Compressible Viscous Flows in a Symmetric Domain with Complete Slip Boundary

This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.


Description and Related Works
In this work, we consider the isentropic compressible Navier-Stokes system in the following, which models the motion of viscous gases (or fluids) in a bounded domain Ω ⊂ R 3 , ∂ t ρ + div (ρu) = 0 in Ω, ∂ t (ρu) + div (ρu ⊗ u) + ∇P = div S(u) in Ω, (1.1) where ρ, u, P, S(u) represent the density, the velocity, the pressure potential, and the viscous tensor respectively. Moreover, the flow is assumed to be Newtonian. For simplicity, the pressure potential and the viscous tensor are taken in the following forms, P = P (ρ) = ρ γ , S(u) = µ ∇u + ∇u ⊤ + λdiv uI 3 , γ > 1, µ, λ > 0, where I 3 is the 3 × 3 identity matrix and µ, λ are the viscosity coefficients.
(1.1) can be complemented with various boundary conditions. In this work, the associated boundary condition is taken as the complete slip boundary condition, i.e., u · n = 0, τ · S(u) n = 0, on Γ = ∂Ω, (1.2) where τ , n denote the tangential and normal vectors on the boundary Γ. (1.1) and (1.2) are given with the initial data (ρ, u)(x, 0) = (ρ 0 , u 0 )(x), Moreover, we take Ω = B 1 being the ball centred at the origin with radius 1. Indeed, the geometry of Ω would be an important factor in our problem. On one hand, as it is illustrated in the study of the stationary problem [10], the shape of the boundary Γ is an important factor in determining the steady states of (1.1) with the boundary condition (1.2). On the other hand, as pointed out in [7], the Korn's inequality for a vector field V with the tangency boundary condition (u · n = 0 on Γ = ∂Ω) only holds when Ω is a non-axisymmetric domain. Therefore the Korn's inequality (1.4), which gives the dissipation and plays an important role in the global analysis of (1.1), no longer applies directly in our setting.
We shall resolve such issue in this work (see (2.16)). As it will be pointed out, (1.1) with (1.2) admits a class of steady states with non-trivial velocity, which satisfy the equations (1.6) withρ = constant. In short, instead of external forces, such non-trivial steady states are consequences of the self-rotation and the geometry of occupied domain. The main goal in this article is to investigate the stability of these steady states.
It should be emphasised that, most of the available stability theories are subject to the no-slip boundary condition ( u| ∂Ω = 0). However, the Navierslip boundary condition is actually more appropriate when studying many phenomena such as hurricanes and tornadoes (see, [2]). The general Navierslip boundary condition is of the form u · n = 0, τ · S(u) n = B(u · τ ), on Γ = ∂Ω. (1.5) Such kinds of boundary conditions were first introduced by Navier [25]. The first work on the mathematical rigorous analysis was due to Solonnikov anď Sčadilov [27] on the linearized stationary equations. A simplified form of (1.5) is by taking B(u · τ ) = −κ −1 u · τ with κ being a constant. The most studied case in the literatures is when κ > 0, corresponding to slip with friction. In this situation, κ is called the slip length. The case κ < 0 corresponds to the case in which the boundary wall accelerates the fluids (see [8]). Our focus is on the case when the boundary does not provide friction or acceleration to the fluids, i.e. B(u · τ ) = 0. Such boundary is called the complete slip boundary (see [5]). For a homogeneous incompressible flow (div u = 0, ρ = constant), Chen and Qian in [5] demonstrated that when κ → 0 + , the weak solutions to the incompressible Navier-Stokes equations converge to the solution for the problem subject to the no-slip boundary condition ( u| ∂Ω = 0) for almost all time. Also, as κ → +∞, the solutions converge to a solution for the problem subject to the complete slip boundary condition. Recently, Ding, Li, Xin [8] show that some instability may occur when κ < 0 in two spatial dimensional setting. The instability is in the following sense. ∃ǫ > 0 such that ∀δ > 0, there exists an initial perturbation of the steady state (u s = 0) that will grow larger than the size ǫ in a suitable function space even thought such perturbation is smaller than the size δ initially. Indeed, a critical value of the viscosity coefficient depending on κ serves as the threshold of stability and instability, and the instability would occur if and only if the viscosity coefficient is smaller than the critical value. The problem with complete slip boundary in an axisymmetric domain was studied by Watanabe in [31]. It is shown that the global weak solution would converge to the projection of initial velocity on the rigid body motion (1.12) in L 2 sense.
For a compressible flow, Hoff [14] proved the local-in-time existence of smooth solutions to the Navier-Stokes equations with the boundary condition (1.5). In [16], Huang, Li, Xin established the global dynamic property to the Stokes approximation of Navier-Stokes equations with the no-stick boundary condition in two dimensional setting. It is shown the solution (ρ, u) would converge the the equilibria (ρ s , 0) in L α × W 1,β space as time grows up. Recently, H. Li and X. Zhang in [21] establish the nonlinear stability of Couette flows with the moving condition on the top and the Navier-slip boundary condition (1.5) in which B(u · τ ) = −κ −1 u · τ, κ > 0 on the bottom. The asymptotic stability of the trivial steady state (ρ s = constant, u s = 0) to the problem with frictional boundary (κ > 0) was also demonstrated by Zajaczkowski in [35].
