On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals

This work is concerned with the solvability of a Navier-Stokes/$Q$-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence of local in time strong solution to the system with the anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.


Introduction
Nematic liquid crystal is a sort of material which may flow as a conventional liquid while the molecules are oriented in a crystal-like way. One of the successful continuum theories modeling nematic liquid crystals is the Q-tensor theory, also referred to as Landau-de Gennes theory, which uses a 3×3 traceless and symmetric matrix-valued function Q(x) as order parameters to characterize the orientation of molecules near material point x (cf. [11]). The matrix Q, also called Q-tensor, can be interpreted as the second momentum of a number density function where f (x, m) corresponds to the number density of liquid crystal molecules which orient along the direction m near material point x. The configuration space for Q-tensor will be denoted by The classic Landau-de Gennes theory associates to each Q(x) a free energy of the following form: Q). (1.2) Here and in the sequel, we shall adopt the Einstein's summation convention by summing over repeated greek letters. In (1.2), F b (Q) is the bulk energy, describing the isotropic-nematic phase transition while F e (∇Q, Q) is the elastic energy which characterizes the distortion effect. The parameters a, b, c are temperature dependent constants with b, c > 0, and L 1 , L 2 , L 3 , L 4 are elastic coefficients. In the sequel, we shall call F(Q, ∇Q) isotropic if L 2 = L 3 = L 4 = 0 and anisotropic if at least one of L 2 , L 3 , L 4 does not vanish. This work is devoted to the later case. Note that the term in (1.2) corresponding to L 4 is cubic and will lead to severe analytic difficulties: it is shown in [4] that F(Q, ∇Q) with L 4 = 0 is not bounded from below. In this work, we follow [5,22] and assume: In order to introduce the sytem under consideration, we need some notation. For any Q ∈ Q defined by (1.1), S Q (M ) will be a linear operator acting on any 3 × 3 matrix M Here A : B = tr AB T and A · B denotes the usual matrix product of A, B and the 'dot' will be sometimes omitted if it is clear from the context. The parameter ξ is a constant depending on the molecular details of a given liquid crystal and measures the ratio between the tumbling and the aligning effect that a shear flow would exert over the liquid crystal directors. Concerning the hydrodynamic part, for any vector field u, its gradient ∇u can be written as the sum of the symmetric and anti-symmetric parts: where D(u) = 1 2 (∇u + (∇u) T ), W (u) = 1 2 (∇u − (∇u) T ).
(1.6c) H(Q) is the molecular field, defined as the variational derivative of (1.2) and is written as the sum of the bulk part and the elastic part: The operator L and J can be written explicitly by We note that the operator defined via (1.8) can not be considered as a perturbation of L 1 ∆ as we only assumes (1.3). Actually, one of the key results in this work is Lemma 2.5 below, showing that (1.8) fulfills the strong Legendre condition.
The coupled system (1.5) has been recently studied by several authors. For the case ξ = 0, which corresponds to the situation when the molecules only tumble in a shear flow but are not aligned by the flow, the existence of global weak solutions to the Cauchy problem in R d with d = 2, 3 is proved in [18]. Moreover, solutions with higher order regularity and the weak-strong uniqueness for d = 2 is discussed. Later, these results are generalized in [19] to the case when |ξ| is sufficiently small. Large time behavior of the solution to the Cauchy problem in R 3 with ξ = 0 is recently discussed in [10]. The global well-posedness and long-time behavior of system with nonzero ξ in the two-dimensional periodic setting are studied in [7].
In [16], the authors considered Beris-Edwards system with anisotropic elastic energy (1.2) (with L 2 + L 3 > 0 and L 4 = 0). They proved the existence of global weak solutions as well as the existence of a unique global strong solution for the Cauchy problem in R 3 provided that the fluid viscosity is sufficiently large. In [8,17], the weak solution of the gradient flow generated by the general Landau-de Gennes energy (1.2) with L 4 = 0 is established for small initial data.
