Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system

In this paper, we consider a hydrodynamic $Q$-tensor system for nematic liquid crystal flow, which is derived from Doi-Onsager molecular theory by the Bingham closure. We first prove the existence and uniqueness of local strong solution. Furthermore, by taking Deborah number goes to zero and using the Hilbert expansion method, we present a rigorous derivation from the molecule-based $Q$-tensor theory to the Ericksen-Leslie theory.

theory derived from viewpoints of statistical mechanics, and the later two are macroscopic theories based on continuum mechanics.
Notations and conventions. The Einstein convention will be assumed throughout the paper. We introduce the following notations for the space of symmetric traceless tensors ) . (1. 2) The space Q is endowed with the inner product Q 1 , Q 2 def = Q 1 : Q 2 = Q 1ij Q 2ij . The set Q is a five-dimensional linear subspace of R 3×3 . We define the matrix norm on Q as |Q| def = trQ 2 = Q ij Q ij . In terms of this norm, the Sobolev space is defined as with k being a non-negative integer and α being a multi-index. For two tensors A, B ∈ Q we denote (A · B) ij = A ik B kj and A : B = A ij B ij . We denote (M : Q) ij = M ijkl Q kl where M is the fourth-order tensor and Q ∈ Q. In addition, n 1 ⊗ n 2 ⊗ · · · ⊗ n k denotes the tensor product of k vectors n 1 , n 2 ,· · · , n k , and we usually omit the symbol ⊗ for simplicity. We use f ,i to denote ∂ i f for simplicity and I to denote the 3 × 3 order identity tensor.
1.1. The Ericksen-Leslie theory. The hydrodynamic theory of liquid crystals, established by Ericksen [8] and Leslie [16] in the 1960's, is a system coupling the time evolution equation of the fluid velocity v = v(t, x) with the director equation describing the motion of the director field n = n(t, x) ∈ S 2 . The general Ericksen-Leslie system takes the form v t + v · ∇v = −∇p + ∇ · σ, (1.3)
1.2. The Q-tensor theory. The most general continuum theory for the nematic liquid crystals is the celebrated Landau-de Gennes theory which can describe uniaxial and biaxial liquid phases. In this phenomenological theory, the detailed nature of molecular interactions and molecular structures is ignored, and the state of the nematic liquid crystals is described by a macroscopic tensor value order parameter Q(x), which is a symmetric and traceless 3×3 matrix, i.e. Q ∈ Q. Physically, it can be interpreted as the second-order traceless moment of the orientational distribution function f , that is, ( 1.12) Under this interpretation, the so-called physical constraint is that the eigenvalues of Q should satisfy namely, Q ∈ Q phy . The nematic liquid crystal is called isotropic at x when Q(x) = 0. When Q(x) has two equal non-zero eigenvalues, it is called uniaxial and Q(x) can be written as When Q(x) has three distinct eigenvalues, it is called biaxial and Q(x) can be written as Q(x) = s nn − 1 3 I + r(n n − 1 3 I), n, n ∈ S 2 , n · n = 0, s, r ∈ R.
The classic Landau-de Gennes energy functional, being a nonlinear functional of Q and its spatial derivatives, takes the following general form (1.14) where a, b, c are material-dependent and temperature-dependent non-negative constants and L i (i = 1, 2, 3, 4) are material dependent elastic constants. We refer to [6,20] for more details.
The energy (1.14) can not ensure Q to satisfy the natural physical constraint (1.13). For this reason, based on the mean-field Maier-Saupe energy, Ball-Majumdar [4] proposed an energy functional, which will diverge if Q ∈ Q phy . There are many works to study the equilibrium solutions of the classic Landau-de Gennes model, for example, one may see [4,19] and the references therein. So far, there are two types of dynamic Q-tensor theories to describe the flow of nematic liquid crystal. The first type models are obtained by variational methods under physical considerations, such as Beris-Edwards model [5] and Qian-Sheng's model [24]. Let F(Q, ∇Q) be the total free energy, and define The dynamical Q-tensor model of this types can be written in the following general form: where v is the fluid velocity, D rot (µ Q ) is the rotational diffusion term, F (Q, D) and Ω·Q−Q·Ω are induced by the deformation part and and rotation part of the velocity gradient respectively. In addition, σ d is the distortion stress, σ a is the anti-symmetric part of orientationalinduced stress, σ s = γF (Q, µ Q ) which conjugates to F (Q, D) (γ is a constant), is the symmetric stress induced by the orientation of molecules, and σ dis is an additional dissipation stress. In Beris-Edwards's model and Qian-Sheng's model, module some constants, σ a and σ d are the same, i.e., In Beris-Edwards's model, the other terms are given by In Qian-Sheng's model, they are given by . When taking F(Q, ∇Q) = F LG (Q, ∇Q), for the well-posedness results of the Beris-Edwards's model on whole space and bounded domain, we refer to [21,22,13] and [1,2].
