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

Abstract

We study the incompressible limit of weak solutions for the compressible flows of liquid crystals under strong stratification on bounded domains.

1. 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: 𝜕_𝑡𝜚+div_𝑥𝜕𝜚𝐮=0,_𝑡𝜚𝐮+div_𝑥𝜚𝐮⊗𝐮+∇_𝑥𝑝𝜚=𝜇Δ𝐮,−𝜈div_𝑥1∇𝐝⊗∇𝐝−2|∇𝐝|²+𝐹(𝐝)𝕀₃𝜕+𝜚∇𝐺,_𝑡𝐝+𝐮⋅∇𝐝=𝜃Δ𝐝−𝑓(𝐝),𝐝(0,𝑥)=𝐝₀(𝑥), (1)(2)(3) where 𝐮 is the vector field, 𝜚 is the density, 𝐝 is the direction field for the averaged macroscopic molecular orientations, 𝐺 is a given potential, and 𝜇,𝜈,𝜃 are viscosities. The smooth vector 𝑓(𝐝) and the smooth scalar function 𝐹(𝐝) are related by 𝑓(𝐝)=∇_𝐝𝐹(𝐝), (4) where 𝑓(𝐝) represents the penalty function and 𝐹(𝐝) is the bulk part of the elastic energy verifying: there exists 𝐶>0 such that if |𝐝|≥𝐶, 𝑓(𝐝)⋅𝐝≥0. (5) Note that the condition (5) is required for the global weak solution. We can see an example as follows: 𝐹(𝐝)=14𝜎²₀|𝐝|²−1²(𝐝)=1,𝑓2𝜎²₀|𝐝|²−1𝐝, (6) where 𝜎₀ is a constant. Finally, the pressure 𝑝 is defined as follows 𝑝(0)=0,𝑝∈𝐶²[0,∞),𝑝^′(𝑟)>0∀𝑟≥0,lim_𝑟→∞𝑝^′(𝑟)𝑟^𝛾−1=𝑞>0, (7) and the function 𝜚̃ is the unique positive solution of the following problem: 𝑝₀∇𝜚̃=𝜚̃∇𝐺,𝑝₀=𝑝^′(0). (8) We use the following scaling used in Feireisl et al. [1] and Wang and Yu [2]: 𝑡≈𝜖𝑡,𝑥≈𝑥,𝜚≈𝜚_𝜖,𝐮≈𝜖𝐮_𝜖,𝐝≈𝐝_𝜖, (9) and for the viscosity coefficients, 𝜇≈𝜖𝜇_𝜖,𝜈≈𝜖²𝜈_𝜖,𝜃≈𝜖𝜃_𝜖, (10) with the convergence of the viscosity coefficients 𝜇_𝜖→𝜇,𝜈_𝜖→𝜈,𝜃_𝜖→𝜃. (11) Finally for the pressure, we use 𝑝_𝜖𝜚=1𝜖^𝛼𝑝𝜖^𝛼𝜚,𝛼∈(2,3). (12) Following the above scalings, the system (1)–(7) reads 𝜕_𝑡𝜚_𝜖+div_𝑥𝜚_𝜖𝐮_𝜖𝜕=0,_𝑡𝜚_𝜖𝐮_𝜖+div_𝑥𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖+1𝜖²∇_𝑥𝑝_𝜖𝜚_𝜖=𝜇_𝜖Δ𝐮_𝜖−𝜈_𝜖div_𝑥∇𝐝_𝜖⊗∇𝐝_𝜖−12∇𝐝_𝜖²+𝐹𝐝_𝜖𝕀₃+1𝜖²𝜚_𝜖𝜕∇𝐺,_𝑡𝐝_𝜖+𝐮⋅∇𝐝_𝜖=𝜃_𝜖Δ𝐝_𝜖−𝑓𝐝_𝜖. (13)(14)(15)

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 𝜖→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, 10–12]. Formally, we will investigate the limit 𝜚_𝜖→𝜚̃,𝐮_𝜖→𝐔,𝐝_𝜖→𝐝, (16) as 𝜖 tends to 0 in the suitable sense such that the given limit {𝜚̃,𝐔,𝐝} represents a solution of the following system: div_𝑥̃𝜕𝜚𝐔=0,_𝑡𝜚̃𝐔+div_𝑥𝜚̃𝐔⊗𝐔+∇_𝑥𝑃1=𝜇Δ𝐔−2|∇𝐝|²+𝐹(𝐝)𝕀₃,𝜕_𝑡𝐝+𝐔⋅∇𝐝=𝜃Δ𝐝−𝑓(𝐝). (17)

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.

2. Weak Solutions

2.1. An Initial-Boundary-Value Problem

Let Ω⊂ℝ³ be a bounded domain with the boundary of class 𝐶^∞𝐮⋅𝐧|_𝜕Ω=0, (18) where 𝐧 stands for the outer normal vector. We also propose the boundary condition on the direction vector 𝐝𝐝|_𝜕Ω=𝐝₀. (19) 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.

