On the Incompressible Limit for the Compressible Flows of Liquid Crystals under Strong Stratification on Bounded Domains

and Applied Analysis 3 (ii) The balance of momentum holds in distributional sense, namely,


Introduction
Liquid crystals flows can be found in the natural world and in technological applications with a variety of examples such as many proteins, cell membranes, and solutions of soap and various related detergents, as well as the tobacco mosaic virus.We here consider the compressible flows of liquid crystals: (u) + div  (u ⊗ u) + ∇   () = Δu, − div  (∇d ⊗ ∇d − ( 1 2 |∇d| 2 +  (d)) where u is the vector field,  is the density, d is the direction field for the averaged macroscopic molecular orientations,  is a given potential, and , ,  are viscosities.The smooth vector (d) and the smooth scalar function (d) are related by where (d) represents the penalty function and (d) is the bulk part of the elastic energy verifying: there exists  > 0 such that if |d| ≥ , Note that the condition (5) is required for the global weak solution.We can see an example as follows: 2 , (d) = 1 2 2 0 (|d| 2 − 1) d, (6) where  0 is a constant.Finally, the pressure  is defined as follows and the function is the unique positive solution of the following problem: We use the following scaling used in Feireisl et al. [1] and Wang and Yu [2]: and for the viscosity coefficients,  ≈   , ≈ 2   , ≈  , (10) with the convergence of the viscosity coefficients   → ,   → ,   → .(11) Finally for the pressure, we use   () = 1    (  ) ,  ∈ (2,3) .(12) Following the above scalings, the system (1)- (7) reads We now notice that the global-in-time existence solutions for system ((1)-( 3)) have been studied by Wang and Yu [3] and Liu and Qing [4].For the case of d = 0, incompressible limit problems have been investigated by many authors, starting with the work by Klainerman and Majda [5] for the Euler equations and Lions and Masmoudi [6] for the isentropic Navier Stokes equations.Similar results in the spirit of the analysis presented by Lions and Masmoudi [6] are the recent progress by Feireisl and Novotný [7,8] and Kwon and Trivisa [9] for the full Navier-Stokes Fourier system.For the liquid crystals, there is one recent progress by Wang and Yu [2] based on the spectral analysis and Duhamel's principle to control difficulties arising in the boundary of bounded domains.There are some results of incompressible limit problems of Navier Stoke Fourier system under strong stratification by Feireisl and Novotný [7] on bounded domain and by Feireisl et al. on unbounded domains, which is extended to full magnetohydrodynamic flows on bounded domain by Novotný et al. [10] and by Lee et al. [11].
For this kind of application, the polymer precursor leads to a stratified structure of a solidified film of polymer and so we have a natural question: is such fluid almost incompressible such that it is strongly stratified when  → 0 as the different models of compressible fluid?In this paper, we derive the rigorous result of the incompressible limit of the flow of liquid crystals with the similar idea used in the previous results [7,[10][11][12].Formally, we will investigate the limit as  tends to 0 in the suitable sense such that the given limit {, U, d} represents a solution of the following system: Finally, in this paper, we will use the many parts of the presentation of Feireisl and Novotný [7] and Feireisl et al. [12] without modification.
The outline of this paper is as follows: In Section 2 we present two initial-boundary-value problems and introduce the notion of weak solutions for the compressible fluid of liquid crystals.In Section 3 we present the main results of the article on the low Mach number problems under strong stratification on bounded domains.In Section 4, we present the proof of the low Mach number problem for bounded domains.

Weak Solutions
where n stands for the outer normal vector.We also propose the boundary condition on the direction vector d Notice that Wang and Yu showed the global weak solution of the system (1)-( 3) with the Dirichlet boundary condition on bounded domains but there will be no problem with the boundary conditions ( 18) and ( 19) for existence result.

