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

Authors: Young-Sam Kwon
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 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:(1)∂tϱ+divx(ϱu)=0,(2)∂t(ϱu)+divx(ϱu⊗u)+∇xp(ϱ)=μΔu,-νdivx(∇d⊗∇d-(12|∇d|2+F(d))𝕀3)+ϱ∇G,(3)∂td+u·∇d=θ(Δd-f(d)),d(0,x)=d0(x),
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
(4)f(d)=∇dF(d),
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,
(5)f(d)·d≥0.
Note that the condition (5) is required for the global weak solution. We can see an example as follows:
(6)F(d)=14σ02(|d|2-1)2,f(d)=12σ02(|d|2-1)d,
where σ0 is a constant. Finally, the pressure p is defined as follows
(7)p(0)=0,p∈C2[0,∞),p′(r)>0∀r≥0,limr→∞p′(r)rγ-1=q>0,
and the function ϱ~ is the unique positive solution of the following problem:
(8)p0∇ϱ~=ϱ~∇G,p0=p′(0).
We use the following scaling used in Feireisl et al. [1] and Wang and Yu [2]:
(9)t≈ϵt,x≈x,ϱ≈ϱϵ,u≈ϵuϵ,d≈dϵ,
and for the viscosity coefficients,
(10)μ≈ϵμϵ,ν≈ϵ2νϵ,θ≈ϵθϵ,
with the convergence of the viscosity coefficients
(11)μϵ→μ,νϵ→ν,θϵ→θ.
Finally for the pressure, we use
(12)pϵ(ϱ)=1ϵαp(ϵαϱ),α∈(2,3).
Following the above scalings, the system (1)–(7) reads
(13)∂tϱϵ+divx(ϱϵuϵ)=0,(14)∂t(ϱϵuϵ)+divx(ϱϵuϵ⊗uϵ)+1ϵ2∇xpϵ(ϱϵ)=μϵΔuϵ-νϵdivx(∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀3)+1ϵ2ϱϵ∇G,(15)∂tdϵ+u·∇dϵ=θϵ(Δdϵ-f(dϵ)).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 whenϵ→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
(16)ϱϵ→ϱ~,uϵ→U,dϵ→d,
as ϵ tends to 0 in the suitable sense such that the given limit {ϱ~,U,d} represents a solution of the following system:
(17)divx(ϱ~U)=0,∂t(ϱ~U)+divx(ϱ~U⊗U)+∇xP=μΔU-(12|∇d|2+F(d)𝕀3),∂td+U·∇d=θ(Δd-f(d)).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 Section2 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

LetΩ⊂ℝ3 be a bounded domain with the boundary of class C∞(18)u·n|∂Ω=0,
where n stands for the outer normal vector. We also propose the boundary condition on the direction vector d(19)d|∂Ω=d0.
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 densityϱ is a nonnegative function, ϱ∈L∞(0,T;Lγ(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
(20)∫0T‍∫Ω‍(ϱB(ϱ)∂tφ+ϱB(ϱ)u·∇xφ-b(ϱ)divxuφ)dxdt=-∫Ω‍ϱ0B(ϱ0)φ(0,·)dxholds for any test functionφ∈𝒟([0,T)×Ω¯) and any b such that
(21)b∈L∞∩C[0,∞),B(ϱ)=B(1)+∫1ϱ‍b(z)z2dz.(ii)
The balance of momentum holds in distributional sense, namely,(22)∫0T‍∫Ω‍(1ϵ2ϱu·∂tφ→+ϱu⊗u:∇xφ→+1ϵ2pϵ(ϱ)divxφ→-νϵΠ:∇φ→+μϵ∇u:∇φ→+1ϵ2∇G·φ→)dxdt=-∫Ω‍ϱ0u0·φ→(0,·)dxfor any test functionφ→∈𝒟([0,T);𝒟(Ω¯;ℝ3)) satisfying φ→·n|∂Ω=0, where Π is defined by
(23)Π=∇d⊗∇d-(12|∇d|2+F(d))𝕀3.(iii)
The total energy of the system holds:(24)E(t)+∫0t‍∫Ω‍(μϵ|∇u|2+νϵθϵ|Δd-f(d)|2)dxdt≤E(0)holds for a.e.t∈(0,T), where
(25)E(t)=∫Ω‍(12ϱ|u|2+1ϵ2Hϵ(ϱ)+νϵ2|d|2+νϵF(d)-ϱG12)dxwith(26)Hϵ(ϱ)=ϱ∫1ϱ‍pϵ(ϱ)z2dz.(iv)
The equation of direction field verifies(27)∫0T‍∫Ω‍d·∂tφ→dxdt+∫0T‍∫Ω‍(u·d)divxφ→+(∇u·d)·φ→-θ(∇d+f(d))·φ→dxdt=0,for allφ→∈[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 ϱ~ϵ must satisfy
(28)∇pϵ(ϱ~ϵ)=ϱ~ϵ∇G,p0=p′(0).
From the statistic equation (28), we can easily derive
(29)p0log(ϱ~(x))+Q(ϵαϱ~(x))-Q(ϵαϱ∞)=G(x)+p0log(ϱ∞)
with
(30)Q′(r)={P′(r)-p0r,ifr>0,P′′(0),ifr=0.We now introduce the weak solutions of the target system.Definition 2.
A couple{ϱ~,U,d} is said to be a weak solution of the target system of the compressible flows of liquid crystals with the potential G∈W1,∞(Ω),∇G∈L1(Ω), supplemented with the boundary conditions
(31)U·n=0,d=d0,
on ∂Ω which belongs to C∞, and the initial conditions
(32)U(0,·)=U0,d(0,·)=d0,
with U∈L2(Ω),  d0∈H1(Ω),  d0|∂Ω∈H3/2(Ω) if the following conditions hold:(i)
U∈L∞(0,T;L2(Ω;ℝ3))∩L2(0,T;H1(Ω;ℝ3)),divx(ϱ~U)=0 a.e. on (0,T)×Ω, U·n|∂Ω=0 in the sense of traces, and the integral identity
(33)∫0T‍∫Ω‍(ϱ~U·∂tφ+(ϱ~U⊗U):∇xφ-μ∇xU:∇xφ)dxdt=-∫0T‍∫Ω‍(∇d⊗∇d):∇φdxdt-∫Ω‍U·φ(0,·)dxholds for any test function(34)φ∈𝒟((0,T)×Ω;ℝ3),divxφ=0inΩ,φ·n|∂Ω=0,(ii)
d∈L2(0,T;H2(Ω;ℝ3))∩L∞(0,T;H1(Ω;ℝ3)), and the integral identity
(35)∫0T‍∫Ω‍d·∂tφ→dxdt+∫0T‍∫Ω‍((U·d)divxφ→+(∇U·d)·φ→-θ(∇d+f(d))·φ→)dxdt=0holds for allφ→∈[𝒟([0,T)×Ω)]3.


