Global well-posedness for the two-dimensional equations of nonhomogeneous incompressible liquid crystal flows with nonnegative density

In this paper, the authors first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in two dimensions if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. We also present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtained, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition (cf. [19])


Introduction
Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional liquid and those of a solid crystal (cf. [12]). The hydrodynamic theory of liquid crystals was first developed by Ericken and Leslie during the period of 1958 through 1968 (see [9,10,20,21]). Since then, many remarkable developments have been made from both theoretical and applied aspects, however, many physically important and mathematically fundamental problems still remain open. In this paper, we consider a simplified Ericken-Leslie model for the nonhomogeneous incompressible nematic liquid crystals in two dimensions: (ρu) t + div(ρu ⊗ u) + ∇P = ∆u − ∇d · ∆d, (1.2) divu = 0, (1.3) d t + u · ∇d = ∆d + |∇d| 2 d, (1.4) where ρ : R 2 × [0, ∞) → R + is the density of the fluid, u : R 2 × [0, ∞) → R 2 is the velocity field of the fluid, P : R 2 × [0, ∞) → R is the pressure of the fluid, and d : R 2 × [0, ∞) → S 2 (the unit sphere in R 3 , i.e. |d| = 1) represents the averaged macroscopic/continuum molecular orientations.
Though system (1.1)-(1.4) is a simplified version of the Ericksen-Leslie model, but it still retains the most interesting mathematical properties without losing the basic nonlinear structure of the original Ericksen-Leslie model [9,10,20,21]. Roughly speaking, the system (1.1)-(1.4) is a system of the nonhomogeneous Navier-Stokes equations for incompressible flows coupled with the equation for heat flow of harmonic maps, and thus, its mathematical analysis is full of challenges. In particular, if ρ = Const., then it turns into the following homogeneous system which models the incompressible flows of nematic liquid crystal u t + u · ∇u + ∇P = µ∆u − ∇d · ∆d, There has been a lot of literature on the mathematical studies of (1.5)-(1.7) and (1.8), see, for example, [14,13,23,24,25,26,33,35] and [3,5,4,6,32], respectively. In the following, we briefly recall some related mathematical results of the liquid crystal flows. In a series of papers, Lin [23] and Lin-Liu [24,25] initiated the mathematical analysis of (1.5)-(1.7) in 1990s. More precisely, to relax the nolinear constraint |d| = 1, they proposed an approximate model of Ericksen-Leslie system with variable length by Ginzburg-Landau functionals, that is, the equation (1.7) with |d| = 1 is replaced by d t + u · ∇d = ∆d + 1 ε 2 1 − |d| 2 d. (1.9) In [23,24], the authors proved the global existence of classical and weak solutions of (1.5), (1.6), (1.9) in dimensions two and three, respectively. The partial regularity of suitable weak solutions was also studied in [25]. However, as pointed out in [24], the vanishing limit of ε → 0 is an open and challenging problem. Indeed, in contrast with (1.9), it is much more difficult to deal with the nonlinear term |∇d| 2 d with |d| = 1 appearing on the right-hand side of (1.4) or (1.7) from the mathematical point of view. In two independent papers [13] and [26], Hong and Lin-Lin-Wang showed the global existence of weak solutions of (1.5)-(1.7) in dimensions two, and proved that the solutions are smooth away from at most finitely many singular times which is analogous to that for the heat flows of harmonic maps (see [3,32]). The global existence of smooth solution with small initial data of (1.5)-(1.7) was also proved [26,33] and [35,22] in dimensions two and three, respectively. For the approximate nonhomogeneous equations (1.1)-(1.3) and (1.9), the global existence of weak solutions with generally large initial data was proved in [28,16], and the global regularity of the solution with strictly positive density was studied in [8]. As aforementioned, the nonlinear term |∇d| 2 d with |d| = 1 will cause serious difficulty in the mathematical analysis of liquid crystal flows. Recently, Wen and Ding [34] established the local existence and uniqueness of strong solutions of (1.1)- (1.4) in the case that the initial density may contain vacuum states (i.e. ρ 0 ≥ 0). Moreover, if the initial density has a positive lower bound (i.e. ρ 0 ≥ ρ > 0) which indicates that there is absent of vacuum initially, the global strong solutions with small initial data was also obtained in [34].
As that for the density-dependent Navier-Stokes equations (see [7,27]), the possible presence of vacuum is one of the major difficulties when the problems of global existence, uniqueness and regularity of solutions are involved. Therefore, in the present paper we aim to investigate the global regularity of (1.1)-(1.4) when the initial density may contain vacuum.
We consider the Cauchy problem of (1.1)-(1.4) with the following initial data: 10) and the far-field behavior at infinity: whereρ > 0 is a given positive constant and e ∈ S 2 is a given unit vector (i.e. |e| = 1).  Then, our first result concerning the global strong solutions with small data can be stated in the following theorem. . (1.14) It is worth mentioning that the smallness condition (1.14) stated in Theorem 1.1 implies that (ρ 0 , u 0 ) can be arbitrarily large if ∇d 0 L 2 is chosen to be suitably small. This is analogous to the one in [33]. Moreover, as a result, we see that the strong solution to the Cauchy problem of nonhomogeneous Navier-Stokes equations (i.e. d = Const.) with large initial data, which may contain vacuum, exists globally on R 2 × [0, T ] for all 0 < T < ∞. Thus, Theorem 1.1 also generalizes the result due to Huang-Wang [14].
The proof of Theorem 1.1 is mainly based on a critical Sobolev inequality of logarithmic type which was recently proved by Huang-Wang (cf. [15]) and is originally due to Brezis-Wainger [2] (see also [29,31]). However, it is remarkable that the arguments in [15] actually depend on the size of the domain considered and cannot be applied directly to the case of the whole space. Thus, some new ideas have to be developed. The main difference lies in the proof of Lemma 3.3, where, instead of ρ 1/2 u t L 2 and ρ 1/2 u · ∇u L 2 , we use the material derivative ρ 1/2u L 2 for some technical reasons. We also note here that the strictly positive far-field conditionρ > 0 plays an important role in our analysis. The strongly nonlinear terms |∇d| 2 d and ∇d · ∆d in (1.2) and (1.4) will also cause some additional difficulties.
For the generally large initial data, it is still an interesting and open problem whether the strong solution blows up or not in finite time. In [26] and [14], the authors proved respectively that the following blowup criteria for the two-dimensional equations of (1.5)-(1.7): where 0 < T * < ∞ is the maximal time of the existence of a strong solution to (1.5)-(1.7). Motivated by the proofs of Theorem 1.1, we can prove the following mechanism for possible breakdown of strong solutions, which is a natural extension of the ones in [26,14].
Based on a frequency localization argument combined with the concentration-compactness approach, Lei-Li-Zhang [19] recently proved the following interesting rigidity theorem for the approximate harmonic maps.