Some recent progresses have also been made on the analysis of certain modified versions of Beris-Edwards system. In [23], when ξ = 0 and the polynomial bulk energy is replaced by a singular potential derived from molecular Maier-Saupe theory, the author proved, under periodic boundary conditions, the existence of global weak solutions in space dimension two and three. Moreover, the existence and uniqueness of global regular solutions for dimension two is obtained. In [12,13], the authors derived a nonisothermal variants of (1.5) and proved the existence of global weak solutions in the case of a singular potential under periodic boundary conditions for general ξ. We also mention that a rigorous derivation of the general Ericksen-Leslie system from the small elastic limit of Beris-Edwards system (with arbitrary ξ) is recently given in [22] using the Hilbert expansion.
In the aforementioned works, the domain under consideration is either the whole space or the tori. The initial-boundary value problems of (1.5) have been also investigated by several authors, see for instance [1,2,14,15], in which the existences of weak solutions has been studied. In addition, in [1,14], the authors proved the existence of local in time solution with higher order time regularity for (1.5) through different approaches. However, the higher order spacial regularity is not obtained due to the lack of effective energy estimate in the presence of inhomogenous boundary condition for Q.
The main goal of the presented work is to improve the results in [1,14] to nature regularity in space variable. This gives a full answer to the construction of local in time strong solution of (1.5) in the presence of inhomogenous boundary condition for Q.
We shall consider the initial-boundary conditions Note that such a result requires a compatibility condition on the initial data Q 0 . To see that, we write (1.5) in an abstract form for all (ϕ, Ψ) ∈ H 1 0,σ (Ω)×L 2 (Ω; Q). Note that the functional spaces used here are defined in Section 2. Since (1.9b) specifies a time-independent boundary condition, it follows that ∂ t Q| ∂Ω = 0 which leads to the compatibility condition that the trace of the second component on the right-hand side of (1.10) vanishes on ∂Ω. This motives to define the admissible class for the initial data It is not hard to see I is not empty. For example, for any Q solving H(Q) = 0 and any u ∈ H 2 0 (Ω), we have (u, Q) ∈ I . We note that such a compatibility condition is nature in the sense that it can not be disregarded by changing the function spaces unless one considers very weak solution.
The main result of this paper can be stated as follows.
Theorem 1.1. Assume the coefficients of elastic energy satisfy (1.3). Then for any (u 0 , Q 0 ) ∈ I with Q 0 | ∂Ω ∈ H 5/2 (∂Ω), there exists some T > 0 such that the system (1.5) and (1.9) has a unique solution (1.12) Theorem 1.1 essentially improves the spatial regularity of solution obtained in [1] and generalizes their result to the case of anisotropic elastic energy. This is accomplished by the crucial observation that the terms containing third order derivatives on Q in (1.5a) can be eliminated and the system can be reduced into a Stokes-type system with positive definite viscosity coefficient. Moreover, under the general assumption (1.3), the operator L defined by (1.8) is strongly elliptic. This fact leads to W 2,p -estimates for the solution so that we can work with the general case of anisotropic energy rather than the isotropic energy case (L 2 = L 3 = 0). The strong ellipticity of L is proved in Lemma 2.5 by an explicit construction of the coefficient matrix, which involves a fairly sophisticated anisotropic tensor of order six. We also mention that the local in time strong solution constructed here is valid for any ξ ∈ R.
The rest parts of the work is organized as follows. In Section 2, we introduce notation and analytic tools that will be used throughout the paper. The most important results involve the solvability theorem on the generalized Stokes system, due to Solonnikov [21], as well as Lemma 2.5 on the analysis of the operator L. In Section 3 an abstract evolution equation that incorporates (1.5), (1.9) and a compatibility condition is introduced and the functional analytic framework is established. The core part, Section 4, is devoted to the proof of Theorem 1.1 by showing that the abstract evolution equation has a local in time solution. This is accomplished by proving the existence of a local in time solution that is regular in temporal variable in the first stage, following the method in [1], and then using the structure of (1.5) to eliminate the higher order terms in the additional stress tensors of (1.5a) and recast it into a generalized Stokes system. Afterwards, the spatial regularity of (1.5) with initial-boundary condition (1.9) is improved using the L p -estimate of the generalized Stokes system together with bootstrap arguments.

