Global existence of weak solutions of the nematic liquid crystal flow in dimensions three
aa r X i v : . [ m a t h . A P ] A ug GLOBAL EXISTENCE OF WEAK SOLUTIONS OF THE NEMATIC LIQUIDCRYSTAL FLOW IN DIMENSIONS THREE
FANGHUA LIN AND CHANGYOU WANG
Abstract.
For any bounded smooth domain Ω ⊂ R (or Ω = R ), we establish the global existence of aweak solution ( u, d ) : Ω × [0 , + ∞ ) → R × S of the initial-boundary value (or the Cauchy) problem of thesimplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for anyinitial and boundary (or Cauchy) data ( u , d ) ∈ H × H (Ω , S ), with d (Ω) ⊂ S (the upper hemisphere).Furthermore, ( u, d ) satisfies the global energy inequality (1.4). Introduction
In this paper, we consider the following simplified Ericksen-Leslie system modeling the hydrodynamicsof nematic liquid crystals in dimensions three: for a bounded smooth domain Ω ⊂ R (or Ω = R ) and0 < T ≤ ∞ , ( u, P, d ) : Ω × [0 , T ) → R × R × S solves ∂ t u + u · ∇ u − ν ∆ u + ∇ P = − λ ∇ · ( ∇ d ⊙ ∇ d ) , in Ω × (0 , T ) , ∇ · u = 0 , in Ω × (0 , T ) ,∂ t d + u · ∇ d = γ (∆ d + |∇ d | d ) , in Ω × (0 , T ) , (1.1)along with the initial and boundary condition: (cid:26) ( u, d ) = ( u , d ) in Ω × { } , ( u, d ) = (0 , d ) on ∂ Ω × (0 , + ∞ ) , (1.2)for a given data ( u , d ) : Ω → R × S , with ∇ · u = 0. Here u : Ω → R represents the velocityfield of the fluid, d : Ω → S (the unit sphere in R ) is a unit vector field representing the macroscopicorientation of the nematic liquid crystal molecules, and P : Ω → R represents the pressure function. Theconstants ν, λ, and γ are positive constants representing the viscosity of the fluid, the competition betweenkinetic and potential energy, and the microscopic elastic relaxation time for the molecular orientation fieldrespectively. ∇· denotes the divergence operator in R , and ∇ d ⊙ ∇ d denotes the symmetric 3 × ∇ d ⊙ ∇ d ) ij = h∇ i d, ∇ j d i , ≤ i, j ≤ . Throughout this paper, we denote h v, w i or v · w as the innerproduct in R for v, w ∈ R .The system (1.1) is a simplified version of the celebrated Ericksen-Leslie model for the hydrodynamics ofnematic liquid crystals developed by Ericksen and Leslie during the period of 1958 through 1968 [5, 9, 3].The full Ericksen-Leslie system reduces to the Oseen-Frank model of liquid crystals in the static case. It is amacroscopic continuum description of the time evolution of the materials under the influence of fluid velocityfield u and the macroscopic description of the microscopic orientation field d of rod-like liquid crystals. Thecurrent form of system (1.1) was first proposed by Lin [10] back in the late 1980’s. From the mathematicalpoint of view, (1.1) is a system strongly coupling the non-homogeneous incompressible Navier-Stokes equationand the transported heat flow of harmonic maps to S . Lin-Liu [13, 14] have initiated the mathematicalanalysis of (1.1) by considering its Ginzburg-Landau approximation or the so-called orientation with variabledegrees in the terminology of Ericksen. Namely, the Dirichlet energy E ( d ) = 12 ˆ |∇ d | for d : R → S isreplaced by the Ginzburg-Landau energy E ǫ ( d ) = ˆ |∇ d | + 14 ǫ (1 − | d | ) ( ǫ >
0) for d : R → R . Hence(1.1) is replaced by ∂ t d + u · ∇ d = γ (∆ d + 1 ǫ (1 − | d | ) d ) . (1.3) Date : July 5, 2018.
Key words and phrases.
Hydrodynamic flow, nematic liquid crystal, global weak solution.
Lin-Liu have proved in [13, 14] (i) the existence of a unique, global smooth solution in dimension two andin dimension three under large viscosity ν ; and (ii) the existence of suitable weak solutions and their partialregularity in dimension three, analogous to the celebrated regularity theorem by Caffarelli-Kohn-Nirenberg[2] (see also [11]) for the three-dimensional incompressible Navier-Stokes equation.As already pointed out by [13, 14], it is a very challenging problem to study the issue of convergence ofsolutions ( u ǫ , P ǫ , d ǫ ) to (1.1) -(1.1) -(1.3) as ǫ tends to 0. In particular, the existence of global Leray-Hopftype weak solutions to the initial and boundary value problem of (1.1) has only been established recently byLin-Lin-Wang [15] in dimension two, see also Hong [7] and Xu-Zhang [27] for related works.Because of the super-critical nonlinear term ∇ · ( ∇ d ⊙ ∇ d ) in (1.1) , it has been an outstanding openproblem whether there exists a global Leray-Hopf type weak solution to (1.1) in R for any initial data( u , d ) ∈ L (Ω , R ) × H (Ω , S ) with ∇ · u = 0. We would like to mention that Wang [26] has recentlyobtained the global (or local) well-posedness of (1.1) for initial data ( u , d ) belonging to possibly the largestspace BMO − × BMO with ∇ · u = 0, which is an invariant space under parabolic scaling associated with(1.1), with small norms.In this paper, we are interested in the global existence of weak solutions to (1.1) for large initial data.Since the exact values of ν, λ, γ don’t play roles in this paper, we henceforth assume ν = λ = γ = 1 . Before stating our theorems, we need to introduce some notations. For b ∈ [ − , S b = (cid:8) y = ( y , y , y ) ∈ S : y ≥ b (cid:9) , and let S = S denote the upper hemisphere. Set H = Closure of C ∞ (Ω , R ) ∩ (cid:8) v : ∇ · v = 0 (cid:9) in L (Ω , R ) , J = Closure of C ∞ (Ω , R ) ∩ (cid:8) v : ∇ · v = 0 (cid:9) in H (Ω , R ) , and H (Ω , S ) = (cid:8) d ∈ H (Ω , R ) : d ( x ) ∈ S a . e . x ∈ Ω (cid:9) . In this context, we are able to prove
Theorem 1.1.
For any u ∈ H and d ∈ H (Ω , S ) with d (Ω) ⊂ S , there exists a global weak solution ( u, d ) : Ω × [0 , + ∞ ) → R × S to the initial and boundary value problem of (1.1) and (1.2) such that (i) u ∈ L ∞ t L x ∩ L t H x (Ω × [0 , + ∞ ) , R ) . (ii) d ∈ L ∞ t H x (Ω , S ) and d ( x, t ) ≥ a.e. ( x, t ) ∈ Ω × (0 , + ∞ ) . (iii) ( u, d ) satisfies the global energy inequality: for L -a.e. ≤ t < + ∞ , ˆ Ω ( | u | + |∇ d | )( t ) + 2 ˆ t ˆ Ω ( |∇ u | + | ∆ d + |∇ d | d | ) ≤ ˆ Ω ( | u | + |∇ d | ) . (1.4) Remark 1.2.
From the proof of Theorem 1.1, it is clear that the weak solution ( u, d ) obtained in Theorem1.1 enjoys the property that for L a.e. t ∈ (0 , + ∞ ), d ( t ) ∈ H (Ω , S ) is a suitable approximated harmonicmap to S with tension field τ ( t ) = ( ∂ t d + u · ∇ d )( t ) ∈ L (Ω , R ) (see the definition 5.1).Based on Theorem 6.1 and Theorem 7.1, we also establish the following compactness property for a classof weak solutions to (1.1) that contains those solutions constructed by Theorem 1.1. Theorem 1.3.
For any < a ≤ and < T ≤ + ∞ , assume that ( u k , d k ) : Ω × (0 , T ] → R × S − a is asequence of weak solutions of (1.1), that satisfies sup k ≥ h sup ≤ t ≤ T ˆ Ω ( | u k | + |∇ d k | ) + ˆ T ˆ Ω ( |∇ u k | + | ∆ d k + |∇ d k | d k | ) i < + ∞ , (1.5) and for L a.e. t ∈ (0 , + ∞ ) , d k ( t ) ∈ H (Ω , S ) is a suitable approximated harmonic map with tension field τ k ( t ) = ( ∂ t d k + u k · ∇ d k )( t ) ∈ L (Ω , R ) . Then there exists a weak solution ( u, d ) : Ω × (0 , T ] → R × S − a of (1.1) such that, after passing to possible subsequences, u k → u, ∇ d k → ∇ d in L (Ω × [0 , T ]) . (1.6) D NEMATIC LIQUID CRYSTAL FLOW 3
The proof of Theorem 1.1 is very delicate. The weak solution ( u, d ) to (1.1) is obtained as a weak limitof a sequence of weak solutions ( u ǫ , d ǫ ) to the Ginzburg-Landau approximated equation of (1.1) (i.e., theequations (1.1) , (1.1) , and (1.3) as ǫ tends to zero. The key ingredient is to show that ∇ d ǫ subsequentiallyconverges to ∇ d in L (Ω × (0 , + ∞ )), or equivalently the subsequential L -compacteness of ∇ d ǫ . This isachieved by showing(i) d ǫ ≥ d ǫ ( t ) enjoys slice almost energy monotonicity property for L -a.e. t > t > d ǫ ( t ) enjoys both regularity estimate and H -precompactness property underthe small energy condition, and(iv) utilizing the range assumption of d ǫ to rule out the defect measures generated during the blow-up analysisof d ǫ ( t ) for points at good time slices where the small energy condition may not hold. As a consequence, weactually show that at any good time slice t , the small energy condition holds everywhere.It is in step (iv) that we need to adapt and extend the blow-up techniques the authors have developed forthe heat flow of harmonic maps in [17, 18, 19]. Remark 1.4.
For general initial data d ∈ H (Ω , S ) (i.e., without the assumption d ( x ) ≥ x ∈ Ω),our blow-up analysis scheme in this paper seems to suggest that defect measures ν may result during theconvergence procedure of ( u ǫ , d ǫ ) to ( u, d ) as ǫ →
0. The defect measure ν represents a transported versionof curvature motion of generalized curves, and ( u, d ) is a weak solution of the nematic liquid crystal flow(1.1) away from the support of ν , which is the energy concentration set of the convergence. This energyconcentration set may correspond to dark threads that appear in the study of liquid crystal flows. Webelieve that, motivated by earlier results on the heat flow of harmonic maps [17, 18, 19], ( u, d ) and ν isa weak solution of the nematic liquid crystal flow (1.1) coupled with transported versions of generalized1-varifold flows in Brakke’s sense. We plan to investigate these issues in a forthcoming paper.The paper is written as follows. In section 2, we will establish some preliminary estimates of (1.3) bythe weak maximum principle. In section 3, we will establish a slice almost monotonicity inequality of (1.3).In section 4, we will prove an δ -compactness property for weak solutions to (1.3). In section 5, we willestablish an δ -regularity for suitable approximated harmonic map to S . In section 6, we will establish H -precompactness for certain solutions of (1.3). In section 7, we will establish H -precompactness for suitableapproximated harmonic maps to S − a . In section 8, we will prove both Theorem 1.1 and Theorem 1.3.2. Maximum principle on the transported Ginzburg-Landau heat flow
In this section, we will establish two pointwise estimates for the transported Ginzburg-Landau heat flowby the weak maximum principle.For ǫ >
0, consider the initial-boundary value problem of the transported Ginzburg-Landau heat flow: ∂ t d ǫ + u ǫ · ∇ d ǫ = ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ in Ω × (0 , + ∞ ) , ∇ · u ǫ = 0 in Ω × (0 , + ∞ ) ,d ǫ = g ǫ on (Ω × { } ) ∪ ( ∂ Ω × (0 , + ∞ )) . (2.1) Lemma 2.1.
