Global Existence and Stability for a Hydrodynamic System in the Nematic Liquid Crystal Flows

In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solutions. In order to figure out the relation between the solution obtained here and weak solution of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.


Introduction
In this paper, we study the following hydrodynamical system modeling the flow of nematic liquid crystal in R n : (1.4) where u and P denote the velocity field and the pressure of the flow, respectively, d denotes the (averaged) macroscopic/continuum molecule orientation field, ν, λ, γ are positive constants, and f (d) is a Ginzburg-Landau approximation function. The notation ∇d ⊙ ∇d denotes the n × n matrix whose (i, j)-th entry is given by ∂ i d · ∂ j d (1 ≤ i, j ≤ n). As in [8] and [14], we assume f (d) = 0 for simplicity. Besides, since the size of the viscosity constants ν, λ and γ do not play a special role in our discussion, we assume that they all equal to the unit.
The above system (1.1)-(1.4) describes the time evolution of nematic liquid crystal materials (cf. [12]). Equation (1.1) is the conservation of linear momentum (the force balance equation). Equation (1.2) is the conservation of angular momentum, in which the left hand side represents the kinematic transport by the flow field, while the right hand side represents the internal relaxation due to the elastic energy. Finally, equation (1.3) represents the incompressibility of the fluid. This system was first introduced by Lin [12] as a simplified version of the liquid crystal model proposed by Ericksen in [3] and Leslie in [10], and retained most of the interesting mathematical properties of the liquid crystal model. Some basic results concerning the mathematical theory of this system were obtained by Lin and Liu in [14] and [15]. More precisely, in [14] they proved global existence of weak solutions by using the modified Galerkin method combined with some compactness argument. Moreover, they also proved global existence of strong solutions if the initial data is sufficiently small (or if the viscosity ν is sufficiently large). In [15] they proved that the one-dimensional space-time Hausdorff measure of the singular set of "suitable" weak solutions is zero. Recently, by using the maximal regularity of Stokes equations and the parabolic equations, Hu and Wang [8] proved global existence of strong solutions to the system (1.1)-(1.4) for initial data belonging to Besov spaces of positive-order under the smallness assumption. Here we also refer the reader to see [4], [6], [11], [13], [16], [17], [7], [19] and the references therein for more details of the physical background of this problem and some different models of similar equations.
As in the work of Hu and Wang [8], we let F = ∇d. Then, taking the gradient of (1.2), noticing the facts that F ⊙ F = F T F (F T denotes the transpose of F ) and for all i, k = 1, 2, · · · , n, and applying the Leray-Hopf projector P to eliminate the pressure P , we see that the system (1.1)-(1.4) can be reduced into the following system: where F 0 = ∇d 0 . Recall that P = I + ∇(−∆) −1 div, i.e., P is the n × n matrix pseudodifferential operator in R n with the symbol (δ ij − ξiξj |ξ| 2 ) n i,j=1 , where I represents the unit operator and δ ij is the Kronecker symbol. Later on we shall consider the Cauchy problem (1.5)-(1.7).
The purpose of this paper is to prove global existence and stability of solutions to the problem (1.5)-(1.7) in the critical Besov spaceḂ −1+n/p p,q (R n ) of negative-order. It is easy to verify that (1.5) and (1.6) have the same scaling property as the Navier-Stokes equations (which are equations obtained by putting d = 0 in (1.1)-(1.3)), namely, if (u, F ) is a solution of (1.5) and (1.6) with initial data (u 0 , F 0 ), then for any δ > 0, by letting we see that (u δ , F δ ) is a solution of (1.5) and (1.6) with initial data (δu 0 (δx), δF 0 (δx)). The so-called critical space for the equations (1.5) and (1.6) is a function space of (u, F ) such that the norm in it is invariant under the scaling (1.8). The so-called self-similar solutions are solutions satisfying the scaling relation u(t, x ∈ R n and t ≥ 0). Obviously, if (u, F ) is a self-similar solution, then we must have Such initial data do not belong to any Lebesgue and Sobolev spaces due to their strong singularity at x = 0 as well as slow decay as |x| → ∞, however, they belong to some homogeneous Besov spaces with negative order. This is the reason why we study the problem (1.5)-(1.7) in the critical Besov space of negative-order. By making use of Chemin-Lerner's time-space estimates of the heat equation and the Fourier localization technique, we shall prove that when initial data (u 0 , F 0 ) belongs to the critical Besov spaceḂ −1+n/p p,q (R n ) for some suitable p and q, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we get existence of self-similar solutions; see Section 3. In order to figure out the relation between the solution obtained here and the weak solution studied by Lin and Liu in [14], we shall prove a stability result, which yields in a direct way that all global weak solutions with the same initial data must coincide with solutions obtained here, which is called the weak-strong uniqueness.
Then the global solution (u, F ) constructed in Theorem 1.1 is a self-similar solution. Theorem 1.3. (Blow-up criterion) Under the hypotheses of Theorem 1.1, we denote by T * the maximum existence time. If T * < ∞, then for any 2 ≤ p < 2n and 2 < q < ∞ satisfying For initial data (u 0 , F 0 ) ∈Ḃ −1+n/p p,q (R n ), by Theorem 1.1, there exists T > 0 such that the system (1.5)-(1.7) has a unique solution (u, F ) ∈ ∩ 1<q≤∞ L q (0, T ;Ḃ −1+n/p+2/q p,r (R n )). If (u 0 , F 0 ) is additionally in the space L 2 (R n ), then it is not difficult to see that (u, F ) is also a weak solution (in the standard sense, see [14]). A natural question is the following: Do all weak solutions coincide with the one we obtained in Theorem 1.1? In order to answer this question, we establish the following stability theorem. Theorem 1.4. (Stability) Let n ≥ 2, 2 ≤ p < ∞, 2 < q < ∞ and n p + 2 q > 1. Assume that (u 0 , F 0 ) and (ũ 0 ,F 0 ) be two vector fields in L 2 (R n ) such that div u 0 = 0 and divũ 0 = 0, and (ũ,F ) and (u, F ) be two weak solutions associated with initial data (ũ 0 ,F 0 ) and It is clear that if u 0 =ũ 0 and F 0 =F 0 , then Theorem 1.4 implies that w = E = 0, i.e., u =ũ and F =F . Hence, we have the following weak-strong uniqueness result for the system (1.5)-(1.7).
, then all weak solutions associated with initial data (u 0 , F 0 ) must coincide with (u, F ) on the time interval [0, T ).
Organization of the paper. In Section 2, we recall some basic facts about Littlewood-Paley decomposition and Besov spaces. In Section 3, we present the proof of Theorem 1.1, which yields existence of global self-similar solutions. In Section 4, we prove Theorem 1.3. Section 5 is devoted to the proof of Theorem 1.4.

