GLOBAL STRONG SOLUTION FOR THE INCOMPRESSIBLE FLOW OF LIQUID CRYSTALS WITH VACUUM IN DIMENSION TWO

. This paper is devoted to the study of the initial-boundary value problem for density-dependent incompressible nematic liquid crystal ﬂows with vacuum in a bounded smooth domain of R 2 . The system consists of the Navier-Stokes equations, describing the evolution of an incompressible viscous ﬂuid, coupled with various kinematic transport equations for the molecular orienta-tions. Assuming the initial data are suﬃciently regular and satisfy a natural compatibility condition, the existence and uniqueness are established for the global strong solution if the initial data are small. We make use of a critical Sobolev inequality of logarithmic type to improve the regularity of the solution. Our result relaxes the assumption in all previous work that the initial density needs to be bounded away from zero.

1. Introduction. In this paper, we establish the well-posedness of the Ericksen-Leslie model of nematic liquid crystals formulated by Ericksen in [10,11,12] and Leslie [20] in the 1960's. A simplified version of the Ericksen-Leslie model was introduced by Lin [22] and successfully analyzed by Lin-Liu [23]- [25] who used a modified Galerkin approach, and by Shkoller [31] who relied on a contraction mapping argument coupled with appropriate energy estimates. When the Ossen-Frank energy configuration functional reduces to the Dirichlet energy functional, the hydrodynamic equations of liquid crystals in R 2 can be written as follows (see [21,22,26]): (ρu) t + ∇ · (ρu ⊗ u) + ∇P = µ u − λ∇ · (∇d ∇d), where ρ ∈ R and u ∈ R 2 represent the density and the velocity field of the flow, d ∈ S 2 (the unit sphere in R 3 ) denotes the unit-vector field that represents the macroscopic molecular orientation of the liquid crystal material, P ∈ R is the for all T ∈ (0, ∞) with large initial data under the assumption that u 0 ∈ L 2 (Ω), d 0 ∈ H 1 (Ω) with d 0 | ∂Ω ∈ H 3 2 (∂Ω) in the two-dimensional and three-dimensional cases. For any fixed ε, the existence and uniqueness of global classical solution was also obtained if u 0 ∈ H 1 (Ω) and d 0 ∈ H 2 (Ω) either for dimension two or for dimension three when the fluid viscosity µ is large enough. Taking the limit of ε → 0 after discussing the cases for each ε. This technique has been successfully used in a lot of other places (see [24,25]). It also fits well with the general theory of Landau's order parameter (see [20]). The partial regularity of the weak solution was investigated in [25] (and also [6,9,28]), similar to the classical theorem by Caffarelli-Kohn-Nirenburgh [3] on the Navier-Stokes equations that asserts the one-dimensional parabolic Hausdorff measure of the singular set of any suitable weak solution is zero. For the density-dependent (nonhomogeneous) case, Liu [27] proved the global existence of weak solutions and classical solutions to the system of Smectic-A liquid crystals under the general condition of the initial density ρ 0 satisfying 0 < α ≤ ρ 0 ≤ β. The global existence of weak solutions in dimension three was established by Liu-Zhang [29] if ρ 0 ∈ L 2 (Ω). Later Jiang-Tan [17] pointed out that the condition on initial density can be weaken to belong to L γ (Ω) for any γ ≥ 3 2 . With the Ginzburg-Landau penalty function, the global weak solution and large-time behavior to the compressible flow of liquid crystals were obtained in [33]. However, they cannot get the estimates uniformly for ε, and therefore cannot take the limit ε → 0.