## 2.1. An Initial-Boundary-Value Problem

LetΩ⊂ℝ3 be a bounded domain with the boundary of class C∞(18)u·n|∂Ω=0,
where n stands for the outer normal vector. We also propose the boundary condition on the direction vector d(19)d|∂Ω=d0.
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 densityϱ is a nonnegative function, ϱ∈L∞(0,T;Lγ(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
(20)∫0T‍∫Ω‍(ϱB(ϱ)∂tφ+ϱB(ϱ)u·∇xφ-b(ϱ)divxuφ)dxdt=-∫Ω‍ϱ0B(ϱ0)φ(0,·)dxholds for any test functionφ∈𝒟([0,T)×Ω¯) and any b such that
(21)b∈L∞∩C[0,∞),B(ϱ)=B(1)+∫1ϱ‍b(z)z2dz.(ii)
The balance of momentum holds in distributional sense, namely,(22)∫0T‍∫Ω‍(1ϵ2ϱu·∂tφ→+ϱu⊗u:∇xφ→+1ϵ2pϵ(ϱ)divxφ→-νϵΠ:∇φ→+μϵ∇u:∇φ→+1ϵ2∇G·φ→)dxdt=-∫Ω‍ϱ0u0·φ→(0,·)dxfor any test functionφ→∈𝒟([0,T);𝒟(Ω¯;ℝ3)) satisfying φ→·n|∂Ω=0, where Π is defined by
(23)Π=∇d⊗∇d-(12|∇d|2+F(d))𝕀3.(iii)
The total energy of the system holds:(24)E(t)+∫0t‍∫Ω‍(μϵ|∇u|2+νϵθϵ|Δd-f(d)|2)dxdt≤E(0)holds for a.e.t∈(0,T), where
(25)E(t)=∫Ω‍(12ϱ|u|2+1ϵ2Hϵ(ϱ)+νϵ2|d|2+νϵF(d)-ϱG12)dxwith(26)Hϵ(ϱ)=ϱ∫1ϱ‍pϵ(ϱ)z2dz.(iv)
The equation of direction field verifies(27)∫0T‍∫Ω‍d·∂tφ→dxdt+∫0T‍∫Ω‍(u·d)divxφ→+(∇u·d)·φ→-θ(∇d+f(d))·φ→dxdt=0,for allφ→∈[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 ϱ~ϵ must satisfy
(28)∇pϵ(ϱ~ϵ)=ϱ~ϵ∇G,p0=p′(0).
From the statistic equation (28), we can easily derive
(29)p0log(ϱ~(x))+Q(ϵαϱ~(x))-Q(ϵαϱ∞)=G(x)+p0log(ϱ∞)
with
(30)Q′(r)={P′(r)-p0r,ifr>0,P′′(0),ifr=0.We now introduce the weak solutions of the target system.Definition 2.
A couple{ϱ~,U,d} is said to be a weak solution of the target system of the compressible flows of liquid crystals with the potential G∈W1,∞(Ω),∇G∈L1(Ω), supplemented with the boundary conditions
(31)U·n=0,d=d0,
on ∂Ω which belongs to C∞, and the initial conditions
(32)U(0,·)=U0,d(0,·)=d0,
with U∈L2(Ω),  d0∈H1(Ω),  d0|∂Ω∈H3/2(Ω) if the following conditions hold:(i)
U∈L∞(0,T;L2(Ω;ℝ3))∩L2(0,T;H1(Ω;ℝ3)),divx(ϱ~U)=0 a.e. on (0,T)×Ω, U·n|∂Ω=0 in the sense of traces, and the integral identity
(33)∫0T‍∫Ω‍(ϱ~U·∂tφ+(ϱ~U⊗U):∇xφ-μ∇xU:∇xφ)dxdt=-∫0T‍∫Ω‍(∇d⊗∇d):∇φdxdt-∫Ω‍U·φ(0,·)dxholds for any test function(34)φ∈𝒟((0,T)×Ω;ℝ3),divxφ=0inΩ,φ·n|∂Ω=0,(ii)
d∈L2(0,T;H2(Ω;ℝ3))∩L∞(0,T;H1(Ω;ℝ3)), and the integral identity
(35)∫0T‍∫Ω‍d·∂tφ→dxdt+∫0T‍∫Ω‍((U·d)divxφ→+(∇U·d)·φ→-θ(∇d+f(d))·φ→)dxdt=0holds for allφ→∈[𝒟([0,T)×Ω)]3.