The second type is derived from the molecular kinetic theory by closure approximations. In such models, the evolution of Q is derived from the evolution of probability density function f by relation (1.12). However, one have to approximate the higher order moment such as by using Q. This process is called closure approximation. There are various kinds of closure approximation and then they lead to different models in Q-tensor form, which are summarized in [10,11]. However, these models do not obey energy dissipation law. In [12], based on Doi's kinetic theory, the authors proposed a Q-tensor model with energy dissipation law by using the Bingham closure. In this paper, we are mainly concerned this model. Before introducing it, we first give a brief description of the Bingham closure. For a given configuration distribution function f (m) satisfying the Bingham closure is to use the quasi-equilibrium distribution (also called the Bingham distribution) to approximate f . Here, B Q ∈ Q depends on Q and is determined by the following relation By Proposition 2.1, B Q can be uniquely determined for Q ∈ Q phy . Then, the fourth-order moment and the sixth-order moment of f are approximated by Now we introduce the dynamic Q-tensor model presented in [12]. For given free energy functional F(Q, ∇Q), define We introduce the following two operators klij )]∂A kl .
Based on the Doi-Onsager's molecular theory, making use of the aforementioned Bingham closure approximation, the new Q-tensor model is given as following [12]: where De and Re are called Deborah number and Reynolds number respectively, and γ ∈ (0, 1) is a constant. The small parameter √ ε characterizes the typical interaction distance, which is usually at the scale of molecule length. The term N Q (µ Q ) represents the translational diffusion. An important feature of this model is that (1.20)-(1.22) obeys the following basic energy dissipative law (see [12]) In [12], the energy functional is also derived from Onsager's molecular theory.
where the bulk energy F b (Q) and the elastic distortion energy F e are respectively given by where Q (4) = Q (4) (Q) is the fourth order symmetric traceless moment of the Bingham distribution f Q . Namely, The difference between Q (4) and M Q is not. The bulk energy F b is equivalent to the penalized energy derived by Ball-Majumdar in [4]. Thus, the order parameter tensor Q should satisfy the physical constraint (1.13).
The parameters appearing in the system (1.20)-(1.22) have clear physical significance but not are phenomenological. In [12], the coefficients L i (i = 0, 1, · · · , 5) are also explicitly calculated in terms of physical molecular parameters. The parameter ε appears in the elastic energy F e due to the fact that the ratios between the coefficients of F e and the ones in F b are at the order of square of molecule length. Another important feature of the moleculebased Q-tensor system (1.20)-(1.22) is that the translational and rotational diffusions are still maintained.

1.3.
Motivations and main results. The connection between different level of liquid crystal theories is a problem of both physical and mathematical importance. Based on a formal asymptotical expansion, Kuzzu-Doi [15] and E-Zhang [7] derived the Ericksen-Leslie equation from the Doi-Onsager equations by taking small Deborah number limit for spacial homogeneous case and inhomogeneous case respectively. Wang-Zhang-Zhang rigorously justified this limit in [27] before the first singularity time of the Ericksen-Leslie system. In [29], they also presented a rigorous derivation from Beris-Edwards model to Ericksen-Leslie model. In [12], it is proposed a systematic study on the modeling for liquid crystals in both static and dynamic cases. They derived a Q-tensor model from Onsager's molecular theory and Doi's kinetic theory, which is introduced in the previous subsection, and also derived Oseen-Frank model and Ericksen-Leslie model.
The main aim of this paper is to prove the local well-posedness for strong solution of the molecule-based Q-tensor model, and also to show that the strong solution will converges to the solution of Ericksen-Leslie system under the limit of Deborah number De → 0.