As for the Cauchy problems and initial boundary valued problems with no-slip boundary for the compressible Navier-Stokes equations, there are rich literatures available . We shall only mention a few. In the absence of vacuum (ρ ≥ ρ > 0), the local and global well-posedness of classical solutions have been widely discussed. The uniqueness of both viscous and inviscid compressible flows was studied by Serrin [26]. Itaya [18] and Tani [28] showed the local-in-time existence of classical solutions. Matsumura and Nishida [23,24] first established the global well-posedness of classical solutions with a small pertuerbation of a uniform non-vacuum state. In the present of vacuum, as pointed out by Xin in [32] (also Xin, Yan in [33]), some singular behaviour may occur. With small initial energy, Huang, Li, Xin [17] constructed the global smooth solutions for isentropic compressible viscous flows in R 3 . For more stability and instability problems, see [12,13,30,19].
To study (1.1) with (1.2), we start with the stability theory as the first step. In this work, we will show the existence of rigid motions as steady solutions which rotate with uniform angular velocity. Unfortunately, such rotating profile may contain vacuum when the angular velocity is large. Indeed, the vacuum would appear around the symmetric axis of the domain. Thus it is supposed to be a vacuum interface problem. Moreover, the density profile admits physical vacuum across the vacuum interface, which contains singularities and is not suited in our functional framework (see [22]). We will study such vacuum interface problem in the future and focus on the nonvacuum problem here. To study the stability of the rigid motion, we would make use of the conservation of angular momentum (Lemma 3) to establish a Korn's type inequality (2.16), which would play important roles in global analysis. This is inspired by the study of free boundary problems in [34]. Moreover, it is introduced a spherical frame which matches the geometry of Ω. With such structure, we would be able to define some differential operators (2.21), which separate the normal derivatives and tangential derivatives. The benefit of doing so is that it avoids applying the partition of unity of the domain and works without introducing local charts. Therefore, we can calculate in the entire domain at once. The differential operators in this work are natural to the spherical domain and it is convenient to separate div S(u) on the right of the momentum equation (1.1) 2 into tangential and normal directions (see (3.17)).
The rest of this work would be organised as follows. In the next section, we shall construct the steady state which we are going to study. In Section 2, we present the main tools that will be used in this work, including the Korn's type inequality, the differential operators and the classical elliptic estimates on the Stokes' problem. The equations and the main theorem in the perturbation variables would be given in Section 2.5. The main energy estimates are listed in Section 3. To illustrate the program, the estimates on the temporal derivatives and lower order spatial derivatives will be recorded in Section 3.1. Then we move onto the higher order spatial derivatives and interior estimates. With such blocks in hands, we chain them together in Section 4 and show the asymptotic stability. In particular, we demonstrate the nonlinearities can be indeed controlled by the energy in Section 4.2.
Remark It is assumed inf x∈Ωρ (x) > 0. From (1.10), the minimum ofρ is achieved on the axis {x 1 = x 2 = 0}. For a fixedω > 0 and the fixed domain Ω, it follows that the total mass of the fluid inside Ω has a lower bound Mω > 0 in order to avoid vacuum. In reality, when the total mass drops below Mω, there would be a vacuum area inside Ω and a vacuum interface.
Indeed, from (1.10), the density profile across the vacuum interface would admit physical vacuum [22]. Similar phenomena would occur when the total mass is fixed and the angular velocity increases. We leave such vacuum problems as future works. In this work, it is devoted to study the stability of the steady state (1.11). In fact, we have the following informal statement of our theorem. Theorem 1.1 (Informal Statement) Provided the angular velocity |ω| is small though, the steady state (ρ,ū) given by (1.11) to the compressible Navier-Stokes equations (1.1) with the complete slip boundary condition (1.2) is asymptotically stable in the following sense. For an initial perturbation with the size less than ǫ 1 for some ǫ 1 > 0 in some appropriate function space (defined in (4.16) and (4.20)), there is a globally defined classical solution to (1.1) and the solution converges to the steady state as time grows up exponentially. The perturbation is taken such that it preserves the angular momentum (see (2.35)).
We shall state the theorem in perturbation variables later in Section 2.5.