## 3. Main Results

In this section we mention the main result as follows.Theorem 3.
LetΩ⊂ℝ3 be a bounded domain with a boundary of class C∞ and {ϱϵ,uϵ,dϵ} a family of weak solutions to the compressible of liquid crystals system verifying (5) in the sense of Definition 1 with G∈W1,∞(Ω),∇G∈L1(Ω). Assume that the initial condition is as follows:
(36){ϱ0,ϵ(1)}ϵ>0boundedin(L2∩L∞)(Ω),withϱ0,ϵ(1):=ϱ0,ϵ-ϱ~ϵ,(37){u0,ϵ}ϵ>0boundedin(L2∩L∞)(Ω;ℝ3),{d0,ϵ}ϵ>0boundedin(L2∩L∞)(Ω;ℝ3).
Then, up to subsequence,(38)ϱϵ→ϱ~a.e.in(0,T)×Ω,uϵ→Ua.e.in(0,T)×Ω,dϵ→da.e.in(0,T)×Ω,
where {ϱ~,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
(39)U(0)=Pϱ~[U0],d(0,·)=d0,
where the Helmholtz projection Pϱ~=I-Qϱ~ and Qϱ~ is defined in (81).


## 4. Proof of Theorem3

### 4.1.  Uniform Bounds

In this section we are going to derive some estimates on the sequence{ϱϵ,uϵ,dϵ}ϵ>0. Multiplying (13) by H′(ϱϵ)-H′(ϱ~ϵ) and adding the energy inequality (24), it follows that
(40)∫Ω‍(12ϱϵ|uϵ|2+1ϵ2(Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-Hϵ′(ϱ~ϵ)(ϱϵ-ϱ~ϵ))+νϵ2|dϵ|2+νϵF(dϵ))dx+∫0t‍∫Ω‍(μϵ|∇uϵ|2+νϵθϵ|Δdϵ-f(dϵ)|2)dxdt≤E0,ϵ,
where
(41)E0,ϵ=∫Ω‍(12ϱ0,ϵ|u0,ϵ|2+1ϵ2(Hϵ(ϱ0,ϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱ0,ϵ-ϱ~ϵ))+νϵ2|d0,ϵ|2+νϵF(d0,ϵ))dx.
For convenient presentation, we introduce the set of the essential and residual values
(42)g=[g]ess+[g]res,
where [g]ess=χ(ϱϵ)g, [g]res=(1-χ(ϱϵ))g and χ is defined as follows:
(43)χ(r)=1∀r∈[ϱ_2,2ϱ¯],χ(r)=0otherwise,
where ϱ~ϵ is the solution of (28) and
(44)ϱ_=infϵ>0infx∈Ωϱ~ϵ(x),ϱ¯=supϵ>0supx∈Ωϱ~ϵ(x).
Notice that two assumptions of pressure in (7) and (12) imply that Hϵ is a strict convex. Thus, thanks to (36), we get that E0,ϵ is uniformly bounded for ϵ→0. Consequently, from the energy balance (40), we obtain
(45)esssupt∈(0,T)∥ϱϵuϵ(t)∥L2(Ω;ℝ3)≤C,(46)esssupt∈(0,T)∥dϵ(t)∥L2(Ω;ℝ3)≤C,(47)esssupt∈(0,T)∥(Δdϵ-f(dϵ))(t)∥L2(Ω;ℝ3)≤C,(48)∥∇xuϵ∥L2((0,T)×Ω)≤C,(49)esssupt∈(0,T)∥[Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)]ess∥L1(Ω)≤ϵ2C,(50)esssupt∈(0,T)∥[Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)]res∥L1(Ω)≤ϵ2C.Using the estimate of (49) implies
(51)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵϵ]ess∥L2(Ω)≤C.
Note that the static equation (29) holds
(52)∥ϱ~ϵ-ϱ~∥C(Ω¯)≤ϵαC,ϱ~ϵ=ϱ~=ϱ∞inΩ-supp[F].
Due to the estimate of (51) and the convergence of (52), it follows that
(53)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵ]ess∥L2(Ω)≤C.In order that we now derive the estimate of the velocity, we first verify that(54)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵ]res∥L2(Ω)≤C.
To do this, we use the following inequality due to two assumptions of pressure (7) and (12),
(55)∂2Hϵ(ϱ)∂ϱ2=p′(ϵαϱ)ϱ≥C(1ϱ+ϵ2α/3ϱ1/3),
which deduce, thanks to (49),
(56)esssupt∈(0,T)[1]res≤ϵ2C,(57)esssupt∈(0,T)∥[ϱϵlog(ϱϵ)]res∥L1(Ω)≤Cϵ2,(58)esssupt∈(0,T)∥[ϱϵ]res∥L5/3(Ω)≤Cϵ(6-2α)/5.
Finally, we get
(59)esssupt∈(0,T)∥[ϱϵ]resϵ∥L6/5(Ω)≤Cϵ1/6,
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(60)esssupt∈(0,T)∥[uϵ]ess∥L2(Ω)≤Csupt>0∫Ω‍ϱϵ|uϵ|2dx≤C,∥[uϵ]res∥L2(Ω)2≤C∫Ω‍|ϱϵ-ϱ~||uϵ|2dx≤ϵC∥uϵ∥Lp(Ω)2≤ϵC∥∇uϵ∥L2(Ω)2,
for a certain p>1 where we have used the Sobolev embedding inequality. Thus we get
(61)∥uϵ∥L2((0,T)×Ω)≤C,
and the estimates in (48) and (61) imply
(62)∥uϵ∥L2(0,T;H1(Ω;ℝ3))≤C.