2.2. Weak Solutions

Definition 1. We say that a quantity {𝜚,𝐮,𝐝} is a weak solution of the compressible flows of liquid crystals (13)–(15) supplemented with the initial data {𝜚₀,𝐮₀,𝐝₀} provided that the following hold.(i) The density 𝜚 is a nonnegative function, 𝜚∈𝐿^∞(0,𝑇;𝐿^𝛾(Ω)), the velocity field 𝐮∈𝐿²(0,𝑇;𝐻¹(Ω;ℝ³)), 𝜚|𝐮|²∈𝐿^∞(0,𝑇;𝐿¹(Ω)), and 𝐮 represents a renormalized solution of (1) on a time-space cylinder (0,𝑇)×Ω, that is, the integral identity 𝜚𝐵𝜚^𝑇_0Ω𝜕_𝑡𝜑+𝜚𝐵𝜚𝐮⋅∇_𝑥𝜑−𝑏𝜚div_𝑥𝐮𝜑d𝑥d𝑡=−_Ω𝜚₀𝐵𝜚₀𝜑(0,⋅)d𝑥 (20) holds for any test function 𝜑∈𝒟([0,𝑇)×Ω) and any 𝑏 such that 𝑏∈𝐿^∞[),𝐵𝜚(1)+∩𝐶0,∞=𝐵^𝜚₁𝑏(𝑧)𝑧²d𝑧. (21)(ii)The balance of momentum holds in distributional sense, namely, ^𝑇_0Ω𝜚𝐮⋅𝜕_𝑡^→𝜑+𝜚𝐮⊗𝐮∶∇_𝑥^→1𝜑+𝜖²𝑝_𝜖𝜚div_𝑥^→𝜑−𝜈_𝜖Π∶∇^→𝜑+𝜇_𝜖∇𝐮∶∇^→1𝜑+𝜖²∇𝐺⋅^→𝜑d𝑥d𝑡=−_Ω𝜚₀𝐮₀⋅^→𝜑(0,⋅)d𝑥 (22) for any test function ^→𝜑∈𝒟([0,𝑇);𝒟(Ω;ℝ³)) satisfying ^→𝜑⋅𝐧|_𝜕Ω=0, where Π is defined by 1Π=∇𝐝⊗∇𝐝−2|∇𝐝|²+𝐹(𝐝)𝕀₃. (23)(iii)The total energy of the system holds: 𝐸(𝑡)+^𝑡_0Ω𝜇_𝜖|∇𝐮|²+𝜈_𝜖𝜃_𝜖Δ𝐝−𝑓(𝐝)²(0)d𝑥d𝑡≤𝐸 (24) holds for a.e. 𝑡∈(0,𝑇), where 𝐸(𝑡)=_Ω12𝜚|𝐮|²+1𝜖²𝐻_𝜖𝜚+𝜈_𝜖2|𝐝|²+𝜈_𝜖𝐹(𝐝)−𝜚𝐺d𝑥 (25) with 𝐻_𝜖𝜚=𝜚^𝜚₁𝑝_𝜖𝜚𝑧²d𝑧. (26)(iv)The equation of direction field verifies ^𝑇_0Ω𝐝⋅𝜕_𝑡^→+𝜑d𝑥d𝑡_𝑇0Ω(𝐮⋅𝐝)div_𝑥^→𝜑+(∇𝐮⋅𝐝)⋅^→𝜑−𝜃∇𝐝+𝑓(𝐝)⋅^→𝜑d𝑥d𝑡=0, (27) for all ^→𝜑∈[𝒟([0,𝑇)×Ω)]³.

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 𝜚̃_𝜖 must satisfy ∇𝑝_𝜖𝜚̃_𝜖=𝜚̃_𝜖∇𝐺,𝑝₀=𝑝^′(0). (28) From the statistic equation (28), we can easily derive 𝑝₀log𝜚̃(𝑥)+𝑄𝜖^𝛼𝜚̃(𝑥)−𝑄𝜖^𝛼𝜚_∞=𝐺(𝑥)+𝑝₀log𝜚_∞ (29) with 𝑄^′(𝑟)=𝑃^′(𝑟)−𝑝₀𝑟𝑃,if𝑟>0,_′′(0),if𝑟=0. (30)

We now introduce the weak solutions of the target system.

Definition 2. A couple {𝜚̃,𝐔,𝐝} is said to be a weak solution of the target system of the compressible flows of liquid crystals with the potential 𝐺∈𝑊^1,∞(Ω),∇𝐺∈𝐿¹(Ω), supplemented with the boundary conditions 𝐔⋅𝐧=0,𝐝=𝐝₀, (31) on 𝜕Ω which belongs to 𝐶^∞, and the initial conditions 𝐔(0,⋅)=𝐔₀,𝐝(0,⋅)=𝐝₀, (32) with 𝐔∈𝐿²(Ω),  𝐝₀∈𝐻¹(Ω),  𝐝₀|_𝜕Ω∈𝐻^3/2(Ω) if the following conditions hold:(i) 𝐔∈𝐿^∞(0,𝑇;𝐿²(Ω;ℝ³))∩𝐿²(0,𝑇;𝐻¹(Ω;ℝ³)),  div_𝑥(𝜚̃𝐔)=0 a.e. on (0,𝑇)×Ω, 𝐔⋅𝐧|_𝜕Ω=0 in the sense of traces, and the integral identity 𝜚̃^𝑇_0Ω𝐔⋅𝜕_𝑡𝜑+𝜚̃𝐔⊗𝐔∶∇_𝑥𝜑−𝜇∇_𝑥𝐔∶∇_𝑥𝜑d𝑥d𝑡=−^𝑇_0Ω(∇𝐝⊗∇𝐝)∶∇𝜑d𝑥d𝑡−_Ω𝐔⋅𝜑(0,⋅)d𝑥 (33) holds for any test function 𝜑∈𝒟(0,𝑇)×Ω;ℝ³,div_𝑥𝜑=0inΩ,𝜑⋅𝐧|_𝜕Ω=0, (34)(ii) 𝐝∈𝐿²(0,𝑇;𝐻²(Ω;ℝ³))∩𝐿^∞(0,𝑇;𝐻¹(Ω;ℝ³)), and the integral identity ^𝑇_0Ω𝐝⋅𝜕_𝑡^→+𝜑d𝑥d𝑡_𝑇0Ω(𝐔⋅𝐝)div_𝑥^→𝜑+(∇𝐔⋅𝐝)⋅^→𝜑−𝜃∇𝐝+𝑓(𝐝)⋅^→𝜑d𝑥d𝑡=0 (35) holds for all ^→𝜑∈[𝒟([0,𝑇)×Ω)]³.