For < T < + ∞ , assume u ǫ ∈ L ([0 , T ] , J ) and g ǫ ∈ H (Ω , R ) satisfies | g ǫ ( x ) | ≤ , a . e . x ∈ Ω . Suppose d ǫ ∈ L ([0 , T ] , H (Ω , R )) , with (1 − | d ǫ | ) ∈ L (Ω × [0 , T ]) , solves (2.1). Then | d ǫ ( x, t ) | ≤ for a.e. ( x, t ) ∈ Ω × [0 , T ] .Proof. For any k >
1, define v kǫ : Ω × [0 , T ] → R + by letting v kǫ ( x, t ) = k − | d ǫ ( x, t ) | > k, | d ǫ ( x, t ) | − < | d ǫ ( x, t ) | ≤ k, | d ǫ ( x, t ) | ≤ . F. LIN AND C. WANG
Then direct calculations imply that v kǫ satisfies ∂ t v kǫ + u ǫ · ∇ v kǫ = ∆ v kǫ − χ { < | d ǫ |≤ k } (cid:0) |∇ d ǫ | + 1 ǫ ( | d ǫ | − | d ǫ | (cid:1) ≤ ∆ v ǫ in Ω × [0 , T ] (2.2)in the weak sense. Since v kǫ = 0 on (Ω × { } ) ∪ ( ∂ Ω × (0 , T ]) and 0 ≤ v kǫ ≤ k − × [0 , T ], we canmultiply the (2.2) by v kǫ and integrate it over Ω × [0 , s ] for any 0 < s ≤ T to obtain ˆ Ω | v kǫ ( s ) | + 2 ˆ s ˆ Ω |∇ v kǫ | ≤ − ˆ s ˆ Ω u ǫ · ∇ ( | v kǫ | ) = 0 , where we have used the fact that ∇ · u ǫ = 0 in the last step. Hence it follows that v kǫ = 0 a.e. in Ω × [0 , T ]and hence | d ǫ | ≤ × [0 , T ]. (cid:3) Lemma 2.2.
For < T < + ∞ , assume u ǫ ∈ L ([0 , T ] , J ) and g ǫ ∈ H (Ω , R ) satisfies | g ǫ ( x ) | ≤ g ǫ ( x ) ≥ , a . e . x ∈ Ω . If d ǫ ∈ L ([0 , T ] , H (Ω , R )) , with (1 − | d ǫ | ) ∈ L (Ω × [0 , T ]) , solves (2.1), then | d ǫ ( x, t ) | ≤ d ǫ ( x, t ) ≥ , a . e . ( x, t ) ∈ Ω × [0 , T ] . Proof.
By Lemma 2.1, we have that | d ǫ | ≤ × [0 , T ] and hence0 ≤ ǫ (1 − | d ǫ | ) ≤ ǫ . Define e d ǫ ( x, t ) = e − tǫ d ǫ ( x, t ). Then we have ∂ t e d ǫ + u ǫ · ∇ e d ǫ − ∆ e d ǫ = (cid:0) ǫ (1 − | d ǫ | ) − ǫ (cid:1) e d ǫ ≡ c ǫ ( x, t ) e d ǫ , where c ǫ ( x, t ) := (cid:0) ǫ (1 − | d ǫ | ) − ǫ (cid:1) ( x, t ) satisfies c ǫ ≤ . e . Ω × [0 , T ] . Define ( e d ǫ ) − = − min (cid:8) e d ǫ , (cid:9) in Ω × [0 , T ]. Then we have ∂ t ( e d ǫ ) − + u ǫ · ∇ ( e d ǫ ) − − ∆( e d ǫ ) − = c ǫ ( x, t )( e d ǫ ) − . (2.3)Since e d ǫ = e − tǫ g ǫ ≥ × { } ) ∪ ( ∂ Ω × [0 , T ]) , it follows that ( e d ǫ ) − = 0 on (Ω × { } ) ∪ ( ∂ Ω × [0 , T ]) . Multiplying (2.3) by ( e d ǫ ) − , integrating the resulting equation over Ω × [0 , s ] for 0 < s ≤ T , and using thefact that ∇ · u ǫ = 0 and c ǫ ≤
0, we obtain ˆ Ω | ( e d ǫ ) − | ( s ) + 2 ˆ s ˆ Ω |∇ ( e d ǫ ) − | = − ˆ s ˆ Ω u ǫ · ∇| ( e d ǫ ) − | + 2 ˆ s ˆ Ω c ǫ ( x, t ) | ( e d ǫ ) − | = 2 ˆ s ˆ Ω c ǫ ( x, t ) | ( e d ǫ ) − | ≤ . Hence it follows that ( e d ǫ ) − = 0 a.e. Ω × [0 , T ]. This implies that d ǫ ≥ . e . Ω × [0 , T ] . This completes the proof. (cid:3)
D NEMATIC LIQUID CRYSTAL FLOW 5 Monotonicity formula for approximated Ginzburg-Landau equation
In this section, we will derive the monotonicity formula for approximated Ginzburg-Landau equationswith L -tension fields in Ω ⊂ R . Lemma 3.1.
Let d ǫ ∈ H (Ω , R ) be a solution of the approximated Ginzburg-Landau equation: ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = τ ǫ in Ω . (3.1) Assume | d ǫ | ≤ a.e. Ω and τ ǫ ∈ L (Ω) . Then, for any x ∈ Ω and < r ≤ R < dist( x , ∂ Ω) , it holds Φ ǫ ( R ) ≥ Φ ǫ ( r ) + 12 ˆ B R ( x ) \ B r ( x ) | x − x | (cid:16)(cid:12)(cid:12) ∂d ǫ ∂ | x − x | (cid:12)(cid:12) + (1 − | d ǫ | ) ǫ (cid:17) , (3.2) where Φ ǫ ( ρ ) = 1 ρ ˆ B ρ ( x ) (cid:0) e ǫ ( d ǫ ) − h ( x − x ) · ∇ d ǫ , τ ǫ i (cid:1) + 12 ˆ B ρ ( x ) | x − x || τ ǫ | (3.3) for ρ > , and e ǫ ( d ǫ ) = (cid:0) |∇ d ǫ | + ǫ (1 − | d ǫ | ) (cid:1) denotes the (modified) Ginzburg-Landau energy densityof d ǫ .Proof. Since | d ǫ | ≤ (cid:13)(cid:13)(cid:13) τ ǫ + 1 ǫ ( | d ǫ | − d ǫ (cid:13)(cid:13)(cid:13) L (Ω) ≤ (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) + | Ω | ǫ . Hence by the W , -estimate, we have that d ǫ ∈ W , (Ω).For simplicity, assume x = 0 ∈ Ω and write d for d ǫ . Multiplying the equation (3.1) by x · ∇ d andintegrating over B ρ ⊂ Ω yields ρ ˆ ∂B ρ (cid:16)(cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) + (1 − | d | ) ǫ (cid:17) + ˆ B ρ e ǫ ( d ) − ρ ˆ ∂B ρ e ǫ ( d ) = ˆ B ρ h τ ǫ , x · ∇ d i . This implies ddρ h ρ ˆ B ρ ( e ǫ ( d ) − h τ ǫ , x · ∇ d i ) i = 1 ρ h ρ ˆ ∂B ρ e ǫ ( d ) − ˆ B ρ e ǫ ( d ) + ˆ B ρ h τ ǫ , x · ∇ d i i − ρ ˆ ∂B ρ h τ ǫ , x · ∇ d i = 1 ρ ˆ ∂B ρ (cid:16)(cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) + (1 − | d | ) ǫ (cid:17) − ρ ˆ ∂B ρ h τ ǫ , x · ∇ d i . By H¨older’s inequality, we have (cid:12)(cid:12)(cid:12) ρ ˆ ∂B ρ h τ ǫ , x · ∇ d i (cid:12)(cid:12)(cid:12) ≤ ρ ˆ ∂B ρ (cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) + 12 ρ ˆ ∂B ρ | τ ǫ | . Thus we obtain ddρ h ρ ˆ B ρ ( e ǫ ( d ) − h τ ǫ , x · ∇ d i ) i ≥ ρ ˆ ∂B ρ (cid:16)(cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) + (1 − | d | ) ǫ (cid:17) − ρ ˆ ∂B ρ | τ ǫ | . Integrating this inequality over r ≤ ρ ≤ R yields (3.2). (cid:3) δ -compactness property of approximated Ginzburg-Landau equation In this section, we will prove an δ -regularity property for approximated Ginzburg-Landau equations with L -tension fields in Ω ⊂ R .First we need to recall some notations. For 1 ≤ p < + ∞ , 0 ≤ q ≤
3, and an open set U ⊂ R , we definethe Morrey space M p,q ( U ) by M p,q ( U ) := n f ∈ L p loc ( U ) (cid:12)(cid:12)(cid:12) (cid:13)(cid:13) f (cid:13)(cid:13) qM p,q ( U ) ≡ sup B r ⊂ U r − q ˆ B r | f | p < + ∞ o . (4.1) F. LIN AND C. WANG
Now we consider approximated Ginzburg-Landau equations with L -tension fields. For 0 < ǫ ≤
1, let d ǫ ∈ H (Ω , R ) be a sequence of solutions to∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = τ ǫ in Ω , (4.2)with uniformly bounded Ginzburg-Landau energies and L -norms of τ ǫ , i.e.,sup <ǫ ≤ E ǫ ( d ǫ ) = ˆ Ω (cid:0) |∇ d ǫ | + 14 ǫ (1 − | d ǫ | ) (cid:1) ≤ L < + ∞ , (4.3)and sup <ǫ ≤ (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) ≤ L < + ∞ . (4.4)After taking a possible subsequence, we may assume that there exists d ∈ H (Ω , S ) such that d ǫ ⇀ d in H (Ω) and strongly in L (Ω) , as ǫ → d ǫ to d in H under the smallness conditionof renormalized Ginzburg-Landau energies. More precisely, we have Lemma 4.1.