In this paper, to avoid some tedious technical difficulties, we will only consider the case when the translational diffusion N Q (µ Q ) = 0 and the coefficients L 0 = 1, L 3 = L 4 = L 5 = 0. Then Then, the corresponding molecule-based Q-tensor system becomes : where σ d is defined by It not hard to see that ∇ · σ d (Q, Q) differs from µ Q : ∇Q with only pressure terms. When α > α * , the bulk energy function F b has stable uniaxial critical points Q = S 2 (nn − 1 3 I) for any n ∈ S 2 , which correspond to nematic phase. Here, S 2 = S 2 (α) is a increasing function of α for α > α * , see the precise definition in (2.19). Throughout this paper, we always assume α > α * and L 1 > 0, L 1 + 2L 2 > 0. Thus, it is known from Lemma 2.2 in [29] that for some constant c 0 > 0. We first state the following the local well-posedness result.

If the initial data satisfies
for all x ∈ R 3 , then there exists T > 0 and a unique solution (v, Q) of the Q-tensor system and Q(t, x) ∈ Q phy,δ/2 .
Next, we consider the small Deborah number limit De → 0. To obtain the full Ericksen-Leslie system, we have to take De = O(ε) as in [12]. For simplicity, we choose De = ε. Then the system can be written as: We define the coefficient in Ericksen-Leslie theory as: and and the elastic constants in Oseen-Frank energy are given by (1.40) Here S 4 = S 4 (α) is also a constant related to α, see the definition in (2.19). For a given direction field n(t, x), we define where the ψ 1 and ψ 2 are constants depending on α. H n (Q) is the linearized operator of B Q − αQ around the local critical point S 2 (nn − 1 3 I). The detailed motivation of the above definitions will be explained in Section 4.
The second main result of this paper is stated as follows. Let Q 0 (t, x) = S 2 n(t, x)n(t, x) − I and the functions Q 1 , Q 2 , Q 3 , v 1 , v 2 are determined by Proposition 5.2. Assume that the initial data (Q ε I , v ε I ) takes the form Then there exists ε 0 > 0 and E 1 > 0 such that for all ε < ε 0 , the system (1.35)-(1.37) has a unique solution Here and H ε n (Q) = H n (Q) + εL(Q).
Remark 1.1. It can be observed from [7] that the Leslie coefficients of Ericksen-Leslie system derived from the Doi-Onsager system have same forms as (1.38)-(1.39) except for γ 1 . The only difference is due to the Bingham closure approximation.
The remaining sections of this paper are organized as follows. In Section 2, the important properties of the Bingham closure and the critical point are presented. Section 3 is devoted to the proof for the existence of the local strong solution of the molecule-based Q-tensor system. In Section 4, we present some important linearized operators which will be used in deriving the Ericksen-Leslie system from the molecule-based Q-tensor system. In Section 5, by using the Hilbert expansion method, we present a rigorous derivation from the molecule-based Q-tensor theory to the Ericksen-Leslie theory.

The Bingham closure and the critical points
This section is mainly concerned to the important properties of the Bingham closure and the critical points.
2.1. The Bingham closure and Bingham map. The Bingham closure plays an important role in the system (1.15)- (1.17). For this, one should find B Q ∈ Q such that for a given Q ∈ Q phy . The following proposition tells us that B Q can be uniquely defined for any Q ∈ Q phy . We call this map from Q ∈ Q phy to B Q ∈ Q Bingham map.
Proposition 2.1 (Existence and uniqueness of B Q ). For a given Q ∈ Q phy , there exists a unique B Q ∈ Q such that (2.1) holds.
Proof. A sketched proof is given in [4]. Here we give a detailed proof for completeness.
Define ω : Q → R as: Obviously, ω(B) depends only on its eigenvalues. From the fact that we know ω(B) is convex. Then we can define its convex conjugate by Legendre transformation: with domain X defined by We will prove that X = Q phy . For this, we need an elementary inequality: To prove it, we can assume B is diagonal without loss of generality. Suppose Q = q 1 n 1 ⊗ n 1 + q 2 n 2 ⊗ n 2 + q 3 n 3 ⊗ n 3 with n i · n j = δ i j .
or for the later case We know that the measure of A is positive in each case. Therefore, Therefore, for any Q ∈ Q phy , there exists B ∈ Q such that (2.7) We let B Q = B, then the existence of B Q is proved. Since ω(B) is convex, we can deduce that (∇ B ω)(B 1 ) = (∇ B ω)(B 2 ) for B 1 = B 2 , which implies the uniqueness.