3. Main Results

In this section we mention the main result as follows.

Theorem 3. Let Ω⊂ℝ³ be a bounded domain with a boundary of class 𝐶^∞ and {𝜚_𝜖,𝐮_𝜖,𝐝_𝜖} a family of weak solutions to the compressible of liquid crystals system verifying (5) in the sense of Definition 1 with 𝐺∈𝑊^1,∞(Ω),∇𝐺∈𝐿¹(Ω). Assume that the initial condition is as follows: 𝜚⁽¹⁾_0,𝜖𝜖>0𝑏𝑜𝑢𝑛𝑑𝑒𝑑𝑖𝑛𝐿²∩𝐿^∞(Ω),𝑤𝑖𝑡ℎ𝜚⁽¹⁾_0,𝜖𝜚∶=^0,𝜖−𝜚̃𝜖,𝐮_0,𝜖𝜖>0𝑏𝑜𝑢𝑛𝑑𝑒𝑑𝑖𝑛𝐿²∩𝐿^∞Ω;ℝ³,𝐝_0,𝜖𝜖>0𝑏𝑜𝑢𝑛𝑑𝑒𝑑𝑖𝑛𝐿²∩𝐿^∞Ω;ℝ³. (36)(37)
Then, up to subsequence, 𝜚_𝜖→̃()𝐮𝜚a.e.𝑖𝑛0,𝑇×Ω,_𝜖()𝐝→𝐔a.e.𝑖𝑛0,𝑇×Ω,_𝜖→𝐝a.e.𝑖𝑛(0,𝑇)×Ω, (38) where {𝜚̃,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 𝐔(0)=𝐏_̃𝜚𝐔₀,𝐝(0,⋅)=𝐝₀, (39) where the Helmholtz projection 𝐏_̃𝜚=𝐈−𝐐_̃𝜚 and 𝐐_̃𝜚 is defined in (81).

4. Proof of Theorem 3

4.1. Uniform Bounds

In this section we are going to derive some estimates on the sequence {𝜚_𝜖,𝐮_𝜖,𝐝_𝜖}_𝜖>0. Multiplying (13) by 𝐻^′(𝜚_𝜖)−𝐻^′(𝜚̃_𝜖) and adding the energy inequality (24), it follows that _Ω12𝜚_𝜖𝐮_𝜖²+1𝜖²𝐻_𝜖𝜚_𝜖−𝐻_𝜖𝜚̃_𝜖−𝐻^′_𝜖𝜚̃_𝜖𝜚_𝜖−𝜚̃_𝜖+𝜈_𝜖2𝐝_𝜖²+𝜈_𝜖𝐹𝐝_𝜖+d𝑥_𝑡0Ω𝜇_𝜖∇𝐮_𝜖²+𝜈_𝜖𝜃_𝜖Δ𝐝_𝜖−𝑓𝐝_𝜖²d𝑥d𝑡≤𝐸_0,𝜖, (40) where 𝐸_0,𝜖=_Ω12𝜚_0,𝜖𝐮_0,𝜖²+1𝜖²𝐻_𝜖𝜚_0,𝜖−𝐻_𝜖𝜚̃_𝜖−𝐻^′𝜚̃_𝜖𝜚_0,𝜖−𝜚̃_𝜖+𝜈_𝜖2𝐝_0,𝜖²+𝜈_𝜖𝐹𝐝_0,𝜖d𝑥. (41) For convenient presentation, we introduce the set of the essential and residual values 𝑔=𝑔_ess+𝑔_res, (42) where [𝑔]_ess=𝜒(𝜚_𝜖)𝑔, [𝑔]_res=(1−𝜒(𝜚_𝜖))𝑔 and 𝜒 is defined as follows: 𝜒(𝑟)𝜚=1∀𝑟∈2,2𝜚,𝜒(𝑟)=0otherwise, (43) where 𝜚̃_𝜖 is the solution of (28) and 𝜚=inf_𝜖>0inf_𝑥∈Ω𝜚̃_𝜖(𝑥),𝜚=sup_𝜖>0sup_𝑥∈Ω𝜚̃_𝜖(𝑥). (44) Notice that two assumptions of pressure in (7) and (12) imply that 𝐻_𝜖 is a strict convex. Thus, thanks to (36), we get that 𝐸_0,𝜖 is uniformly bounded for 𝜖→0. Consequently, from the energy balance (40), we obtain esssup_{𝑡∈(0,𝑇)}√𝜚_𝜖𝐮_𝜖(𝑡)_{𝐿2(Ω;ℝ3)}≤𝐶,esssup_{𝑡∈(0,𝑇)}𝐝_𝜖(𝑡)_{𝐿2(Ω;ℝ3)}≤𝐶,esssup_{𝑡∈(0,𝑇)}(Δ𝐝_𝜖−𝑓(𝐝_𝜖))(𝑡)_{𝐿2(Ω;ℝ3)}∇≤𝐶,_𝑥𝐮_{𝜖𝐿2((0,𝑇)×Ω)}≤𝐶,esssup_{𝑡∈(0,𝑇)}𝐻_𝜖𝜚_𝜖−𝐻_𝜖𝜚̃_𝜖−𝐻^′𝜚̃_𝜖𝜚_𝜖−𝜚̃_{𝜖ess𝐿1(Ω)}≤𝜖²𝐶,esssup_{𝑡∈(0,𝑇)}[𝐻_𝜖(𝜚_𝜖)−𝐻_𝜖(𝜚̃_𝜖)−𝐻^′(𝜚̃_𝜖)(𝜚_𝜖−𝜚̃_𝜖)]_res𝐿1(Ω)≤𝜖²𝐶. (45)(46)(47)(48)(49)(50)