2.1.
Notations. Throughout this paper, the Einstein's summation convention will be adopted. That is, we shall sum over repeated greek letters. For any 3 × 3 martrix A, B ∈ R 3×3 , their usual matrix product will be denoted by A · B or even shortly by AB if it is clear from the context. The Frobenius product of two matrices corresponds to A : In tensor analysis, the Levi-Civita symbol {ε ijk } 1≤i,j,k≤3 and Kronecker symbol {δ i j } 1≤i,j≤3 are very useful to deal with operations involving inner and wedge product: for any a, b ∈ R 3 , their inner and wedge products are given by respectively. The following identity is well known: For any vector field u, its divergence and curl can be calculated by 2.2. Function spaces and the generalized Stokes system. Throughout this work, Ω ⊂ R 3 will be a bounded domain with smooth boundary and Ω T = Ω × (0, T ) will denote the parabolic cylinder. Standard notation for the Lebesgue and Sobolev spaces L p (Ω) and W s,p (Ω) as well as where L 2 σ (Ω) denotes the space of solenoidal vector field and its orthogonal space is given by The Helmholtz projection (also referred to as Leray projection), i.e., the orthogonal projection L 2 (Ω; R 3 ) → L 2 σ (Ω), is denoted by P σ . The readers can refer to [20] for its basic properties. For any We end this subsection by the following result due to Solonnikov [21], which is crucial in the discussion of the spatial regularity during the proof of Theorem 1.1.
, and the strong Legendre condition, i.e. there exists two positive constants Λ > λ > 0 such that Moreover, the following estimate holds for some constant C that is independent of v 0 and f : The result is still valid when p = 3/2 and v 0 ≡ 0.
In the sequel, we shall also need the stationary version of the above result when p = 3 2 : 6 (Ω) and there exists two positive constants Λ > λ > 0 such that then for any f ∈ L 3/2 (Ω; has a unique solution (v, P ) with v ∈ W 2,3/2 (Ω), ∇P ∈ L 3/2 (Ω) and the following estimate holds In the above two inequalities, C only depends on the continuous modulus of A kℓ ij (x) and geometric information of Ω.
2.3. Abstract parabolic equation. Following the method in [1], we shall prove the regular in time solution with the aid of the following result: Proposition 2.3. Suppose that V and H are two separable Hilbert spaces such that the embedding V ֒→ H is injective, continuous, and dense. Fix T ∈ (0, ∞). Suppose that a bilinear form a(·, ·) : V × V → R is given which satisfies for all φ, ψ ∈ V the following assumptions: (a) there exists a constant c > 0, independent of φ and ψ, with Then there exists a representation operator L : V → V ′ with a(φ, ψ) = Lφ, ψ V ′ ,V , which is continuous and linear. Moreover, for all f ∈ L 2 ((0, T ); V ′ ) and y 0 ∈ H, there exists a unique solution solving the equation 2.4. Anisotropic Laplacian. We consider the following bilinear form for Ψ, Φ ∈ H 1 (Ω; Q).