Preliminaries
We first recall some basic notions and preliminary results used in the proof of our main results. Let S(R n ) be the Schwartz space and S ′ (R n ) be its dual. Given f ∈ S(R n ), the Choose two non-negative functions φ, ψ ∈ S(R n ) supported, respectively, in D 1 and D 2 such that Then the dyadic blocks ∆ j and S j can be defined as follows: is a frequency projection to the annulus {|ξ| ∼ 2 j }, and S j = k≤j−1 ∆ k is a frequency projection to the ball {|ξ| ≤ 2 j }.
Remark 2.1. The above definition does not depend on the choice of the couple (φ, ψ). Recall that if either s < n p or s = n p and q = 1, then (Ḃ s p,q (R n ), · Ḃs p,q ) is a Banach space.
Let us now state some basic properties for the homogeneous Besov spaces.
There exists a constant C independent of f and j such that for all 1 ≤ p ≤ q ≤ ∞, the following estimate holds: ) Let k ∈ Z + and |α| = k for multi-index α. There exists a constant C k depending only on k such that for all s ∈ R and 1 ≤ p, r ≤ ∞, the following estimate holds: We now recall the definition of the Chemin-Lerner space L r (0, T ;Ḃ s p,q (R n )): The following product between functions will enable us to estimate nonlinear terms appeared in (1.5) and (1.6). [18]) Let 1 ≤ p, q, r, q 1 , q 2 ≤ ∞, s 1 , s 2 < n p , s 1 + s 2 > 0 and 1 q = 1 q1 + 1 q2 . Then there exists a positive constant C depending only on s 1 , s 2 , p, q, r, q 1 , q 2 and n such that Notations: The product of Banach spaces X × Y will be equipped with the usual norm (f, g) X ×Y = f X + g Y , and if X = Y, we use (f, g) X to denote by (f, g) X ×X . For two n × n matrixes A = (a ij ) n i,j=1 and B = (b ij ) n i,j=1 , we denote A : B = n i,j=1 a ij b ij . Throughout the paper, C stands for a generic constant, and its value may change from line to line.

Local and global existence of solution
In this section we prove Theorem 1.1. Let p and r be as in Theorem 1.1, i.e., 2 ≤ p < 2n and 1 ≤ r ≤ ∞, and let 1 < q ≤ ∞. We choose a number 2 < q 1 ≤ 2q such that 2 q1 + n p > 3 2 . For a constant T > 0 to be specified later, we denote X T = L q1 (0, T ;Ḃ −1+n/p+2/q1 p,r (R n )). In order to establish the desired estimates in X T , let us recall the solvability of the Cauchy problem of the heat equation: (3.1) ) Let s ∈ R and 1 ≤ p, q 1 , r ≤ ∞, and let T > 0 be a real number. Assume that u 0 ∈Ḃ s p,r (R n ) and f ∈ L q1 (0, T ;Ḃ s+2/q1−2 p,r (R n )). Then the Cauchy problem (3.1) has a unique solution Moreover, there exists a constant C > 0 depending only on n such that for any q 1 ≤ q ≤ ∞, Besides, if u 0 belongs to the closure of S(R n ) inḂ s p,r (R n ), then u ∈ C([0, T ),Ḃ s p,r (R n )).
We also recall an existence and uniqueness result for an abstract operator equation in a generic Banach space. For the proof we refer the reader to see Lemarié-Rieusset [9].  9]) Let X be a Banach space and B : X × X → X is a bilinear bounded operator, · X being the X -norm. Assume that for any u 1 , u 2 ∈ X , we have Then for any y ∈ X such that y X ≤ ε < 1 4C0 , the equation u = y +B(u, u) has a solution u in X . Moreover, this solution is the only one such that u X ≤ 2ε, and depends continuously on y in the following sense: if y X ≤ ε, u = y + B( u, u) and u X ≤ 2ε, then Now for given (u, F ) ∈ X T , we define by G(u, F ) = (ū,F ), where (ū,F ) is a solution of the following linear equations: Proposition 3.3. Let (u, F ) ∈ X T . Then we have (ū,F ) ∈ X T . In addition, the following estimates hold: Proof. We prove only the results forū, it can be done analogous forF . By the Duhamel principle, (3.3) can be transformed into the following equivalent integral equations: Since we have assumed 2 ≤ p < 2n, 2 < q 1 < ∞ and n p + 2 q1 > 3 2 , we can apply Lemma 2.5 by choosing s 1 = −1 + n p + 2 q1 and s 2 = −2 + n p + 2 q1 and Lemma 2.3 to obtain that u · ∇u L q 1 /2 (0,T ;Ḃ −3+n/p+4/q 1 . (3.9) Hence, from Proposition 3.1 we know u ∈ X T . Moreover, by using the boundedness of P in the homogeneous Besov spaces, (3.8) and (3.9), we get This proves Proposition 3.3.
The Proposition 3.3 implies that G is well-defined and maps X T into itself. Moreover, from (3.6) and (3.7), we know there exists a constant C 0 > 0 such that for all (u, F ) ∈ X T and (ū,F ) = G(u, F ), we have the following estimate: Case 1. (The small initial data). Taking T = ∞ and denoting X = X ∞ . From Proposition 3.1, there exists a constant C 1 such that we can rewrite (3.11) as follows: Now if we choose ε > 0 sufficiently small such that C 1 (u 0 , F 0 ) Ḃ −1+n/p p,r ≤ ε < 1 4C0 , that is to say, (u 0 , F 0 ) Ḃ −1+n/p p,r ≤ ε C1 < 1 4C0C1 , then by Proposition 3.2, the system (1.5)-(1.7) has a global solution.
Hence, for any q1 2 ≤ q ≤ ∞, we have Finally, we consider the uniqueness of solution. Note that in Proposition 3.2 we obtained only a partial answer to the uniqueness problem of solution, i.e., in the closed ball B 2ε , the solution of (1.5)-(1.7) is unique. Now we intend to get rid of this restrictive condition.
Let (u, F ) and (ũ,F ) be two solutions of (1.5)-(1.7) in X T associated with initial data (u 0 , F 0 ) and (ũ 0 ,F 0 ), respectively. Set w = u −ũ and E = F −F . Then (w, E) satisfies the following equations: As the proof of Proposition 3.3, we can prove that Hence, by Proposition 3.1, we get Denoting M (T ) := C 0 u XT + ũ XT + F XT + F XT . By the Lebesgue dominated convergence theorem, we know that M (T ) is a continuous nondecreasing function vanishing at zero. Hence, if we choose T 1 sufficiently small such that M ( (3.15) Repeating the above procedure to the interval [0, T 1 ), [T 1 , 2T 1 ), . . . enables us to conclude that there exists a constant C such that This implies the uniqueness result immediately.