Weak Solutions
Definition 1.We say that a quantity {, u, d} is a weak solution of the compressible flows of liquid crystals (13)-(15) supplemented with the initial data { 0 , u 0 , d 0 } provided that the following hold.
(i) The density  is a nonnegative function,  ∈  ∞ (0, ;   (Ω)), the velocity field u ∈  2 (0, ;  1 (Ω; R 3 )), |u| 2 ∈  ∞ (0, ;  1 (Ω)), and u represents a renormalized solution of (1) on a time-space cylinder (0, ) × Ω, that is, the integral identity holds for any test function  ∈ D([0, ) × Ω) and any  such that (ii) The balance of momentum holds in distributional sense, namely, for any test function ⃗  ∈ D([0, ); D(Ω; R 3 )) satisfying ⃗  ⋅ n| Ω = 0, where Π is defined by (iii) The total energy of the system holds: holds for a.e. ∈ (0, ), where with (iv) The equation of direction field verifies Let us now discuss the static states which are solutions of system ( 13)-(15) with vanishing velocity field.In the present setting, the positive density  must satisfy From the statistic equation ( 28), we can easily derive with We now introduce the weak solutions of the target system.
Definition 2. A couple {, U, d} is said to be a weak solution of the target system of the compressible flows of liquid crystals with the potential  ∈  1,∞ (Ω), ∇ ∈  1 (Ω), supplemented with the boundary conditions on Ω which belongs to  ∞ , and the initial conditions with div  (U) = 0 a.e. on (0, ) × Ω, U ⋅ n| Ω = 0 in the sense of traces, and the integral identity holds for any test function )), and the integral identity

Main Results
In this section we mention the main result as follows.
Theorem 3. Let Ω ⊂ R 3 be a bounded domain with a boundary of class  ∞ and {  , u  , d  } a family of weak solutions to the compressible of liquid crystals system verifying (5) in the sense of Definition 1 with  ∈  1,∞ (Ω), ∇ ∈  1 (Ω).Assume that the initial condition is as follows: Then, up to subsequence,   → a.e. in (0, ) × Ω, where {, U, d} solves a weak solution of the incompressible flows of liquid crystals in the sense of Definition 2 with the boundary condition U ⋅ n| Ω = 0 and the initial data where the Helmholtz projection P = I − Q and Q is defined in (81).

Proof of Theorem 3
4.1.Uniform Bounds.In this section we are going to derive some estimates on the sequence {  , u  , d  } >0 .Multiplying (13) by   (  )−  (  ) and adding the energy inequality (24), it follows that where For convenient presentation, we introduce the set of the essential and residual values where [] ess = (  ), [] res = (1 − (  )) and  is defined as follows: where  is the solution of (28) and Notice that two assumptions of pressure in ( 7) and ( 12) imply that   is a strict convex.Thus, thanks to (36), we get that  0, is uniformly bounded for  → 0. Consequently, from the energy balance (40), we obtain ess sup ess sup ess sup ess sup ess sup Using the estimate of (49) implies ess sup Note that the static equation (29 Due to the estimate of (51) and the convergence of (52), it follows that ess sup In order that we now derive the estimate of the velocity, we first verify that ess sup To do this, we use the following inequality due to two assumptions of pressure ( 7) and ( 12), which deduce, thanks to (49), ess sup ess sup ess sup Finally, we get ess sup where we have used the above two estimates (57) and ( 58) and so the estimate (56) together with (59) shows (54).
We now derive the estimate of the velocity.Indeed, it is easy to show with a simple computation that ess sup up to a subsequence of { > 0}.In accordance with (53) and (54), we obtain   → in  ∞ (0, ;   (Ω)) , for a certain  > 0, (64) and so we can take the limit of  in the continuity equation (20) to get for all  ∈  ∞  ((0, ) × Ω).

Convergence of Moment Equation.
To begin with, using two estimates (63) and (64), for a certain  > 0. (66) Hence for a certain  > 1.Actually, we do not know U ⊗ U = U ⊗ U due to the oscillations of the gradient component of the velocity field and we postpone this part to handle the oscillations of the gradient component in the next section.
We also need to get the uniform estimates for the directional field {d  } >0 .Multiplying d  on (27) yields and so applying the maximum principle for weak solutions provides which implies where we have used (47), the basic elliptic theory, and the following Gagliardo-Nirenberg inequality:     ∇d in the sense of distribution.
We are now able to identify the limit problem of the moment equation ( 22).To do this, we first rewrite the moment equation as follows: Abstract and Applied Analysis From the previous estimates, we get for any test function where we have here used ( 8), (53), and (54).It remains to show that for any The proofs of ( 78) and ( 79) are provided in the next two sections.

