assume γ > and the condition (1.3), and ( ρ , m , d ) satisfies (1.9), (1.10),(1.11). Then there exists a global finite energy weak solution ( ρ ǫ , u ǫ , d ǫ ) : Ω × [0 , + ∞ ) → R + × R × R to the system (4.1), under the initial and boundary condition (1.2), such that (i) d ǫ = ( d ǫ , d ǫ , d ǫ ) ∈ L ∞ ((0 , ∞ ) , H (Ω , R )) , with | d ǫ | ≤ and d ǫ ≥ for a.e. ( x, t ) ∈ Ω × (0 , ∞ ) . (ii) ( ρ ǫ , u ǫ , d ǫ ) satisfies the global energy inequality ddt F ǫ ( t ) + ˆ Ω (cid:0) µ |∇ u ǫ | + e µ |∇ · u ǫ | + |△ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ | (cid:1) ( t ) ≤ in D ′ ((0 , + ∞ )) , where F ǫ ( t ) := ˆ Ω (cid:16) ρ ǫ | u ǫ | + aρ γǫ γ − (cid:0) |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) (cid:1)(cid:17) ( t ) . Proof.
The existence of finite energy weak solutions has been established by Wang-Yu [35], whichuses a three level approximation scheme similar to that of compressible Navier-Stokes equation by[4] and [6]. It consists of Faedo-Galerkin approximation, artificial viscosity, and artificial pressure.The reader can consult the proof of [35] Theorem 2.1 for the detail.Here we only indicate the proof of (i). Let ǫ > α > δ >
0, and 0 < T < + ∞ , we first approximate the initial data ( ρ , m , d )by (cid:0) ρ ,δ , m ,δ , d ,δ (cid:1) ∈ C (Ω , R + × R × R ) such that the following conditions hold: δ ≤ ρ ,δ ≤ δ − in Ω , ∂ρ ,δ ∂ν (cid:12)(cid:12) ∂ Ω = 0 , and ρ ,δ → ρ in L γ (Ω) ,m ,δ → m in L γγ +1 (Ω) , | m ,δ | ρ ,δ → | m | ρ χ { ρ > } in L (Ω) , | d ,δ ( x ) | ≤ , d ,δ ( x ) ≥ . e . x ∈ Ω , d ,δ → d in H (Ω , R ) , (4.3)as δ → u ∈ C ([0 , T ] , C (Ω , R )), with u (cid:12)(cid:12) t =0 = u ,δ ≡ m ,δ ρ ,δ , let d δ = d δ ([ u ]) ∈ C ([0 , T ] , C (Ω , R ))be the unique solution of (see [35] Lemma 3.1 and Lemma 3.2): ( ∂ t d + u · ∇ d = ∆ d + ǫ (1 − | d | ) d in Q T ,d = d ,δ on ∂ p Q T . (4.4)Since | d ,δ ( x ) | ≤ d ,δ ( x ) ≥ x ∈ Ω, it follows from Lemma 2.2 and Lemma 2.3 that d δ satisfies | d δ ( x, t ) | ≤ d δ ( x, t ) ≥ , ∀ ( x, t ) ∈ Q T . (4.5)Now let ρ α,δ = ρ α,δ ([ u ]) ∈ C ([0 , T ] , C (Ω)) be the unique solution of the problem: ∂ t ρ + ∇ · ( ρu ) = α ∆ ρ in Q T ,ρ ( x,
0) = ρ ,δ ( x ) in Ω , ∂ρ∂ν = 0 on ∂ Ω × (0 , T ) . (4.6) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 9
While for u , it involves to employ first the Galerkin method and then the fixed point theorem tosolve u = u α,δ ([ u ]) to the problem: for some β > max { , γ } , ∂ t ( ρ α,δ u ) + ∇ · ( ρ α,δ u ⊗ u ) + a ∇ (cid:0) ρ γα,δ (cid:1) + δ ∇ (cid:0) ρ βα,δ (cid:1) + α ∇ u · ∇ ρ α,δ = L u − ∇ · h ∇ d δ ⊙ ∇ d δ − (cid:0) |∇ d δ | + ǫ (1 − | d δ | ) (cid:1) I i , in Q T ,u = u ,δ on ∂ p Q T . (4.7)Since the global weak solution ( ρ ǫ , u ǫ , d ǫ ) to the system (4.1), under the initial and boundary condi-tion (1.2), constructed in [35], was obtained as a strong limit of ( ρ α,δ , u α,δ , d δ ) in L γ ( Q T ) × L ( Q T ) × L ([0 , T ] , H (Ω , R )) for any 0 < T < + ∞ , as viscosity coefficients α → δ →
0. It is readily seen that d ǫ satisfies the property (ii). (cid:3) Existence of global weak solutions
In this section, we will prove Theorem 1.1 by studying in depth the convergence of sequences ofsolutions ( ρ ǫ , u ǫ , d ǫ ), constructed by Theorem 4.1, as ǫ → + . Proof of Theorem 1.1 .To prove the existence of global finite energy weak solutions to (1.1), let ( ρ ǫ , u ǫ , d ǫ ) : Ω × [0 , + ∞ ) → R + × R × R , 0 < ǫ ≤
1, be a family of finite energy weak solutions to the system (4.1), under theinitial and boundary condition (1.2), constructed by Theorem 4.1. Since | d | = 1 and d ≥ ρ ǫ , u ǫ , d ǫ ) satisfies all these properties in Theorem 4.1. In particular, it follows from (4.2) thatsup ǫ> h sup
0, for any 0 < T < + ∞ .We will prove that ( ρ, u, d ) is a global finite energy weak solution to (1.1) and (1.2). The proofwill be divided into several subsections.5.1. d ǫ → d strongly in L ([0 , T ] , H (Ω)) . This will be achieved by applying Theorem 3.1, similarto that of [18]. First it follows from the equation (4 . and the inequality (5.1) that ∂ t d ǫ ∈ L ([0 , T ] , L (Ω)) + L ([0 , T ] , L (Ω)) so that ∂ t d ǫ ∈ L ([0 , T ] , H − (Ω)) andsup <ǫ ≤ (cid:13)(cid:13) ∂ t d ǫ (cid:13)(cid:13) L (0 ,T ; H − (Ω)) < + ∞ . (5.3)By Aubin-Lions’ lemma, we conclude that d ǫ → d in L ( Q T ) and ∇ d ǫ ⇀ ∇ d in L ([0 , T ] , L (Ω)) . (5.4)By Fatou’s lemma, (5.1) implies that ˆ T lim inf ǫ → ˆ Ω (cid:12)(cid:12) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12) ≤ E (0) . (5.5) For sufficiently large Λ > , define the set of good time slice, G T Λ , by G T Λ := n t ∈ [0 , T ] (cid:12)(cid:12)(cid:12) lim inf ǫ → ˆ Ω (cid:12)(cid:12) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12) ( t ) ≤ Λ o , and the set of bad time slices, B T Λ , by B T Λ := [0 , T ] \ G T Λ = n t ∈ [0 , T ] (cid:12)(cid:12)(cid:12) lim inf ǫ → ˆ Ω (cid:12)(cid:12) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12) ( t ) > Λ o . It is easy to see from (5.5) that (cid:12)(cid:12)(cid:12) B T Λ (cid:12)(cid:12)(cid:12) ≤ E (0)Λ . (5.6)By (5.1) and (5.6), we obtain ˆ B T Λ ˆ Ω h |∇ d ǫ − ∇ d | + 1 ǫ (1 − | d ǫ | ) ( t ) i ≤ C (cid:12)(cid:12) B T Λ (cid:12)(cid:12) sup 1; Ω), Theorem 3.1 implies that there exists d ( t ) ∈ Y ( E (0) , Λ , 1; Ω) suchthat after passing to a subsequence, d ǫ ( t ) → d ( t ) strongly in H (Ω) and ǫ (1 − | d ǫ ( t ) | ) → L (Ω).Now we want to show that, after passing to a subsequence, ∇ d ǫ → ∇ d in L (Ω × G T Λ ) . (5.8)This can be done similarly to Claim 8.2 of [18]. Here we provide it. Suppose (5.8) were false. Thenthere exist a subdomain e Ω ⊂⊂ Ω, δ > 0, and ǫ i → ˆ e Ω × G T Λ |∇ ( d ǫ i − d ) | ≥ δ . (5.9)Note that from (5.4) we have lim ǫ i → ˆ Ω × G T Λ | d ǫ i − d | = 0 . (5.10)By Fubini’s theorem, (5.9), and (5.10), we have that there exists t i ∈ G T Λ such thatlim ǫ i → ˆ Ω | d ǫ i ( t i ) − d ( t i ) | = 0 , (5.11)and ˆ e Ω (cid:12)(cid:12) ∇ ( d ǫ i ( t i ) − d ( t i )) (cid:12)(cid:12) ≥ δ T . (5.12)It is easy to see that (cid:8) d ǫ i ( t i ) (cid:9) ⊂ X ( E (0) , Λ , 1; Ω) and (cid:8) d ( t i ) (cid:9) ⊂ Y ( E (0) , Λ , 1; Ω). It follows fromTheorem 3.1 and Theorem 3.2 that there exist d , d ∈ Y ( E (0) , Λ , 1; Ω) such that d ǫ i ( t i ) → d and d ( t i ) → d in L (Ω) ∩ H ( e Ω) . This and (5.12) imply that ˆ e Ω (cid:12)(cid:12) ∇ ( d − d ) (cid:12)(cid:12) ≥ δ T . (5.13)On the other hand, from (5.11), we have that ˆ Ω | d − d | = 0 . (5.14) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 11 It is clear that (5.13) contradicts (5.14). Hence (5.8) is proven. Similar to Lemma 4.1, Claim 4.4 in[18], we also have ˆ e Ω × G T Λ ǫ (1 − | d ǫ | ) → ǫ → . (5.15)Combining (5.7), (5.8), with (5.15), we obtainlim ǫ → h k d ǫ − d k L ([0 ,T ] ,H ( e Ω)) + ˆ e Ω × [0 ,T ] (1 − | d ǫ | ) ǫ i ≤ C Λ − . (5.16)Since Λ > d ǫ → d in L ([0 , T ] , H (Ω)) and (1 − | d ǫ | ) ǫ → L ([0 , T ] , L (Ω)) . (5.17)5.2. ρ ǫ u ǫ → ρu in the sense of distributions. By (5.1), √ ρ ǫ is bounded in L ∞ ([0 , T ] , L γ (Ω))and √ ρ ǫ u ǫ is bounded in L ∞ ([0 , T ] , L (Ω)). Thus ρ ǫ u ǫ is bounded in L ∞ ([0 , T ] , L γγ +1 (Ω)) and ∂ t ρ ǫ = −∇ · ( ρ ǫ u ǫ ) is bounded in L ∞ ([0 , T ] , W − , γγ +1 (Ω)) . Applying [26] Lemma C.1, we have ρ ǫ → ρ in C ([0 , T ] , L γ weak (Ω)) . (5.18)Since L γ (Ω) ⊂ H − (Ω) is compact, we conclude that ρ ǫ → ρ in C ([0 , T ] , H − (Ω)) . (5.19)Thus we show that ρ ǫ u ǫ → ρu in D ′ ( Q T ) . (5.20)5.3. Higher integrability estimates of ρ ǫ . There exist θ > C > γ and T such that for any 0 < ǫ ≤ 1, it holds ˆ T ˆ Ω ρ γ + θǫ ≤ C. (5.21)By Theorem 4.1, ( ρ ǫ , u ǫ ) is a renormalized solution of (4.1) . Let ( ρ ǫ , u ǫ ) : R × (0 , T ) → R + × R be the extension of ( ρ ǫ , u ǫ ) from Ω such that ( ρ ǫ , u ǫ ) = (0 , 0) in R \ Ω. Then ( ρ ǫ , u ǫ ) satisfies, inthe sense of distributions, that ∂ t ( b ( ρ ǫ )) + ∇ · ( b ( ρ ǫ ) u ǫ ) + (cid:0) b ′ ( ρ ǫ ) ρ ǫ − b ( ρ ǫ ) (cid:1) ∇ · u ǫ = 0 in R × (0 , T ) , (5.22)for any bounded function b ∈ C ((0 , + ∞ )) ∩ C ([0 , + ∞ )) (see, e.g., [6]).As in [4], [6] and [35], we can employ suitable approximations so that (5.22) also holds for b ( ρ ǫ ) = ρ θǫ for 0 < θ < 1. Note that ρ θǫ ∈ L γθ ( Q T ) . For m ≥ 1, let S m ( f ) = η m ∗ f denote thestandard mollification of f ∈ L ( R ). Then we have ∂ t (cid:0) S m ( ρ θǫ ) (cid:1) + ∇ · (cid:0) S m ( ρ θǫ ) u ǫ (cid:1) − (1 − θ ) S m (cid:0) ρ θǫ ∇ · u ǫ (cid:1) = q m in R × (0 , T ) , (5.23)where q m = ∇ · ( S m ( ρ θǫ ) u ǫ ) − S m ( ∇ · ( ρ θǫ u ǫ )) . By virtue of [26] Lemma 2.3, ρ θǫ ∈ L ∞ ([0 , T ] , L γθ (Ω)), and u ǫ ∈ L ([0 , T ] , H (Ω)), we have thatlim m →∞ (cid:13)(cid:13) q m (cid:13)(cid:13) L ([0 ,T ] ,L λ ( R )) = 0 , with 1 λ = θγ + 12 , (5.24)provided θ < γ . As in [4] and [26], define the (inverse of divergence) operator B : n f ∈ L p (Ω) (cid:12)(cid:12) ˆ Ω f = 0 o W ,p (Ω , R ) such that for any 1 < p < + ∞ , ( ∇ · B ( f ) = f in Ω , B ( f ) = 0 on ∂ Ω , (cid:13)(cid:13) B ( f ) (cid:13)(cid:13) W ,p (Ω) ≤ C ( p ) (cid:13)(cid:13) f (cid:13)(cid:13) L p (Ω) . (5.25)Set ˛ Ω f = 1 | Ω | ˆ Ω f. For ϕ ∈ C ∞ ((0 , T )), with 0 ≤ ϕ ≤ 1, let φ ( x, t ) = ϕ ( t ) B h S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) i ( x, t ) . By (5.1), we see that for sufficiently small θ , h S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) i ∈ C ([0 , T ] , L p (Ω)) , ∀ p ∈ (1 , + ∞ ) . By (5.25) and the Sobolev embedding theorem, we have that φ ∈ C ( Q T ). Thus we can test theequation (4 . by φ and obtain a ˆ T ˆ Ω ϕ ( t ) ρ γǫ S m ( ρ θǫ )= a ˆ T ϕ ( t ) (cid:0) ˆ Ω ρ γǫ (cid:1)(cid:0) ˛ Ω S m ( ρ θǫ ) (cid:1) − ˆ T ˆ Ω ϕ ′ ( t ) ρ ǫ u ǫ B (cid:2) S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) (cid:3) + ˆ T ˆ Ω ϕ ( t )( µ ∇ u ǫ − ρ ǫ u ǫ ⊗ u ǫ ) ∇B (cid:2) S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) (cid:3) + ˆ T ˆ Ω ϕ ( t ) e µ ∇ · u ǫ ∇ · B (cid:2) S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) (cid:3) +(1 − θ ) ˆ T ˆ Ω ϕ ( t ) ρ ǫ u ǫ B (cid:2) S m ( ρ θǫ ∇ · u ǫ ) − ˛ Ω S m ( ρ θǫ ∇ · u ǫ ) (cid:3) + ˆ T ˆ Ω ϕ ( t ) (cid:0) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:1) · ∇ d ǫ B (cid:2) S m ( ρ θǫ ) − ˛ Ω S m ( ρ θǫ ) (cid:3) − ˆ T ˆ Ω ϕ ( t ) ρ ǫ u ǫ B (cid:2) ∇ · ( S m ( ρ θǫ ) u ǫ ) (cid:3) + ˆ T ˆ Ω ϕ ( t ) ρ ǫ u ǫ B (cid:2) q m − ˛ Ω q m (cid:3) = X i =1 L mi + ˆ T ˆ Ω ϕ ( t ) ρ ǫ u ǫ B (cid:2) q m − ˛ Ω q m (cid:3) . Since ρ ǫ u ǫ is bounded in L ∞ ([0 , T ] , L γγ +1 (Ω)) and q m satisfies (5.24), it follows from (5.25), theSobolev embedding theorem, and the H¨older inequality thatlim m →∞ ˆ T ˆ Ω ϕ ( t ) ρ ǫ u ǫ B (cid:2) q m − ˛ Ω q m (cid:3) = 0 . Hence, after taking m → ∞ , we have ˆ (0 ,T ) × Ω ϕρ γ + θǫ ≤ lim sup m →∞ X i =1 L mi . (5.26)Now we estimate L m , · · · , L m as follows. OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 13 (1) For L m , we have that (cid:12)(cid:12) lim m →∞ L m (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) a ˆ T ˆ Ω ϕρ γǫ ( ˛ Ω ρ θǫ ) (cid:12)(cid:12)(cid:12) ≤ C (cid:16) ˆ T ˆ Ω ρ γǫ (cid:17)(cid:13)(cid:13)(cid:13) ρ ǫ (cid:13)(cid:13)(cid:13) θL ∞ ([0 ,T ] ,L γ (Ω)) is uniformly bounded.(2) For L m , we have that | L m | = (cid:12)(cid:12)(cid:12) ˆ T ˆ Ω ϕ ′ ( t ) ρ ǫ u ǫ B (cid:2) q m − ˛ Ω q m (cid:3)(cid:12)(cid:12)(cid:12) ≤ C ˆ T (cid:13)(cid:13) ρ ǫ u ǫ (cid:13)(cid:13) L γγ +1 (Ω) (cid:13)(cid:13) B (cid:2) q m − ˛ Ω q m (cid:3)(cid:13)(cid:13) L γγ − (Ω) dt ≤ C ˆ T (cid:13)(cid:13) B (cid:2) q m − ˛ Ω q m (cid:3)(cid:13)(cid:13) W ,λ (Ω) dt ≤ C ˆ T (cid:13)(cid:13) q m − ˛ Ω q m (cid:13)(cid:13) L λ (Ω) dt ≤ C (cid:13)(cid:13)(cid:13) q m (cid:13)(cid:13)(cid:13) L ([0 ,T ] ,L λ (Ω)) → m → + ∞ , provided θ < γ − .