Using the estimate of (49) implies esssup_{𝑡∈(0,𝑇)}𝜚_𝜖−𝜚̃_𝜖𝜖_ess𝐿²_(Ω)≤𝐶. (51) Note that the static equation (29) holds 𝜚̃_𝜖−𝜚̃_𝐶(Ω)≤𝜖^𝛼𝐶,𝜚̃_𝜖=𝜚̃=𝜚_∞inΩ−supp[𝐹]. (52) Due to the estimate of (51) and the convergence of (52), it follows that esssup_{𝑡∈(0,𝑇)}𝜚_𝜖−𝜚̃𝜖_ess𝐿²_(Ω)≤𝐶. (53)

In order that we now derive the estimate of the velocity, we first verify that esssup_{𝑡∈(0,𝑇)}𝜚_𝜖−𝜚̃𝜖_res𝐿²_(Ω)≤𝐶. (54) To do this, we use the following inequality due to two assumptions of pressure (7) and (12), 𝜕²𝐻_𝜖𝜚𝜕𝜚²=𝑝^′𝜖^𝛼𝜚𝜚1≥𝐶𝜚+𝜖^2𝛼/3𝜚^1/3, (55) which deduce, thanks to (49), esssup_{𝑡∈(0,𝑇)}[1]_res≤𝜖²𝐶,esssup_{𝑡∈(0,𝑇)}[𝜚_𝜖log(𝜚_𝜖)]_res𝐿1(Ω)≤𝐶𝜖²,esssup_{𝑡∈(0,𝑇)}[𝜚_𝜖]_{res𝐿5/3(Ω)}≤𝐶𝜖^{(6−2𝛼)/5}. (56)(57)(58) Finally, we get esssup_{𝑡∈(0,𝑇)}[𝜚_𝜖]_res𝜖_𝐿^6/5_(Ω)≤𝐶𝜖^1/6, (59) 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 esssup_{𝑡∈(0,𝑇)}[𝐮_𝜖]_ess𝐿2(Ω)≤𝐶sup_𝑡>0Ω𝜚_𝜖𝐮_𝜖²𝐮d𝑥≤𝐶,_{𝜖res2𝐿2(Ω)}≤𝐶_Ω𝜚_𝜖−𝜚̃𝐮_𝜖²d𝑥≤𝜖𝐶𝐮_𝜖²_𝐿𝑝(Ω)≤𝜖𝐶∇𝐮_𝜖²_𝐿2(Ω), (60) for a certain 𝑝>1 where we have used the Sobolev embedding inequality. Thus we get 𝐮_{𝜖𝐿2((0,𝑇)×Ω)}≤𝐶, (61) and the estimates in (48) and (61) imply 𝐮_{𝜖𝐿2(0,𝑇;𝐻1(Ω;ℝ3))}≤𝐶. (62)

4.2. Convergence of Anelastic Constraint

We will use the uniform estimate (62) to deduce 𝐮_𝜖→𝐔weaklyin𝐿²0,𝑇;𝑊^1,2Ω;ℝ³, (63) up to a subsequence of {𝜖>0}. In accordance with (53) and (54), we obtain 𝜚_𝜖→𝜚̃in𝐿^∞0,𝑇;𝐿^𝑝(Ω),foracertain𝑝>0, (64) and so we can take the limit of 𝜖 in the continuity equation (20) to get 𝜚̃^𝑇_0Ω𝐔⋅∇_𝑥𝜑d𝑥d𝑡=0, (65) for all 𝜑∈𝐶^∞_𝑐((0,𝑇)×Ω).