Lemma 2.4. For any f ∈ H −1 (Ω; Q), there exists a unique Q ∈ H 1 0 (Ω; Q) such that a(Q, Φ) = f, Φ for any Φ ∈ H 1 0 (Ω; Q), and there exists a constant C depending on the geometry of Ω such that . This lemma can be proved using the construction in the proof of Lemma 2.5 below. However, we present a simpler proof here : Proof of Lemma 2.4. One can verify that a(Q, Φ) = L(Q), Φ H −1 ,H 1 0 where L is the operator defined by (1.8). In order to apply Lax-Milgram theorem to deduce the existence of solution to (2.7), we need to show that a(·, ·) is coercive in H 1 0 (Ω; Q): for some λ > 0. Note that, it suffices to prove the above inequality for smooth functions that vanishes on ∂Ω. Actually, for any Q ∈ H 1 0 (Ω; Q), we choose Q n ∈ C ∞ 0 (Ω; Q) such that Q n → Q strongly in H 1 0 (Ω; Q). Then the conclusion follows from the continuity of a(·, ·). Now we focus on the proof of (2.8) for Q ∈ C ∞ 0 (Ω). It follows from (2.1) and (2.2) that Therefore by Q| ∂Ω = 0, The above two formula together implies Therefore, it is easy to see that The validity of the L p -estimate requires the verification of the strong Legendre condition for L in (1.7). To this end, we consider the following second order operator defined for Q ∈ H 2 (Ω; R 3×3 ): Lemma 2.5. Let p > 1 be fixed. For any F ∈ L p (Ω; R 3×3 ) and g ∈ W 2−1/p,p (∂Ω; R 3×3 ), there exists a unique Q ∈ W 2,p (Ω; R 3×3 ) that solves LQ = F with boundary condition Q| ∂Ω = g. Moreover, there exists C > 0 depending only on Ω such that Especially, when F, g ∈ Q, we have Q ∈ W 2,p (Ω; Q) satisfying LQ = F .
Proof. It suffices to verify the strong Legendre condition (see (2.3) for instance) for L and then the conclusion follows from standard theory of elliptic system (cf. [3, Chapter IV] or [9, Chapter 10] ).
To this end, we first note that L can be written as (2.9) To verify the strong Legendre condition for A ℓk (ij)(i ′ j ′ ) , we need to compute To this end, we define a new tensor by ζ . Then it is easy to verify that Thus we have, for the case L 2 + L 3 ≤ 0, In the case when L 2 + L 3 ≥ 0, we can write

This yields
and we conclude that L satisfies the strong Legendre condition in both cases. To prove the 'especially' part, we first note that L(Q) = L(Q) for Q ∈ Q. Consequently, if Q ∈ Q solves L(Q) = F for some F with image in Q, then L(Q) = F . On the other hand, if F ∈ Q, and Q ∈ R 3×3 solves L(Q) = F , then we have  8) is an isomorphism.

Abstract form of the system
The task of this section is to setup the functional analytic framework for (1.5) and (1.9). We first remark that the Beris-Edward system (1.5) obeys the basic energy dissipation law This can be formally done by first testing equation (1.5a) by the velocity field u and testing (1.5c) by H(Q) in (1.7), then simple integration by parts lead to: In addition, if M is symmetric, then where S and A are symmetric and anti-symmetric parts of P respectively.
Proof. Note that the space consisting of all k by k matrices under the Frobenius product A : B = tr AB T = A ij B ij is a Hilbert space and it allows the following direct product decomposition: Actually, any M can be uniquely written as the sum of two orthogonal parts Then direct calculations implies the identity (3.4). Formula (3.5) is trickier and can be proved using (3.4): The typical situation for the application of (3.5) is when P = ∇u = D(u) + W (u) for some vector field u: Let (u 0 , Q 0 ) ∈ I , defined by (1.11). As usual, the first equation in (1.5) will be formulated by testing with divergence-free vector fields, or equivalently, by applying the Leray's projector (Ω) and div : L 2 (Ω; R 3×3 ) → H −1 (Ω; R 3 ) are defined in Section 2. The idea is to rewrite the nonlinear system (3.6) as an abstract evolutionary equation in a suitable Banach space. With the notation introduced in Section 2, we define the linearized operator at the initial director field Q 0 by and nonlinear part is given by as well as the homogeneous initial-boundary conditions. Due to the inhomogenous boundary conditions, the operator N Q 0 is defined on an affine space. For the purpose of applying classical result in functional analysis and operator theory, we shall rewrite it as an nonlinear operator between two Banach spaces. To this end, we denote the stationary version of (3.7) by Then it follows from the linearity of (3.7) and the assumption that (u 0 , Q 0 ) is time-independent that, equation (3.9) is equivalent to (3.10)

Proof of Theorem 1.1
The first step towards the proof of the local in time existence of strong solutions is to construct a regular in time solution, following the method in [1]. The following result establishes the invertibility of the linear operator equation. Note that we are seeking a solution of the linear equation in X 0 , i.e., a solution with homogeneous initial and boundary conditions.  7) is a bounded linear operator X 0 → Y 0 and for every (f, g) ∈ Y 0 , the operator equation has a unique solution (u, Q) ∈ X 0 satisfying where C L is independent of T ∈ (0, 1]. In particular L Q 0 : X 0 → Y 0 is invertible and L −1 Q 0 is a bounded linear operator with norm independent of T ∈ (0, 1].