Notations
Through out this work, conventionally, for any quantities A, B, where the constant C > 0 may depend on sup x∈Ωρ , inf x∈Ωρ , Ω, µ, λ but is independent ofω, ρ, u. Similarly, For a constant 0 < ω < 1, the corresponding value C ω would denote a positive value satisfying 1 ≤ C ω ≤ 1/ω. In the meantime, for a vector w = (w 1 , w 2 , w 3 ) ⊤ , the associated differential operator is defined as, where ∂ i = ∂ x i represents the spatial derivative for i = 1, 2, 3. Also, the commutator operator is defined by where A, B may stand for functions or differential operators. Notice [·, ·] is bilinear. The Sobolev norm in Ω and on the boundary Γ = ∂Ω is denoted as where ∇,∇ denote the differential operators in Ω and on the boundary Γ respectively.

Korn's Inequality
The following form of Korn's inequality is from [34, Lemma 5.1].

Lemma 1 (Korn's Inequality) For
provided the right hand side is finite.
For the sake of completeness, we show the proof here.
Proof Denote the energy of the symmetric part of any vector U = (U 1 , U 2 , U 3 ) ⊤ as , Introduce a decomposition of V , Then direct calculation yields It follows from the original Korn's inequality (see [11,15]), It remains to estimate |b| 2 . From (2.2), we consider the equations The non-degeneracy of the coefficient matrix Therefore, applying the Poincaré inequality together with (2.5) and (2.6), it holds, (2.9) (2.1) follows from (2.7), (2.8) and (2.9). In addition, the following lemma consists of the L 2 estimate of the orthogonal complement of V with respect to the rigid motions (1.12).

Lemma 2 (Poincaré-Morrey Inequality) For
The proof contains a compactness argument. We refer the proof to [31,Lemma 4.2]. See also [10]. As a corollary, one can derive the following Poincaré inequality.