4.3. Convergence of Moment Equation

To begin with, using two estimates (63) and (64), 𝜚_𝜖𝐮_𝜖→𝜚̃𝐔weakly−(∗)in𝐿^∞0,𝑇;𝐿^𝑝Ω;ℝ³,foracertain𝑝>0. (66) Hence 𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖→𝜚̃𝐔⊗𝐔weaklyin𝐿^∞0,𝑇;𝐿^𝑞Ω;ℝ³, (67) for a certain 𝑞>1. Actually, we do not know 𝜚𝐔⊗𝐔=𝜚̃𝐔⊗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_𝜖}_𝜖>0. Multiplying 𝐝_𝜖 on (27) yields 𝜕_𝑡𝐝_𝜖²−Δ𝐝_𝜖²+𝐮_𝜖⋅∇𝐝_𝜖²≤0, (68) and so applying the maximum principle for weak solutions provides 𝐝_𝜖𝜖>0boundedin𝐿^∞([0,𝑇]×Ω), (69) which implies ∇𝐝_𝜖𝜖>0boundedin𝐿²[0,𝑇];𝐻²(Ω), (70) where we have used (47), the basic elliptic theory, and the following Gagliardo-Nirenberg inequality: ∇𝐝_{𝜖𝐿4(Ω)}≤𝐶Δ𝐝_𝜖𝐿^1/2_2(Ω)𝐝_𝜖𝐿^1/2_∞(Ω)+𝐶𝐝_{𝜖𝐿∞(Ω)}. (71) and thus we derive 𝜕_𝑡𝐝_𝜖𝜖>0boundedin𝐿^𝑝0,𝑇;𝑊^−1,2Ω;ℝ³, (72) for a certain 𝑝>1. Applying the Aubin-Lions lemma applied to (27) together with (47), (70), (72), and (61) implies 𝐝_𝜖→𝐝strongly𝐿²0,𝑇;𝐿²Ω;𝐻¹(Ω). (73) Thus we get the boundary condition 𝐝|_𝜕Ω=𝐝₀ and ∇𝐝_𝜖⊗∇𝐝_𝜖−12∇𝐝_𝜖²+𝐹𝐝_𝜖𝕀1→∇𝐝⊗∇𝐝−2|∇𝐝|²+𝐹(𝐝)𝕀 (74) 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: 𝜕_𝑡𝜚_𝜖𝐮_𝜖+div_𝑥𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖+1𝜖²∇_𝑥𝑝_𝜖𝜚_𝜖−𝑝₀𝜚̃_𝜖+𝑝₀∇𝜚_𝜖−𝜚̃𝜖²+𝜚̃−𝜚_𝜖𝜖²=𝜇_𝜖Δ𝐮_𝜖−𝜈_𝜖div_𝑥∇𝐝_𝜖⊗∇𝐝_𝜖−12∇𝐝_𝜖²+𝐹𝐝_𝜖𝕀₃, (75) From the previous estimates, we get 𝜚̃^𝑇_0Ω𝐔⋅𝜕_𝑡𝜑+𝜚̃𝐔⊗𝐔∶∇_𝑥𝜑=d𝑥d𝑡_𝑇0Ω𝜇∇_𝑥𝐔∶∇_𝑥𝜑−𝜈(∇𝐝⊗∇𝐝)∶∇_𝑥𝜑−d𝑥d𝑡_Ω𝜚̃𝐔₀⋅𝜑d𝑥 (76) for any test function 𝜑∈𝐶^∞_𝑐[0,𝑇]×Ω;ℝ³,div_𝑥𝜑=0, (77) if we show 1𝜖^2𝑇_0Ω𝑝_𝜖𝜚_𝜖−𝑝₀𝜚_𝜖d𝑥d𝑡→0as𝜖→0, (78) where we have here used (8), (53), and (54). It remains to show that 𝜚̃^𝑇_0Ω𝐔⊗𝐔∶∇_𝑥𝜑d𝑥d𝑡=^𝑇_0Ω𝜚̃𝐔⊗𝐔∶∇_𝑥𝜑d𝑥d𝑡 (79) for any 𝜑∈𝐶^∞_𝑐(0,𝑇)×Ω;ℝ³,div_𝑥𝜑=0. (80) The proofs of (78) and (79) are provided in the next two sections.