For any L , L > , there exist δ > and r > such that for < ǫ ≤ , if d ǫ ∈ H (Ω , R ) ,with | d ǫ | ≤ a.e. Ω , is a family of solutions of (4.2) satisfying (4.3) and (4.4), and r ˆ B r ( x ) e ǫ ( d ǫ ) ≤ δ , (4.5) for some x ∈ Ω and < r ≤ min { r , dist( x , ∂ Ω) } , then after passing to subsequences, d ǫ → d in H ( B r ( x ) , R ) as ǫ → .Proof. For simplicity, assume x = 0 ∈ Ω. For any fixed x ∈ B r and 0 < ǫ ≤ r , define b d ǫ ( x ) = d ǫ ( x + ǫx ) : B → R . Then we have ∆ b d ǫ = − (1 − | b d ǫ | ) b d ǫ + b τ ǫ in B , where b τ ǫ ( x ) = ǫ τ ǫ ( x + ǫx ). Since | b d ǫ | = | d ǫ | ≤ (cid:13)(cid:13) ∆ b d ǫ (cid:13)(cid:13) L ( B ) ≤ (cid:13)(cid:13) (1 − | b d ǫ | ) b d ǫ (cid:13)(cid:13) L ( B ) + (cid:13)(cid:13) b τ ǫ (cid:13)(cid:13) L ( B ) ≤ C + ǫ (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) ≤ C + L . Thus b d ǫ ∈ W , ( B ) and (cid:13)(cid:13) b d ǫ (cid:13)(cid:13) W , ( B ) ≤ C h(cid:13)(cid:13) b d ǫ (cid:13)(cid:13) L ( B ) + (cid:13)(cid:13) ∆ b d ǫ (cid:13)(cid:13) L ( B ) i ≤ C (1 + L ) . By Sobolev’s embedding theorem, we conclude that b d ǫ ∈ C ( B ) and h b d ǫ i C ( B ) ≤ C (cid:13)(cid:13)(cid:13) b d ǫ (cid:13)(cid:13)(cid:13) W , ( B ) ≤ C (1 + L ) . Scaling back to the original scales, this implies that (cid:12)(cid:12) d ǫ ( x ) − d ǫ ( y ) (cid:12)(cid:12) ≤ C (1 + L ) (cid:16) | x − y | ǫ (cid:17) , ∀ x, y ∈ B ǫ ( x ) . Now we have
Claim 4.1 . | d ǫ ( x ) | ≥ for x ∈ B r .Suppose that the claim were false. Then there exists x ∈ B r such that | d ǫ ( x ) | < . Then for any θ ∈ (0 ,
1) and x ∈ B θ ǫ ( x ), it holds | d ǫ ( x ) − d ǫ ( x ) | ≤ C (cid:16) | x − x | ǫ (cid:17) ≤ Cθ < , provided θ < C . Hence we have | d ǫ ( x ) | ≤ , ∀ x ∈ B θ ǫ ( x ) , D NEMATIC LIQUID CRYSTAL FLOW 7 so that 1 θ ǫ ˆ B θ ǫ ( x ) (1 − | d ǫ | ) ǫ ≥ (cid:0) (cid:1) (cid:12)(cid:12) B θ ǫ ( x ) (cid:12)(cid:12) θ ǫ ≥ (cid:0) (cid:1) θ (cid:12)(cid:12) B (cid:12)(cid:12) . (4.6)On the other hand, by the monotonicity inequality (3.2) we have1 θ ǫ ˆ B θ ǫ ( x ) (1 − | d ǫ | ) ǫ ≤ θ ǫ ˆ B θ ǫ ( x ) e ǫ ( d ǫ ) ≤ C r ˆ B r ( x ) e ǫ ( d ǫ ) + C ˆ B r ( x ) | x − x || τ ǫ | ≤ C ( δ + r (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) ) ≤ C ( δ + L r ) . (4.7)It is clear that (4.6) contradicts (4.7), provided r > δ > | d ǫ | ≥ in B r , we can perform the polar decomposition of d ǫ by d ǫ = f ǫ ω ǫ , where f ǫ := | d ǫ | : B r → [ 12 ,
1] and ω ǫ := d ǫ | d ǫ | : B r → S . Denote the cross product in R by × . It is readily seen that ∇ d ǫ = ( ∇ f ǫ ) ω ǫ + f ǫ ∇ ω ǫ , ∇ d ǫ × d ǫ = f ǫ ∇ ω ǫ × ω ǫ in B r . Hence we have that for any subset U ⊂ B r , it holds (cid:13)(cid:13) ∇ f ǫ (cid:13)(cid:13) M , ( U ) + (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , ( U ) ≤ C (cid:13)(cid:13) ∇ d ǫ (cid:13)(cid:13) M , ( U ) . Now we have
Claim 4.2 . For any 2 < p < ∇ ω ǫ ∈ L p ( B r ) and (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L p ( B r ) ≤ C (cid:16)(cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) + (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L ( B r ) (cid:17) . (4.8)Let d and d ∗ denote exterior derivative and co-exterior derivative respectively. It follows directly from(4.2) that d ∗ ( dd ǫ × d ǫ ) = τ ǫ × d ǫ in B r . (4.9)For any ball B r ⊂ B r , let e f ǫ : R → R and e ω ǫ : R → R be extensions of f ǫ and ω ǫ in B r such that e f ǫ = f ǫ in B r , ≤ e f ǫ ≤ R , (cid:13)(cid:13) ∇ e f ǫ (cid:13)(cid:13) L ( R ) ≤ C (cid:13)(cid:13) ∇ f ǫ (cid:13)(cid:13) L ( B r ) , e ω ǫ = ω ǫ in B r , | e ω ǫ | ≤ R , (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) L ( R ) ≤ C (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) , (cid:13)(cid:13) ∇ e f ǫ (cid:13)(cid:13) M , ( R ) ≤ C (cid:13)(cid:13) ∇ f ǫ (cid:13)(cid:13) M , ( B r ) , (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) M , ( R ) ≤ C (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , ( B r ) . (4.10)Set e d ǫ = e f ǫ e ω ǫ in R . Applying the Hodge decomposition theorem to d e d ǫ × e d ǫ = e f ǫ d e ω ǫ × e ω ǫ , we concludethat there exist G ǫ ∈ ˙ H ( R , M × ) and H ǫ ∈ ˙ H ( R , ∧ ( M × )) such that d e d ǫ × e d ǫ = dG ǫ + d ∗ H ǫ , dH ǫ = 0 in R , (4.11)and (cid:13)(cid:13) ∇ G ǫ (cid:13)(cid:13) L ( R ) + (cid:13)(cid:13) ∇ H ǫ (cid:13)(cid:13) L ( R ) . (cid:13)(cid:13) d e d ǫ × e d ǫ (cid:13)(cid:13) L ( R ) . (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) L ( R ) . (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) . (4.12)Taking the exterior derivative of both sides of the equation (4.11), we have∆ H ǫ = d ( d e d ǫ × e d ǫ ) = d e d ǫ × d e d ǫ = X i,j =1 ∂ e d ǫ ∂x i × ∂ e d ǫ ∂x j dx i ∧ dx j in R . (4.13)Multiplying (4.13) by H ǫ and applying integration by parts, we have ˆ R |∇ H ǫ | = − ˆ R d e d ǫ × d e d ǫ · H ǫ = − ˆ R d e d ǫ × e d ǫ · d ∗ H ǫ = − ˆ R d e ω ǫ × ( e f ǫ e ω ǫ ) · d ∗ H ǫ . F. LIN AND C. WANG
Applying the duality between the Hardy space H ( R ) and the BMO space BMO( R ) (see [4] [6] or [20]),we then obtain ˆ R |∇ H ǫ | . (cid:13)(cid:13) d e ω ǫ · d ∗ H ǫ (cid:13)(cid:13) H ( R ) (cid:13)(cid:13) e f ǫ e ω ǫ (cid:13)(cid:13) BMO( R ) . (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) L ( R ) (cid:13)(cid:13) ∇ H ǫ (cid:13)(cid:13) L ( R ) (cid:13)(cid:13) ∇ ( e f ǫ e ω ǫ ) (cid:13)(cid:13) M , ( R ) . (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) h(cid:13)(cid:13) ∇ e f ǫ (cid:13)(cid:13) M , ( R ) + (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) M , ( R ) i . (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) h(cid:13)(cid:13) ∇ f ǫ (cid:13)(cid:13) M , ( B r ) + (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , ( B r ) i . (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L ( B r ) (cid:13)(cid:13) ∇ d ǫ (cid:13)(cid:13) M , ( B r ) , where we have used the Poincar´e inequality to estimate (cid:13)(cid:13) e f ǫ e ω ǫ (cid:13)(cid:13) BMO( R ) . (cid:13)(cid:13) ∇ ( e f ǫ e ω ǫ ) (cid:13)(cid:13) M , ( R ) . h(cid:13)(cid:13) ∇ e f ǫ (cid:13)(cid:13) M , ( R ) + (cid:13)(cid:13) ∇ e ω ǫ (cid:13)(cid:13) M , ( R ) i among the first three inequalities. Utilizing the energy monotonicity inequality (3.2), we obtain (cid:13)(cid:13) ∇ d ǫ (cid:13)(cid:13) M , ( B r ) . n r ˆ B r e ǫ ( d ǫ ) + r ˆ B r | τ ǫ | o . δ + L r . Thus we have ˆ R |∇ H ǫ | ≤ C ( δ + L r ) ˆ B r |∇ ω ǫ | . (4.14)To estimate G ǫ , first observe that by taking the co-exterior derivative d ∗ of both sides of the equation (4.11),we have ∆ G ǫ = τ ǫ × d ǫ in B r . (4.15)Decompose G ǫ = G ( ǫ, + G ( ǫ, , where G ( ǫ, ∈ H ( B r , M × ) solves ( ∆ G ( ǫ, = τ ǫ × d ǫ in B r G ( ǫ, = 0 on ∂B r , and G ( ǫ, ∈ H ( B r , M × ) solves ( ∆ G ( ǫ, = 0 in B r G ( ǫ, = G ǫ on ∂B r . By the standard elliptic theory, we have that ˆ B r |∇ G ( ǫ, | . r ˆ B r | τ ǫ | , and ˆ B θr |∇ G ( ǫ, | . θ ˆ B r |∇ G ǫ | . θ ˆ B r |∇ ω ǫ | , ∀ < θ < . Combining these two estimates together yields ˆ B θr |∇ G ǫ | . θ ˆ B r |∇ ω ǫ | + r ˆ B r | τ ǫ | , ∀ < θ < . (4.16)Putting (4.14) and (4.16) together and using (4.11), we obtain that1 θr ˆ B θr |∇ ω ǫ | ≤ C (cid:0) θ + θ − ( δ + L r ) (cid:1) r ˆ B r |∇ ω ǫ | + Cθ − r ˆ B r | τ ǫ | , (4.17)for any B r ⊂ B r and 0 < θ < α ∈ (0 , ), first choose θ = θ ∈ (0 ,
1) such that Cθ ≤ θ α , then choose δ ∈ (0 ,
1) such that Cδ ≤ θ α +10 , and finally choose r > CL r ≤ θ α +10 . Thus it follows from (4.17) that1 θ r ˆ B θ r |∇ ω ǫ | ≤ θ α n r ˆ B r |∇ ω ǫ | o + C θ r ˆ B r | τ ǫ | , ∀ B r ⊂ B r . (4.18) D NEMATIC LIQUID CRYSTAL FLOW 9
Iterating (4.18) finitely many times yields1 r ˆ B r |∇ ω ǫ | ≤ (cid:0) rr (cid:1) α n r ˆ B r |∇ ω ǫ | o + C r ˆ B r | τ ǫ | , ∀ B r ⊂ B r . (4.19)Taking supremum over all balls B r ⊂ B r , we obtain that for any α ∈ (0 , ], it holds (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , − α ( B r ) ≤ C ( r ) h ˆ B r |∇ ω ǫ | + ˆ B r | τ ǫ | i . (4.20)It follows from (4.20) that d ( dd ǫ × d ǫ ) = d ( f ǫ dω ǫ × ω ǫ ) ∈ M , − α ( B r ) and (cid:13)(cid:13) d ( dd ǫ × d ǫ ) (cid:13)(cid:13) M , − α ( B r ) . (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , − α ( B r ) + (cid:13)(cid:13) ∇ f ǫ (cid:13)(cid:13) M , ( B r ) (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) M , − α ( B r ) ≤ C ( r ) h ˆ B r |∇ ω ǫ | + ˆ B r | τ ǫ | i . (4.21)Based on (4.21), we can repeat both the extension and the Hodge decomposition as in (4.10), (4.11), (4.12),and (4.13). Since H ǫ ( x ) = ˆ R | x − y | d ( d e d ǫ × e d ǫ )( y ) dy, ∀ x ∈ R , we have |∇ H ǫ ( x ) | . ˆ R | d ( d e d ǫ × e d ǫ ) | ( y ) | x − y | dy = I ( | d ( d e d ǫ × e d ǫ ) | )( x ) , ∀ x ∈ R , where I ( f )( x ) := ˆ R | f ( y ) || x − y | dy, f ∈ L ( R ) , is the Riesz potential of f of order 1.Since we can construct the extension e d ǫ of d ǫ such that (cid:13)(cid:13) d ( d e d ǫ × e d ǫ ) (cid:13)(cid:13) M , − α ( R ) ≤ C (cid:13)(cid:13) d ( dd ǫ × d ǫ ) (cid:13)(cid:13) M , − α ( B r ) , we can apply Morrey space estimates of Riesz potentials (see [1]) to conclude that ∇ H ǫ ∈ M − α − α , − α ∗ ( R )and (cid:13)(cid:13)(cid:13) ∇ H ǫ (cid:13)(cid:13)(cid:13) M − α − α , − α ∗ ( R ) . (cid:13)(cid:13)(cid:13) d ( d e d ǫ × e d ǫ ) (cid:13)(cid:13)(cid:13) M , − α ( R ) . C ( r ) h ˆ B r |∇ ω ǫ | + ˆ B r | τ ǫ | i . (4.22)Since lim α ↑ − α − α = 3, it follows from (4.22) that ∇ H ǫ ∈ L p ( B r ) for any 2 < p <
3, and (cid:13)(cid:13)(cid:13) ∇ H ǫ (cid:13)(cid:13)(cid:13) L p ( B r ) ≤ C (cid:13)(cid:13)(cid:13) ∇ H ǫ (cid:13)(cid:13)(cid:13) M − α − α , − α ∗ ( R ) ≤ C ( r ) h ˆ B r |∇ ω ǫ | + ˆ B r | τ ǫ | i . (4.23)On the other hand, applying W , -estimate of the equation (4.15) we conclude that G ǫ ∈ W , ( B r ), and (cid:13)(cid:13) ∇ G ǫ (cid:13)(cid:13) L ( B r ) . (cid:13)(cid:13) ∇ G ǫ (cid:13)(cid:13) H ( B r ) . (cid:13)(cid:13) ∇ G ǫ (cid:13)(cid:13) L ( B r ) + (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L ( B r ) ≤ C ( r ) n ˆ B r ( |∇ d ǫ | + | τ ǫ | ) o . (4.24)Combining (4.23) and (4.24) yields that ∇ ω ǫ ∈ L p ( B r ) for any 2 < p <
3, and the estimate (4.8) holds.
