A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

In this paper, we establish a new blowup criterion for the strong solution to the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{3} $\end{document} . Specifically, we obtain the blowup criterion in terms of \begin{document}$ \|P\|_{L^\infty_t BMO_{x}} $\end{document} and \begin{document}$ \|\nabla d\|_{L^s_t L^\infty_x} $\end{document} , for any \begin{document}$ s>3 $\end{document} . The appearance of vacuum could be allowed.

In this paper, we will consider the following initial-boundary conditions: (ρ, u, d) where ν is the unit outer normal vector of ∂Ω.
To begin with, let us briefly review some previous works. In dimension one, Ding-Lin-Wang-Wen [4] and Ding-Wang-Wen [5] have proven the existence of global strong solution and weak solution respectively. In dimension two, Liu-Zheng-Li-Liu [15] obtained the local existence of strong solution to the Cauchy problem. In dimension two or three, Jiang-Jiang-Wang [11] has proven the global existence of weak solution to the initial-boundary problem with large initial energy. Liu-Qing [14], Wang-Yu [17] and Lin-Lai-Wang [12] also studied the existence of weak solution to the three-dimensional case. Under the condition that the initial data is close to a constant equilibrium state in H N (R 3 ) (N ≥ 3), Gao-Tao-Yao [6] obtained the global existence and uniqueness of classical solution. While Huang-Wang-Wen [8] proved the local existence of strong solution provided that the initial data (ρ 0 , u 0 , d 0 ) was sufficiently regular.
But, whether the strong solutions could exist globally in two or more dimensions is still an outstanding open problem. It is necessary to study the blowup mechanism of the nematic liquid crystal flow system (1.1)-(1.3). For three dimension, Huang-Wang-Wen [8] built up the following blowup criterion (1.8) under the assuption 7µ > 9λ, lim sup T ↑T * ρ L ∞ (0,T ;L ∞ ) + ∇d L 3 (0,T ;L ∞ ) = ∞, (1.8) where 0 < T * < +∞ is the maximum existence time for strong solution. At the same time, Huang-Wang-Wen [9] gave another blowup criterion lim sup where D(u) = ∇u+(∇u) T 2 is the deformation tensor. While Huang-Wang [10] obtained a Serrin-type blowup criterion lim sup where 2 si + n ri ≤ 1, n < r i ≤ ∞, i = 1, 2 and n is the spatial dimension. For two dimensions, Gao-Tao-Yao [7] established a Serrin-type blowup criterion for initialboundary value problem of (1.1)-(1.3) as follows, More recently, Liu-Wang [13] gave a blowup criterion of the strong solution to the 2D Cauchy problem in terms of density only lim sup Inspired by the papers [8] and [9] on nematic liquid crystal flows and the paper [2] on the MHD system and the paper [3] on a viscous liquid-gas two-phase flow model, we will establish in this paper a new blowup criterion for the strong solution to the initial and boundary problem (1.1)-(1.3) subject to (1.6) and (1.7). The blowup criterion we obtain is in terms of P L ∞ t BM Ox and ∇d L s t L ∞ x , for any s > 3. Before stating our main result, let us introduce some notations and the definition of strong solution. Notations.
Our main results are as follows.
Proof of (1.15). With the fact that P is globally Lipschitz continuous, we have Therefore, we finish the proof of (1.15).
We will prove Theorem 1.2 by the contradiction argument. Assume that (1.14) were false, that is, there is some s 0 , which can be infinitely close to but bigger than 3, and a positive constant M such that lim sup Since Ω is bounded, (1.16) implies that, there is some s 0 , which can be infinitely close to but bigger than 3 such that lim sup for any 1 ≤ p < ∞ and some positive constant M 1 . We aim to prove that lim sup where C is a generic positive constant which may depend on µ, λ, Ω, ρ 0 , u 0 , d 0 , M , M 1 , T * and C i (i = 1, 2, 3, 4, 5). (1.18) together with Theorem 1.3 in Huang-Wang-Wen [8] implies that T * is not the maximum existence time, which is the desired contradiction. The paper is organized as follows. In Section 2, we will present two important fundamental results. In Section 3, we will prove Theorem 1.3.

2.
Preliminaries. For completeness, we will firstly present the existence and uniqueness of local strong solution to the initial and boundary value problem (1.1)-(1.3), (1.6) together with (1.7) in the following theorem. The proof can be found in Huang-Wang-Wen [8].