In this paper, we are interested in the existence and uniqueness of global strong solution (ρ, u, d, P ) of the initial-boundary value problem of system (1) in a bounded smooth domain Ω ⊂ R 2 with the initial and boundary conditions: with d 0 ∈ S 2 being given with compatibility, ∇ · u 0 = 0 in Ω and d 0 ∈ C 1 (Ω) satisfying ∇d 0 = 0 on the boundary ∂Ω (see [15]). Note that the Dirichlet boundary condition for the velocity field implies non-slip on the boundary arising from viscous flows are known to stick to the kinematic boundary. Compared with the Ginzburg-Landau approximation problem, |∇d| 2 in (1d) brings us some new difficulties. Since the strong solutions of a harmonic map must be blowing up at finite time (see Chang-Ding-Ye [4] for the heat flow of harmonic maps), one cannot expect that there exists a global strong solution to system (1)-(3) with general initial data. For the homogeneous case of system (1)-(3), both the regularity and existence of global weak solutions were established by Lin-Lin-Wang [26]. More explicitly, they obtained both interior and boundary regularity theorem for such a flow under smallness conditions, and the existence of global weak solutions that are smooth away from at most finitely many singular times in any bounded smooth domain of R 2 . The reader is referred to Theorems 1.2, 1.3 in [26] for the details. Meanwhile, Hong [14] also showed the global existence of weak solution to this system in two dimensional space. Wang [32] established a global well-posedness theory for the incompressible liquid crystals for rough initial data, provided that Assuming that the initial density ρ 0 has a positive bound from below and under smallness conditions on the initial data, Wen-Ding [34] got the global existence and uniqueness of the strong solution to (1)-(3) in Sobolev spaces in dimension two. For nonhomogeneous Navier-Stokes equations with initial density being bounded from above and below by some positive constants, Paicu-Zhang-Zhang [30] established the global well-posedness of solution when u 0 ∈ H s (R 2 ) for s > 0 in dimension two, or u 0 ∈ H 1 (R 3 ) satisfying u 0 L 2 ∇u 0 L 2 being sufficiently small in dimension three. This result improves the work in [8], which required u 0 ∈ H 2 (R d ) for the existence and uniqueness of local solution, and a smallness condition prescribed on the fluctuation of the initial density for the global solution.
Throughout this paper, we denote L q = L q (Ω), W k,q = W k,q (Ω) and H k = W k,2 and introduce . We make the assumptions: Observing the system (1) has the following scaling invariant property. If ρ(x, t), u(x, t), d(x, t) and P (x, t) solve (1), then, for each l > 0, ρ l = ρ(lx, l 2 t), u l = lu(lx, l 2 t), d l = d(lx, l 2 t) and P l = l 2 P (lx, l 2 t) also solve (1). One can encode this property by assigning a scaling dimension to each quantity as follows: each x i has dimension 1, time variable t has dimension 2, ρ has dimension 0, u has dimension −1, d has dimension 0, P has dimension −2, ∂ xi has dimension −1 and ∂ t has dimension −2. This dimension analysis of equations enable us to build up various quantities, so called "energy"'s, associated with the solutions of (1) and is very important to the discussion of the partial regularity property of the solutions.
Note that ∇ · (∇d ∇d) = ∇ · (∇d) ∇d = ∇( |∇d| 2 2 ) + (∇d) ∆d, by using the fact ∇ · u = 0 in Ω, we rewrite (1) as The paper is written in the following way. In Section 2, we recall three useful lemmas and state our main result. In Section 3, we establish the basic energy law governing the system and employ the method that involves using higher order energy estimates to establish that the local strong solution does not blow up in finite time. In other words, the local strong solution can be extended to be a global one. And, the proof of the uniqueness property for the strong solutions is a standard procedure.
Throughout of the whole paper, sometimes, we make use of A B in place of A ≤ C 0 B, where C 0 stands for a "harmless" constant whose exact meaning depends on the context, and A ≈ B means that A B and B A.

2.
Useful lemmas and the main results. The following result is quite standard in dimension two (as a matter of fact, it is a straightforward generalization of the one presented in [19], also see Lemma 2.4 in [34] and Ladyzenskaya [18]): If Ω is a bounded smooth domain in R 2 , then the following inequality is true for every function f ∈ H 1 : Next, let us recall the classical regularity theory for Stokes equations (see [13] for its proof): where the constant C depends on Ω and q. Moreover, if F ∈ (H 1 ) 2 , then where the constant C depends only on Ω.
Finally, we state a critical Sobolev inequality of logarithmic type, which is originally due to Brezis-Wainger [2]. The reader is referred to Section 2 in [16] for the proof.
where the constant C depends only on Ω and q, and is independent of s, t .
Our main result about the global existence of a unique strong solution for the initial boundary value problem reads: Let Ω ⊂ R 2 be a bounded smooth domain. Assume the initial data (ρ 0 , u 0 , d 0 ) satisfies the regularity (4) and the compatibility condition Remark 1. We would like to point out if d is a constant map, then (1) reduces to the nonhomogeneous Navier-Stokes equations. A straightforward application of Theorem 2.4 is that there exists a unique global strong solution enjoying the above regularity property for the 2D density-dependent Navier-Stokes system with vacuum, which erases the assumption that the initial density is strictly positive (see [1]).