The map from Q phy to Q which satisfies (2.7) is a diffeomorphism, and so is its inverse. We denote them by B = B(Q) : Q phy → Q and Q = Q(B) : Q → Q phy respectively. For Λ, δ > 0, we introduce compact subsets of Q as The next proposition tells us that B(Q) maps a compact subset of Q phy to a compact subset of Q.
Proof. We only have to consider the case Q and B are both diagonal. (2.10) Therefore, we have This concludes the proof of the proposition.
Proof. It is straightforward to calculate that for any non-zero E ∈ Q, it holds Thus, the Jacobian ∇ B Q(B) is positive definite. Together with the fact that Q(B) is a smooth function of B, we know the inverse B(Q) is also smooth.
We give some estimates related to the Bingham map.
The above lemma is a direct consequence of Proposition 2.3 and Lemma 6.2 by using change of variables.
The first assertion is a direct consequence of Proposition 2.3. The second one can be induced by Proposition 2.3 and Lemma 6.4.
Q is a smooth function of B Q = B(Q), it shares the same estimates with B(Q). Now we give some properties for the operator M Q : Q . Note that M Q is defined not only for the symmetric matrix, and M Q (A) is not necessarily symmetric even if A is symmetric. The following Lemma 2.3 gives some basic properties of M Q , which proof can be found in [12].
Lemma 2.4. For any δ > 0, there is a positive constant C δ depending on δ such that if Q(x) ∈ Q phy,δ , A ∈ R 3×3 , it holds for any multiple index a, Proof. With Lemma 6.3, Lemma 2.2 and Remark 2.1, direct computation shows that (2.15) can be deduced by the same argument with Lemma 6.3.
Lemma 2.5. For any δ > 0 and k ∈ N * , there exist constants Proof. From Lemma 6.1, we have that Then the conclusion can be deduced from Lemma 2.2 and Remark 2.1.

2.2.
The energy functional and critical points. The bulk part of free energy density functional takes the following form A direct calculation yields that We say that a tensor Q 0 is a critical point of the bulk free energy density functional The critical points are completely classified in [18,9].
Then there holds
In the sequel, we always choose α > α * , and η = η 1 (α) corresponding the stable equilibrium solution. We also introduce some important constants used in this paper. All of them only depend on the parameter α.

We define
where P k (x) is the k-th order Legendre polynomial. Particularly, Then we have An important fact induced by Proposition 2.4 is that (2.20) The relation and the inequalities will play important roles in Section 4. Their proofs can be found in [27], and we omit them here.

Existence and uniqueness of the local strong solution for the dynamical Q-tensor systems
This section is devoted to the proof for the existence of the local strong solution of the system (1.28) -(1.30). For s ≥ 2, we define the space: If (Q, v) ∈ X, then by Sobolev imbedding, we have The proof of Theorem 1.1 is based on iterative argument and a closed energy estimate.
3.1. Linearized system and iteration scheme. First of all, we take Assuming that (Q (n) , v (n) ) ∈ X(δ, T, C 0 ) has been constructed, we construct (Q (n+1) , v (n+1) ) by solving the following linearized system: with initial data: The existence of (Q (n+1) , v (n+1) ) is ensured by the classical parabolic theory, see [3] for example. Now we prove that (Q (n+1) , v (n+1) ) ∈ X, for a suitably chosen T > 0. Define the energy functional Obviously, we have , v (n) ). We will prove the following closed energy estimates: for some small ν > 0. The proof is split into three steps.
Step 1. L 2 energy estimate for Q (n+1) − Q * From Lemma 2.2, we have Therefore, by making L 2 inner product to (3.2) with Q (n+1) − Q * , we get (3.7) Step 2. L 2 estimates for (∇Q (n+1) , v (n+1) ) In this step and the next step, a key point is that we will use the self-adjointness of M Q n . By making L 2 inner product to (3.2) with L(Q (n+1) ), we get Thus, we obtain from (3.8)-(3.9) that Step 3.