The proof of Theorem 1.3
Let 2 ≤ p < 2n, 1 ≤ r ≤ ∞, 2 < q < ∞ such that n p + 2 q > 3 2 . Assume that (u, F ) L q (0,T ;Ḃ −1+n/p+2/q p,r ) < ∞. By the embedding relation (2.4) and the proof of Theorem 1.1 we see that It suffices to prove that if (u, F ) L q (0,T ;Ḃ −1+n/p+2/q p,r ) < ∞, then T * > T . In other words, if T * < ∞, then (u, F ) L q (0,T ;Ḃ −1+n/p+2/q p,r ) = ∞. To this end, for any t ∈ [0, T ), we take (u(x, t), F (x, t)) as a new initial data of the problem (1.5)-(1.7), and split u( Similarly, we split F (x, t) := F 1 (x, t) + F 2 (x, t). Since 2 ≤ p < 2n, by using the properties of the Besov spaces, there exists a sufficiently large constant N ∈ N such that On the other hand, if we chooseT > t such that then we can obtain (e t∆ u 2 , e t∆ F 2 ) Xt+T ε ≤ ε 2 . This result together with (4.1), by Proposition 3.2, yield that there exists a constant T ε depending only on ε and M such that for any t ∈ [0, T ), the problem (1.5)-(1.7) has a solution on the time interval [t, t + T ε ). By the uniqueness we know that all solutions obtained in this way are equal in their common existence interval, so that the solution can be extended to the time interval [0, T + T ε ). That is to say T * > T , we complete the proof of Theorem 1.3.

Stability and weak-strong uniqueness
The aim of this section is to prove Theorem 1.4. Let us recall the definition of weak solutions to the system (1.5)-(1.7).
is the usual homogeneous Sobolev space.
(2) (u, F ) satisfies the system (1.5)-(1.7) in the distributional sense, i.e., divu = 0 in the distributional sense and for all v ∈ C ∞ 0 (R n × (0, T )) and G ∈ C ∞ 0 (R n × (0, T )) with divv = 0, we have (3) The following energy inequality holds: Remark 5.1. Formally, taking v = u and G = F , and adding them together, we get which implies the above energy inequality. Here we have used the fact AB : C = A : CB T = B : A T C for any three n × n matrixes A, B and C, , and we denote by (ũ,F ) and (u, F ) two weak solutions in the space WS associated with the initial data (ũ 0 ,F 0 ) and (u 0 , F 0 ), respectively. Assume that (u, F ) ∈ L q (0, T ;Ḃ −1+n/p+2/q p,q (R n )), where 2 ≤ p < ∞ and 2 < q < ∞ satisfying n p + 2 q > 1. Obviously, the above energy inequality yields that (ũ(t),F (t)) 2 Note that by (5.1) and (5.2), Here (·|·) denotes the scalar product in L 2 (R 2 ). In order to prove Proposition 5.2, we need to introduce the following lemma.
We shall use Lemma 1.1 in [5] to prove Lemma 5.3.

Lemma 5.4. ([5])
Let n ≥ 2, 2 ≤ p < ∞ and 2 < q < ∞ such that n p + 2 q > 1. Then for every T > 0, the trilinear form is continuous. In particular, the following estimate holds:  [5] in detail, we find that the special structure of the Navier-Stokes equations (div u = 0) was not used, and (5.5) holds in both scalar and vector cases.
The proof of Lemma 5.3.
We split the proof into the following two steps.