### 4.2. Convergence of Anelastic Constraint

We will use the uniform estimate (62) to deduce
(63)uϵ→UweaklyinL2(0,T;W1,2(Ω;ℝ3)),
up to a subsequence of {ϵ>0}. In accordance with (53) and (54), we obtain
(64)ϱϵ→ϱ~inL∞(0,T;Lp(Ω)),foracertainp>0,
and so we can take the limit of ϵ in the continuity equation (20) to get
(65)∫0T‍∫Ω‍ϱ~U·∇xφdxdt=0,
for all φ∈Cc∞((0,T)×Ω).


### 4.3. Convergence of Moment Equation

To begin with, using two estimates (63) and (64),
(66)ϱϵuϵ→ϱ~Uweakly-(*)inL∞(0,T;Lp(Ω;ℝ3)),foracertainp>0.
Hence
(67)ϱϵuϵ⊗uϵ→ϱ~U⊗U¯weaklyinL∞(0,T;Lq(Ω;ℝ3)),
for a certain q>1. Actually, we do not know ϱU⊗U¯=ϱ~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{dϵ}ϵ>0. Multiplying dϵ on (27) yields
(68)∂t|dϵ|2-Δ|dϵ|2+uϵ·∇|dϵ|2≤0,
and so applying the maximum principle for weak solutions provides
(69){dϵ}ϵ>0boundedinL∞([0,T]×Ω),
which implies
(70){∇dϵ}ϵ>0boundedinL2([0,T];H2(Ω)),
where we have used (47), the basic elliptic theory, and the following Gagliardo-Nirenberg inequality:
(71)∥∇dϵ∥L4(Ω)≤C∥Δdϵ∥L2(Ω)1/2∥dϵ∥L∞(Ω)1/2+C∥dϵ∥L∞(Ω).
and thus we derive
(72){∂tdϵ}ϵ>0boundedinLp(0,T;W-1,2(Ω;ℝ3)),
for a certain p>1. Applying the Aubin-Lions lemma applied to (27) together with (47), (70), (72), and (61) implies
(73)dϵ→dstronglyL2(0,T;L2(Ω;H1(Ω))).
Thus we get the boundary condition d|∂Ω=d0 and
(74)∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀→∇d⊗∇d-(12|∇d|2+F(d))𝕀
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:
(75)∂t(ϱϵuϵ)+divx(ϱϵuϵ⊗uϵ)+1ϵ2∇x(pϵ(ϱϵ)-p0ϱ~ϵ)+p0∇(ϱϵ-ϱ~ϵ2)+(ϱ~-ϱϵϵ2)=μϵΔuϵ-νϵdivx((12|∇dϵ|2+F(dϵ))∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀3),
From the previous estimates, we get
(76)∫0T‍∫Ω‍(ϱ~U·∂tφ+ϱ~U⊗U¯:∇xφ)dxdt=∫0T‍∫Ω‍(μ∇xU:∇xφ-ν(∇d⊗∇d):∇xφ)dxdt-∫Ω‍(ϱ~U)0·φdx
for any test function
(77)φ∈Cc∞([0,T]×Ω;ℝ3),divxφ=0,
if we show
(78)1ϵ2∫0T‍∫Ω‍(pϵ(ϱϵ)-p0ϱϵ)dxdt→0asϵ→0,
where we have here used (8), (53), and (54). It remains to show that
(79)∫0T‍∫Ω‍ϱ~U⊗U¯:∇xφdxdt=∫0T‍∫Ω‍[ϱ~U⊗U]:∇xφdxdt
for any
(80)φ∈Cc∞((0,T)×Ω;ℝ3),divxφ=0.
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 areflexive space Lq, 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):
(81)φ(t,x)=ψ(t)ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx],ψ∈𝒟(0,T).
We will write the moment equation (2) with simple computations again:
(82)1ϵ2∫0T‍∫Ω‍ψpϵ(ϱϵ)b(ϱϵ)dxdt=1ϵ2|Ω|∫0T‍∫Ω‍ψpϵ(ϱϵ)dx(∫Ω‍b(ϱϵ)dx)dt-1ϵ2∫0T‍∫Ω‍ψϱϵ∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍ψΠϵ·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt+Jϵ,
where
(83)Πϵ∶=-νϵdivx(∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀3),
and Jϵ is defined by
(84)Jϵ=∫0T‍∫Ω‍ψμϵ∇xuϵ:∇ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍ψϱϵuϵ⊗uϵ:∇ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍∂tψϱϵuϵ·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt+∫0T‍∫Ω‍ψϱϵuϵ·ℬ[divx(b(ϱϵ)uϵ)]dxdt+∫0T‍ψ∫Ω‍ϱϵuϵ·ℬ[1|Ω|∫Ω‍(ϱϵb′(ϱϵ)-b(ϱϵ))divxuϵ-1|Ω|∫Ω‍(b(ϱϵ)-b′(ϱϵ)ϱϵ)divxuϵdx]dxdt.