Pressure Estimate.
The next challenge is to establish uniform bounds on the pressure as well as on the internal energy in terms of a reflexive space   , with  > 1.We here use the Bogovski operator.Let us take a test function and adapt this test function in the moment equation ( 2): ∈ D (0, ) . (81) We will write the moment equation ( 2) with simple computations again: where and   is defined by We notice that all estimates in (84) are uniformly bounded due to the uniform estimates in the previous section if we take a special  verifying with a certain  ∈ (0, 1).In virtue of (57), it is easy to see ess sup and so we get for  < 1/ and the first integral of the right hand side of (82) is uniformly bounded.We now need to control the first and second integrals of the right hand side of (82).To do this, let us rewrite the first term into the following form with using the static problem (8) where the last integral is uniformly bounded due to (86).Following estimates (53), (56), and (87) together with the  estimates for B, one gets the uniform boundedness of the first and second terms of the right hand side of (88).
On the other hand, the estimates in ( 86) and (87) together with the   -estimates for B yield that the third term of the right hand side of (82) is uniformly bounded.Consequently, we deduce that for a certain  ∈ (0, 1).We first split the integration of (78) into two parts to show (78): For the first one of the right hand side, we get where we have used the Taylor expansion of degree 2. We also use (57) and (89) to prove the second part of the right hand side.

Convergence of the Convective Term.
In this section, our aim is to show that (79) holds.Before we prove (79), we will introduce the Helmholtz decomposition and the following material may be found in most of the text book of fluid mechanics.Let us denote  2 1/ by the Hilbert space with the inner product ⟨v, and  1,2 by the completion of  ∞  (Ω) with respect to the norm ‖∇  ‖  2 (Ω).Theorem 4. For v ∈  2  1/ (Ω : R 3 ), a vector function v : Ω → R 3 is written as where and Ψ is uniquely determined as the following Neumann problem: where n is the outward unit normal to Ω.
We now write Let us first show To do this, we adapt the following test function to the moment equation ( 22): Taking into account the uniform estimates obtained in the previous sections, it follows that precompact in [0, ] and so the uniform estimates (45) and (64) with using the Sobolev embedding  5/4 (Ω) → [ 1,2 (Ω)] * imply that in  weak ([0, ] ;  5/4 (Ω; R 3 )) . (100) Notice that where we have here used (100).In virtue of ( 64) and (101), one gets Since the Helmholtz projections map continuously the spaces   (Ω; R 3 ) and  1, (Ω; R 3 ) into itself for any 1 < , it is easily seen that In order to show (79), we should prove lim for any We next study the acoustic equations.The acoustic equations are used to describe the time evolution of fast acoustic waves in the compressible models in order to handle the oscillation of Q[  u  ].To begin with, we rewrite (1) and (2): where the previous estimates provide for a certain  > 1 and We now use the method of spectral analysis of the wave operator.Let us consider the eigenvalue problem: with the boundary condition Thus, the eigenvalue problem ( 112) and ( 113) can be written into the following form: with Following the eigenvalue problem (114) and (115), it is well known that there is an orthonormal basis {A , } ∞,  =0,=1 of real eigenfunctions of the weighed Lebesgue space  2  1/ (Ω) corresponding to eigenvalues Λ , such that where   stands for the multiplicity of Λ  .We also see that 2  1/ (Ω) 3 ] defined by (117) We now take  =  1 ()A , and  =  2 ()W ±, as test functions to the system (107), and then we obtain (Ω) → 0, as  → ∞ in  2 (0, ;  2 1/ (Ω; R 3 )) where we here have used the Parseval identity and (121) (see section 6.6 of [7]).Thus it is sufficient to show (105) for Q , instead of Q .Observe that, thanks to (114) and (121), it is easy to see which converges to 0 as  tends to 0 where we can see this proof in section 6.6 of Feireisl and Novotný [7].