Claim 4.3 . There is a map ω ∈ H ( B r , S ) such that after taking possible subsequences, ω ǫ → ω in H ( B r ) as ǫ → f ǫ ⇀ H ( B r ) and there exists ω ∈ H ( B r , S ) such that ω ǫ ⇀ ω in H ( B r ). We may also assume that τ ǫ ⇀ τ in L (Ω) for some τ ∈ L (Ω , R ). We also recall that (4.20) and Morrey’s decay lemma (see [22]) imply ω ǫ ∈ C ( B r ) and (cid:2) ω ǫ (cid:3) C ( B r ) ≤ C ( r ) (cid:16) k∇ ω ǫ k L ( B r ) + k τ ǫ k L ( B r ) (cid:17) ≤ C. (4.25) Hence we can assume that lim ǫ → (cid:13)(cid:13) ω ǫ − ω (cid:13)(cid:13) L ∞ ( B r ) = 0 . (4.26)It follows from Claim 4.2 and (4.8) that for any 2 < p < k∇ ω ǫ k L p ( B r ) ≤ C. (4.27)Direct calculations imply that ω ǫ satisfies the equation∆ ω ǫ = e τ ǫ := −|∇ ω ǫ | ω ǫ − f − ǫ ∇ f ǫ · ∇ ω ǫ + ( τ ǫ − h τ ǫ , ω ǫ i ω ǫ ) , in B r . (4.28)Since f ǫ ≥ in B r , we have (cid:13)(cid:13) e τ ǫ (cid:13)(cid:13) L pp +2 ( B r ) ≤ C ( r ) h k τ ǫ k L ( B r ) + k∇ f ǫ k L ( B r ) k∇ ω ǫ k L p ( B r ) i ≤ C ( r , p, L , L ) . It follows from the W , pp +2 -theory that ω ǫ ∈ W , pp +2 ( B r ) and (cid:13)(cid:13) ω ǫ (cid:13)(cid:13) W , pp +2 ( B r ) ≤ C (cid:16) k∇ ω ǫ k L ( B r ) + (cid:13)(cid:13) e τ ǫ (cid:13)(cid:13) L pp +2 ( B r ) (cid:17) ≤ C ( r , p, L , L ) . (4.29)It follows from (4.27), (4.29), and the compact embedding of W , pp +2 ⊂ W , pp +2 that ∇ ω ǫ → ∇ ω in L ( B r ). Claim 4.4 . After passing to possible subsequences, f ǫ → H ( B r ).To see this, we need to estimate ˆ B r |∇ f ǫ | . First, observe that f ǫ satisfies∆(1 − f ǫ ) − ǫ (1 − f ǫ ) f ǫ = −|∇ ω ǫ | f ǫ − τ ǫ · ω ǫ , in B r . (4.30)By Fubini’s theorem, there exists r ∈ ( r , r ) such that ˆ ∂B r e ǫ ( d ǫ ) dH ≤ r ˆ B r e ǫ ( d ǫ ) . (4.31)Since | d ǫ | ≤ B r , it is readily seen that for any 2 < q < + ∞ , (cid:13)(cid:13) − | d ǫ | (cid:13)(cid:13) L q ( B r ) ≤ (cid:13)(cid:13) − | d ǫ | (cid:13)(cid:13) q L ( B r ) (cid:13)(cid:13) − | d ǫ | (cid:13)(cid:13) q − q L ∞ ( B r ) ≤ Cǫ q (cid:16) ˆ B r e ǫ ( d ǫ ) (cid:17) q ≤ Cǫ q . (4.32)Multiplying (4.30) by (1 − f ǫ ) and integrating the resulting equation over B r and using ≤ | f ǫ | ≤ ˆ B r |∇ f ǫ | + ˆ B r ǫ (1 − f ǫ ) f ǫ (1 + f ǫ )= ˆ ∂B r (1 − f ǫ ) ∂f ǫ ∂r dH + ˆ B r |∇ ω ǫ | f ǫ (1 − f ǫ ) + ˆ B r τ ǫ · ω ǫ (1 − f ǫ ) ≤ ˆ ∂B r (1 − | d ǫ | ) (cid:12)(cid:12) ∂d ǫ ∂r (cid:12)(cid:12) dH + ˆ B r |∇ ω ǫ | (1 − | d ǫ | ) + ˆ B r | τ ǫ | (1 − | d ǫ | ) . ǫ r ˆ B r e ǫ ( d ǫ ) + (cid:13)(cid:13) ∇ ω ǫ (cid:13)(cid:13) L p ( B r ) (cid:13)(cid:13) − | d ǫ | (cid:13)(cid:13) q L q ( B r ) + ǫ (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L ( B r ) (cid:16) ˆ B r e ǫ ( d ǫ ) (cid:17) , (4.33)where p ∈ (2 ,
3) and q = pp − ∈ (3 , + ∞ ). Hence, by (4.8) and (4.32) we have ˆ B r (cid:0) |∇ f ǫ | + (1 − | d ǫ | ) ǫ (cid:1) ≤ Cǫ q → , as ǫ → . (4.34)Combining claim 4.3 with claim 4.4, we see that d ǫ = f ǫ ω ǫ → d = ω in H ( B r ). The proof is nowcomplete. (cid:3) D NEMATIC LIQUID CRYSTAL FLOW 11 δ -regularity for suitable approximated harmonic map to S In this section, we will introduce the notion of suitable approximated harmonic maps to S with L -tensionfields. Then we will establish the sequential compactness property for such approximated harmonic mapsunder the energy smallness condition.Recall that a map d ∈ H (Ω , S ) is called an approximated harmonic map with L -tension field, if thereexists τ ∈ L (Ω , R ) such that ∆ d + |∇ d | d = τ in Ω , (5.1)holds in the sense of distributions. Definition 5.1.
An approximated harmonic map d ∈ H (Ω , S ) , with tension field τ ∈ L (Ω , R ) , is calleda suitable approximated harmonic map, if ddt (cid:12)(cid:12)(cid:12) t =0 ˆ Ω (cid:0) |∇ ( d ◦ F t ) | + h τ, d ◦ F t i (cid:1) = 0 (5.2) holds for any F t ( x ) = x + tY ( x ) , where Y = ( Y , Y , Y ) ∈ C (Ω , R ) . Direct calculations imply that (5.2) is equivalent to ˆ Ω (cid:16)(cid:10) ∂d∂x i , ∂d∂x j (cid:11) ∂Y i ∂x j − |∇ d | Y + (cid:10) τ, Y · ∇ d (cid:11)(cid:17) = 0 , ∀ Y ∈ C ∞ (Ω , R ) . (5.3) Remark 5.2.
An approximated harmonic map d ∈ H (Ω , S ), with L -tension field τ , is a suitable approx-imated harmonic map, if d ∈ W , (Ω , S ). In fact, (5.3) can be obtained by the Pohozaev argument, namelymultiplying (5.1) by Y · ∇ d and integrating the resulting equation over Ω.For suitable approximated harmonic maps, we have the following energy monotonicity inequality. Lemma 5.3.
Assume d ∈ H (Ω , S ) is a suitable approximated harmonic map with tension field τ ∈ L (Ω , R ) . Then 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) , (5.4) for x ∈ Ω and < r ≤ R < d( x , ∂ Ω) , where Ψ 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 | . Proof.