Theorem 2.1. Assume that P : [0, +∞) → R is a locally Lipschitz continuous function, ρ 0 ∈ W 1,q (Ω) for some q ∈ (3, 6], u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω), ∇d 0 ∈ H 2 (Ω) and |d 0 | = 1 in Ω. If, in additions, the following compatibility condition Next we would like to introduce a variant of the Brezis-Waigner's inequality which will paly a crucial role when we establish the L ∞ t L q x -estimate of ∇P . (see [16]) . Then there exists a constant C depending on p and the Lipschitz property of the domain Ω such that Here BM O(Ω) denotes the John-Nirenberg's space of bounded mean oscillation whose norm is defined by with the semi-norm is the ball with center x and radius r and d is the diameter of Ω. For a measurable subset E of R N , |E| denotes its Lebesgue measure and 3. Proof of Theorem 1.2. The proof of Theorem 1.2 is based on several lemmas. for any 1 ≤ p < ∞.
With (1.17), we can obtain the standard energy estimate as follows. The proof can be found in Huang-Wang-Wen [8], for simplicity, we omit the detail.
Then we can derive the following estimate of ∇ 2 d L 2 (0,T * ;L 2 ) .
Proof. Applying (3.2) and (1.16), we see that Taking H 2 regularity estimate for d with Neumann boundary condition, we have Therefore, the proof is completed.
In order to obtain a high order estimate of u, we would adopt the approach by decomposing the velocity into two parts (see [16]). Specifically, we denote u = w+v, where v = L −1 ∇(P (ρ)) is the solution of the Lamé system: We can use Proposition 2.1 of [16] and (1.17) to obtain that Moreover from [1] and (1.16), we find Next, we aim to prove Lemma 3.3 in the following, which plays an important role in proving Theorem 1.2. , there exists some r > 5, such that for any 0 ≤ t < T * , Since the proof for Lemma 3.3 is somewhat complicated, for better illustration, we will first prove Lemmas 3.4-3.6 and based on these three lemmas we can finally prove Lemma 3.3.
Lemma 3.4. Under the assumptions of Theorem 1.2 and (1.17), given any r > 5, which can be infinitely close to 5, there is a constant C such that for any 0 ≤ t < T * , Proof. Multiplying (3.5) 1 by w t , integrating over Ω, and using integration by parts and Cauchy's inequality, we have We then estimate I 1 and I 3 respectively. First of all, we can easily use Cauchy's inequality to show that (3.11) As for I 1 , different from what are shown in Huang-Wang-Wen [8], we need to overcome the difficulty arising from the lack of the assumption for ρ L ∞ . By Minkowski inequality, we obtain that Before estimating I 11 , we want to emphasize that given any r > 5, we can choose p 1 ∈ ( 2r r−2 , 6) and ). Thanks to this observation, we can prove that where we have used Hölder's inequality, Young's inequality, interpolation inequality, (3.1) and (3.6). Since p 2 < 2, we can then use Hölder's inequality, Proposition 2.1 in [16] and (3.5) to obtain Substituting (3.14) into (3.13), we find that for any ε > 0, and (3.17) Putting (3.15), (3.16) and (3.17) into (3.12), we finish the estimate for I 1 by choosing ε sufficiently small,  where m(r) = 2(2r−1)(r−2)

3(r+1)
. Clearly m(5) = 3 and m (r) > 0 for r > 5, which means that for the given constant s 0 > 3 in (1.17), we have 3 < m(r) ≤ s 0 when r is sufficiently close to 5. On the other hand, given λ < 7µ 9 , we find that if r is close enough to 5, then It turns out that we can choose some r > 5, which can be infinitely close to 5, such that 3 < m(r) ≤ s 0 and the above inequality holds. Meanwhile we let ε sufficiently small such that Lemma 3.6. Under the assumptions of Theorem 1.2, given any r > 5, which can be infinitely close to 5, we have that for any 0 ≤ t < T * , where we could take n = 6r r+1 . Proof. The proof of this lemma can be divided into two steps. The first is to estimate |∇d| n dx and the second is to estimate |∇ 2 d| 2 dx.