Corollary 1
The same vector V as in Lemma 2 would satisfy the following Proof This form of Poincaré inequalities can be found in [3]. However, a new proof is provided here. One can rewrite where it has been used the fact Ω P S V dx = 0. Therefore, the Poincaré inequality and (2.10) then yield An interesting property of (1.1) comes from the complete slip boundary (1.2). More precisely, in the symmetric domain Ω = B 1 , the flow admits conservation of angular momentum. where ϕ i are defined in (2.3) withx = 0.
Proof Take inner product of (1.1) 2 with ϕ i (i = 1, 2, 3), and record the resulting after integration by parts, (2.13) Notice ϕ i · n = (x × e i ) · x |x| = 0, u · n = 0 on Γ, which means both ϕ i and u are in the tangential direction on Γ. Therefore all the boundary integrations above vanish. In the meantime, ∇ϕ i is an anti-symmetric matrix, while S(u) is symmetric. Hence div ϕ i = tr (∇ϕ i ) = 0, u·(u·∇)ϕ i = u·(∇ϕ)u = 0, S(u) : (2.14) Now we would able to derive the Korn's type inequality which is our first important block in this work.
Lemma 4 (Korn's Type Inequality) For any smooth solution to (1.1) with Moreover, the last term Ω u dx on the right can be dropped in (2.16).
Therefore, after plugging in the decomposition of u−ū, we have the following system of equations (i = 1, 2, 3 ), By using the Poincaré inequality and (2.17), (2.16) follows by chaining the above inequalities. In addition, the identity by applying (2.10). Similarly, we have the following Lemma 5 Under the same assumptions as in Lemma 4, Generally, for any integer k ≥ 1, In particular,the last terms on the right of (2.18) and (2.19) can be dropped.

Spherical Differential Frame
Here, we introduce the decomposition of differential operators ∇ = (∂ x 1 , ∂ x 2 , ∂ x 3 ) near the boundary Γ into two kinds of operators, corresponding to tangential and normal derivatives respectively. To begin with, define the following cut-off function, satisfying |∇ψ(x)| ≤ 8 and Our differential operators are defined as In the following, we study the properties of these differential operators. First, the commutator of two differential operators is also a differential operator. More precisely, Lemma 6 (Commutator) For any function f : R 3 −→ R and any vector fields This finishes the proof.
In the meantime, the following lemma shows that ∇ can be indeed decomposed into ∇ T and ∇ N in the boundary subdomain, in the sense that the estimates of L 2 norms of ∇ T f, ∇ N f would be sufficient to obtain the corresponding estimate of ∇f .

Lemma 7
For any smooth function f : where ∇(ψf ) L 2 (Ω) can be replaced by ∇f L 2 (B 1/2 ) . Here and in the following, ∇ T f (·) stands for the sum of We adopt this convention through the rest of this work.
Proof First we separate Ω into the interior and boundary subdomains, Then (2.24) follows from (2.23) and the property of ψ in (2.20). Thus it remains to show (2.23). It is sufficient to show that in Ω\B 1/2 , the rank of {ϕ 1 , ϕ 2 , ϕ 3 , N} is equal to three. Notice, in the boundary subdomain, x 2 1 + x 2 2 + x 2 3 ≥ 1/4 and thus at least one of |x 1 |, |x 2 |, |x 3 | is no less than 1/4. Without loss of generality, we only consider the case when This finishes the proof. For higher order derivatives, it also admits the following lemma.
Lemma 8 and ∇ k f L 2 (B 1/2 ) respectively. In these inequalities, ∇ N and ∇ T can be interchanged.
Proof We show (2.25) only. (2.26) can be proved via a similar argument. Notice, Then, apply (2.23) repeatly, where in the second inequality (2.22) has been applied. Again, after applying (2.22) again, it holds Thus this finishes the proof of (2.25). The next lemma concerns the calculus on the boundary Γ. In fact, ∇ T | Γ is a differential operator on the boundary.

Similarly
, Proof Without lost of generality, we show the lemma for To do so, we introduce the following coordinate representation of Γ, This finishes the proof.