4.4. 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 𝐿^𝑞, with 𝑞>1. We here use the Bogovski operator. Let us take a test function and adapt this test function in the moment equation (2): 𝜑(𝑡,𝑥)=𝜓(𝑡)ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖,().d𝑥𝜓∈𝒟0,𝑇 (81) We will write the moment equation (2) with simple computations again: 1𝜖^2𝑇_0Ω𝜓𝑝_𝜖𝜚_𝜖𝑏𝜚_𝜖=1d𝑥d𝑡𝜖²|Ω|^𝑇_0Ω𝜓𝑝_𝜖𝜚_𝜖d𝑥_Ω𝑏𝜚_𝜖−1d𝑥d𝑡𝜖^2𝑇_0Ω𝜓𝜚_𝜖𝑏𝜚∇𝐺⋅ℬ_𝜖−1|Ω|_Ω𝑏𝜚_𝜖d𝑥d𝑥d𝑡−^𝑇_0Ω𝜓Π_𝜖⋅ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖d𝑥d𝑥d𝑡+𝐽_𝜖, (82) where Π_𝜖∶=−𝜈_𝜖div_𝑥∇𝐝_𝜖⊗∇𝐝_𝜖−12∇𝐝_𝜖²+𝐹𝐝_𝜖𝕀₃, (83) and 𝐽_𝜖 is defined by 𝐽_𝜖=^𝑇_0Ω𝜓𝜇_𝜖∇_𝑥𝐮_𝜖∶∇ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖−d𝑥d𝑥d𝑡_𝑇0Ω𝜓𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖∶∇ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖−d𝑥d𝑥d𝑡_𝑇0Ω𝜕_𝑡𝜓𝜚_𝜖𝐮_𝜖⋅ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖+d𝑥d𝑥d𝑡_𝑇0Ω𝜓𝜚_𝜖𝐮_𝜖⋅ℬdiv_𝑥𝑏𝜚_𝜖𝐮_𝜖+d𝑥d𝑡_𝑇0𝜓_Ω𝜚_𝜖𝐮_𝜖⋅ℬ𝜚_𝜖𝑏^′𝜚_𝜖−𝑏𝜚_𝜖div_𝑥𝐮_𝜖−1|Ω|_Ω𝑏𝜚_𝜖−𝑏^′𝜚_𝜖𝜚_𝜖div_𝑥𝐮_𝜖d𝑥d𝑥d𝑡. (84) We notice that all estimates in (84) are uniformly bounded due to the uniform estimates in the previous section if we take a special 𝑏 verifying 𝑏𝜚+𝜚𝑏^′𝜚≤𝐶𝜚^𝛾, (85) with a certain 𝛾∈(0,1). In virtue of (57), it is easy to see esssup_{()𝑡∈0,𝑇Ω}𝑏𝜚_𝜖d𝑥≤𝐶𝜖², (86) and so we get 𝑏(𝜚_𝜖)^𝑞_𝐿𝑞(Ω)≤𝜚_𝜖^𝛾𝑞_res𝐿1(Ω)≤𝜚_𝜖log𝜚_𝜖^𝛾𝑞_res𝐿1(Ω)≤𝐶𝜖², (87) for 𝛾<1/𝑞 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) 1𝜖²_Ω𝜚_𝜖∇𝐺⋅ℬ𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖=1d𝑥d𝑥𝜖_Ω𝜚_𝜖−𝜚̃𝜖_ess𝑏𝜚∇𝐺⋅ℬ_𝜖−1|Ω|_Ω𝑏𝜚_𝜖+1d𝑥d𝑥𝜖_Ω𝜚_𝜖−𝜚̃𝜖_res𝑏𝜚∇𝐺⋅ℬ_𝜖−1|Ω|_Ω𝑏𝜚_𝜖−𝑝d𝑥d𝑥₀𝜖²_Ω𝜚̃𝑏𝜚_𝜖−1|Ω|_Ω𝑏𝜚_𝜖d𝑥d𝑥, (88) where the last integral is uniformly bounded due to (86). Following estimates (53), (56), and (87) together with the 𝐿^𝑝-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 𝐿^𝑝-estimates for ℬ yield that the third term of the right hand side of (82) is uniformly bounded. Consequently, we deduce that ^𝑇_0Ω𝑝_𝜖𝜚_𝜖res𝜚^𝜈_𝜖d𝑥d𝑡≤𝐶𝜖², (89) for a certain 𝜈∈(0,1). We first split the integration of (78) into two parts to show (78): 1𝜖^2𝑇_0Ω𝑝₀𝜚_𝜖−𝑝_𝜖𝜚_𝜖=1d𝑥d𝑡𝜖^2𝑇_{0𝜚𝜖≤𝑀}𝑝₀𝜚_𝜖−𝑝_𝜖𝜚_𝜖+1d𝑥d𝑡𝜖^2𝑇_{0𝜚𝜖>𝑀}𝑝₀𝜚_𝜖−𝑝_𝜖𝜚_𝜖d𝑥d𝑡. (90) For the first one of the right hand side, we get 1𝜖^2𝑇_{0𝜚𝜖≤𝑀}𝑝₀𝜚_𝜖−𝑝_𝜖𝜚_𝜖≤𝜖d𝑥d𝑡_𝛼−2𝑀²2sup_{0≤𝑟≤𝑀}𝑃^′′(𝑟)|Ω|, (91) 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.

4.5. 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 𝐿²_1/𝜚̃ by the Hilbert space with the inner product ⟨𝐯,𝐰⟩_1/𝜚̃=_Ω𝐯⋅𝐰d𝑥𝜚̃ (92) and 𝐷^1,2 by the completion of 𝐶^∞_𝑐(Ω) with respect to the norm ‖∇_𝑥𝜑‖_𝐿²(Ω).

Theorem 4. For 𝐯∈𝐿²_1/𝜚̃(Ω∶ℝ³), a vector function 𝐯∶Ω→ℝ³ is written as 𝐯=𝐏_̃𝜚[𝐯]+𝐐_̃𝜚[𝐯], (93) where 𝐐_̃𝜚[𝐯]=𝜚̃∇_𝑥Ψ, (94) and Ψ is uniquely determined as the following Neumann problem: div_𝑥𝜚̃∇_𝑥Ψ=div_𝑥̃𝐯𝑖𝑛Ω,𝜚∇_𝑥Ψ⋅𝐧|_𝜕Ω=0,_ΩΨd𝑥=0, (95) where 𝐧 is the outward unit normal to 𝜕Ω.