Proposition 1.1 ([19, Theroem 1.5]) For given positive constants
Then there exists a positive constant δ 0 ∈ (0, 1), which depends only on C 0 and ε, such that As an immediate consequence of Theorem 1.2 and Proposition 1.1, we can remove the smallness restriction (1.14) on the initial data and prove the following existence theorem of global strong solutions with large initial data, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition. This extends the Lei-Li-Zhang's result (cf. [19]) to the case of nonhomogeneous incompressible liquid crystal flows with initial vacuum. The rest of the paper is organized as follows. In Sect. 2, we state some known inequalities and facts which will be used later. The proof of Theorem 1.1 will be done in Sect. 3, based on the local existence theorem and the global a priori estimates. In Sect. 4, we outline the proof of Theorems 1.2 and 1.3.

Preliminaries
In this section, we list some useful lemmas which will be frequently used in the next sections. We first recall the well-known Ladyzhenskaya and Sobolev inequalities (see, for example, [17,1]).
We will also use the following Poincaré type inequality, which shows that the velocity u actually belongs to L 2 -space even that the vacuum states may appear.
Proof. Indeed, by virtue of Hölder and (2.2), we have for any q ≥ 2 that which proves (2.4) immediately.
Next, to improve the regularity of the velocity, we need to use the following estimates of the Stokes equations (see, for example, [11,18]).