We now turn to the estimate of the higher order derivative for Q (n+1) , These terms can be estimated as following: For the estimate of the higher order derivative for v (n+1) , we have Estimating them term by term, we obtain Thus we get Combining (3.7), (3.10), (3.11) and (3.12), we know that it holds for ν > 0 small enough. By Gronwall's inequality, we get for any t ∈ [0, T ]. Then if we take T 0 > 0 such that C(δ, C 0 )T 0 ≤ ln(1+C 0 )−ln(1+E s (Q I , v I )), then sup Thus, together with the assumption Q I ∈ Q phy,δ , it yields that Q (n+1) ∈ Q phy,δ/2 for t ∈ [0, T 0 ], if we choose T 0 sufficiently small. Then we obtain (Q (n+1) , v (n+1) ) ∈ X(δ, T, C 0 ) for T ≤ T 0 .

3.2.
Convergence of the sequence. In this subsection, we are going to show that the approximate solution sequence {(v ( ) , Q ( ) )} ∈N is a Cauchy sequence. We set By taking the difference between the equations for (v ( +1) , Q ( +1) ) and (v ( ) , Q ( ) ), we find that where From Proposition 2.5, we have Similar to the proof of (3.10), we can deduce that there exist ν > 0 small enough and C(δ, C 0 , ν) > 0, such that We denote Thus, we get By the uniform bounds and interpolation, we have for any s ∈ (0, s), Thus we (Q, v) is a classical solution of (1.28)-(1.30). The uniqueness of (Q, v) is guaranteed by the same energy estimate as we have done to the prove the convergence of {(Q (n) , v (n) )}. Moreover, by the standard regularity argument for parabolic system, we have that We omit the details here. This completes the proof of Theorem 1.1.

Some linearized operators
In this section, we study some important linearized operators which will be used in deriving the Ericksen-Leslie system from the molecule-based Q-tensor system (1. We can also introduce the linearized operator of B(Q) aroundQ, which is actually Q −1 Q , since Q(B) and B(Q) are inverse functions of each other.
The following proposition shows that QQ is a self-adjoint and positive operator. Q and the fact that which concludes the proof.
We are particularly interested in the linearized operators around the equilibrium tensor Q 0 = S 2 (nn − 1 3 I), where S 2 are introduced in Section 2. We denote Q Q 0 (B) by Q n (B) for B 0 = η(nn − 1 3 I). For use of convenience, we calculate Q n explicitly.
For the equilibrium tensor Q 0 , the distribution function f Q 0 and the order parameter tensor M (4) Q 0 can be written as Q 0 ,ijkl =S 4 n i n j n k n l + S 2 − S 4 7 (n i n j δ kl + n i n k δ jl + n i n l δ jk + n j n k δ il + n j n l δ ik + n k n l δ ij ) + ( Substituting (4.1) and (4.2) to the linear operator To calculate Q −1 n explicitly, we may assume that where ψ i (1 ≤ i ≤ 3) are constants. Then we have Therefore, the coefficients ψ i (1 ≤ i ≤ 3) satisfy By (4.4) and the definitions of S 2 and S 4 (see (2.19)), we get that Thus, the coefficients ψ 1 , ψ 2 , ψ 3 can be uniquely determined.
Another important linear operator is the linearized operator H n (Q) of B(Q) − αQ around Q 0 , which is given by plays an important role in next sections. First, we introduce a two-dimensional subspace of Q as Q in n = nn ⊥ + n ⊥ n : n ⊥ ∈ V n , where V n := {n ∈ R 3 |n ⊥ · n = 0}, and let Q out n be the orthogonal complement of Q in n in Q. The following proposition gives a characterization on the kernel space and non-negativity of H n : There exists a positive constant c 0 such that for any Q ∈ Q out n , Proof. (i) From (4.5) and (4.7), the linearized operator H n can be written as where ψ 1 , ψ 2 , ψ 3 are given by (4.6). By (4.4) and definitions of S 2 and S 4 , we have Together with (4.6), we know ψ 2 + ψ 3 = (ξ 2 + ξ 3 ) −1 = α. Thus, we get This yields the assertions in (i) by observing (4.10) (ii) From the assertion in (i) and (4.10), we have Together with the fact that Q n is a bounded operator, we only need to prove that for some positive constant c 0 and any B ∈ Q out n . From (4.3), we have B, Q n (B) =ξ 1 |nn : B| 2 + 2ξ 2 |n · B| 2 + ξ 3 |B| 2 , and Q n (B), Q n (B) = 2 3 (ξ 1 + 2ξ 2 ) 2 − 2ξ 2 2 + 2ξ 2 ξ 3 |nn : B| 2 + (2ξ 2 2 + 4ξ 2 ξ 3 )|n · B| 2 + ξ 2 3 |B| 2 .