We now write 𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖=𝐏_̃𝜚𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖+𝐐_̃𝜚𝜚_𝜖𝐮_𝜖⊗𝐏_̃𝜚𝐮_𝜖+𝐐_̃𝜚𝜚_𝜖𝐮_𝜖⊗𝐐_̃𝜚𝐮_𝜖. (96) Let us first show 𝐏_̃𝜚𝜚_𝜖𝐮_𝜖→𝐏_̃𝜚𝜚̃𝐔=𝜚̃𝐔in𝐿¹(0,𝑇)×Ω;ℝ³. (97) To do this, we adapt the following test function to the moment equation (22): 𝜑()=𝜑(𝑡)𝑡,𝑥𝜚̃𝐏_̃𝜚𝜚̃Ψ,Ψ∈𝐶^∞_𝑐Ω;ℝ³,Ψ⋅𝐧|_𝜕Ω=0,𝜓∈𝐶^∞_𝑐(0,𝑇). (98) Taking into account the uniform estimates obtained in the previous sections, it follows that 𝑡∈[0,𝑇]↦_Ω𝐏_̃𝜚𝜚_𝜖𝐮_𝜖⋅Ψd𝑥 (99) precompact in 𝐶[0,𝑇] and so the uniform estimates (45) and (64) with using the Sobolev embedding 𝐿^5/4(Ω)↪[𝑊^1,2(Ω)]^∗ imply that 𝐏_̃𝜚𝜚_𝜖𝐮_𝜖→𝐏_̃𝜚𝜚̃𝐔=𝜚̃𝐔in𝐶_weak[0,𝑇];𝐿^5/4Ω;ℝ³. (100) Notice that ^𝑇_0Ω𝐏_̃𝜚𝜚_𝜖𝐮_𝜖⋅𝐏_̃𝜚𝜚̃𝐮_𝜖d𝑥𝜚̃=d𝑡_𝑇0Ω𝐏_̃𝜚𝜚_𝜖𝐮_𝜖⋅𝐮_𝜖→d𝑥d𝑡_𝑇0Ω𝐏_̃𝜚̃=𝜚𝐔⋅𝐔d𝑥d𝑡_𝑇0Ω𝜚̃²|𝐔|²d𝑥𝜚̃d𝑡, (101) where we have here used (100). In virtue of (64) and (101), one gets 𝐏_̃𝜚𝜚_𝜖𝐮_𝜖→𝜚̃𝐔in𝐿²(0,𝑇)×Ω;ℝ³. (102) Since the Helmholtz projections 𝐯→𝐏_̃𝜚[𝐯],𝐯→𝐐_̃𝜚[𝐯] (103) map continuously the spaces 𝐿^𝑝(Ω;ℝ³) and 𝑊^1,𝑝(Ω;ℝ³) into itself for any 1<𝑝, it is easily seen that 𝐐_̃𝜚𝜚_𝜖𝐮_𝜖→0weakly−(∗)in𝐿^∞0,𝑇;𝐿^5/4Ω;ℝ³. (104) In order to show (79), we should prove lim_𝜖→0^𝑇_0Ω𝐐_̃𝜚𝜚_𝜖𝐮_𝜖⊗𝐐_̃𝜚𝜚_𝜖𝐮_𝜖∶∇_𝑥𝜑𝜚̃d𝑥𝜚̃d𝑡=0, (105) for any 𝜑∈𝐶^∞_𝑐(0,𝑇)×Ω;ℝ³,div_𝑥𝜑=0,𝜑⋅𝐧|_𝜕Ω=0. (106)

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[𝜚_𝜖𝐮_𝜖]. To begin with, we rewrite (1) and (2): 𝜖𝜕_𝑡𝑋_𝜖+1𝜚̃div_𝑥𝐕_𝜖=0,𝜖𝜕_𝑡𝐕_𝜖+𝑝₀𝜚̃∇_𝑥𝑋_𝜖=𝜖div_𝑥𝐆¹_𝜖+∇𝐆²_𝜖, (107) where the previous estimates provide 𝐆¹_𝜖isboundedin𝐿^𝑝((0,𝑇)×Ω)^3×3𝐆²_𝜖isboundedin𝐿^𝑝((0,𝑇)×Ω)³, (108) for a certain 𝑝>1 and 𝑋_𝜖=𝜚_𝜖−𝜚̃𝜖𝜚̃,𝐕_𝜖=𝜚_𝜖𝐮_𝜖𝐆¹_𝜖=𝜇_𝜖∇𝐮_𝜖−𝜚_𝜖𝐮_𝜖⊗𝐮_𝜖−𝜈_𝜖∇𝐝_𝜖⊗∇𝐝_𝜖𝐆²_𝜖=12∇𝐝_𝜖²+𝐹𝐝_𝜖−𝐻_𝜖𝜚_𝜖−𝐻_𝜖𝜚̃_𝜖−𝐻^′𝜚̃_𝜖𝜚_𝜖−𝜚̃_𝜖. (109)(110)(111) We now use the method of spectral analysis of the wave operator. Let us consider the eigenvalue problem: 𝜚̃∇_𝑥𝐀𝜚̃=𝜆𝐖,𝑝₀div_𝑥𝐖=𝜆𝐀,inΩ, (112) with the boundary condition 𝐖⋅𝐧|_𝜕Ω=0. (113) Thus, the eigenvalue problem (112) and (113) can be written into the following form: −div_𝑥𝜚̃∇_𝑥𝐀𝜚̃𝜚̃𝐀=Λ𝜚̃,inΩ, (114) with ∇_𝑥𝐀𝜚̃⋅𝐧|_𝜕Ω=0,𝜆²=−Λ𝑝₀. (115) Following the eigenvalue problem (114) and (115), it is well known that there is an orthonormal basis {𝐀_𝑗,𝑚}^∞,𝑚𝑗_{𝑗=0,𝑚=1} of real eigenfunctions of the weighed Lebesgue space 𝐿²_1/𝜚̃(Ω) corresponding to eigenvalues Λ_𝑗,𝑚 such that 𝑚₀=1,Λ_0,1,𝐀_0,1=𝜚̃,0<Λ_1,1=⋯=Λ_0,𝑚1=Λ₁<Λ_2,1=⋯=Λ_2,𝑚2=Λ₂<⋯, (116) where 𝑚_𝑗 stands for the multiplicity of Λ_𝑗. We also see that {𝐖_𝑗,𝑚}^∞,𝑚𝑗_{𝑗=0,𝑚=1} is an orthonormal basis in 𝐐[𝐿²_1/𝜚̃(Ω)³] defined by 𝐖_±𝑗,𝑚=∓𝑖𝑝₀Λ_𝑗𝜚̃∇_𝑥𝐀_𝑗,𝑚𝜚̃,𝑗=1,2,...,𝑚,𝑚=1,...,𝑚_𝑗. (117) We now take 𝜑=𝜓₁(𝑡)𝐀_𝑗,𝑚 and 𝜙=𝜓₂(𝑡)𝐖_±𝑗,𝑚 as test functions to the system (107), and then we obtain 𝜖𝜕_𝑡𝑋_{𝜖𝑗,𝑚}−𝑝₀Λ_𝑗div_𝑥𝐕_{𝜖𝑗,𝑚}=0,𝜖𝜕_𝑡𝐕_{𝜖𝑗,𝑚}+Λ_𝑗𝑋_{𝜖𝑗,𝑚}=𝜖𝐵_{𝜖𝑗,𝑚}, (118) for 𝑗=1,2,..., and 𝑚=1,...,𝑚_𝑗, where [𝑋_𝜖]_𝑗,𝑚,  [𝐕_𝜖]_𝑗,𝑚 are defined by 𝑋_{𝜖𝑗,𝑚}=_Ω𝑋_𝜖𝐀_𝑗,𝑚𝐕d𝑥,_{𝜖𝑗,𝑚}=𝑖𝑝_0Ω𝐕_𝜖⋅𝐖_𝑗,𝑚d𝑥, (119) and the estimates in (108) show that 𝐵_{𝜖𝑗,𝑚𝜖>0}isboundedin𝐿¹(0,𝑇)foranyfix𝑗,𝑚. (120) Moreover, in virtue of section 5 in [7] for a finite number of modes, we set 𝐐_̃𝜚,𝑁𝜚̃𝐙=−𝑖𝑝_{0𝑗,0<Λ𝑗}^𝑚_≤𝑁^𝑗_𝑚=1𝜚̃𝐙_𝑗,𝑚𝐖_𝑗,𝑚. (121)