Assume x = 0. For 0 < r < d(0 , ∂ Ω) and 0 < ǫ < r , let η ǫ ( x ) = η ( | x | ) ∈ C ∞ ( B r ) be such that0 ≤ η ǫ ≤ η ǫ ≡ B (1 − ǫ ) r , and η ≡ B r . Substituting Y ( x ) = η ǫ ( x ) x into (5.3) and then sending ǫ to zero, we obtain − ˆ B r |∇ d | + ˆ B r h τ, x · ∇ d i − r ˆ ∂B r (cid:12)(cid:12) ∂d∂r (cid:12)(cid:12) + r ˆ ∂B r |∇ d | = 0 . This implies ddρ (cid:16) r ˆ B r ( 12 |∇ d | − h τ, x · ∇ d i ) (cid:17) = 1 r h r ˆ ∂B r |∇ d | − ˆ B r |∇ d | + ˆ B r h τ, x · ∇ d i i − r ˆ ∂B r h τ, x · ∇ d i = 1 r ˆ ∂B r (cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) − r ˆ ∂B r h τ, x · ∇ d i ≥ r ˆ ∂B r (cid:12)(cid:12) ∂d∂ | x | (cid:12)(cid:12) − r ˆ ∂B r | τ | . Integrating this inequality over [ r, R ] implies (5.4). (cid:3)
With the monotonicity inequality (5.4), we have the following small energy regularity result.
Lemma 5.4.
For any L , L > , there exist δ > , r > such that if d ∈ H (Ω , S ) is a suitableapproximated harmonic map with tension field τ , that satisfies E ( d ) := 12 ˆ Ω |∇ d | ≤ L , (cid:13)(cid:13) τ (cid:13)(cid:13) L (Ω) ≤ L , (5.5) and r ˆ B r ( x ) |∇ d | ≤ δ , (5.6) for some x ∈ Ω and < r ≤ min (cid:8) r , d( x , ∂ Ω) (cid:9) , then d ∈ C ∩ W , ( B r , S ) , and (cid:2) d (cid:3) C ( B r ) + (cid:13)(cid:13) d (cid:13)(cid:13) W , ( B r ) ≤ C ( r , δ , L , L ) . (5.7) Proof.
Since the proof is similar to that of Lemma 4.1, we only sketch it here. First, multiplying both sidesof (5.1) by × d yields that d satisfies div( ∇ d × d ) = τ × d in Ω . (5.8)Then by repeating the argument of the claim 4.2 lines by lines, we can obtain that for any α ∈ (0 , ), thereexists θ ∈ (0 ,
1) such that1 θ r ˆ B θ r |∇ d | ≤ θ α n r ˆ B r |∇ d | o + C θ r ˆ B r | τ | , ∀ B r ⊂ B r . (5.9)Iterating (5.9) finitely many times, we obtain1 r ˆ B r |∇ d | ≤ (cid:0) rr (cid:1) α n r ˆ B r |∇ d | o + C r ˆ B r | τ | , ∀ B r ⊂ B r . (5.10)Taking supremum over all balls B r ⊂ B r , we obtain that for any α ∈ (0 , ], it holds (cid:13)(cid:13) ∇ d (cid:13)(cid:13) M , − α ( B r ) ≤ C ( r ) h ˆ B r |∇ d | + ˆ B r | τ | i . (5.11)This, combined with Morrey’s lemma (see [22]), implies that d ∈ C ( B r ) and (cid:2) d (cid:3) C ( B r ) ≤ C ( r ) (cid:16) k∇ d k L ( B r ) + k τ k L ( B r ) (cid:17) ≤ C ( r , δ , L , L ) . (5.12)To see the interior W , -regularity of d , we proceed as follows. Let η ∈ C ∞ ( B r ) be a cut-off function of B r , i.e., 0 ≤ η ≤ η ≡ B r , and |∇ η | ≤ r . Define d , d : R → R by d ( x ) = ˆ R ( η |∇ d | d )( y ) | x − y | dy, d ( x ) = ˆ R ( η τ )( y ) | x − y | dy, and d : B r → R by d = d − d − d . Then by Morrey space estimates of Riesz potentials as in claim 4.2 of Lemma 4.1, we have that ∇ d ∈ M − α − α , − α ∗ ( R ), ∇ d ∈ L ( R ), and (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) M − α − α , − α ∗ ( R ) ≤ C (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) M , − α ( B r ) ≤ C ( r , δ , L , L ) , (5.13)and (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) L ( R ) ≤ C (cid:13)(cid:13) τ (cid:13)(cid:13) L ( B r ) . (5.14)Since lim α ↑ − α − α = + ∞ , (5.13) implies that ∇ d ∈ L q ( B r ) for any q ∈ (1 , + ∞ ), and (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) L q ( B r ) ≤ C ( q, r , δ , L , L ) . (5.15)Since ∆ d = 0 in B r , it follows from the standard theory that ∇ d ∈ L ( B r ) and (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) L ( B r ) ≤ C (cid:16)(cid:13)(cid:13) ∇ d (cid:13)(cid:13) L ( B r ) + (cid:13)(cid:13) ∇ d (cid:13)(cid:13) L ( B r ) + (cid:13)(cid:13) ∇ d (cid:13)(cid:13) L ( B r ) (cid:17) ≤ C ( r , δ , L , L ) . (5.16) D NEMATIC LIQUID CRYSTAL FLOW 13
Putting (5.15), (5.14), and (5.16) together yields that ∇ d ∈ L ( B r ) and (cid:13)(cid:13)(cid:13) ∇ d (cid:13)(cid:13)(cid:13) L ( B r ) ≤ C ( r , δ , L , L ) . Now we can apply the standard L -estimate to conclude that d ∈ W , ( B r ) with the desired estimate. (cid:3) H pre-compactness for certain approximated Ginzburg-Landau equation In this section, we will consider the set of solutions to approximated Ginzburg-Landau equation withranges in (cid:8) y = ( y , y , y ) ∈ R : | y | ≤ , y ≥ − a (cid:9) , with uniformly bounded energies and uniformlybounded L -tension fields. We will show that it is precompact in H (Ω) and uniformly bounded in H (Ω).For any 0 < a ≤ L , and L >
0, define the set X ( L , L , a ; Ω) consisting of maps d ǫ ∈ H (Ω , R ),0 < ǫ ≤
1, that are solutions of ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = τ ǫ in Ω (6.1)such that the following properties hold:(i) | d ǫ | ≤ d ǫ ≥ − a for a.e. x ∈ Ω.(ii) E ǫ ( d ǫ ) = ˆ Ω e ǫ ( d ǫ ) dx ≤ L .(iii) (cid:13)(cid:13) τ ǫ (cid:13)(cid:13) L (Ω) ≤ L .We have Theorem 6.1.
For any a ∈ (0 , , L > , and L > , the set X ( L , L , a ; Ω) is precompact 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 amap d ∈ H (Ω , S ) such that after passing to possible subsequences, d ǫ → d in H (Ω , R ) .Proof. For 0 < ǫ i ≤
1, let { d ǫ i } ⊂ X ( L , L , a ; Ω) be a sequence of maps. Assume that there are ǫ ∈ [0 , d ∈ H (Ω , R ) such that ǫ i → ǫ and d ǫ i ⇀ d in H (Ω , R ) and τ ǫ i ⇀ τ in L (Ω , R ) as i → + ∞ .We divide the proof into two cases. Case 1 : ǫ >
0. Since (cid:13)(cid:13)(cid:13) ǫ i (1 − | d ǫ i | ) d ǫ i (cid:13)(cid:13)(cid:13) L ∞ (Ω) ≤ ǫ , we have (cid:13)(cid:13)(cid:13) ∆ d ǫ i (cid:13)(cid:13)(cid:13) L (Ω) ≤ (cid:13)(cid:13)(cid:13) ǫ i (1 − | d ǫ i | ) d ǫ i (cid:13)(cid:13)(cid:13) L (Ω) + (cid:13)(cid:13) τ ǫ i (cid:13)(cid:13) L (Ω) ≤ Cǫ − | Ω | + L . By W , -estimate we conclude that { d ǫ i } is a bounded sequence in W , (Ω). Hence we have that d ǫ i → d in H (Ω , R ) as i → + ∞ . Case 2 : ǫ = 0. Then it is easy to see that d ∈ H (Ω , S ). We may assume that there exists nonnegativeRadon measures ν and µ in Ω such that e ǫ i ( d ǫ i ) dx ⇀ µ := 12 |∇ d | dx + ν as convergence of Radon measures in Ω for i → + ∞ .Let δ > ⊂ Ω byΣ := \
0, and sufficiently large i ≥ r ( d ǫ i ; x ) ≤ δ − η , ∀ i ≥ i . This, combined with (6.4), implies that(1 − r ) 1 r ˆ B r ( x ) e ǫ i ( d ǫ i ) ≤ Φ r ( d ǫ i ) + cL r ≤ δ − η + cL r ≤ δ (1 − r ) , ∀ i ≥ i . D NEMATIC LIQUID CRYSTAL FLOW 15 provide r = r ( η , δ ) > r ˆ B r ( x ) e ǫ i ( d ǫ i ) ≤ δ , ∀ i ≥ i . Applying Lemma 4.1, we conclude that d ǫ i → d in H ( B r ( x ) , R ) and (1 − | d ǫ i | ) ǫ i → L ( B r ( x )).Hence ν ≡ B r ( x ). It also follows from (4.8) and (5.12) that d ∈ C ∩ W ,p ( B r ( x )) for all 2 < p < x ∈ Ω \ Σ is arbitrary, Claim 6.2 follows.Denote the singular set of d bysing( d ) := n x ∈ Ω : d is discontinuous at x o . Then we have
Claim 6.3 . Σ = supp( ν ) ∪ sing( d ).It is easy to see from claim 6.2 that supp( ν ) ∪ sing( d ) ⊂ Σ. If x / ∈ supp( ν ) ∪ sing( d ), then there exists r > ν ( B r ( x )) = 0 and d ∈ C ( B r ( x )). By Lemma 4.1, we have that d ∈ H (Ω , S ) is anapproximated harmonic map with tension field τ , i.e.,∆ d + |∇ d | d = τ . (6.8)For small ǫ >
0, assume r > B r ( x ) ( d ) ≤ ǫ. Then by the standard hole filling argument (see [20]), there exists θ ∈ (0 , ) such that1 θ r ˆ B θ r ( x ) |∇ d | ≤ r ˆ B r ( x ) |∇ d | + Cr . Iterating this inequality then implies that there exists α ∈ (0 , ) such that for any 0 < r ≤ r ,1 r ˆ B r ( x ) |∇ d | ≤ (cid:0) rr (cid:1) α r ˆ B r ( x ) |∇ d | + Cr.
In particular, we have lim r → r ˆ B r ( x ) |∇ d | = 0 , so that Θ ( µ, x ) = 0 and hence x / ∈ Σ. This proves claim 6.3.