Therefore we get where the coefficients are given by for B ∈ Q out n . If β 1 > 2β 3 , the assertion is apparently true. If β 1 < 2β 3 , it is direct to check that for traceless matrix B Some further tedious calculations give that This concludes the proof.
We denote by P in the projection operator from Q to Q in n and by P out the projection operator from Q to Q out n . By direct computation we have |Q − (nn ⊥ + n ⊥ n)| 2 = |Q| 2 − 2|Q · n| 2 + 2|Q : nn| 2 + |n ⊥ − (I − nn) · Q · n| 2 .
Direct computation gives that which imply As Q n and J n are self-adjoint on Q, we also have In summary, we get

Rigorous derivation from the Q-tensor theory to the Ericksen-Leslie theory
In this section, by making the Hilbert expansion for the solution of the molecule-based Q-tensor systems (1.28)-(1.30), we present a rigorous derivation from the molecule-based Q-tensor theory to the Ericksen-Leslie theory. 5.1. The Hilbert expansion. Let (Q ε , v ε ) be a solution of the system (1.35)-(1.37). We perform the following so-called Hilbert expansion: are independent of ε and will be determined in what follows. (Q R , v R ) are called the remainder term which depend upon ε.
Substituting the above expansion to (1.35)-(1.37), and expanding all the terms with respect to ε, we can get several systems of equations to solve (Q i , v i )(0 ≤ i ≤ 2) and Q 3 by collecting all the terms of the same order with respect to ε. In [27] and [29], the expanding can be performed directly as it involves only polynomials of variables. In contrast, the dependence of B and M B on Q is much more complicated here.
First, we make the following formal expansion for Z Q ε and B Q ε : are independent of ε and B R , Z R are the linear funtions of Q R . All the terms with higher order of ε are put in ε 4 R B and ε 4 R Z . To perform the Hilbert expansion, we have to write B i , Z i and B R , Z R in terms of Q i , Q R explicitly. By viewing Z Q ε as a function of B ε , we have: where By the expression of Q ε , we have where Noting the definition of linear operator Q Q 0 and from (5.6) we can duduce that Thanks to the invertibility of Q Q 0 , we know that B i can be explicitly given by Q j (0 ≤ j ≤ i), and B R is linearly depend on Q R . Similarly, we next make the expansion for M (4) R + ε 4 R M (4) . (5.11) • The zero-order term in ε • The first-order term in ε • The second-order term in ε Here, F 1 , F 2 , G 1 , G 2 are defined as following: 1 , The equation of O(ε −1 ) (5.13) is equivalent to B 0 − αQ 0 = 0. Thanks to Proposition 2.4, Q 0 takes the form for some n(t, x) ∈ S 2 . The evolution of n(t, x) is determined by the O(1) system (5.14)-(5.16). At first glance, this system is not closed since it involves Q 1 which is unknown. However, if we project (5.14) into the subspace Q in n = Ker H n , then Q 1 is vanished in (5.14) by Proposition 4.2. In addition, if we project (5.14) into the subspace Q out n = (Ker H n ) ⊥ , then we can solve H n (Q 1 ) in terms of (Q 0 , v 0 ). Thus Q 1 can also be eliminated in (5.15). Actually, the following proposition shows that the system (5.14)-(5.16) implies (n, v 0 ) satisfies the Ericksen-Leslie system with coefficients depending on the molecule parameters. One can see the detailed proof in [12]. In the next subsections, we will show how to solve Q i (1 ≤ i ≤ 3) and v j (1 ≤ j ≤ 2) from (5.17-5.22). The whole procedure is very similar to the one used in [27] and [29].

5.2.
Existence of the Hilbert expansion. Assume that (v 0 , n) is a solution of the systems Notice that we can solve Q ⊥ 1 by the equation (5.14) and have Q ⊥ 1 ∈ C([0, T ]; H k−1 ). In order to solve (v 1 , Q 1 ), we need to derive a closed system for (v 1 , Q 1 ) from (5.17)- (5.19). We will also show that this system is linear and have a closed energy estimate, which implies the solution (v 1 , Q 1 ) will not blow up in In what follows, we denote by L(Q 1 , v 1 ) the terms which only depend on (Q 1 , v 1 ) (not their derivatives) linearly with the coefficients belonging to C([0, T ]; H k−1 ). We also use R ∈ C([0, T ]; H k−3 ) to denote the terms depending only on n, v 0 and Q ⊥ 1 .