Embedding Theories and the Stokes Problem
We start with the trace theorem. The following is from [1].
If kp = 3, then the above relation holds for p ≤ q < ∞. In particular, Proof See [1, Theorem 5.36] and [4]. Meanwhile, we shall record the following form of trace theorem.
Lemma 11 (Trace Theory: Fractional Sobolev space) For n ≥ 1, and u ∈ H n (Ω), it holds, Proof We shall apply the following fact about the trace operator (see [9, Theorem 1]): the trace operator T r : H s (Ω) → H s−1/2 (∂Ω) is bounded for 1/2 < s < 3/2. Then (2.32) is a consequence of the fact the rank of {ϕ 1 , ϕ 2 , ϕ 3 } is equal to two and therefore In the meantime, we shall record the classical regularity theory for the Stokes problem. Consider the Stokes problem, Ω is a domain with a smooth boundary Γ. The following is from [30,19], then there exists unique u ∈ H n (Ω), P ∈ H n−1 (Ω)(up to constants) solving (2.33). Moreover, Proof See [20,29].

Perturbed Formulation and Poincaré Inequality
We aim to study the stability of (1.11). It is assumed the initial data (ρ 0 , u 0 ) in (1.3) satisfies, where (ρ,ū) is given in (1.11) with someω > 0. In particular, it is assumed ρ(0) > 0 so thatρ admits uniform lower and upper bounds. Moreover, ∇ kū ω for any k ≥ 0. Now it is time to introduce the perturbed formulation of (1.1) around the steady solution (1.11). Define q := ρ −ρ, v := u −ū. Also, write Notice, from the definition of R, is at least quadratic in q, ∇q. From (1.1) and (1.6), we easily derive the system of the perturbation variables (q, v), (2.38) It should be noticed in the definition of F 2 , it has been used the fact In addition, we have the following Poincaré inequality for q and v.
In terms of (q, v), our stability theorem can be stated as follows.
In particular, there is a constants ǫ 1 > 0 (given in Lemma 29) such that the classical solution (q, v) to (2.37) exists globally with the given initial .16) and (4.20)) for L ≥ 3. Moreover, the following inequalities hold for the energy functionals, for some positive constant σ > 0. Consequently, the initial perturbation would decay to zero as time grows up and therefore (ρ, u)( Remark Roughly speaking, the energy functionalĒ L consists of the H s (Ω) norms of ∂ l t q, ∂ l t v for some s, l ≥ 0. On the other hand, the functionalĒ L consists of the anisotropic H s (Ω) norms. In particular,Ē L mainly includes L 2 (Ω) norms of the tangential and interior derivatives. However, the relation E L ≤Ē L does not hold as one can easily check.

Energy Estimates
To investigate the stability theory, we shall build some blocks in order to describe the propagation of the initial regularities. Thought out this section, the nonnegative integers l, m, n are not specified, and would be addressed with appropriate values in the next section.

On Temporal and Lower Order Spatial Derivatives
To begin with, it is shown the estimate of the temporal derivatives of (q, v) in this section. Moreover, we present the estimates of the first tangential derivatives and the estimates of normal derivatives through a Stokes problem in terms of the differential frames introduced in Section 2.3.
Applying the temporal derivative ∂ l t for l = 0, 1, · · · to (2.37), it results in the following system, where by using the Leibniz's rule, We record the following energy identity.