Claim 6.4 . For H a.e. x ∈ Σ, Θ ( ν ; x ) = lim r → r ν ( B r ( x ))exists and δ ≤ Θ ( ν ; x ) ≤ C , and ν = Θ ( ν, · ) H LΣ.Since d ∈ H (Ω , S ), by Federer-Ziemmer’s theorem [21] that for H a.e. x ∈ ΩΘ (cid:0) |∇ d | , x (cid:1) := lim r → r ˆ B r ( x ) |∇ d | = 0 . (6.9)The conclusions of claim 6.4 then follow from this. See [10] or [20] for the detail. We have not used thecondition (i) in the definition of X ( L , L , a ; Ω) during the proof of the above claims. Next we employ thiscondition to show that ν ≡ Claim 6.5 . H (Σ) = 0 and ν ≡ d ǫ → d in H (Ω , R ).Suppose that H (Σ) >
0. Then, as in [10] and [18], since Θ ( ν, · ) is H -measurable, it is approximatelycontinuous for H a.e. x ∈ Σ. This, combined with (6.9) and the 1-rectifiability of Σ, implies that thereexists x ∈ Σ such thati) Θ ( ν, · ) is H -approximately continuous at x , and δ ≤ Θ ( ν, x ) ≤ C .ii) Σ has 1-dimensional tangent plane T x Σ at x .iii) Θ ( |∇ d | , x ) = 0. For simplicity, assume x = 0 ∈ Σ and T x Σ = (cid:8) (0 , , x ) : x ∈ R (cid:9) . For r i →
0, define e d i ( x ) = d ǫ i ( r i x ) and e τ i ( x ) = r i τ ǫ i ( r i x ) for x ∈ Ω i ≡ r − i Ω, and e ǫ i = ǫ i r i . Then we have∆ e d i + 1 e ǫ i (1 − | e d i | ) e d i = e τ i in Ω i . (6.10)By following the blow-up scheme outlined in [17], we can assume that after passing to possible subsequences,there exists a tangent measure µ ∗ of µ at 0 such that e e ǫ i ( e d i ) dx ⇀ µ ∗ , as convergence of Radon measures on R , and e d i ⇀ constant in H ( R , R ) . Moreover, µ ∗ = Θ ( ν, H L (cid:8) (0 , , x ) : x ∈ R (cid:9) . (6.11)Since e d i is a solution of the equation (6.10), e d i also satisfies the energy monotonicity formula (3.2). Inparticular, we haveΦ R ( e d i ; x ) ≥ Φ r ( e d i ; x ) + 12 ˆ B R ( x ) \ B r ( x ) | y − x | − (cid:16)(cid:12)(cid:12) ∂ e d i ∂ | y − x | (cid:12)(cid:12) + ( | − | e d i | ) e ǫ i (cid:17) , (6.12)for x ∈ Ω i and 0 < r ≤ R < d( x, ∂ Ω i ), whereΦ r ( e d i ; x ) := 1 r ˆ B r ( x ) (cid:0) e e ǫ i ( e d i ) − h ( y − x ) · ∇ e d i , e τ i i (cid:1) + 12 ˆ B r ( x ) | y − x || e τ i | , x ∈ Ω i , < r < d( x, ∂ Ω i ) . Since (cid:12)(cid:12)(cid:12) r − ˆ B r ( x ) h ( y − x ) · ∇ e d i , e τ i i (cid:12)(cid:12)(cid:12) ≤ Cr i and ˆ B r ( x ) | y − x || e τ i | ≤ Cr i , it follows that lim i →∞ Φ r ( e d i , x ) = 1 r µ ∗ ( B r ( x )) , (6.13)for x ∈ R and r >
0. It is clear that (6.13), (6.11), and (6.12) imply thatlim i →∞ ˆ B R ( x ) \ B r ( x ) | y − x | − (cid:16)(cid:12)(cid:12) ∂ e d i ∂ | y − x | (cid:12)(cid:12) + ( | − | e d i | ) e ǫ i (cid:17) = 0 , (6.14)holds for any x = (0 , , x ) ∈ T Σ and 0 < r < R . Applying (6.14) to two center points (0 , ,
0) and (0 , , i →∞ ˆ B × [ − , (cid:16)(cid:12)(cid:12) ∂ e d i ∂x (cid:12)(cid:12) + ( | − | e d i | ) e ǫ i (cid:17) = 0 . (6.15)Recall that the condition (i) of X ( L , L , a ; Ω) implies | e d i | ≤ , e d i ≥ − a. (6.16)Now we indicate how to produce a nontrivial harmonic map ω : R → S with finite energy. Define f i : [ − , → R + by f i ( t ) = ˆ B (cid:16)(cid:12)(cid:12) ∂ e d i ∂x (cid:12)(cid:12) + ( | − | e d i | ) e ǫ i (cid:17) ( x, t ) dx,g i : [ − , → R + by g i ( t ) = ˆ B e e ǫ i ( e d i )( x, t ) dx, and h i : [ − , → R + by h i ( t ) = ˆ B | e τ i | ( x, t ) dx. By Fubini’s theorem and (6.15), we havelim i →∞ (cid:13)(cid:13) f i (cid:13)(cid:13) L ([ − , = 0 and lim sup i →∞ (cid:13)(cid:13) h i (cid:13)(cid:13) L ([ − , ≤ L . D NEMATIC LIQUID CRYSTAL FLOW 17
Thus by the weak L -estimate of the Hardy-Littlewood maximal function we have that for any β > E β ⊂ [ − , ], with | E β | ≥ − β , such thatlim i →∞ sup
0) and λ i → + such that ˆ B λi ( x i ) e e ǫ i ( e d i )( x, dx = δ C (3) = max z ∈ B ˆ B λi ( z ) e e ǫ i ( e d i )( x, dx, (6.20)where C (3) > b d i ( x, x ) = e d i (( x i ,
0) + λ i ( x, x )) , ( x, x ) ∈ b Ω i := λ − i (cid:0) Ω i \ { ( x i , } (cid:1) . Then b d i solves ∆ b d i + 1 b ǫ i (1 − | b d i | ) b d i = b τ i in b Ω i , (6.21)where b ǫ i = e ǫ i λ i and b τ i ( x, x ) = λ i e τ i ( λ i x, λ i x ). It follows from (6.17), (6.18), and (6.20) thatlim i →∞ sup
2] so that b d ( x, x ) = b d ( x ) is independent of x . By (6.24) and (6.19), we have that δ C (3) ≤ ˆ R |∇ b d | ≤ Θ( ν, . (6.27) Hence b d ∈ ˙ H ( R , S ). Moreover, it follows from (6.23) and (6.27) that b h is a nontrivial smooth harmonicmap from R to S with finite energy, i.e.,∆ b d + (cid:12)(cid:12) ∇ b d (cid:12)(cid:12) b d = 0 in R . On the other hand, it follows from (6.16) that b d i ( x, x ) ≥ − a for ( x, x ) ∈ R × [ − , b d ( x ) ≥ − a, x ∈ R . In particular, deg( b d ) = 0. Since any nontrivial harmonic map from R to S with finite energy has non-zerodegree, we conclude that b d = constant. This yields the desired contradiction. Hence the conclusion of claim6.5 holds true. Claim 6.6 . Σ = ∅ , and d ∈ W , (Ω , S − a ).Suppose Σ = ∅ and x ∈ Σ. By the definition of Σ, we have thatlim i →∞ Φ r ( d ǫ i , x ) ≥ δ , ∀ r > . (6.28)It follows from claim 6.5 e ǫ i ( d ǫ i ) dx ⇀ |∇ d | dx as convergence of Radon measures as i → ∞ . In particular, d ǫ i → d in H (Ω , R ). Assume τ ǫ i ⇀ τ in L (Ω , R ). Then we know that d is an approximated harmonic map to S − a with tension field τ :∆ d + |∇ d | d = τ in Ω , (6.29)and d satisfies the energy monotonicity formula:Ψ R ( d ; x ) ≥ Ψ r ( d ; x ) + 12 ˆ B R ( x ) \ B r ( x ) | y − x | − (cid:12)(cid:12) ∂d ∂ | y − x | (cid:12)(cid:12) , (6.30)for x ∈ Ω and 0 < r ≤ R < d( x, ∂ Ω), whereΨ r ( d ; x ) := 1 r ˆ B r ( x ) (cid:0) |∇ d | − h ( y − x ) · ∇ d , τ i (cid:1) + 12 ˆ B r ( x ) | y − x || τ | . (6.31)It follows from (6.28) and (6.30) thatΨ( d , x ) := lim R ↓ Ψ R ( d , x ) ≥ δ . (6.32)For r i →
0, define the blow-up sequence of d at x , d i ( x ) = d ( x + r i x ) : B → S − a . Then we havelim i →∞ ˆ B |∇ d i | = lim i →∞ Ψ r i ( d , x ) ≥ δ . It is clear that d i is an approximated harmonic map with tension field τ i ( x ) = r i τ ( r i x ) such thati) d i ( B ) ⊂ S − a .ii) E ( d i ) = 12 ˆ B |∇ d i | ≤ C . iii) d i satisfies the energy monotonicity inequality (6.30), with d and τ replaced by d i and τ i .iv) k τ i k L ( B ) ≤ C √ r i . Hence { d i } ⊂ Y ( C , C √ r i , a ; B ). It follows from Theorem 7.1 below that there exists a harmonic map ω ∈ H ( B , S − a ) such that d i → ω in H ( B , R ) so that12 ˆ B |∇ ω | ≥ δ . Moreover, (6.30) implies that ˆ B (cid:12)(cid:12) ∂ω∂ | x | (cid:12)(cid:12) = 0 , so that ω ( x ) = ω ( x | x | ) is homogeneous of degree zero and ω : S → S − a is a nontrivial harmonic map. Thisis impossible. Hence Σ = ∅ and hence Lemma 5.4 implies d ∈ W , (Ω , S ). The proof is complete. (cid:3) D NEMATIC LIQUID CRYSTAL FLOW 19 H precompactness of suitable approximated harmonic map to S For 0 < a ≤ L , L >
0, define the set Y ( L , L , a ; Ω) consisting of maps d ∈ H (Ω , S ) that aresuitable approximated harmonic maps, i.e.,∆ d + |∇ d | d = τ in Ω (7.1)that satisfy, in addition to (5.4), the following properties:(i) d ( x ) ≥ − a for a . e . x ∈ Ω.(ii) E ( d ) = 12 ˆ Ω |∇ d | dx ≤ L .(iii) (cid:13)(cid:13) τ (cid:13)(cid:13) L (Ω) ≤ L . Theorem 7.1.