Next let us turn to the estimate of |∇ 2 d| 2 dx. Again thanks to |∇d| 2 = −d·∆d, we have Then we multiply (3.24) by ∇d t , integrate it by parts over Ω and use (3.26) to obtain 1 2 What left is to estimate the last term on the right hand side of the above inequality. Before doing so, we need to find the upper bound of ∇ 3 d 2 L 2 . By applying H 3estimate of elliptic equations to (1.3) and taking use of (3.26), we have (3.28) Now we come back to estimate the last term on the right hand side of (3.27). Taking use of Hölder's inequality and Nirenberg's interpolation inequality, we have where we have used the fact that 12r 7r−5 < 2, 4r 2r−5 < 4 due to r > 5. putting (3.28) into (3.29) and choosing ε sufficiently small, we have (3.30) Substituting (3.30) into (3.27), using (3.6), and choosing ε sufficiently small, we obtain d dt Based on Lemmas 3.4-3.6, we will prove Lemma 3.3 in the following.
In what follows, we will estimate √ ρu L ∞ (0,T ;L 2 ) and ∇u L 2 (0,T ;L 2 ) . Before that, we will present Corollary 2, which will be used in the later proof.
Proof. (1.2) can be written as ρu + ∇(P (ρ)) = Lu − ∇d · ∆d. (3.39) Differentiating (3.39) with respect to t, we obtain Testing (3.40) byu, we derive 1 2 d dt ρ|u| 2 dx + µ|∇u| 2 + (µ + λ)|divu| 2 dx = ((P (ρ)) t divu + u ⊗ ∇(P (ρ)) : ∇u) dx + (µ + λ) div (∇div u ⊗ u) − ∇div (u · ∇u) ·u dx + µ (div (∆u ⊗ u) − ∆(u · ∇u)) ·u dx + (u ⊗ (∆d · ∇d)) : ∇u dx (3.41) By the same calculations as that in Huang-Wang-Wen [8], we have As for J 1 , we should be careful due to the lack of the assumption about the upper bound of ρ. Taking use of Equation (1.1), (1.5) and (3.1), we find that Substituting the aforementioned inequalities for J i , i = 1, 2, 3, 4, 5 into (3.41) and using Cauchy inequality as well as Sobolev's inequality, we have In order to derive the estimate of ∇d t 2 L 2 , we differentiate (1.3) with respect to t, multiply the resulting equation by d tt and then integrate it over Ω to obtain where we have used ∂dt ∂ν ∂Ω = 0. By the same calculations as that in Huang-Wang-Wen [8], we can easily estimate K 1 , which is given by When we come to estimate K 2 , we can not just follow the steps as that in Huang-Wang-Wen [8]. In Corollary 2, difficulty arises because we can not derive the estimate for ∇u L 2 (0,T ;L 6 ) . By the definition ofu, Hölder's inequality, Sobolev's inequality, Corollary 2 and Young's inequality, we estimate K 2 as follows Putting the estimates of K 1 and K 2 into (3.43), and then combining the resulting inequality together with (3.42), we have d dt What left is to estimate the first two terms on the right hand side of (3.42). Firstly by applying H 3 -estimate of elliptic equations to (1.3) as well as taking use of Lemma3.3, Corollary 2 and Nirenberg's interpolation inequality, we have (3.45) Next let us get started to estimate ∇u 4 L 4 . Observe that w satisfies Lw = ρu + ∆d · ∇d.
Proof. It follows from (3.45), (3.49) and Lemma 3.7 that sup 0≤t<T * By Sobolev's inequality, we see that From (3.46) and (3.1), we obtain Therefore, by Lemma 3.7 and Sobolev's inequality, we have We complete the proof.
With Corollary 3, we are ready to prove the upper bound for ∇P L q with q > 3, which is important to derive the desired contradiction. Unlike the proof for lemma 3.7 of [2] where the upper bound for ∇ρ L q was obtained due to the given boundedness for ρ L ∞ t BM Ox in the contradiction arguments, here we have to derive the upper bound in terms of the pressure gradient. It turns out that the Brezis-Waigner's inequality (2.2) and the BM O estimate for the Lamé system play important parts. Proof. Multiplying the equation (1.1) by P (ρ) and then differentiating with respect to x j , and then multiplying both sides of the resulting equation by q|∇P | q−2 ∂ j P , we get (3.54) Integrating (3.54) over Ω, by (1.5), we obtain d dt |∇P | q dx ≤ C (1 + ρ|P | 1 |P | )|∇u| |∇P | q + C ρ|P | |∇div u| |∇P | q−1 ≤ C |∇u||∇P | q + C P |∇div u||∇P | q−1 (3.55) ≤ C ∇u L ∞ ∇P q L q + C P ∇ 2 u L q ∇P q−1 L q .