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
(85)|b(ϱ)|+|ϱb′(ϱ)|≤Cϱγ,
with a certain γ∈(0,1). In virtue of (57), it is easy to see
(86)esssupt∈(0,T)∫Ω‍b(ϱϵ)dx≤Cϵ2,
and so we get
(87)∥b(ϱϵ)∥Lq(Ω)q≤∥[ϱϵ]resγq∥L1(Ω)≤∥[ϱϵlogϱϵ]resγq∥L1(Ω)≤Cϵ2,
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)
(88)1ϵ2∫Ω‍ϱϵ∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx=1ϵ∫Ω‍[ϱϵ-ϱ~ϵ]ess∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx+1ϵ∫Ω‍[ϱϵ-ϱ~ϵ]res∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx-p0ϵ2∫Ω‍ϱ~[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx,
where the last integral is uniformly bounded due to (86). Following estimates (53), (56), and (87) together with the Lp-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 Lp-estimates for ℬ yield that the third term of the right hand side of (82) is uniformly bounded. Consequently, we deduce that
(89)∫0T‍∫Ω‍[pϵ(ϱϵ)]resϱϵνdxdt≤Cϵ2,
for a certain ν∈(0,1). We first split the integration of (78) into two parts to show (78):
(90)1ϵ2∫0T‍∫Ω‍|p0ϱϵ-pϵ(ϱϵ)|dxdt=1ϵ2∫0T‍∫ϱϵ≤M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt+1ϵ2∫0T‍∫ϱϵ>M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt.
For the first one of the right hand side, we get
(91)1ϵ2∫0T‍∫ϱϵ≤M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt≤ϵα-2M22sup0≤r≤MP′′(r)|Ω|,
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 L1/ϱ~2 by the Hilbert space with the inner product
(92)〈v,w〉1/ϱ~=∫Ω‍v·wdxϱ~
and D1,2 by the completion of Cc∞(Ω¯) with respect to the norm ∥∇xφ∥L2(Ω).Theorem 4.
Forv∈L1/ϱ~2(Ω:ℝ3), a vector function v:Ω→ℝ3 is written as
(93)v=Pϱ~[v]+Qϱ~[v],
where
(94)Qϱ~[v]=ϱ~∇xΨ,
and Ψ is uniquely determined as the following Neumann problem:
(95)divx(ϱ~∇xΨ)=divxvinΩ,ϱ~∇xΨ·n|∂Ω=0,∫Ω‍Ψdx=0,
where n is the outward unit normal to ∂Ω.We now write(96)ϱϵuϵ⊗uϵ=Pϱ~[ϱϵuϵ]⊗uϵ+Qϱ~[ϱϵuϵ]⊗Pϱ~[uϵ]+Qϱ~[ϱϵuϵ]⊗Qϱ~[uϵ].
Let us first show
(97)Pϱ~[ϱϵuϵ]→Pϱ~[ϱ~U]=ϱ~UinL1((0,T)×Ω;ℝ3).
To do this, we adapt the following test function to the moment equation (22):
(98)φ(t,x)=φ(t)ϱ~Pϱ~[ϱ~Ψ],Ψ∈Cc∞(Ω¯;ℝ3),Ψ·n|∂Ω=0,ψ∈Cc∞(0,T).
Taking into account the uniform estimates obtained in the previous sections, it follows that
(99)t∈[0,T]↦∫Ω‍Pϱ~[ϱϵuϵ]·Ψdx
precompact in C[0,T] and so the uniform estimates (45) and (64) with using the Sobolev embedding L5/4(Ω)↪[W1,2(Ω)]* imply that
(100)Pϱ~[ϱϵuϵ]→Pϱ~[ϱ~U]=ϱ~UinCweak([0,T];L5/4(Ω;ℝ3)).
Notice that
(101)∫0T‍∫Ω‍Pϱ~[ϱϵuϵ]·Pϱ~[ϱ~uϵ]dxϱ~dt=∫0T‍∫Ω‍Pϱ~[ϱϵuϵ]·uϵdxdt→∫0T‍∫Ω‍Pϱ~[ϱ~U]·Udxdt=∫0T‍∫Ω‍ϱ~2|U|2dxϱ~dt,
where we have here used (100). In virtue of (64) and (101), one gets
(102)Pϱ~[ϱϵuϵ]→ϱ~UinL2((0,T)×Ω;ℝ3).
Since the Helmholtz projections
(103)v→Pϱ~[v],v→Qϱ~[v]