For any a ∈ (0 , , L > , and L > , the set Y ( L , L , a ; Ω) is precompact in H (Ω , S ) .In particular, if { d i } ⊂ Y ( L , L , a ; Ω) is a sequence of approximated harmonic maps, with tension fields { τ i } , then there exist τ ∈ L (Ω , R ) and an approximated harmonic map d ∈ Y ( L , L , a ; Ω) , with tensionfield τ , such that after passing to possible subsequences, d i → d in H (Ω , S ) and τ i ⇀ τ in L (Ω , R ) .Moreover, d ∈ W , (Ω , S ) .Proof. The proof is almost identical to that of Theorem 6.1. Here we only sketch it. Suppose that d i ⇀ d in H (Ω), but not strongly in H (Ω). Then there exists a Radon measure ν ≥ ν
0) such that12 |∇ d i | dx ⇀ µ := 12 |∇ d | dx + ν as convergence of Radon measures as i → ∞ . Define the concentration setΣ = \
0, with the help of Lemma 5.3 and Lemma 5.4, the same argument as in Theorem 6.1 yields(i) Σ is a 1-dimensional rectifiable, closed set, with H (Σ) > C > L and L such thatΣ = n x ∈ Ω : δ ≤ Θ ( µ, x ) ≤ C o , where Θ ( µ, x ) = lim r → Θ r ( µ, x ) (cid:0) = lim r → r µ ( B r ( x )) (cid:1) is the 1-dimensional density of µ at x .(iii) Σ = supp( ν ) ∪ sing( d ).(iv) For H a.e. x ∈ Σ, Θ (cid:0) |∇ d | dx, x (cid:1) = lim r → r ˆ B r ( x ) |∇ d | = 0 , and Θ ( ν, x ) = lim r → r ν (cid:0) B r ( x ) (cid:1) exists and equals to Θ ( µ, x ).As in claim 6.5, we can choose a generic point x ∈ Σ such thata) Θ ( |∇ d | dx, x ) = 0.b) Θ ( ν, · ) is H -approximately continuous at x and δ ≤ Θ ( ν, x ) ≤ C .c) Σ has 1-dimensional tangent plane T x Σ at x .Then we perform the blow-up procedure of d i at x exactly as what we did in claim 6.5 (we leave the detailto interested readers). As a consequence, we will obtain a harmonic map ω : R → S such that0 < ˆ R |∇ ω | < + ∞ , ω ( x ) ≥ − a, ∀ x ∈ R . This is impossible. Hence ν ≡ d i → d in H (Ω). Since Y ( L , L , a ; Ω) is closed under H (Ω)-convergence, we conclude that d ∈ Y ( L , L , a ; Ω).Now we want to show that Θ (cid:0) |∇ d | dx, x (cid:1) = 0 , ∀ x ∈ Ω . (7.2) For, otherwise, there exists x ∈ Ω and λ i → e d i ( x ) = d ( x + λ i x ) : B → S − a satisfies12 ˆ B |∇ e d i | = 12 λ i ˆ B λi ( x ) |∇ d | → Θ (cid:0) |∇ d | dx, x (cid:1) > i → ∞ . It is easy to see that { e d i } ⊂ Y ( L , L √ λ i , a ; B ). The compactness of Y ( L , L √ λ i , a ; B ) implies thatthere exists a harmonic map e d ∈ H ( B , S ), with e d ( x ) ≥ − a for x ∈ B , such that e d i → e d in H ( B ).Moreover, it follows from the monotonicity inequality (5.4) for e d i that ˆ B (cid:12)(cid:12) ∂ e d∂ | x | (cid:12)(cid:12) = 0 . Hence e d ( x ) = e d (cid:0) x | x | (cid:1) : S → S − a is a nontrivial harmonic map, which is impossible. This proves (7.2) andhence Lemma 5.4 yields d ∈ W , (Ω , S ). (cid:3) Global weak solutions of (1.1) and proofs of Theorem 1.1 and Theorem 1.3
In this section, we will utilize the existence of global solutions to the Ginzburg-Landau approximation(8.1) of the nematic liquid crystal flow (1.1) and the compactness Theorem 6.1 to show the existence ofglobal weak solutions to (1.1).For ǫ >
0, consider the modified Ginzburg-Landau approximation of (1.1): ∂ t u + u · ∇ u − ∆ u + ∇ P = −∇ · ( ∇ d ⊙ ∇ d ) in Ω × (0 , + ∞ ) , ∇ · u = 0 in Ω × (0 , + ∞ ) ,∂ t d + u · ∇ d = ∆ d + ǫ (1 − | d | ) d in Ω × (0 , + ∞ ) . (8.1)First, we have the following result on the existence of global solutions to (8.1). Theorem 8.1.
For any ǫ > , u ∈ H , and d ∈ H (Ω , S ) , there exists a global weak solution ( u ǫ , d ǫ ) :Ω × [0 , + ∞ ) → R × R of the equation (8.1) under the initial and boundary condition (1.2) that satisfies (i) | d ǫ | ≤ a.e. ( x, t ) ∈ Ω × [0 , + ∞ ) . (ii) the global energy inequality: there exists a measure zero set E ⊂ (0 , + ∞ ) such that for any ≤ t , t ∈ R \ E with t < t , ˆ Ω (cid:0) | u ǫ | + |∇ d ǫ | + 12 ǫ (1 − | d ǫ | ) (cid:1) ( t ) + 2 ˆ t t ˆ Ω (cid:16) |∇ u ǫ | + (cid:12)(cid:12) ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12) (cid:17) ≤ ˆ Ω (cid:0) | u ǫ | + |∇ d ǫ | + 12 ǫ (1 − | d ǫ | ) (cid:1) ( t ) . (8.2)(iii) If, in addition, d ( x ) ≥ a.e. x ∈ Ω , then d ǫ ( x, t ) ≥ a.e. ( x, t ) ∈ Ω × (0 , + ∞ ) .Proof. The existence is based on the Galerkin method and the energy method. The reader can refer to theproof presented by [13] §
2. The properties (i) and (iii) follow from Lemma 2.1 and Lemma 2.2. (cid:3)
Now we would like to study the convergence of sequence of solutions ( u ǫ , d ǫ ) constructed by Theorem 8.1as ǫ tends to zero. Proof of Theorem 1.1 . Since | d | = 1 and d ≥
0, it follows from Theorem 8.1 that for ǫ >
0, there existsglobal weak solutions ( u ǫ , d ǫ ) : Ω × [0 , + ∞ ) → R × R of (8.1) that satisfies all the three properties (i), (ii),and (iii). It follows from (8.2) thatsup ǫ> sup
Hence, by Aubin-Lions’ Lemma [25] we have that after taking possible subsequences, there exists u ∈ L ∞ t L x ∩ L t H x (Ω × R + , R ) and d ∈ L ∞ t H x (Ω × R + , S ) such that( u ǫ , d ǫ ) → ( u, d ) in L (Ω × R + ) , ( ∇ u ǫ , ∇ d ǫ ) ⇀ ( ∇ u, ∇ d ) in L t L x (Ω × R + ) . (8.5)It follows from (8.3) and Fatou’s lemma that ˆ ∞ lim inf ǫ → ˆ Ω (cid:12)(cid:12)(cid:12) ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12)(cid:12) ≤ C . (8.6)For Λ >>
1, define G T Λ := n t ∈ [0 , T ] : lim inf ǫ → ˆ Ω (cid:12)(cid:12)(cid:12) ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12)(cid:12) ( t ) ≤ Λ o , and B T Λ := [0 , T ] \ G T Λ = n t ∈ [0 , T ] : lim inf ǫ → ˆ Ω (cid:12)(cid:12)(cid:12) ∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ (cid:12)(cid:12)(cid:12) ( t ) > Λ o . Then by the weak L -estimate, we have (cid:12)(cid:12)(cid:12) B T Λ (cid:12)(cid:12)(cid:12) ≤ C Λ . (8.7)Now we have Claim 8.1 . For any t ∈ G T Λ , set τ ǫ ( t ) := (cid:0) ∆ d ǫ + ǫ (1 − | d ǫ | ) d ǫ (cid:1) ( t ). Then there exists τ ( t ) ∈ L (Ω , R ) suchthat, after passing to possible subsequences, τ ǫ ( t ) ⇀ τ ( t ) in L (Ω), and d ǫ ( t ) → d ( t ) in H (Ω) , e ǫ ( d ǫ ( t )) dx ⇀ |∇ d ( t ) | dx as convergence of Radon measures in Ω . (8.8)In particular, d ( t ) ∈ H (Ω , S ) is a suitable approximated harmonic map, with L -tension field τ ( t ).Since t ∈ G T Λ , it follows from the definition that lim inf ǫ → (cid:13)(cid:13) τ ǫ ( t ) (cid:13)(cid:13) L (Ω) ≤ Λ and hence there exists τ ( t ) ∈ L (Ω , R ) such that, after passing to possible subsequences, τ ǫ ( t ) ⇀ τ ( t ) in L (Ω). To show (8.8), recallthat by (8.3) we can assume, after passing to possible subsequences, d ǫ ( t ) ⇀ d ( t ) in H (Ω) and there existsa nonnegative Radon measure ν t in Ω such that e ǫ ( d ǫ ( t )) dx ⇀ |∇ d ( t ) | dx + ν t as convergence of Radon measures in Ω. It is easy to check from the definition of G T Λ that for any t ∈ G T Λ , { d ǫ ( t ) } ⊂ X ( C , Λ , a ; Ω) with a = 1. Hence Theorem 6.1 implies that ν t ≡ , d ǫ ( t ) → d ( t ) in H (Ω) , and 1 ǫ (cid:0) − | d ǫ ( t ) | (cid:1) → L (Ω) . (8.9)Hence d ( t ) ∈ H (Ω , S ) is an approximated harmonic map with tension field τ ( t ) ∈ L (Ω , R ). To see d ( t )is a suitable approximated harmonic map, observe that the same calculations as in Lemma 3.1 apply to Y · ∇ d ǫ ( t ) for any Y ∈ C ∞ (Ω , R ). Hence we obtain ˆ Ω (cid:16)(cid:10) ∂d ǫ ( t ) ∂x i , ∂d ǫ ( t ) ∂x j (cid:11) ∂Y i ∂x j − (cid:0) |∇ d ǫ ( t ) | + 14 ǫ (1 − | d ǫ ( t ) | ) (cid:1) div Y + (cid:10) τ ǫ ( t ) , Y · ∇ d ǫ ( t ) (cid:11)(cid:17) = 0 . (8.10)After sending ǫ →
0, (8.10), combined with (8.9), implies that d ( t ) satisfies the identity (5.3) and hence d ( t )is a suitable approximated harmonic map. The Claim 8.1 is proven. Claim 8.2 . For any subdomain e Ω ⊂⊂ Ω, it holds thatlim ǫ → ˆ e Ω × G T Λ |∇ ( d ǫ − d ) | = 0 . (8.11)We prove (8.11) by contradiction. Suppose (8.11) were false. Then there exist a subdomain e Ω ⊂⊂ Ω, δ > ǫ i → ˆ e Ω × G T Λ |∇ ( d ǫ i − d ) | ≥ δ . (8.12)Note that from (8.5) we have lim ǫ i → ˆ Ω × G T Λ | d ǫ i − d | = 0 . (8.13) By Fubini’s theorem, (8.12), and (8.13), 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 , (8.14)and ˆ e Ω (cid:12)(cid:12) ∇ ( d ǫ i ( t i ) − d ( t i )) (cid:12)(cid:12) ≥ δ T . (8.15)It is easy to see that (cid:8) d ǫ i ( t i ) (cid:9) ⊂ X ( C , Λ ,
1; Ω) and (cid:8) d ( t i ) (cid:9) ⊂ Y ( C , Λ ,
1; Ω). It follows from Theorem 6.1and Theorem 7.1 that there exist d , d ∈ Y ( C , Λ ,