3. Proof of Theorem 2.4. Note that the local existence of a unique strong solution with vacuum to (1)-(3) in a bounded domain of R 2 can be established following the procedure of Wen-Ding [34] or Choe-Kim [5]. Therefore, this paper is devoted to showing global estimates for the density, velocity and the molecule orientational direction vector. We give some a priori estimates globally in time to get Theorem 2.4. The method here is to find a new energy inequality involving the following quantities: We can then show without much difficulty that T * is not the maximum time provided (6) holds, i.e., we can extend the strong solution beyond T * , which is the desired contradiction. With the regularities of the solution we get, the proof of the uniqueness by the energy estimate of the difference between two different solutions is straightforward(see Step 4 in [34]). Now we outline the proof of Theorem 2.4 into several steps: Step 1. (L ∞ bound for ρ). As (5a) is a transport equation for ρ, it follows easily from Proposition 3.1 in [7] that Step 2. (Basic energy inequality). We derive the energy inequality: multiplying (1b) by u, integrating over Ω, then multiplying (1d) by (∆d + |∇d| 2 d) and integrating over Ω and finally, by adding two results above and using (1a), noticing that that, by taking advantage of |d| = 1, and that, as d t = 0 on ∂Ω, we get the identity SOLUTION TO LIQUID CRYSTAL FLOWS IN TWO DIMENSIONS   4913 for all t ∈ (0, T * ), which implies for a.a. t ∈ (0, T ), 0 < T < T * .
Multiplying (5b) by u t and integrating over Ω, it yields Let us estimate the terms on right-hand of (9). By virtue of Hölder's inequality and Young's inequality, we have where (7) is used, and Here we have used u | ∂Ω = 0, and the fact that ∆d L 2 ≈ ∇ 2 d L 2 given ∇d 0 = 0 on the boundary ∂Ω. Finally, we conclude Similarly, multiplying (5d) by ∆d t integrating over Ω, using Hölder's inequality and Young's inequality, we obtain, bearing in mind the facts that |d| = 1, u = 0 and ∇d 0 = 0 on ∂Ω, that Next, taking advantage of Lemma 2.1, we have Setting then, (8) implies that for any 0 < t ≤ T , Multiplying (11) byCE 0 + λ, integrating the result with respect to time, we have, for every 0 ≤ s < T < T * , where C 0 is the constant included in (11). Integrating (10) with respect to time, for every 0 ≤ s < T < T * , we have Noticing that, using integration by parts, one can easily get which means, by using Lemma 2.1, that Finally, taking advantage of the estimates (13), (14) and (16), we conclude that, for every 0 ≤ s < T < T * , where C 0 is the constant included in (11). Recalling the energy balance (8) that sup t ∇d 2 L 2 ≤ 1 λ E 0 , one can easily from inequality (17) to get here C depends onC, C 0 , E 0 , λ and the regularity parameters. Denote by the scaling argument used in Section 1, we can see the new energy ∇u(t) 2 L 2 + ∆d(t) 2 L 2 contained in Ψ(t) is of higher order than E(t). Higher order estimates of the density, velocity and the molecule orientational can be done in a standard way once (u, ∇d) H 1 is uniformly bounded with respect to time. Now, for every 0 ≤ s < T < T * , one can easily from (12) and (18) to get To continue, we need to get estimates on the norms u L 2 (s,T ;L ∞ ) and ∇d L 2 (s,T ;L ∞ ) . From Lemma 2.3, we have + ∇d 2 L 2 (s,T ;H 1 ) )(ln + u L 2 (s,T ;W 1,4 ) + ln + ∇d L 2 (s,T ;W 1,4 ) ). (20) Applying the operator ∇ to (5d), we get γ∆(∇d) = ∇d t + ∇(u · ∇d) − γ∇(|∇d| 2 d).
It follows from the classical regularity theory for Stokes equations, as indicated by Lemma 2.2, and for elliptic equations that

XIAOLI LI
where, similar to C 3 , the constant C 4 depends only on C 2 .
Differentiating (5b) with respect to time, taking inner product with u t and integrating by parts, bearing in mind (1a) and (1c), we derive Now we will estimate the right-hand side of (31) term by term, using Hölder's inequality and Cauchy's inequality, we get Putting all the estimates above together and combining with (31), it yields to Similarly, as for (5d), we simply take t-derivative, multiply the result by ∆d t , integrate over Ω and use integration by parts to obtain Now, we have, by (32) and (33), In accordance with (27), (29), (30), then it comes directly from Gronwall's inequality that With (34), if the inequality (25) is reconsidered, one has Step 6. (Estimates for ∇ρ L ∞ (0,T ;H 1 ) and (u, ∇d) L 2 (0,T ;H 3 ) ). First, we prove u enjoys higher regularity in space. By using Lemmas 2.1,2.2 and (7), one has u W 2,4