We first note 𝐐_̃𝜚[𝜚̃𝐮_𝜖]−𝐐_̃𝜚,𝑁[𝜚̃𝐮_𝜖]²_𝐿21/𝜚̃^(Ω;ℝ3)≤1𝑁div_𝑥𝜚̃𝐮_𝜖²_𝐿21/𝜚̃(Ω)→0, (122) as 𝑁→∞ in 𝐿²(0,𝑇;𝐿²_1/𝜚̃(Ω;ℝ³)) 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_̃𝜚,𝑁 instead of 𝐐_̃𝜚. Observe that, thanks to (114) and (121), it is easy to see Ψ_𝜖=1𝑝₀^𝑁𝑚_𝑗=1^𝑗_𝑚=1𝐕_{𝜖𝑗,𝑚}Λ_𝑗𝐀_𝑗,𝑚𝜚̃,−div_𝑥𝜚̃∇_𝑥Ψ_𝜖=1𝑝₀^𝑁𝑚_𝑗=1^𝑗_𝑚=1Λ_𝑗𝐕_{𝜖𝑗,𝑚}𝐀_𝑗,𝑚. (123)

We are now ready to show (105) and so we now use (118) and (123) to rewrite the oscillation part 𝐐_̃𝜚[𝜚_𝜖𝐮_𝜖] as lim_𝜖→0^𝑇_0Ω𝐐_̃𝜚𝜚_𝜖𝐮_𝜖⊗𝐐_̃𝜚𝜚_𝜖𝐮_𝜖∶∇_𝑥𝜑𝜚̃d𝑥d𝑡=lim_𝜖→0^𝑇_0Ω𝜚̃∇_𝑥Ψ_𝜖⊗∇_𝑥Ψ_𝜖∶∇_𝑥𝜑𝜚̃d𝑥d𝑡=−lim_𝜖→0^𝑇_0Ωdiv_𝑥𝜚̃∇_𝑥Ψ_𝜖∇_𝑥Ψ_𝜖∶∇_𝑥𝜑𝜚̃𝜖d𝑥d𝑡=−𝑝²₀lim_𝜖→0^𝑇_0Ω^𝑁𝑚_𝑗=1^𝑗_𝑚=1Λ_𝑗𝑋_{𝜖𝑗,𝑚}×𝐀_𝑗,𝑚𝜚̃∇_𝑥Ψ_𝜖⋅𝜕_𝑡+𝜖𝜑d𝑥d𝑡𝑝²₀lim_𝜖→0^𝑇_0Ω^𝑁𝑚_𝑗=1^𝑗_𝑚=1Λ_𝑗𝑋_{𝜖𝑗,𝑚}×𝐀_𝑗,𝑚𝜚̃𝜕_𝑡∇_𝑥Ψ_𝜖⋅𝜑d𝑥d𝑡 (124) which converges to 0 as 𝜖 tends to 0 where we can see this proof in section 6.6 of Feireisl and Novotný [7].

Acknowledgment

The work of Y.-S. Kwon was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0003611).