map continuously the spaces Lp(Ω;ℝ3) and W1,p(Ω;ℝ3) into itself for any 1<p, it is easily seen that
(104)Qϱ~[ϱϵuϵ]→0weakly-(*)inL∞(0,T;L5/4(Ω;ℝ3)).
In order to show (79), we should prove
(105)limϵ→0∫0T‍∫Ω‍Qϱ~[ϱϵuϵ]⊗Qϱ~[ϱϵuϵ]:∇x(φϱ~)dxϱ~dt=0,
for any
(106)φ∈Cc∞((0,T)×Ω¯;ℝ3),divxφ=0,φ·n|∂Ω=0.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 ofQ[ϱϵuϵ]. To begin with, we rewrite (1) and (2):
(107)ϵ∂tXϵ+1ϱ~divxVϵ=0,ϵ∂tVϵ+p0ϱ~∇xXϵ=ϵ(divxGϵ1+∇Gϵ2),
where the previous estimates provide
(108)Gϵ1isboundedinLp((0,T)×Ω)3×3Gϵ2isboundedinLp((0,T)×Ω)3,
for a certain p>1 and
(109)Xϵ=ϱϵ-ϱ~ϵϱ~,Vϵ=ϱϵuϵ(110)Gϵ1=μϵ∇uϵ-(ϱϵuϵ⊗uϵ)-νϵ(∇dϵ⊗∇dϵ)(111)Gϵ2=12|∇dϵ|2+F(dϵ)-(Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)).
We now use the method of spectral analysis of the wave operator. Let us consider the eigenvalue problem:
(112)ϱ~∇x(Aϱ~)=λW,p0divxW=λA,inΩ,
with the boundary condition
(113)W·n|∂Ω=0.
Thus, the eigenvalue problem (112) and (113) can be written into the following form:
(114)-divx[ϱ~∇x(Aϱ~)]=Λϱ~(Aϱ~),inΩ,
with
(115)∇x(Aϱ~)·n|∂Ω=0,λ2=-Λp0.
Following the eigenvalue problem (114) and (115), it is well known that there is an orthonormal basis {Aj,m}j=0,m=1∞,mj of real eigenfunctions of the weighed Lebesgue space L1/ϱ~2(Ω) corresponding to eigenvalues Λj,m such that
(116)m0=1,Λ0,1,A0,1=ϱ~,0<Λ1,1=⋯=Λ0,m1(=Λ1)<Λ2,1=⋯=Λ2,m2(=Λ2)<⋯,
where mj stands for the multiplicity of Λj. We also see that {Wj,m}j=0,m=1∞,mj is an orthonormal basis in Q[L1/ϱ~2(Ω)3] defined by
(117)W±j,m=∓ip0Λjϱ~∇xAj,mϱ~,j=1,2,...,m,m=1,...,mj.
We now take φ=ψ1(t)Aj,m and ϕ=ψ2(t)W±j,m as test functions to the system (107), and then we obtain
(118)ϵ∂t[Xϵ]j,m-p0Λjdivx[Vϵ]j,m=0,ϵ∂t[Vϵ]j,m+Λj[Xϵ]j,m=ϵ[Bϵ]j,m,
for j=1,2,..., and m=1,...,mj, where [Xϵ]j,m,  [Vϵ]j,m are defined by
(119)[Xϵ]j,m=∫Ω‍XϵAj,mdx,[Vϵ]j,m=ip0∫Ω‍Vϵ·Wj,mdx,
and the estimates in (108) show that
(120){[Bϵ]j,m}ϵ>0isboundedinL1(0,T)foranyfixj,m.
Moreover, in virtue of section 5 in [7] for a finite number of modes, we set
(121)Qϱ~,N[ϱ~Z]=-ip0∑j,0<Λj≤N‍∑m=1mj‍[ϱ~Z]j,mWj,m.We first note(122)∥Qϱ~[ϱ~uϵ]-Qϱ~,N[ϱ~uϵ]∥L1/ϱ~2(Ω;ℝ3)2≤1N∥divx(ϱ~uϵ)∥L1/ϱ~22(Ω)→0,
as N→∞ in L2(0,T;L1/ϱ~2(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 Qϱ~,N instead of Qϱ~. Observe that, thanks to (114) and (121), it is easy to see
(123)Ψϵ=1p0∑j=1N‍∑m=1mj‍[Vϵ]j,mΛj(Aj,mϱ~),-divx(ϱ~∇xΨϵ)=1p0∑j=1N‍∑m=1mj‍Λj[Vϵ]j,mAj,m.We are now ready to show (105) and so we now use (118) and (123) to rewrite the oscillation part Qϱ~[ϱϵuϵ] as
(124)limϵ→0∫0T‍∫Ω‍(Qϱ~[ϱϵuϵ]⊗Qϱ~[ϱϵuϵ]):∇x(φϱ~)dxdt=limϵ→0∫0T‍∫Ω‍(ϱ~∇xΨϵ⊗∇xΨϵ):∇x(φϱ~)dxdt=-limϵ→0∫0T‍∫Ω‍divx(ϱ~∇xΨϵ)∇xΨϵ:∇x(φϱ~)dxdt=-ϵp02limϵ→0∫0T‍∫Ω‍∑j=1N‍∑m=1mj‍Λj[Xϵ]j,m×[Aj,m]ϱ~∇xΨϵ·∂tφdxdt+ϵp02limϵ→0∫0T‍∫Ω‍∑j=1N‍∑m=1mj‍Λj[Xϵ]j,m×[Aj,m]ϱ~∂t∇xΨϵ·φdxdt
which converges to 0 as ϵ tends to 0 where we can see this proof in section 6.6 of Feireisl and Novotný [7].