1; Ω) such that d ǫ i ( t i ) → d and d ( t i ) → d in L (Ω) ∩ H ( e Ω) . This and (8.15) imply that ˆ e Ω (cid:12)(cid:12) ∇ ( d − d ) (cid:12)(cid:12) ≥ δ T . (8.16)On the other hand, from (8.14), we have that ˆ Ω | d − d | = 0 . (8.17)It is clear that (8.16) contradicts (8.17). Hence the Claim 8.2 is proven.Now we can use argument as in Lemma 4.1 Claim 4.4 to conclude that ˆ e Ω × G T Λ ǫ (1 − | d ǫ | ) → . (8.18)Combining (8.11) and (8.18) yields (cid:13)(cid:13)(cid:13) d ǫ − d (cid:13)(cid:13)(cid:13) L t H x (cid:0) e Ω × G T Λ (cid:1) + ˆ e Ω × G T Λ ǫ (1 − | d ǫ | ) → . (8.19)On the other hand, it follows from (8.3) and (8.7) that (cid:13)(cid:13)(cid:13) d ǫ − d (cid:13)(cid:13)(cid:13) L H x (cid:0) Ω × B T Λ (cid:1) + ˆ e Ω × B T Λ ǫ (1 − | d ǫ | ) ≤ (cid:16) t> ˆ Ω e ǫ ( d ǫ )( t ) (cid:17)(cid:12)(cid:12)(cid:12) B T Λ (cid:12)(cid:12)(cid:12) ≤ C Λ − . (8.20)Hence we have lim ǫ → h(cid:13)(cid:13) d ǫ − d (cid:13)(cid:13) L H x ( e Ω × [0 ,T ]) + ˆ e Ω × [0 ,T ] ǫ (1 − | d ǫ | ) i ≤ C Λ − . (8.21)Since Λ > ∇ d ǫ → ∇ d strongly in L (Ω × [0 , T ]) and1 ǫ (1 − | d ǫ | ) → L (Ω × [0 , T ]).Since ( u ǫ , d ǫ ) solves the equation (8.1) along with (1.2), it is standard that by utilizing (8.21) and (8.5)we can show that ( u, d ) is a weak solution of the equation (1.1) and (1.2). The global energy inequality(1.4) for ( u, d ) follows from (8.2), with t = 0 and t = t >
0, by sending ǫ to 0, with the help of the lowersemicontinuity and the observation that∆ d ǫ + 1 ǫ (1 − | d ǫ | ) d ǫ = ∂ t d ǫ + u ǫ · ∇ d ǫ ⇀ ∂ t d + u · ∇ d in L (Ω × [0 , T ]) . This completes the proof of Theorem 1.1. (cid:3)
Proof of Theorem 1.3 . The proof is similar to that of Theorem 1.1. Here we only sketch it. First, itfollows from the equation (1.1) and the condition (1.5) that there exists p > < T < + ∞ ,sup k h k ∂ t u k k L (Ω × [0 ,T ])+ L ([0 ,T ] ,H − (Ω))+ L ([0 ,T ] ,W − ,p (Ω)) + k ∂ t d k k L (Ω × [0 ,T ]) i < + ∞ . (8.22)Hence, by Aubin-Lions’ Lemma [25] we have that after taking to possible subsequences, there exists u ∈ L ∞ t L x ∩ L t H x (Ω × [0 , T ] , R ) and d ∈ L ∞ t H x (Ω × [0 , T ] , S ) such that( u k , d k ) → ( u, d ) in L (Ω × [0 , T ]) , ( ∇ u k , ∇ d k ) ⇀ ( ∇ u, ∇ d ) in L t L x (Ω × [0 , T ]) . (8.23) D NEMATIC LIQUID CRYSTAL FLOW 23
It follows from (1.5) and Fatou’s lemma that ˆ ∞ lim inf k →∞ ˆ Ω (cid:12)(cid:12) ∆ d k + |∇ d k | d k (cid:12)(cid:12) ≤ C . (8.24)For Λ >>
1, define G T Λ := n t ∈ [0 , T ] : lim inf k →∞ ˆ Ω (cid:12)(cid:12) ∆ d k + |∇ d k | d k (cid:12)(cid:12) ( t ) ≤ Λ o , and B T Λ := [0 , T ] \ G T Λ = n t ∈ [0 , T ] : lim inf k →∞ ˆ Ω (cid:12)(cid:12) ∆ d k + |∇ d k | d k (cid:12)(cid:12) ( t ) > Λ o . Then by the weak L -estimate, we have (cid:12)(cid:12)(cid:12) B T Λ (cid:12)(cid:12)(cid:12) ≤ C Λ . (8.25)Now we have Claim 8.3 . For L a.e. t ∈ G T Λ , set τ k ( t ) = (∆ d k + |∇ d k | d k )( t ). Then there exists τ ( t ) ∈ L (Ω , R ) suchthat, after passing to subsequences, τ k ( t ) ⇀ τ ( t ) in L (Ω), and d k ( t ) → d ( t ) in H (Ω) . (8.26)In particular, d ( t ) ∈ H (Ω , S − a ) is a suitable approximated harmonic map, with tension field τ ( t ).Since t ∈ G T Λ , we have lim inf k →∞ (cid:13)(cid:13) τ k ( t ) (cid:13)(cid:13) L (Ω) ≤ Λ and hence there exists τ ( t ) ∈ L (Ω , R ) such that, aftertaking a subsequence, τ k ( t ) ⇀ τ ( t ) in L (Ω). To show (8.26), recall that by (1.5) we can assume, afterpassing to subsequences, d k ( t ) ⇀ d ( t ) in H (Ω) and there exists a nonnegative Radon measure ν t in Ω suchthat 12 |∇ d k | ( t ) dx ⇀ |∇ d ( t ) | dx + ν t as convergence of Radon measures in Ω. It is easy to check from the definition that for L a.e. t ∈ G T Λ , { d k ( t ) } is a family of suitable approximated harmonic map such that { d k ( t ) } ⊂ Y ( C , Λ , a ; Ω) with 0 < a ≤ ν t ≡ d k ( t ) → d ( t ) strongly in H (Ω). Hence d ( t ) ∈ H (Ω , S − a ) isa suitable approximated harmonic map. This proves the Claim 8.3. Claim 8.4 . For any subdomain e Ω ⊂⊂ Ω, it holds thatlim k →∞ ˆ e Ω |∇ ( d k − d ) | = 0 . (8.27)Similar to the proof of Claim 8.2, (8.27) can be proven by contradiction. For, otherwise, there exist e Ω ⊂⊂ Ω, δ > k l → ∞ such that ˆ Ω × G T Λ | d k l − d | → , and ˆ e Ω × G T Λ |∇ ( d k l − d ) | ≥ δ . By Fubini’s theorem, there exists { t l } ⊂ G T Λ such that ˆ Ω | d k l ( t l ) − d ( t l ) | → , and ˆ e Ω |∇ ( d k l ( t l ) − d ( t l )) | ≥ δ T . (8.28)Since { d k l ( t l ) } , { d ( t l ) } ⊂ Y ( C , Λ , a ; Ω) for 0 < a ≤
2, it follows from Theorem 7.1 that there exist d , d ∈ Y ( C , Λ , a ; Ω) such that d k l ( t l ) → d , d ( t l ) → d in L (Ω) ∩ H (Ω) . Hence, by (8.28), we obtain ˆ Ω | d − d | = 0 , and ˆ e Ω |∇ ( d − d | ≥ δ T .
This is impossible. Thus we obtain (cid:13)(cid:13)(cid:13) d k − d (cid:13)(cid:13)(cid:13) L t H x (cid:0) e Ω × G T Λ (cid:1) → . (8.29)On the other hand, it follows from (1.5) and (8.25) that (cid:13)(cid:13)(cid:13) d ǫ − d (cid:13)(cid:13)(cid:13) L H x (cid:0) e Ω × B T Λ (cid:1) ≤ (cid:16) sup t> ˆ Ω |∇ d k | ( t ) (cid:17)(cid:12)(cid:12)(cid:12) B T Λ (cid:12)(cid:12)(cid:12) ≤ C Λ − . (8.30) Hence we have lim k →∞ (cid:13)(cid:13) d k − d (cid:13)(cid:13) L H x ( e Ω × [0 ,T ]) ≤ C Λ − . (8.31)Since Λ > d k → d strongly in H (Ω × [0 , T ]). Since( u k , d k ) solves the equation (1.1), it is standard that by utilizing (8.31) and (8.23) we can show that ( u, d )is a weak solution of the equation (1.1). (cid:3) Acknowledgements . The first author is partially supported by NSF grants DMS1065964 and DMS1159313.The second author is partially supported by NSF grants DMS1001115 and DMS 1265574 and NSFC grant11128102.
References [1] D. Adams,
A note on Riesz potentials . Duke Math. J. , 765-778 (1975).[2] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of Navier-Stokes equations.
Comm. PureAppl. Math. , 771-831 (1982).[3] P. G. de Gennes, The Physics of Liquid Crystals. Oxford, 1974.[4] L. C. Evans, Partial regularity for stationary harmonic maps into spheres.
Arch. Rational Mech. Anal. , 101-113(1991).[5] J. L. Ericksen,
Hydrostatic theory of liquid crystals.
Arch. Ration. Mech. Anal. , 371-378 (1962).[6] F. H´elein, Harmonic Maps, Conservation Laws and Moving Frames, second ed., Cambridge Tracts in Math., vol. 150,Cambridge Univ. Press, Cambridge, 2002.[7] M. C. Hong, Global existence of solutions of the simplified Ericksen-Leslie system in dimension two.
Calc. Var. PartialDifferential Equations , no. 1-2, 15-36 (2011).[8] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace.
Acta. Math. , 183-248 (1934).[9] F. M. Leslie, Some constitutive equations for liquid crystals.
Arch. Ration. Mech. Anal. , 265-283.[10] F. H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena.
Comm. Pure Appl.Math. , no. 6, 789-814 (1989).[11] F. H. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem . Comm. Pure Appl. Math. (3) 241-257 (1998).[12] F. H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps . Ann. of Math. (2) (3) 785-829(1999).[13] F. H. Lin, C. Liu,
Nonparabolic dissipative systems modeling the flow of liquid crystals . Comm. Pure Appl. Math.
XLVIII ,501-537 (1995).[14] F. H. Lin, C. Liu,
Partial regularity of the dynamic system modeling the flow of liquid crystals.
Dis. Cont. Dyn. Syst. (1),1-22 (1998).[15] F. H. Lin, J. Y. Lin, C. Y. Wang, Liquid crystal flows in two dimensions.
Arch. Ration. Mech. Anal. , no. 1, 297-336(2010).[16] F. H. Lin, C. Y. Wang,
On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals.
Chin. Ann. Math. Ser. B no. 6, 921-938 (2010).[17] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres . Comm. Anal. Geom. , no. 2, 397-429 (1999).[18] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. II.
Comm. Anal. Geom. , no. 2, 341-375 (2002).[19] F. H. Lin, C. Y. Wang, Harmonic and quasi-harmonic spheres. III. Rectifiability of the parabolic defect measure andgeneralized varifold flows.
Ann. Inst. H. Poincar Anal. Non Linaire , no. 2, 209-259 (2002).[20] 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.[21] H. Federer, W. Ziemmer, The Lebesgue set of a function whose distribution derivatives are p -th power summable . IndianaUniv. Math. J. (1972/73), 139-158.[22] C. B. Morrey, Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band130 Springer-Verlag New York, Inc., New York 1966 ix+506 pp.[23] D. Preiss, Geometry of measures in R n : distribution, rectifiability, and densities . Ann. of Math. (2) (3), 537-643 (1987).[24] E. Stein, Singular integrals and differentiability properties of functions. Princeton Univ. Press, 1970.[25] R. Temam, Navier-Stokes equations. Studies in Mathematics and its Applications, Vol. 2, North Holland, Amsterdam,1977.[26] C. Y. Wang, Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data.
Arch.Ration. Mech. Anal. no. 1, 1-19 (2011).[27] X. Xu, Z. F. Zhang,
Global regularity and uniqueness of weak solution for the 2-D liquid crystal flows.
J. DifferentialEquations no. 2, 1169-1181 (2012).
Courant Institute of Mathematical Sciences, New York University, NY 10012 and NYU-ECNU Institute ofMathematical Sciences, at NYU Shanghai, 3663, North Zhongshan Rd., Shanghai, PRC 200062
E-mail address : [email protected] Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA
E-mail address ::