(3) For L m , we have (cid:12)(cid:12) L m (cid:12)(cid:12) ≤ C ˆ T n(cid:13)(cid:13) u ǫ (cid:13)(cid:13) H (Ω) (cid:13)(cid:13) B (cid:2) ρ θǫ − ˛ ρ θǫ (cid:3)(cid:13)(cid:13) H (Ω) + (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) L γ (Ω) (cid:13)(cid:13) u ǫ (cid:13)(cid:13) L (Ω) (cid:13)(cid:13) B (cid:2) ρ θǫ − ˛ ρ θǫ (cid:3)(cid:13)(cid:13) W , γ γ − (Ω) o dt ≤ C ˆ T (cid:16)(cid:13)(cid:13) u ǫ (cid:13)(cid:13) H (Ω) (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) θL θ (Ω) + (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) L γ (Ω) (cid:13)(cid:13) u ǫ (cid:13)(cid:13) H (Ω) (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) θL γθ γ − (Ω) (cid:17) dt is uniformly bounded, provided θ < min (cid:8) γ , γ − (cid:9) . (4) For L m , we have (cid:12)(cid:12) L m (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ T ˆ Ω e µϕ ∇ · u ǫ ∇ · B (cid:2) ρ θǫ − ˛ Ω ρ θǫ (cid:3)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ T ˆ Ω e µϕ ∇ · u ǫ (cid:0) ρ θǫ − ˛ Ω ρ θǫ (cid:1)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13) u ǫ (cid:13)(cid:13)(cid:13) L ([0 ,T ] ,H (Ω)) (cid:13)(cid:13)(cid:13) ρ ǫ (cid:13)(cid:13)(cid:13) θL ∞ ([0 ,T ] ,L θ (Ω)) is uniformly bounded, provided θ ≤ γ . (5) For L m , we have (cid:12)(cid:12) L m (cid:12)(cid:12) = (1 − θ ) (cid:12)(cid:12)(cid:12) ˆ T ˆ Ω ϕρ ǫ u ǫ B (cid:2) ρ θǫ ∇ · u ǫ − ˛ Ω ρ θǫ ∇ · u ǫ (cid:3)(cid:12)(cid:12)(cid:12) ≤ C ˆ T (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) L γ (Ω) (cid:13)(cid:13) u ǫ (cid:13)(cid:13) L (Ω) (cid:13)(cid:13) B (cid:2) ρ θǫ ∇ · u ǫ − ˛ Ω ρ θǫ ∇ · u ǫ (cid:3)(cid:13)(cid:13) L γ γ − (Ω) dt ≤ C ˆ T (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) L γ (Ω) (cid:13)(cid:13) u ǫ (cid:13)(cid:13) L (Ω) (cid:13)(cid:13) ρ θǫ ∇ · u ǫ (cid:13)(cid:13) L γ γ − (Ω) dt ≤ C ˆ T (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) L γ (Ω) (cid:13)(cid:13) u ǫ (cid:13)(cid:13) H (Ω) (cid:13)(cid:13) ρ ǫ (cid:13)(cid:13) θL γθ γ − (Ω) (cid:13)(cid:13) ∇ · u ǫ (cid:13)(cid:13) L (Ω) dt is uniformly bounded, provided θ < γ − (6) For L m , we have (cid:12)(cid:12) L m (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ˆ T ˆ Ω ϕ (cid:0) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:1) · ∇ d ǫ B (cid:2) ρ θǫ − ˛ Ω ρ θǫ (cid:3)(cid:12)(cid:12)(cid:12) ≤ C (cid:13)(cid:13)(cid:13) △ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:13)(cid:13)(cid:13) L ( Q T ) · (cid:13)(cid:13)(cid:13) ∇ d ǫ (cid:13)(cid:13)(cid:13) L ∞ ([0 ,T ] ,L (Ω)) (cid:13)(cid:13)(cid:13) B (cid:2) ρ θǫ − ˛ Ω ρ θǫ (cid:3)(cid:13)(cid:13)(cid:13) L ([0 ,T ] ,L ∞ (Ω)) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B (cid:2) ρ θǫ − ˛ Ω ρ θǫ (cid:3)(cid:13)(cid:13) W , γθ (Ω) (cid:13)(cid:13)(cid:13) L ([0 ,T ]) ≤ C sup It follows from § § § § ǫ → + in the equation (4.1), ( ρ, u, d )satisfies the following system: ∂ t ρ + ∇ · ( ρu ) = 0 ,∂ t ( ρu ) + ∇ · ( ρu ⊗ u ) + a ∇ ρ γ = L u − ∇ · (cid:2) ∇ d ⊙ ∇ d − |∇ d | I (cid:3) ,∂ t d + u · ∇ d = △ d + |∇ d | d, (5.30)in the sense of distributions, where ρ γ is a weak limit of ρ γǫ in L γ + θγ ( Q T ).It is straightforward that ( ρ, u, d ) satisfies the first two equations of (5.30). To see ( u, d ) solvesthe third equation of (5.30), we employ the standard technique, due to Chen [1], as follows. Let × denote the cross product in R . Then the equation (4 . for ( u ǫ , d ǫ ) can be rewritten as( ∂ t d ǫ + u ǫ · ∇ d ǫ ) × d ǫ = ∆ d ǫ × d ǫ , in D ′ ( Q T ) . After taking ǫ → 0, we have that ( u, d ) satisfies( ∂ t d + u · ∇ d ) × d = ∆ d × d, in D ′ ( Q T ) . (5.31)Since | d | = 1, the equation (5.31) is equivalent to (5 . .In order to identify ρ γ , we need to establish the strong convergence of ρ ǫ to ρ in L γ ( Q T ). To doit, we need to have fine estimates of the effective viscous flux, which has played important rules inthe study of compressible Navier-Stokes equations (see [4] and [26]).For k ≥ 1, define T k ( z ) = kT ( zk ) : R → R , where T ( z ) ∈ C ∞ ( R ) is a concave function such that T ( z ) = ( z, if z ≤ , , if z ≥ . Fine estimates of effective viscous flux H ǫ := aρ γǫ − e µ ∇ · u ǫ . For any fixed k ≥ 1, thereholds lim ǫ → ˆ T ˆ Ω ψφ (cid:0) aρ γǫ − e µ ∇ · u ǫ (cid:1) T k ( ρ ǫ ) = ˆ T ˆ Ω ψφ (cid:0) a ( ρ γ ) − e µ ∇ · u (cid:1) T k ( ρ ) , (5.32)for any ψ ∈ C ∞ ((0 , T )) and φ ∈ C ∞ (Ω). By density arguments, similar to [4], it is not hard to seethat (5.32) remains to be true for φ = ψ = 1.Since ( ρ ǫ , u ǫ ) is a renormalized solution to (4 . in Q T , it is clear that if we extend ( ρ ǫ , u ǫ ) to R by letting it to be zero in R \ Ω, then ( ρ ǫ , u ǫ ) is also a renormalized solution of (4 . in R .Replacing b ( z ) by T k ( z ) in (4 . yields ∂ t (cid:0) T k ( ρ ǫ ) (cid:1) + ∇ · ( T k ( ρ ǫ ) u ǫ ) + (cid:0) T ′ ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:1) ∇ · u ǫ = 0 in D ′ ( R × (0 , T )) . (5.33)Since T k ( ρ ǫ ) is bounded in L ∞ ( Q T ), we have T k ( ρ ǫ ) ⇀ T k ( ρ ) weak ∗ in L ∞ ( Q T ) . This, combined with the equation (5.33), implies that for any p ∈ (1 , + ∞ ) ,T k ( ρ ǫ ) → T k ( ρ ) in C ([0 , T ] , L p weak (Ω)) and in C ([0 , T ] , H − (Ω)) . (5.34)Hence, after sending ǫ → ∂ t T k ( ρ ) + ∇ · ( T k ( ρ ) u ) + (cid:0) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:1) ∇ · u = 0 in D ′ ( Q T ) , (5.35)where (cid:0) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:1) ∇ · u is a weak limit of (cid:0) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:1) ∇ · u ǫ in L ( Q T ).Now we need to estimate the effective viscous flux ( aρ γǫ − e µ ∇ · u ǫ ). Define the operator A =( A , A , A ) by letting A i = ∂ x i △ − 16 J. LIN, B. LAI, AND C. WANG for i = 1 , , 3, where △ − denote the inverse of the Laplace operator on R (see [6]). By the L p regularity theory of the Laplace equation, we have (cid:13)(cid:13) A v (cid:13)(cid:13) W ,s (Ω) ≤ C (cid:13)(cid:13) v (cid:13)(cid:13) L s ( R ) , < s < + ∞ , (cid:13)(cid:13) A v (cid:13)(cid:13) L q (Ω) ≤ C (cid:13)(cid:13) v (cid:13)(cid:13) L s ( R ) , q ≥ s − , (cid:13)(cid:13) A v (cid:13)(cid:13) L ∞ (Ω) ≤ C (cid:13)(cid:13) v (cid:13)(cid:13) L s ( R ) , s > , (5.36)where C > s and Ω . Testing the equation (5.33) by A i [ ϕ ] for ϕ ∈ C ∞ ( Q T ) yields ∂ t (cid:0) A i [ T k ( ρ ǫ )] (cid:1) + ∇ · (cid:0) A i [( T k ( ρ ǫ ) u ǫ )] (cid:1) + A i h ( T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ )) ∇ · u ǫ i = 0 , (5.37)in D ′ ( R × (0 , T )) ∩ L ( R × (0 , T )) . This implies ∂ t (cid:0) A i [ T k ( ρ ǫ )] (cid:1) ∈ L ( R × (0 , T )). Hence we cantest the equation (4 . by ψφ A [ T k ( ρ ǫ )], for φ ∈ C ∞ (Ω) and ψ ∈ C ∞ ((0 , T )), and obtain ˆ T ˆ Ω ψφ ( aρ γǫ − e µ ∇ · u ǫ ) T k ( ρ ǫ )= ˆ T ˆ Ω ψ ( e µ ∇ · u ǫ − aρ γǫ ) ∇ φ A [ T k ( ρ ǫ )]+ µ ˆ T ˆ Ω ψ (cid:8) ∇ φ ∇ u iǫ A i [ T k ( ρ ǫ )] − u iǫ ∇ φ ∇A i [ T k ( ρ ǫ )] + u ǫ ∇ φT k ( ρ ǫ ) (cid:9) − ˆ T ˆ Ω φρ ǫ u ǫ (cid:8) ∂ t ψ A [ T k ( ρ ǫ )] + ψ A [( T k ( ρ ǫ ) − T ′ k ( ρ ǫ ) ρ ǫ ) ∇ · u ǫ ] (cid:9) − ˆ t ˆ Ω ψρ ǫ u iǫ u jǫ ∂ j φ A i [ T k ( ρ ǫ )]+ ˆ T ˆ Ω ψu iǫ (cid:8) T k ( ρ ǫ ) R ij [ φρ ǫ u iǫ ] − φρ ǫ u jǫ R ij [ T k ( ρ ǫ )] (cid:9) − ˆ T ˆ Ω ψ ∇ · n ∇ d ǫ ⊙ ∇ d ǫ − [ 12 |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) ] I o · φ A [ T k ( ρ ǫ )] , (5.38)where R ij = ∂ x j A i is the Riesz transform.Similarly, we can test the equation (5 . by ψφ A i [ T k ( ρ )] and obtain ˆ T ˆ Ω ψφ ( aρ γ − e µ ∇ · u ) T k ( ρ )= ˆ T ˆ Ω ψ (cid:0)e µ ∇ · u − aρ γ (cid:1) ∇ φ A [ T k ( ρ )]+ µ ˆ T ˆ Ω ψ (cid:8) ∇ φ ∇ u i A i [ T k ( ρ )] − u i ∇ φ ∇A i [ T k ( ρ )] + u ∇ φT k ( ρ ) (cid:9) − ˆ T ˆ Ω φρu (cid:8) ∂ t ψ A [ T k ( ρ )] + ψ A [( T k ( ρ ) − T ′ k ( ρ ) ρ ) ∇ · u ] (cid:9) − ˆ t ˆ Ω ψρu i u j ∂ j φ A i [ T k ( ρ )]+ ˆ T ˆ Ω ψu i (cid:8) T k ( ρ ) R ij [ φρu i ] − φρu j R ij [ T k ( ρ )] (cid:9) − ˆ T ˆ Ω ψ ∇ · h ∇ d ⊙ ∇ d − |∇ d | I i · φ A [ T k ( ρ )] . (5.39) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 17 To prove (5.32), it suffices to show that each term in the right hand side of (5.38) converges to thecorresponding term in the right hand side of (5.39). Since the convergence of the first five terms inthe right hand side of (5.38) can be done in the exact same way as in [6] (see also [35]), we onlyindicate how to show the convergence of the last term in the right hand side of (5.38), namely, ˆ T ˆ Ω ψ ∇ · h ∇ d ǫ ⊙ ∇ d ǫ − ( 12 |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) ) I i · φ A [ T k ( ρ ǫ )] → ˆ T ˆ Ω ψ ∇ · h ∇ d ⊙ ∇ d − |∇ d | I i · φ A [ T k ( ρ )] as ǫ → . (5.40)To see this, first observe that T k ( ρ ǫ ) is bounded in L ∞ ( Q T ) and hence we have, by (5.37), that (seealso [6]) A [ T k ( ρ ǫ )] → A [ T k ( ρ )] in C (Ω × [0 , T ]) . (5.41)Secondly, observe that a.e. in Q T , there holds ∇ · h ∇ d ǫ ⊙ ∇ d ǫ − ( 12 |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) ) I i = (cid:0) ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:1) · ∇ d ǫ = (cid:0) ∂ t d ǫ + u ǫ · ∇ d ǫ (cid:1) · ∇ d ǫ . By the energy inequality (5.1), we see that (cid:0) ∂ t d ǫ + u ǫ · ∇ d ǫ (cid:1) is bounded in L ( Q T ) and hence thereexists v ∈ L ( Q T ) such that (cid:0) ∂ t d ǫ + u ǫ · ∇ d ǫ (cid:1) ⇀ v in L ( Q T ) . (5.42)On the other hand, since d ǫ → d in L ([0 , T ] , H (Ω)) and u ǫ ⇀ u in L ([0 , T ] , H (Ω)), we havethat (cid:0) ∂ t d ǫ + u ǫ · ∇ d ǫ (cid:1) → (cid:0) ∂ t d + u · ∇ d (cid:1) in D ′ ( Q T ) . Hence we have v = ∂ t d + u · ∇ d in Q T . (5.43)By (5.41) and the local L -convergence of ∇ d ǫ to ∇ d in Q T , we know that ∇ d ǫ φ A [ T k ( ρ ǫ )] → ∇ dφ A [ T k ( ρ )] in L ( Q T ) . Hence we obtain ˆ T ˆ Ω ψ ∇ · h ∇ d ǫ ⊙ ∇ d ǫ − ( 12 |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) ) I i · φ A [ T k ( ρ ǫ )] → ˆ T ˆ Ω ψ ( ∂ t d + u · ∇ d ) ∇ d · φ A [ T k ( ρ )] as ǫ → . (5.44)Applying the equation (5 . and the fact that h|∇ d | d, ∇ d i = 0 a.e. in Q T , we obtain ˆ T ˆ Ω ψ ( ∂ t d + u · ∇ d ) ∇ d · φ A [ T k ( ρ )]= ˆ T ˆ Ω ψ (∆ d + |∇ d | d ) ∇ d · φ A [ T k ( ρ )]= ˆ T ˆ Ω ψ ∆ d ∇ d · φ A [ T k ( ρ )]= ˆ T ˆ Ω ψ ∇ · h ∇ d ⊙ ∇ d − |∇ d | I i · φ A [ T k ( ρ )] . (5.45)It is easy to see that (5.40) follows from (5.44) and (5.45).In order to show the strong convergence of ρ ǫ , we also need to estimate on the oscillation defectmeasure of ( ρ ǫ − ρ ) in L γ ( Q T ). Estimate of oscillation of defect measures. There exists C > k ≥ ǫ → (cid:13)(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) γ +1 L γ +1 ( Q T ) ≤ lim ǫ → ˆ T ˆ Ω h ρ γǫ T k ( ρ ǫ ) − ρ γ T k ( ρ ) i ≤ C, (5.46)where T k ( ρ ) is a weak ∗ limit of T k ( ρ ǫ ) in L ∞ ( Q T ).Following the lines of argument presented in [6] and using (5.32), we obtainlim sup ǫ → (cid:13)(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) γ +1 L γ +1 ( Q T ) ≤ lim ǫ → ˆ T ˆ Ω h ρ γǫ T k ( ρ ǫ ) − ρ γ T k ( ρ ) i = e µa lim ǫ → ˆ T ˆ Ω h ( ∇ · u ǫ ) (cid:0) T k ( ρ ǫ ) − T k ( ρ ) (cid:1)i ≤ C (cid:16) sup ǫ> (cid:13)(cid:13) ∇ u ǫ (cid:13)(cid:13) L ( Q T ) (cid:17) lim sup ǫ → h(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13) L ( Q T ) + (cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13) L ( Q T ) i ≤ C lim sup ǫ → (cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13) L ( Q T ) , this implies (5.46) by applying Young’s inequality and using γ + 1 > Claim 1 . ( ρ, u ) is a renormalized solution to the equation (5 . . Observe that (5.35) also holds for ( ρ ǫ , u ǫ ) in R provided it is set to be zero in R \ Ω. Hence wehave ∂ t T k ( ρ ) + ∇ · ( T k ( ρ ) u ) + (cid:0) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:1) ∇ · u = 0 in D ′ ( R × (0 , T )) . (5.47)As in the step 3, we can mollify (5.47) and obtain ∂ t (cid:0) S m h T k ( ρ ) i(cid:1) + ∇ · (cid:0) S m h T k ( ρ ) i u (cid:1) + S m h [ T ′ k ( ρ ) ρ − T k ( ρ )] ∇ · u i = q m , (5.48)where q m := ∇ · (cid:0) S m h T k ( ρ ) i u (cid:1) − S m h ∇ · ( T k ( ρ ) u ) i → L ([0 , T ] , L s (Ω)) , as m → ∞ , for any s ∈ [1 , b be a test function in the definition of renormalized solutions of (5 . . Multiplying (5.48)by b ′ (cid:16) S m (cid:2) T k ( ρ ) (cid:3)(cid:17) yields ∂ t (cid:0) b ( S m h T k ( ρ ) i ) (cid:1) + ∇ · (cid:0) b (cid:0) S m h T k ( ρ ) i(cid:1) u (cid:1) + (cid:16) b ′ ( S m h T k ( ρ ) i ) S m h T k ( ρ ) i − b ( S m h T k ( ρ ) i ) (cid:17) ∇ · u = − b ′ ( S m h T k ( ρ ) i ) S m (cid:16) [ T ′ ( ρ ) ρ − T k ( ρ )] ∇ · u (cid:17) + b ′ ( S m h T k ( ρ ) i ) q m . Sending m → + ∞ in the above equation yields that ∂ t (cid:0) b (cid:16) T k ( ρ ) (cid:17)(cid:1) + ∇ · (cid:0) b (cid:16) T k ( ρ ) (cid:17) u (cid:1) + (cid:16) b ′ (cid:16) T k ( ρ ) (cid:17) T k ( ρ ) − b (cid:16) T k ( ρ ) (cid:17)(cid:17) ∇ · u = − b ′ (cid:16) T k ( ρ ) (cid:17)(cid:2) T ′ ( ρ ) ρ − T k ( ρ ) (cid:3) ∇ · u in D ′ (cid:0) R × (0 , T ) (cid:1) . (5.49)On the other hand, for p ∈ [1 , γ ), we have (cid:13)(cid:13) T k ( ρ ) − ρ (cid:13)(cid:13) pL p ( Q T ) ≤ lim inf ǫ → (cid:13)(cid:13) T k ( ρ ǫ ) − ρ ǫ (cid:13)(cid:13) pL p ( Q T ) . (5.50) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 19 On the other hand, we have (cid:13)(cid:13) T k ( ρ ǫ ) − ρ ǫ (cid:13)(cid:13) pL p ( Q T ) ≤ ˆ { ρ ǫ ≥ k } (cid:12)(cid:12) kT ( ρ ǫ k ) − ρ ǫ (cid:12)(cid:12) p ≤ p ˆ { ρ ǫ ≥ k } (cid:12)(cid:12) ρ ǫ (cid:12)(cid:12) p (cid:0) since kT ( ρ ǫ k ) ≤ ρ ǫ (cid:1) ≤ p k − γ + p ˆ { ρ ǫ ≥ k } ρ γǫ ≤ Ck − γ + p → , as k → + ∞ , uniformly in ǫ. (5.51)It follows from (5.50) and (5.51) thatlim k → + ∞ (cid:13)(cid:13)(cid:13) T k ( ρ ) − ρ (cid:13)(cid:13)(cid:13) L p ( Q T ) = 0 , for p ∈ [1 , γ ) . (5.52)For any M > b ′ ( z ) = 0 for z ≥ M , we set Q k,M := n ( x, t ) ∈ Q T (cid:12)(cid:12) T k ( ρ ) ≤ M o . Then ˆ T ˆ Ω (cid:12)(cid:12)(cid:12) b ′ (cid:16) T k ( ρ ) (cid:17)(cid:2) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:3) ∇ · u (cid:12)(cid:12)(cid:12) = ˆ Q k,M (cid:12)(cid:12)(cid:12) b ′ (cid:16) T k ( ρ ) (cid:17)(cid:2) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:3) ∇ · u (cid:12)(cid:12)(cid:12) ≤ sup Q k,M (cid:12)(cid:12)(cid:12) b ′ ( T k ( ρ )) (cid:12)(cid:12)(cid:12) ˆ Q k,M (cid:12)(cid:12)(cid:12)(cid:2) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:3) ∇ · u (cid:12)(cid:12)(cid:12) ≤ sup ≤ z ≤ M (cid:12)(cid:12) b ′ ( z ) (cid:12)(cid:12) lim inf ǫ → ˆ Q k,M (cid:12)(cid:12)(cid:12)(cid:2) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:3) ∇ · u ǫ (cid:12)(cid:12)(cid:12) ≤ C lim inf ǫ → (cid:13)(cid:13) ∇ u ǫ (cid:13)(cid:13) L ( Q T ) (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) L ( Q k,M ) ≤ C lim inf ǫ → (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) − γ L ( Q T ) (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) + γ L γ +1 ( Q k,M ) . (5.53)Now we can estimate (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) L ( Q T ) ≤ k − γ sup ǫ k ρ ǫ k γL γ ≤ Ck − γ → , as k → ∞ . (5.54)On the other hand, since T k ( z ) is a concave function and T ′′ k ( z ) ≤ 0, we have, by Taylor’s expansion,that 0 = T k ( z ) − T ′ k ( z ) z + 12 T ′′ ( ξz ) z for some ξ ∈ (0 , . In particular, we have T ′ k ( z ) z ≤ T k ( z ) and hence (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) ≤ (cid:13)(cid:13) T k ( ρ ǫ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) ≤ (cid:16)(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) + (cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) + (cid:13)(cid:13) T k ( ρ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) (cid:17) ≤ (cid:16)(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13) L γ +1 ( Q T ) + (cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13) L γ +1 ( Q T ) + M | Q k,M | γ +1 (cid:17) . Applying (5.46), we then obtain that there exists C > k such thatlim sup ǫ → (cid:13)(cid:13) T ′ k ( ρ ǫ ) ρ ǫ − T k ( ρ ǫ ) (cid:13)(cid:13) L γ +1 ( Q k,M ) ≤ C (cid:0) M | Q k,M | γ +1 (cid:1) ≤ C. (5.55)Substituting (5.54) and (5.55) into (5.53) yieldslim k →∞ ˆ T ˆ Ω (cid:12)(cid:12)(cid:12) b ′ (cid:16) T k ( ρ ) (cid:17)(cid:2) T ′ k ( ρ ) ρ − T k ( ρ ) (cid:3) ∇ · u (cid:12)(cid:12)(cid:12) = 0 . (5.56) Sending k → ∞ into the equation (5.49) and applying (5.52), (5.56), we conclude that ( ρ, u ) is arenormalized solution of the equation (5.30). This proves Claim 1.5.7. ρ ǫ → ρ strongly in L p ( Q T ) for any ≤ p < γ + θ . Hence ρ γ = ρ γ a.e. in Q T . It sufficesto show that ρ ǫ → ρ in L ( Q T ). This can be done in the exactly same lines as in [6]. Here we sketchit for the readers’ convenience. Let L k ( z ) ∈ C (0 , + ∞ ) ∩ C ([0 , + ∞ )) be defined by L k ( z ) = ( z ln z, ≤ z ≤ k,z ln k + z ´ zk T k ( s ) s ds, z > k. Note that for z large enough, L k ( z ) is a linear function, i.e., for z ≥ k,L k ( z ) = β k z − k, with β k = ln k + ˆ kk T k ( s ) s ds + 23 . Therefore b k ( z ) := L k ( z ) − β k z ∈ C (0 , + ∞ ) ∩ C ([0 , + ∞ )) satisfies b ′ k ( z ) = 0 for z large enough.Moreover, it is easy to see b ′ k ( z ) z − b k ( z ) = T k ( z ) . Since ( ρ ǫ , u ǫ ) is a renormalized solution of the equation (4 . and ( ρ, u ) is a renormalized solutionof the equation (5 . , we can take b ( z ) = b k ( z ) in the definition of the renormalized solutions toget that ∂ t L k ( ρ ǫ ) + ∇ · (cid:0) L k ( ρ ǫ ) u ǫ (cid:1) + T k ( ρ ǫ ) ∇ · u ǫ = 0 , in D ′ ( Q T ) , (5.57)and ∂ t L k ( ρ ) + ∇ · ( L k ( ρ ) u ) + T k ( ρ ) ∇ · u = 0 , in D ′ ( Q T ) . (5.58)Subtracting (5.57) from (5.58) gives ∂ t (cid:0) L k ( ρ ǫ ) − L k ( ρ ) (cid:1) + ∇ · (cid:0) L k ( ρ ǫ ) u ǫ − L k ( ρ ) u (cid:1) + (cid:0) T k ( ρ ǫ ) ∇ · u ǫ − T k ( ρ ) ∇ · u (cid:1) = 0 , (5.59)in D ′ ( Q T ).Since L k ( z ) is a linear function for z sufficiently large, we have that L k ( ρ ǫ ) is bounded in L ∞ ([0 , T ] , L γ (Ω)), uniformly in ǫ . Thus we have L k ( ρ ǫ ) ⇀ L k ( ρ ) weak ∗ in L ∞ ([0 , T ] , L γ (Ω)) , as ǫ → . This, combined with the equation (5.58), implies L k ( ρ ǫ ) ⇀ L k ( ρ ) in C ([0 , T ] , L γ weak (Ω)) ∩ C ([0 , T ] , H − (Ω)) , as ǫ → . (5.60)In particular, we have L k ( ρ ǫ ) , L k ( ρ ) ∈ C ([0 , T ] , L γ weak (Ω)) . (5.61)Hence we can multiply the equation (5.59) by φ ∈ C ∞ (Ω) and integrate the resulting equation over Q t , 0 < t ≤ T , to obtain ˆ Ω [ L k ( ρ ǫ ) − L k ( ρ )]( t ) φ = ˆ t ˆ Ω (cid:8) [ L k ( ρ ǫ ) u ǫ − L k ( ρ ) u ] · ∇ φ + [ T k ( ρ ) ∇ · u − T k ( ρ ǫ ) ∇ · u ǫ ] φ (cid:9) , where we have used the fact that [ L k ( ρ ǫ ) − L k ( ρ )] (cid:12)(cid:12) t =0 = 0. Taking ǫ → ˆ Ω h L k ( ρ ) − L k ( ρ ) i ( t ) φ = ˆ t ˆ Ω h L k ( ρ ) − L k ( ρ ) i u · ∇ φ + lim ǫ → ˆ t ˆ Ω (cid:2) T k ( ρ ) ∇ · u − T k ( ρ ǫ ) ∇ · u ǫ (cid:3) φ. (5.62) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 21 As in [6], we can choose φ = φ m ∈ C ∞ (Ω) in (5.62), which approximates the characteristic functionof Ω, i.e., ( ≤ φ m ≤ , φ m ( x ) = 1 for x ∈ Ω with dist( x, ∂ Ω) ≥ m ,φ m → m → ∞ and |∇ φ m ( x ) | ≤ m for all x ∈ Ω . (5.63)We then obtain that for 0 < t ≤ T , it holds ˆ Ω h L k ( ρ ) − L k ( ρ ) i ( t ) = lim ǫ → ˆ t ˆ Ω (cid:2) T k ( ρ ) ∇ · u − T k ( ρ ǫ ) ∇ · u ǫ (cid:3) . Hence we have ˆ Ω h L k ( ρ ) − L k ( ρ ) i ( t )= ˆ t ˆ Ω T k ( ρ ) ∇ · u − lim ǫ → ˆ t ˆ Ω T k ( ρ ǫ ) ∇ · u ǫ = ˆ t ˆ Ω T k ( ρ ) ∇ · u + 1 e µ lim ǫ → ˆ t ˆ Ω ( aρ γǫ − e µ ∇ · u ǫ ) T k ( ρ ǫ ) − a e µ lim ǫ → ˆ t ˆ Ω ρ γǫ T k ( ρ ǫ )= ˆ t ˆ Ω T k ( ρ ) ∇ · u + 1 e µ ˆ t ˆ Ω (cid:16) aρ γ − e µ ∇ · u (cid:17) T k ( ρ ) − a e µ lim ǫ → ˆ t ˆ Ω ρ γǫ T k ( ρ ǫ ) (cid:0) by (cid:0) . (cid:1) = ˆ t ˆ Ω h T k ( ρ ) − T k ( ρ ) i ∇ · u − a e µ lim ǫ → ˆ t ˆ Ω h ρ γǫ T k ( ρ ǫ ) − ρ γ T k ( ρ ) i ≤ ˆ t ˆ Ω h T k ( ρ ) − T k ( ρ ) i ∇ · u (cid:0) by (5 . (cid:1) ≤ (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) (cid:13)(cid:13)(cid:13) ∇ · u (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) + (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( { ρ ≤ k } ) (cid:13)(cid:13)(cid:13) ∇ · u (cid:13)(cid:13)(cid:13) L ( { ρ ≤ k } ) ≤ C (cid:16)(cid:13)(cid:13)(cid:13) ∇ · u (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) + (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( { ρ ≤ k } ) (cid:17) ≤ C (cid:16)(cid:13)(cid:13)(cid:13) ∇ · u (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) + (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) γ − γ L ( { ρ ≤ k } ) (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) γ +12 γ L γ +1 ( { ρ ≤ k } ) (cid:17) ≤ C (cid:16)(cid:13)(cid:13)(cid:13) ∇ · u (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) + (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) γ − γ L ( { ρ ≤ k } ) (cid:17) , where we have used (5.46) that guarantees (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L γ +1 ( Q T ) ≤ lim inf ǫ → (cid:13)(cid:13)(cid:13) T k ( ρ ǫ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L γ +1 ( Q T ) ≤ C, uniformly in k .Since T k is concave, it follows T k ( ρ ) ≤ T k ( ρ ). By the definition of T k , we also have T k ( ρ ) ≤ ρ .Hence we have (cid:13)(cid:13)(cid:13) T k ( ρ ) − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( { ρ ≤ k } ) ≤ (cid:13)(cid:13)(cid:13) ρ − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( { ρ ≤ k } ) ≤ (cid:13)(cid:13)(cid:13) ρ − T k ( ρ ) (cid:13)(cid:13)(cid:13) L ( Q T ) → k → ∞ (cid:0) by (5 . (cid:1) Since ∇ · u ∈ L ( Q T ), it follows that lim k →∞ (cid:13)(cid:13) ∇ · u (cid:13)(cid:13) L ( { ρ ≥ k } ) = 0 . Therefore we obtain lim k →∞ ˆ Ω h L k ( ρ ) − L k ( ρ ) i ( t ) ≤ , t ∈ (0 , T ) . (5.64) It follows from the definition of L k that ˆ T ˆ Ω (cid:12)(cid:12) L k ( ρ ) − ρ ln ρ (cid:12)(cid:12) ≤ (cid:13)(cid:13)(cid:13) L k ( ρ ) − ρ ln ρ (cid:13)(cid:13)(cid:13) L ( { ρ ≥ k } ) ≤ ˆ ˆ { ρ ≥ k } | ρ ln ρ | → , as k → + ∞ , (5.65)and (cid:13)(cid:13)(cid:13) L k ( ρ ǫ ) − ρ ǫ ln ρ ǫ (cid:13)(cid:13)(cid:13) L ( Q T ) ≤ ˆ ˆ { ρ ǫ ≥ k } (cid:12)(cid:12) L k ( ρ ǫ ) − ρ ǫ ln ρ ǫ (cid:12)(cid:12) ≤ ˆ ˆ { ρ ǫ ≥ k } | L k ( ρ ǫ ) | + | ρ ǫ ln ρ ǫ | ρ γǫ ρ γǫ ≤ C ( δ ) ˆ ˆ { ρ ǫ ≥ k } ρ γǫ ρ γ − − δǫ (cid:0) δ > (cid:1) ≤ Ck − γ +1+ δ → , as k → + ∞ , uniformly in ǫ, (5.66)so that by the lower semicontinuity we havelim k →∞ (cid:13)(cid:13)(cid:13) L k ( ρ ) − ρ ln ρ (cid:13)(cid:13)(cid:13) L ( Q T ) ≤ lim k →∞ lim inf ǫ → (cid:13)(cid:13)(cid:13) L k ( ρ ǫ ) − ρ ǫ ln ρ ǫ (cid:13)(cid:13)(cid:13) L ( Q T ) = 0 . (5.67)Combining (5.64), (5.66), (5.66), with (5.67) implies that ˆ Ω h ρ ln ρ − ρ ln ρ i ( t ) ≤ , t ∈ (0 , T ) . Since ρ ln ρ ≥ ρ ln ρ a.e. in Q T , this implies that ρ ln ρ = ρ ln ρ a.e. in Q T . By the convexity of the function ω ( z ) = z ln z : (0 , + ∞ ) → R , this implies that ρ ǫ → ρ in L ( Q T ) . Since ρ ǫ is bounded in L γ + θ ( Q T ), it follow from a simple interpolation that ρ ǫ → ρ in L p ( Q T ) for any 1 ≤ p < γ + θ. Thus ρ γ = ρ γ a.e. in Q T .The energy inequality (1.6) for ( ρ, u, d ) follows from the energy inequality (4.2) for ( ρ ǫ , u ǫ , d ǫ ). Infact, (4.2) implies that for almost all 0 < t < ∞ , it holds F ǫ ( t ) + ˆ t ˆ Ω (cid:16) µ |∇ u ǫ | + e µ |∇ · u ǫ | + | ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ | (cid:17) ≤ F ǫ (0) = E (0) . (5.68)On the other hand, by the lower semicontinuity, we have that for almost all t ∈ (0 , + ∞ ) E ( t ) + ˆ t ˆ Ω (cid:16) µ |∇ u | + e µ |∇ · u | + | ∆ d + |∇ d | d | (cid:17) ≤ lim inf ǫ → n F ǫ ( t ) + ˆ t ˆ Ω (cid:16) µ |∇ u ǫ | + e µ |∇ · u ǫ | + | ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ | (cid:17)o , (5.69)where we have used the observation that∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = ∂ t d ǫ + u ǫ · ∇ d ǫ ⇀ ∂ t d + u · ∇ d = ∆ d + |∇ d | d in L ( Q t ) . It is clear that (5.68) and (5.69) imply (1.6).After these steps, we conclude that ( ρ, u, d ) is a global finite energy weak solution of the system(1.1), under the initial and boundary condition (1.2), that satisfies the properties (i) of Theorem1.1. OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 23 The property (ii) for ( u, d ) follows from the strong convergence of d ǫ to d in L ((0 , + ∞ ) , H (Ω)).In fact, it is easy to see that d ǫ ∈ L ((0 , + ∞ ) , H (Ω)). For any X ∈ C (Ω , R ) and η ∈ C ((0 , + ∞ )), we can multiply the equation (4 . by η ( t ) X ( x ) · ∇ d ǫ ( x ) and integrate the result-ing equation over Ω × (0 , + ∞ ) and apply the integration by parts a few times to obtain ˆ ∞ η ( t ) ˆ Ω (cid:0) e ǫ ( d ǫ ) ∇ · X − ∇ d ǫ ⊙ ∇ d ǫ : ∇ X (cid:1) = ˆ ∞ η ( t ) ˆ Ω (cid:10) ∂ t d ǫ + u ǫ · ∇ d ǫ , X · ∇ d ǫ (cid:11) , (5.70)where e ǫ ( d ǫ ) := 12 |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) . Since ∂ t d ǫ + u ǫ ∇ d ǫ ⇀ ∂ t d + u · ∇ d in L (Ω × (0 , + ∞ )) , as ǫ → , we obtain, by sending ǫ → ˆ T η ( t ) ˆ Ω (cid:0) |∇ d | ∇ · X − ∇ d ⊙ ∇ d : ∇ X (cid:1) = ˆ T η ( t ) ˆ Ω (cid:10) ∂ t d + u · ∇ d, X · ∇ d (cid:11) . (5.71)The proof of Theorem 1.1 is now complete. (cid:3) Large time behavior of finite energy solutions and proof of corollary 1.2 In this section, we will study the large time asymptotic behavior of the global finite energy weaksolutions obtained in Theorem 1.1 and give a proof of Corollary 1.2. Proof of Corollary 1.2 :First it follows from (1.6) thatesssup t> E ( t ) + ˆ ∞ ˆ Ω (cid:0) µ |∇ u | + | ∆ d + |∇ d | d | (cid:1) ≤ E (0) . (6.1)For any positive integer m , define ( ρ m , u m , d m ) : Q → R + × R × S by ρ m ( x, t ) = ρ ( x, t + m ) ,u m ( x, t ) = u ( x, t + m ) ,d m ( x, t ) = d ( x, t + m ) . Then ( ρ m , u m , d m ) is a sequence of finite energy weak solutions of (1.1) in Q . It follows from (6.1)that (cid:13)(cid:13) ρ m (cid:13)(cid:13) L ∞ ([0 , ,L γ (Ω)) + (cid:13)(cid:13) ρ m u m (cid:13)(cid:13) L ∞ ([0 , ,L (Ω)) + (cid:13)(cid:13) ρ m u m (cid:13)(cid:13) L ∞ ([0 , ,L γ γ +1 (Ω)) + (cid:13)(cid:13) d m (cid:13)(cid:13) L ∞ ([0 , ,H (Ω)) ≤ C ( E (0)) , (6.2)and lim m →∞ ˆ (cid:16)(cid:13)(cid:13) ∇ u m (cid:13)(cid:13) L (Ω) + (cid:13)(cid:13) ∆ d m + |∇ d m | d m (cid:13)(cid:13) L (Ω) (cid:17) = 0 . (6.3)After passing to a subsequence, we may assume that as m → ∞ , ρ m ⇀ ρ ∞ in L γ ( Q ) , u m ⇀ u ∞ in L ([0 , , H (Ω)) , d m ⇀ d ∞ in L ([0 , , H (Ω)) . Applying (6.3) and the Poincar´e inequality, we havelim m →∞ ˆ (cid:13)(cid:13) u m (cid:13)(cid:13) L (Ω) = 0 , and hence u ∞ = 0 a.e. in Q .Sending m → ∞ in (1 . , we see that d ∞ solves ∂ t d ∞ = ∆ d ∞ + |∇ d ∞ | d ∞ in Q . On the other hand, by the lower semicontinuity and (6.3) we have ˆ ˆ Ω (cid:12)(cid:12) ∆ d ∞ + |∇ d ∞ | d ∞ (cid:12)(cid:12) = 0 . Hence ∂ t d ∞ = 0 in Q and d ∞ ( x, t ) = d ∞ ( x ) ∈ H (Ω , S ) is a harmonic map, with d ∞ = d on ∂ Ω.By H¨older’s inequality, (6.2), and (6.3), we havelim m →∞ ˆ (cid:16)(cid:13)(cid:13) ρ m u m (cid:13)(cid:13) L γγ +6 (Ω) + (cid:13)(cid:13) ρ m | u m | (cid:13)(cid:13) L γγ +3 (Ω) (cid:17) = 0 . (6.4)Since ( ρ m , u m , d m ) solves (1 . in Q , we have ∂ t ( ρ m u m ) + ∇ · ( ρ m u m ⊗ u m ) + a ∇ ρ γm = µ ∆ u m + e µ ∇ ( ∇ · u m ) − (∆ d m + |∇ d m | d m ) · ∇ d m in Q , which, after sending m → ∞ and applying (6.2), (6.3), (6.4), and Claim 3 below, implies ∇ ρ γ ∞ = 0 in Q . Hence ρ ∞ is x -independent in Q . On the other hand, since ρ ∞ is a weak solution of ∂ t ρ ∞ + ∇ · ( ρ ∞ u ∞ ) = 0 in Q , so that ∂ t ρ ∞ = 0 and ρ ∞ is t -independent in Q . Thus ρ ∞ is a constant.It remains to show ( ρ m , d m ) → ( ρ ∞ , d ∞ ) in L γ ( Q ) × L ([0 , , H (Ω)). This is divided into twoseparate claims. Claim 2. d m → d ∞ in L ([0 , , H (Ω)). The idea is based on the compactness Theorem 3.2, andthe argument is similar to that given in § § > G Λ = n t ∈ [0 , (cid:12)(cid:12)(cid:12) lim inf m →∞ ˆ Ω (cid:12)(cid:12)(cid:12) ∆ d m + |∇ d m | d m (cid:12)(cid:12) ≤ Λ o , and B Λ = [0 , \ G Λ . From (6.3), we have (cid:12)(cid:12) B Λ (cid:12)(cid:12) ≤ Λ − lim inf m →∞ ˆ ˆ Ω (cid:12)(cid:12)(cid:12) ∆ d m + |∇ d m | d m (cid:12)(cid:12) = 0 . (6.5)Since d m satisfies (1.12) for any X ∈ C (Ω) and η ∈ C ((0 , Z ⊂ G Λ , with | Z | = 0, such that for any t ∈ G Λ \ Z , it holds ˆ Ω (cid:16) ∇ d m ⊙ ∇ d m − |∇ d m | I (cid:17) ( t ) : ∇ X = − ˆ Ω (cid:10) ( ∂ t d m + u m · ∇ d m )( t ) , X · ∇ d m ( t ) (cid:11) = − ˆ Ω (cid:10) (∆ d m + |∇ d m | d m )( t ) , X · ∇ d m ( t ) (cid:11) (cid:0) by (1 . (cid:1) (6.6)It is standard (see [23]) that (6.6) implies that d m ( t ), t ∈ G Λ \ Z , satisfies the almost energymonotonicity inequality (3.3), i.e., x ∈ Ω and 0 < r ≤ R < d( x , ∂ Ω),Ψ R ( d m ( t ) , x ) ≥ Ψ r ( d m ( t ) , x ) + 12 ˆ B R ( x ) \ B r ( x ) | x − x | − (cid:12)(cid:12) ∂d m ( t ) ∂ | x − x | (cid:12)(cid:12) , (6.7)whereΨ r ( d m ( t ) , x ) = 1 r ˆ B r ( x ) (cid:0) |∇ d m | ( t ) − h ( x − x ) · ∇ d m ( t ) , τ m ( t ) i (cid:1) + 12 ˆ B r ( x ) | x − x || τ m ( t ) | , and τ m ( t ) = (∆ d m + |∇ d m | d m )( t ) . From the definition of G Λ , we have (cid:13)(cid:13) τ m ( t ) (cid:13)(cid:13) L (Ω) ≤ Λ , ∀ t ∈ G Λ \ Z. (6.8) OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 25 From (6.2), we see E ( d m ( t )) = 12 ˆ Ω |∇ d m ( t ) | ≤ C ( E (0)) . (6.9)Note also that d m ( x, t ) ≥ . e . x ∈ Ω , ∀ t ∈ G Λ \ Z. (6.10)From (6.7), (6.8), (6.9), and (6.10), we conclude that { d m ( t ) } m ≥ ⊂ Y ( C ( E (0)) , Λ , 0; Ω) for any t ∈ G Λ \ Z . Hence, by Theorem 3.2, we have that { d m } m ≥ is bounded in H (Ω , S ) and precompactin H (Ω , S ).Since ∂ t d m = − u m · ∇ d m + (∆ d m + |∇ d m | d m ) ∈ L ([0 , , L (Ω)) + L ([0 , , L (Ω)) , and sup m ≥ (cid:13)(cid:13)(cid:13) ∂ t d m (cid:13)(cid:13)(cid:13) L ([0 , ,L (Ω))+ L ([0 , ,L (Ω)) ≤ C. We can apply Aubin-Lions’ lemma, similar to § e Ω ⊂⊂ Ω, aftertaking a subsequence, there holdslim m →∞ (cid:13)(cid:13)(cid:13) ∇ ( d m − d ∞ ) (cid:13)(cid:13)(cid:13) L ( e Ω × ( G Λ \ Z )) = 0 . (6.11)On the other hand, by (6.2), we havesup m ≥ (cid:13)(cid:13)(cid:13) ∇ ( d m − d ∞ ) (cid:13)(cid:13)(cid:13) L ( e Ω × ( B Λ ∪ Z )) ≤ C ( E (0)) (cid:12)(cid:12) B Λ ∪ Z (cid:12)(cid:12) = 0 . (6.12)Putting (6.11) and (6.12) together yieldslim m →∞ (cid:13)(cid:13)(cid:13) ∇ ( d m − d ∞ ) (cid:13)(cid:13)(cid:13) L ( e Ω × (0 , = 0 . (6.13)Claim 2 follows from (6.13). Claim ρ m → ρ , ∞ in L γ ( Q ) . To show this claim, first observe that by the same lines of argumentin § ρ ǫ , u ǫ , d ǫ ) replaced by ( ρ m , u m , d m ), we can obtain that there exist θ > C > m such that ˆ ˆ Ω ρ γ + θm ≤ C, ∀ m ≥ . (6.14)From (6.14), we may assume that ρ γm ⇀ ρ γ ∞ in L p ( Q ) , < p ≤ γ + θγ ( Q ) . (6.15)There are two methods to prove that ρ γ ∞ = ρ γ ∞ a.e. in Q and ρ m → ρ ∞ in L γ ( Q ): the firstis to repeat the same lines of arguments given by § § § ρ ǫ , u ǫ , d ǫ ) replaced by( ρ m , u m , d m ); and the second is to apply the div-curl lemma, similar to [5] Proposition 4.1. Herewe sketch it. For simplicity, assume the pressure coefficient a = 1. Let Div and Curl denotethe divergence and curl operators in Q . As pointed out by [6] Remark 1.1, (1.4) also holds for b ( ρ m ) = G ( ρ γm ) when G ( z ) = z α , with0 < α < min (cid:8) γ , θθ + γ (cid:9) . Using the equation (1.4), one can check thatDiv (cid:2) , , , G ( ρ γm ) (cid:3) is precompact in W − ,q ( Q )for some q > . and (6.2), one can checkCurl (cid:2) , , , ρ γm (cid:3) is precompact in W − ,q ( Q ) for some q > G ( ρ γm ) ⇀ G ( ρ γ ∞ ) in L p ( Q ) , and G ( ρ γm ) ρ γm ⇀ G ( ρ γ ∞ ) ρ γ ∞ in L r ( Q ) , with p = 1 α , r = 1 p + 1 p . Then by the div-curl lemma we conclude that G ( ρ γ ∞ ) ρ γ ∞ = G ( ρ γ ∞ ) ρ γ ∞ As G is strictly monotone, this implies G ( ρ γ ∞ ) = G (cid:16) ρ γ ∞ (cid:17) . Since L p is uniformly convex, this impliesthat the convergence in (6.15) is strong in L ( Q ). Hence we have that ρ m → ρ ∞ in L γ ( Q ) . Since ˆ Ω ρ m ( t ) = ˆ Ω ρ for 0 < t < ρ ∞ is constant, it follows that ρ ∞ ≡ | Ω | ˆ Ω ρ (:= ρ , ∞ ).From claim 2, claim3, and (6.2), we can apply Fubini’s theorem to conclude that there exists t m ∈ ( m, m + 1) such that as m → ∞ , (cid:0) ρ ( t m ) , d ( t m ) (cid:1) → (cid:0) ρ , ∞ , d ∞ (cid:1) in L γ (Ω) × H (Ω , S ) , and (cid:13)(cid:13) u ( t m ) (cid:13)(cid:13) H (Ω) → . Hence by Sobolev’s embedding theorem we have that u ( t m ) → L p (Ω) for any 1 < p < 6. Theproof is now complete. (cid:3) Acknowledgements . Lin is partially supported by NSF of China (Grant 11001085, 11371152)and 973 Program (Grant 2011CB808002). Lai is partially supported by NSF of China (Grants11201119 and 11126155). Both Lin and Lai are also partially supported by the Chinese ScholarshipCouncil. Wang is partially supported by NSF grants 1001115 and 1265574, and NSF of Chinagrant 11128102. The work was completed while both Lin and Lai were visiting Department ofMathematics, University of Kentucky. Both of them would like to thank the Department for itshospitality and excellent research environment. References [1] Y. M. Chen, The weak solutions to the evolution problems of harmonic maps. Math. Z. (1989), no. 1, 69-74.