Lemma 5.1. It holds that Proof. The proof can be found in [12].
For any Q ∈ Q, we set where M 1 (Q) and Z 1 (Q) are nonlinear functions with respect to Q, Therefore, note that Q 1 = Q 1 + Q ⊥ 1 , we have where the definition of L(·) is as the above. The next lemma tells us that when we take the projection P in on F 1 , the terms which are nonlinear with respect to Q 1 will vanish.
We are now in a position to derive the systems of (v 1 , Q ). We denote A 1 = P in J n (L(Q 1 )) , A 2 = P out J n (L(Q 1 )) , Taking the projection P in on both sides of (5.17), note that H n (Q 2 ) ∈ Q out n and J n (L(Q 1 )) = J n (L(Q 1 )) + R, from Lemma 5.1 and Lemma 5.2 we get Here we have absorbing P in v 1 · ∇Q 0 into L(v 1 ). Taking the projection P out on both sides of (5.17), we have Substituting (5.25) to (5.18) and together with (5.24), we obtain the following closed system for (v 1 , Q 1 ) Apparently, (5.26)-(5.28) is a linear system of (v 1 , Q 1 ). To prove its solvability, we give a priori estimate for the energy We will prove that there exists a positive constant C such that d dt Without loss of generality, we only prove (5.29) in the case of = 0 and the proof is similar for the general case. When = 0, the corresponding energy is given by First, we get by (5.26) that Meanwhile, we can obtain from (5.26) and (5.27) that For I 2 , I 3 and I 5 , we have Now we turn to estimate I 1 + I 4 . Recalling the fact that for any Q ∈ Q in n (Q out n ), J n (Q) and M Q 0 (Q) belong to Q in n (Q out n ), we have: where we have repeatedly used the symmetry of J n (·) and the self-adjointness of M n (note that M n (·) is not symmetric and J n is not self-adjoint). Similarly, it holds that J n (L(Q 1 )), P in L(Q 1 ) ≥ 0.
Again, we write Q 2 = Q 2 + Q ⊥ 2 with Q 2 ∈ Q in n and Q ⊥ 2 ∈ Q out n . By (5.25) we can solve Q ⊥ 2 as Then, (v 2 , Q 2 ) can be solved in a similar way as (v 1 , Q 1 ). Q 3 can be solved similarly as in (5.38)(unique up to a term in Q in n ). We omit the details and leave them to the interest readers.
To summarize, we have proved: 5.3. The system for the remainder. In this subsection, we focus on the derivation of systems of the remainder and uniform estimates for the remainder. Throughout this subsection, we assume that v i ∈ C([0, T ]; H k−4i ) for i = 0, 1, 2 and Q i ∈ C([0, T ]; H k+1−4i ) for i = 0, 1, 2, 3. We denote by C a constant depending on By Sobolev embedding inequality, for k = 0, 1, 2, we have Proof. First, we have Using Talylor expansion for B Q = B(Q), we get By Lemma 2.5, we can obtain  Similarly, the following inequalities hold ε 2 Together with (5.63)-(5.66), we arrive at Recalling that R denotes good terms with R L 2 + ε ∇R L 2 + ε 2 ∆R L 2 ≤ C(εE)(1 + E + εF ) + εf (E), and Corollary (5.1), we have Taking δ enough small leads to the Proposition 5.3.
6. Appendix 6.1. Some basic estimates in Sobolev spaces. The following product estimates and commutator estimates are well-known, see [25] for example, and frequently used in this paper.
Lemma 6.1. Let s ≥ 0. Then for any multi-index α, β, γ, δ, there holds In particular, we have Lemma 6.2. Let s ≥ 0 and F (·) ∈ C ∞ (R d ) with F (0) = 0. Then Lemma 6.3. Let a be a multiple index. There holds Moreover, if |a| ≥ 2, it holds Lemma 6.4. Let Ω be a convex domain in R d and k ≥ 0 be an integer. F (·) ∈ C ∞ (Ω) and k = max{k, 2}. Then Proof. We may assume that F (0) = 0, since if not, we can consider G(u) = F (u) − u · F (0). By the fact that and for k ≥ 2, Here, we have used the following estimate which is induced by Lemma 6.2: This concludes the proof.