Lemma 14
For any smooth solution (∂ l t q, ∂ l t v) to (3.1), it holds the following energy identities, for any integer l ≥ 0, d dt Proof Take inner product of (3.1) 2 with ∂ l t v and then record the resulting after integration by parts, d dt Similar as before, from (3.1) 3 , ∂ k t v is in the tangential direction of Γ. Therefore, from (3.1) 3 and (2.37) 3 , the boundary integrals in (3.4) where we have applied integration by parts and the boundary condition (2.37) 3 . Thus (3.3) follows after chaining the above identity.
Here and after, a > 0 would denote an integer which might be different from line to line. Next lemma is concerning the estimate of the spatial derivative.
Lemma 16 Under the same assumptions as in Lemma 14, Proof Take inner product of (3.1) 2 with ∂ l+1 t v and then record the resulting after integration by parts. Similar arguments as in the previous lemma yield, Meanwhile, together with (2.39) and (2.40), where it has been making use of the fact, from (3.1) 1 and (2.40) Similarly, Thus chaining these estimates with an appropriately small δ > 0 leads to (3.6).
As the consequence of (3.5) and (3.6), we have the following estimates on the temporal derivatives.
Proposition 1 (Temporal Derivatives) Denote Λ l = Λ l (·) as a polynomial of the following quantities with the property Λ l (0) = 0. Then it shall hold the following estimate. For any 0 < ω < 1, Proof This is a direct consequence of the linear combination c×(3.6)+(3.5).
We only show the the choice of c can be justified. By applying the Cauchy's inequality and (2.1), (2.40), Together with the fact inf x∈Ωρ > 0, (3.8) holds after choosing c > 0 sufficiently small. Notice, in Proposition 1, the estimate (3.7) contains the term Ω ∇∂ l t q 2 dx on the right hand side. To derive an estimate in a consistent form, it is desirable to perform the estimate on the spatial derivatives in the rest of this section.
Here we establish the estimates on the tangential derivatives. Starting with the first order tangential derivative, ∇ T (3.1) can be written as, Take inner product of (3.9) 2 with ∂ l t ∇ T v and then record the resulting after integration by parts, d dt where we have used the boundary condition (2.37) 3 . Meanwhile, from (3.9) 1 , (2.37) 3 , Applying Cauchyś inequality and Poincaré inequality as follows, To estimate the boundary terms, from (3.9) 3 , as consequences of the trace theorem (2.31), Hölder inequality and (2.28), (2.39), (2.40), it holds On the other hand, on Γ, we have the following identities from (3.9) 3 with |v l,1 | ∂ l t ∇ T v . Therefore, the calculus on the boundary (2.28) then yields Indeed, the above calculation indicates the next lemma.
Remark The identities (3.12) on the boundary and the boundary condition (3.9) 3 should be understood as follows. τ stands for one of the tangential vector fields ϕ 1 , ϕ 2 , ϕ 3 (defined in (2.3)), which are smooth and defined globally on Γ. Therefore, ∇ T τ is non-singular and smooth. Moreover, the rank of {ϕ 1 , ϕ 2 , ϕ 3 } is equal to two and hence any tangental vector on Γ can be represented by them. To show that v l,1 τ makes sense and |v l,1 | ∂ l t ∇ T v , we adopt the following representation of tangential vector fields. Indeed, we claim that any tangential vector fields V on Γ can be denoted as To show this is possible, consider a point p = (x 1 , x 2 , x 3 ) on Γ. Without loss of generality, we assume x 1 > 1/4. Then for a neighbourhood W p of p, {ϕ 2 , ϕ 3 } forms a non-degenerate basis of the tangential space and |ϕ 2 | , |ϕ 3 | > 1/4. Then V can be written as Since Γ is compact, one can construct a finite cover {W p } of Γ and corresponding {p; V 1,p , V 2,p , V 3,p }. Then a partition of unity argument would yield (3.15). Notice, V 1 , V 2 , V 3 are not necessarily continuous.
Next, we show how to develop the estimates of the normal derivatives in the spherical differential frame. Just as it is classically done (see, for example, [19]), we shall derive the ordinary differential equation(ODE) satisfied by ∂ l t ∇ N q. In order to do so, taking inner product of (3.1) 2 with N yields, (3.16) where in the last equality we have substituted the following identity, 1≤i,j≤3 1≤i,j,m,n,s≤3 In the meantime, from (3.1) 1

Lemma 18 (Normal Direction Estimate)
The following estimate on ∇ N ∂ l t q holds, Also, div ∂ l t v admits the estimate,

(3.22)
Proof After taking inner product of (3.19) 1 with ∂ l t ∇ N q and recording the resulting after integration by parts, it holds, Meanwhile, for some a > 0. (3.21) then follows from Hölder's inequality, (2.39), (2.40) and the fact inf x∈Ωρ > 0. On the other hand, Similarly, Then together with the fact as the consequence of (2.24) and (2.22), (3.22) follows after chaining the these inequalities. Now it is time to introduce the associated Stokes problem. From (3.1) 2 , (∂ l t v, γρ γ−1 ∂ l t q) satisfies the following Stokes system. (3.24)

Proposition 2
We have obtained the estimate concerning the spatial derivatives as follows, Together with (3.14) and an appropriate choice of ω > 0, (3.27) follows easily.