[2] P. G. de Gennes, The Physics of Liquid Crystals. Oxford, 1974.[3] J. L. Ericksen, Hydrostatic theory of liquid crystal , Arch. Rational Mech. Anal. 9 (1962), 371-378.[4] E. Feireisl, On compactness of solutions to the compressible isentropic Navier-Stokes equations when the densityis not square integrable. Comment Math. Univ. Carolin, (1) (2001), 83-98.[5] E. Feireisl, H. Petzeltov´a, Large-time Behaviour of Solutions to the Navier-Stokes Equations of CompressibleFlow . Arch. Rational Mech. Anal. (1999) 77-96.[6] E. Feireisl, A. Novotny and H. Petzeltov´a, On the existence of globally defined weak solutions to the Navier-Stokesequations. J. Math. Fluid Mech., (2001), 358-392.[7] F. M. Leslie, Some constitutive equations for liquid crystals , Arch. Rational Mech. Anal. 28, 1968, 265-283.[8] R. Hardt, D. Kinderlehrer, F. Lin, Existence and partial regularity of static liquid crystal configurations . Comm.Math. Phys., 105 (1986), 547-570.[9] M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in R . Calc. Var. PartialDifferential Equations (2011), no. 1-2, 15-36.[10] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals: Phase transition and flow phenomena . CPAM, (1989), 789-814.[11] F. H. Lin, C. Liu, Nonparabolic Dissipative Systems Modeling the Flow of Liquid Crystals . CPAM, Vol. XLVIII,501-537 (1995). OMPRESSIBLE NEMATIC LIQUID CRYSTAL FLOW 27 [12] F. H. Lin, C. Liu, Partial Regularity of The Dynamic System Modeling The Flow of Liquid Cyrstals . DCDS,Vol. 2, No. 1 (1998) 1-22.[13] F. H. Lin, J. Y. Lin, C. Y. Wang, Liquid crystal flows in two dimensions , Arch. Rational Mech. Anal., (2010) 297-336.[14] F. H. Lin, C. Y. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematicliquid crystals . Chinese Annals of Mathematics, B (6) (2010), 921-938.[15] Z. Lei, D. Li, X. Y. Zhang, A new proof of global wellposedness of liquid crystals and heat harmonic maps in twodimensions. Proc. Amer. Math. Soc., to appear.[16] F. H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps . Ann. of Math. (2) (3)785-829 (1999).[17] F. H. Lin, C. Y. Wang, The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2008. xii+267 pp.[18] F. H. Lin, C. Y. Wang, Global existence of weak solutions of the nematic liquid crystal flow in dimensions three. Preprint, 2014.[19] F. H. Lin, C. Y. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals , (aninvited suvery article of the special issue edited by A. Majumdar, E. G. Vriga on New Trends in Active LiquidCrystals: Mechanics, Dynamics and Applications”), Philosophical Transactions of Royal Society A, to appear.[20] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres . Comm. Anal. Geom. , no. 2, 397-429 (1999).[21] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. II. Comm. Anal. Geom. , no. 2, 341-375 (2002).[22] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measureand generalized varifold flows. Ann. Inst. H. Poincar´e Anal. Non Lin´eaire , no. 2, 209-259 (2002).[23] F. H. Lin, C. Y. Wang, The analysis of harmonic maps and their heat flows. World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2008. xii+267 pp.[24] A. Morro, Modelling of Nematic Liquid Crystals in Electromagnetic Fields. Adv. Theor. Appl. Mech., Vol. 2(2009), no. 1, 43-58.[25] A. V. Zakharov, A. A. Vakulenko, Orientational dynamics of the compressible nematic liquid crystals inducedby a temperature gradient. Phys. Rev. E (2009), 011708.[26] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol.I, Incompressible Models . Clarendon Press, Oxford,1996.[27] X. G. Liu, J. Qing, Existence of globally weak solutions to the flow of compressible liquid crystals system. DiscreteContin. Dyn. Syst. (2013), no. 2, 757-788.[28] S. J. Ding, J. Y. Lin, C. Y. Wang, H. Y. Wen, Compressible hydrodynamic flow of liquid crystals in 1D . DiscreteContin. Dyn. Syst. (2012), no. 2, 539-563.[29] S. J. Ding, C. Y. Wang, H. Y. Wen, Weak solution to compressible hydrodynamic flow of liquid crystals indimension one . Discrete Contin. Dyn. Syst. Ser. B (2011), no. 2, 357-371.[30] T. Huang, C. Y. Wang, H. Y. Wen, Strong solutions of the compressible nematic liquid crystal flow . J. DifferentialEquations (2012), no. 3, 2222-2265.[31] T. Huang, C. Y. Wang, H. Y. Wen, Blow up criterion for compressible nematic liquid crystal flows in dimensionthree. Arch. Ration. Mech. Anal. (2012), no. 1, 285-311.[32] S. J. Ding, J. R. Huang, F. G. Xia, H. Y. Wen, R. Z. Zi, Incompressible limit of the compressible nematic liquidcrystal flow . J. Funct. Anal. (7) (2013), 1711-1756.[33] F. Jiang, J. Song, D. H. Wang, On multi-dimensional compressible flows of nematic liquid crystals with largeinitial energy in a bounded domain. J. Funct. Anal. (2013), no. 12, 3369-3397.[34] J. Li, Z. Xu, J. Zhang, Global well-posedness with large oscillations and vacuum to the three-dimensional equa-tions of compressible nematic liquid crystal flows . arXiv:1204.4966v1.[35] D. H. Wang, C. Yu, Global weak solution and large-time behavior for the compressible flow of liquid crystals. Arch. Rational Mech. Anal. (2012), 881-915.[36] X. Xu, Z. F. Zhang, Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows . J.Differential Equations (2012), no. 2, 1169-1181. Department of Mathematics, South China University of Technology, Guangdong 510640, P. R. China E-mail address : [email protected] School of Mathematics and Information Sciences, Henan University, Kaifeng 475004, Henan, P. R.China E-mail address : [email protected] Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907,USA E-mail address ::