Lemma 2.3 Consider the following stationary Stokes equations:
Then for any f ∈ W m,p (R 2 ) with m ∈ Z + and p > 1, there exists a positive constant C, depending only on m and p, such that To estimate the L 2 -norm of the gradient of the velocity, we shall apply a critical Sobolev inequality of logarithmic type which was prove by Huang-Wang (cf. [15]) and is originally due to Brezis-Wainger [2] (see also [29,31]). This is the key tool for the proofs of Theorems 1.1-1.3.
Then there exists a positive constant C(q), independent of s, t, such that In the case that the lower bound of the density is nonnegative, the local existence of strong solutions to (1.1)-(1.4), (1.10) and (1.11) was proved in [34]. Indeed, in [34] the authors only considered the case of smooth bounded domains, however, as pointed out in [7], the similar procedure also works for the whole space by means of the standard domain expansion technique. For simplicity, we quote the following local existence theorem of strong solutions without proofs.  (1.11). In order to prove Theorem 1.1, it suffices to prove there actually exists a generic positive constant 0 < M < ∞, depending only on the initial data (ρ 0 , u 0 , d 0 ) and T * , such that holds for any 0 < T < T * . So, by the local existence theorem (see Lemma 2.5) it can be easily shown that the strong solution can be extended beyond T * , which gives a contradiction of T * . Hence, the strong solution exists globally on R 2 × [0, T ] for any 0 < T < ∞. The proof of Theorem 1.1 is therefore complete. The proof of (3.1) is based on a series of lemmas. Throughout the remainder of the paper, for simplicity we denote by C a generic constant which depends only on the initial data and T * , and may change from line to line.
First, it is easy to see from the method of characteristics and (1.1) that for every 0 < T < T * , Moreover, multiplying (1.1) by q|ρ −ρ| q−2 (ρ −ρ) with q ≥ 2, integrating it by parts over (0, t), and using the divergence-free condition (1.3), we find that In view of (1.1)-(1.4), we have the following standard energy estimates.
Lemma 3.1 For every 0 < T < T * , one has Proof. Multiplying (1.2) by u in L 2 and integrating by parts, by (1.3) we know that Due to the fact that |d| = 1, multiplying (1.4) by (∆d+|∇d| 2 d) in L 2 , we obtain after integrating the resulting equations by parts over R 2 that 1 2 which, combined with (3.5), immediately leads to (3.4).
To be continued, we need the following key estimates on ∇ 2 d L 2 (0,T ;L 2 ) .

Lemma 3.2 Assume that the initial data satisfies
then it holds for every T ∈ (0, T * ) that Proof. After integrating by parts, we easily deduce from the identity |d| = 1 that On the other hand, integration by parts, together with the divergence-free condition (1.3), gives where and in what follows the repeated indices denotes the summation over the indices. Putting (3.9), (3.10) into (3.6) and recalling the fact that which, combined with (2.1) and the Cauchy-Schwarz inequality, yields It follows from (3.7) that and thus, by the local existence theorem and the continuity argument we see that there exists a T 1 > 0 such that for any t ∈ [0, T 1 ], SetT sup{T | (3.12) holds}.
By Lemmas 2.4 and 3.2, we can now derive the estimates of ∇u L 2 and ∇ 2 d L 2 which is the most important step among the proofs.

14)
which particularly gives Proof. Letḟ f t + u · ∇f denote the material derivative. Also set then it is easily seen that ∇d · ∆d = div(M (d)).
To prove (3.14), multiplying (1.2) by u t and integrating it by parts over R 2 , we deduce where we have also used (3.2) and Cauchy-Schwarz inequality. As a result, Next, one easily obtains from (1.4) that To deal with the term ∇d t 2 L 2 on the right-hand side of (3.16), we first apply ∇ to both sides of (1.4) to get that ∇d t − ∇∆d = −∇(u · ∇d) + ∇(|∇d| 2 d), (3.18) from which it follows that where we have used (2.1) and (3.4) to get that
Next, we proceed to estimate ρ 1/2 u t L 2 and ∇d t L 2 .
Lemma 3.4 For every 0 < T < T * , one has

28)
and moreover, Proof. Differentiating (1.2) with respect to t gives which, multiplied by u t in L 2 and integrated by parts over R 2 , results in We are now in a position of estimating the right-hand side of (3.30) term by term. First, using (1.1) and integrating by parts, by Lemma 2.1, (3.2) and (3.14) we deduce where we have used Cauchy-Schwarz inequality and the following estimate due to Lemma 2.1 and (3.14): (3.31) Due to (2.4), (3.2) and (3.3), we have 32) and thus, by (2.1), (3.2) and (3.14) the second term I 2 can be bounded as follows: Finally, it is easily seen from (2.3) and (3.14) that Substituting the estimates of I 1 , I 2 and I 3 into (3.30), one obtains To estimate ∇d t L 2 , we differentiate (1.4) with respect to t to get and hence, using Lemma 2.1 and (3.14), we deduce after direct calculations that where we have also used (3.32) and the following estimate due to (2.1), (2.3) and (3.14): . Now, multiplying (3.33) by 2C 1 + 1 and adding it to (3.34), we see that L 2 , which, combined with (3.14), (3.15) and Gronwall's inequality, leads to (3.28), since the compatibility condition stated in (1.13) 2 implies that (ρ 1/2 u t )(x, 0) ∈ L 2 (R 2 ) is well defined.
To conclude, we have proved that there exists a positive constant C, depending on the initial data, T * and M 0 , such that for any 0 < T < T * ,