Proof. In order to apply Proposition 2.3, we define the Hilbert spaces , and equip them with standard product Sobolev norm. The dual spaces of V with respect to pivot space H is , and the dual product is given by according to Lemma 2.6. As a result, the space Y 0 defined by (3.13) can be written by We shall define the bilinear form a(·, ·) on V by a((u, Q), (v, P )) = One can verify that this bilinear form satisfies the hypothesis for applying Proposition 2.3. Especially, coerciveness follows from cancellation law (3.5), In the last step, we employed Corollary 2.6. So there exists a bounded linear operator L : V → V ′ such that L(u, Q), (ϕ, Φ) V ′ ,V = a((u, Q), (ϕ, Φ)).
Moreover, for any (f, g) ∈ Y 0 , the abstract evolution equation has a unique solution (u, Q) satisfying (u, Q) ∈ H 1 ((0, T ); V) and (∂ 2 t u, ∂ 2 t Q) ∈ L 2 (0, T ); V ′ . or shortly (u, Q) ∈ X 0 . Now we need to show that (4.4) is equivalent to (4.1): it is evident that, choosing (ϕ, 0) ∈ V in (4.4) implies the first equation in (4.1). To identify the equation for Q, we choose (0, Φ) ∈ V as test function and deduce from (4.3) that In view of Lemma 2.6, L : V 2 → L 2 (Ω; Q) is bijective and thus Altogether, we have proven that L Q 0 : X 0 → Y 0 is an isomorphism. Since L Q 0 is also a bounded linear operator and the operator norm only depends on Q 0 and geometry of Ω, the boundedness of its inverse operator L −1 Q 0 : Y 0 → X 0 follows from inverse mapping theorem. The assertion that C L is independent of T follows from standard energy estimate and the cancellation law (3.5). Here we omit the details.
The proof of this result can be adapted line by line from [1,Proposition 4.3]. Actually, the proof in [1] is slightly more general since they work with variable viscosity in the fluid equation and mixed boundary condition for the Q-tensor field.
Proof of Theorem 1.1. The proof will be divided into two steps. First, we shall use Proposition 4.1 and 4.2 to prove the existence and uniqueness of a regular in time solution. Based on this, in the second step, owning to a special structure of the system, we improve the spatial regularity of u and also Q and this leads to the strong solution of (1.5).
Step 1: Regularity in time.

We first show that
has a unique fixed-point. By (4.2) and (4.5) we find for all (u hi , Q hi ) ∈ B X 0 (0, R) that Therefore A is a contraction mapping for T ≪ 1. A similar argument shows that A maps B X 0 (0, R) into itself. In fact, by (4.6), we deduce that So we can fix R ≫ 1 large enough and then choose T ≪ 1 small enough in such a way that We conclude from Banach's fixed-point theorem that A possess a unique fixed-point (u h , Q h ) ∈ X 0 and it is a solution of the system (1.5), according to (3.12) and Proposition 3.2.
The argument implies the uniqueness as well. Suppose that there was another solution (û h ,Q h ) in B X 0 (0, R 1 ) with R 1 > R. ChooseT ≤ T and repeat the above argument to show the uniqueness of fixed-points of A , which implies (u h , Q h ) = (û h ,Q h ) on (0,T ) × Ω. Then the uniqueness follows by the continuity argument.
So we end up with an improved estimate f ∈ L ∞ (0, T ; L 2 (Ω)), in contrast to (4.14). So one can argue in the same manner as in the previous step by employing the second part of Corollary 2.2: u L ∞ (0,T ;H 2 (Ω)) ≤ C ( f , u t ) L ∞ (0,T ;L 2 (Ω)) .