On Higher Order Spatial Derivatives
Through the following arguments similar to those in the last section, the estimates involving higher order spatial derivatives would be shown. In particular, the estimates of tangential derivatives are obtained through a high-order version of (3.9). Also, by taking mixed derivatives to (3.19), we shall track the regularities of q and more importantly, the regularities of div v. Then a high-order version of the Stokes problem (3.23) would eventually yield the estimates of v and q. These estimates would play important roles in the global analysis.
Apply ∇ m T to (3.1) and record the resulting system as follows, Applying standard energy estimate to (3.28) then will yield the following lemma.
Proof Taking inner product of (3.28) 2 with ∂ l t ∇ m T v and recording the resulting after integration by parts, d dt Also, followed from the definition of F l,m 1 , F l,m 2 , it holds with |v l,m | ≤ ∂ l t ∇ m T v . Then together with (3.28) 4 From the definition, Therefore, by applying trace embedding inequality (2.31), Cauchy's inequality and Poincaré inequality, Summing up these estimates with an appropriately small δ > 0 and (2.1), it holds, d dt Therefore, (3.29) follows after applying (2.39) and (2.40).
Next it is to derive the estimates of mixed derivatives of q. Applying ∇ m T ∇ n−1 N to (3.19), it holds Then the following estimate holds, Proof After taking inner product of (3.30) 1 with ∂ l t ∇ m T ∇ n N q and recording the resulting after integration, it admits, (3.35) The last estimate we shall obtain in this section is from the following Stokes problem. After applying ∇ m T to (3.23), record the resulting system as follows.
As consequences of the Stokes estimate in Lemma 12, the following lemma indicates the estimates of normal derivatives.
Proof Take inner product of (3.41) 2 with ∂ m ∂ l tṽ and then record the resulting after integration by parts, d dt with m ≥ 1. Moreover, a comparable quantity of D l,m can be written as Then it holds, Also, denotē Notice, the above quantities are monotone increasing in each index l, m, n.

Lemma 24
where G l is defined as Meanwhile, as the consequences of (3.27) and (3.42) with m = 1, by choosing an appropriately small ω > 0, Thus, after summing from j = 1 to j = k, together with (4.10) with m = k and an appropriately small ω > 0, it holds (4.14) Thus we have shown the following .  Similarly, for i ≥ 2, 1 ≤ j ≤ i, from (4.11), Hence, after summing over j, it holds Then (4.15) is the consequence of (4.2) with l = L, (4.18) and summing (4.19) with i from 2 to L.

Estimates on Nonlinearities
In this section, we shall perform the estimates on the nonlinearities. To do so, denote the intermediate energy and dissipation E L , D L as (4.20) (4.21) Then it holds, We shall use (4.22) and (4.23) to manipulate the nonlinear terms on the right hand side of (4.15).
Proof We only show G L , and others can be handled in a similar way. Recall, .
We shall consider the highest order terms only. In the following, let l.o.t denote the terms involving relatively lower order derivatives than others, which might be different from line by line. To evaluate (i), from (2.38), (3.2), Similarly, where C denotes cubic terms. Therefore, Similarly, from (3.10) where the subscripts represent the corresponding norms to bound the terms.
The same notations would be adopted in the following. Meanwhile, the rests in G L,1 1 are bounded in a similar way. Similar arguments then yield, Thus, (ii) P(Ē L )D L . Similarly (iii) P(Ē L )D L . To handle (iv), from (3.20), (3.24) in which the terms on the right hand side have already appeared before. Hence, (iv) P(Ē L )D L . We have shown G L P(Ē L )D L . The estimate on Λ L is the direct consequence of (4.22). Next lemma concerns the quantity analysis ofĒ L andD L . Also, a direct calculation yields,