## 4.1.  Uniform Bounds

In this section we are going to derive some estimates on the sequence{ϱϵ,uϵ,dϵ}ϵ>0. Multiplying (13) by H′(ϱϵ)-H′(ϱ~ϵ) and adding the energy inequality (24), it follows that
(40)∫Ω‍(12ϱϵ|uϵ|2+1ϵ2(Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-Hϵ′(ϱ~ϵ)(ϱϵ-ϱ~ϵ))+νϵ2|dϵ|2+νϵF(dϵ))dx+∫0t‍∫Ω‍(μϵ|∇uϵ|2+νϵθϵ|Δdϵ-f(dϵ)|2)dxdt≤E0,ϵ,
where
(41)E0,ϵ=∫Ω‍(12ϱ0,ϵ|u0,ϵ|2+1ϵ2(Hϵ(ϱ0,ϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱ0,ϵ-ϱ~ϵ))+νϵ2|d0,ϵ|2+νϵF(d0,ϵ))dx.
For convenient presentation, we introduce the set of the essential and residual values
(42)g=[g]ess+[g]res,
where [g]ess=χ(ϱϵ)g, [g]res=(1-χ(ϱϵ))g and χ is defined as follows:
(43)χ(r)=1∀r∈[ϱ_2,2ϱ¯],χ(r)=0otherwise,
where ϱ~ϵ is the solution of (28) and
(44)ϱ_=infϵ>0infx∈Ωϱ~ϵ(x),ϱ¯=supϵ>0supx∈Ωϱ~ϵ(x).
Notice that two assumptions of pressure in (7) and (12) imply that Hϵ is a strict convex. Thus, thanks to (36), we get that E0,ϵ is uniformly bounded for ϵ→0. Consequently, from the energy balance (40), we obtain
(45)esssupt∈(0,T)∥ϱϵuϵ(t)∥L2(Ω;ℝ3)≤C,(46)esssupt∈(0,T)∥dϵ(t)∥L2(Ω;ℝ3)≤C,(47)esssupt∈(0,T)∥(Δdϵ-f(dϵ))(t)∥L2(Ω;ℝ3)≤C,(48)∥∇xuϵ∥L2((0,T)×Ω)≤C,(49)esssupt∈(0,T)∥[Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)]ess∥L1(Ω)≤ϵ2C,(50)esssupt∈(0,T)∥[Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)]res∥L1(Ω)≤ϵ2C.Using the estimate of (49) implies
(51)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵϵ]ess∥L2(Ω)≤C.
Note that the static equation (29) holds
(52)∥ϱ~ϵ-ϱ~∥C(Ω¯)≤ϵαC,ϱ~ϵ=ϱ~=ϱ∞inΩ-supp[F].
Due to the estimate of (51) and the convergence of (52), it follows that
(53)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵ]ess∥L2(Ω)≤C.In order that we now derive the estimate of the velocity, we first verify that(54)esssupt∈(0,T)∥[ϱϵ-ϱ~ϵ]res∥L2(Ω)≤C.
To do this, we use the following inequality due to two assumptions of pressure (7) and (12),
(55)∂2Hϵ(ϱ)∂ϱ2=p′(ϵαϱ)ϱ≥C(1ϱ+ϵ2α/3ϱ1/3),
which deduce, thanks to (49),
(56)esssupt∈(0,T)[1]res≤ϵ2C,(57)esssupt∈(0,T)∥[ϱϵlog(ϱϵ)]res∥L1(Ω)≤Cϵ2,(58)esssupt∈(0,T)∥[ϱϵ]res∥L5/3(Ω)≤Cϵ(6-2α)/5.
Finally, we get
(59)esssupt∈(0,T)∥[ϱϵ]resϵ∥L6/5(Ω)≤Cϵ1/6,
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(60)esssupt∈(0,T)∥[uϵ]ess∥L2(Ω)≤Csupt>0∫Ω‍ϱϵ|uϵ|2dx≤C,∥[uϵ]res∥L2(Ω)2≤C∫Ω‍|ϱϵ-ϱ~||uϵ|2dx≤ϵC∥uϵ∥Lp(Ω)2≤ϵC∥∇uϵ∥L2(Ω)2,
for a certain p>1 where we have used the Sobolev embedding inequality. Thus we get
(61)∥uϵ∥L2((0,T)×Ω)≤C,
and the estimates in (48) and (61) imply
(62)∥uϵ∥L2(0,T;H1(Ω;ℝ3))≤C.


## 4.2. Convergence of Anelastic Constraint

We will use the uniform estimate (62) to deduce
(63)uϵ→UweaklyinL2(0,T;W1,2(Ω;ℝ3)),
up to a subsequence of {ϵ>0}. In accordance with (53) and (54), we obtain
(64)ϱϵ→ϱ~inL∞(0,T;Lp(Ω)),foracertainp>0,
and so we can take the limit of ϵ in the continuity equation (20) to get
(65)∫0T‍∫Ω‍ϱ~U·∇xφdxdt=0,
for all φ∈Cc∞((0,T)×Ω).


## 4.3. Convergence of Moment Equation

To begin with, using two estimates (63) and (64),
(66)ϱϵuϵ→ϱ~Uweakly-(*)inL∞(0,T;Lp(Ω;ℝ3)),foracertainp>0.
Hence
(67)ϱϵuϵ⊗uϵ→ϱ~U⊗U¯weaklyinL∞(0,T;Lq(Ω;ℝ3)),
for a certain q>1. Actually, we do not know ϱU⊗U¯=ϱ~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{dϵ}ϵ>0. Multiplying dϵ on (27) yields
(68)∂t|dϵ|2-Δ|dϵ|2+uϵ·∇|dϵ|2≤0,
and so applying the maximum principle for weak solutions provides
(69){dϵ}ϵ>0boundedinL∞([0,T]×Ω),
which implies
(70){∇dϵ}ϵ>0boundedinL2([0,T];H2(Ω)),
where we have used (47), the basic elliptic theory, and the following Gagliardo-Nirenberg inequality:
(71)∥∇dϵ∥L4(Ω)≤C∥Δdϵ∥L2(Ω)1/2∥dϵ∥L∞(Ω)1/2+C∥dϵ∥L∞(Ω).
and thus we derive
(72){∂tdϵ}ϵ>0boundedinLp(0,T;W-1,2(Ω;ℝ3)),
for a certain p>1. Applying the Aubin-Lions lemma applied to (27) together with (47), (70), (72), and (61) implies
(73)dϵ→dstronglyL2(0,T;L2(Ω;H1(Ω))).
Thus we get the boundary condition d|∂Ω=d0 and
(74)∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀→∇d⊗∇d-(12|∇d|2+F(d))𝕀
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:
(75)∂t(ϱϵuϵ)+divx(ϱϵuϵ⊗uϵ)+1ϵ2∇x(pϵ(ϱϵ)-p0ϱ~ϵ)+p0∇(ϱϵ-ϱ~ϵ2)+(ϱ~-ϱϵϵ2)=μϵΔuϵ-νϵdivx((12|∇dϵ|2+F(dϵ))∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀3),
From the previous estimates, we get
(76)∫0T‍∫Ω‍(ϱ~U·∂tφ+ϱ~U⊗U¯:∇xφ)dxdt=∫0T‍∫Ω‍(μ∇xU:∇xφ-ν(∇d⊗∇d):∇xφ)dxdt-∫Ω‍(ϱ~U)0·φdx
for any test function
(77)φ∈Cc∞([0,T]×Ω;ℝ3),divxφ=0,
if we show
(78)1ϵ2∫0T‍∫Ω‍(pϵ(ϱϵ)-p0ϱϵ)dxdt→0asϵ→0,
where we have here used (8), (53), and (54). It remains to show that
(79)∫0T‍∫Ω‍ϱ~U⊗U¯:∇xφdxdt=∫0T‍∫Ω‍[ϱ~U⊗U]:∇xφdxdt
for any
(80)φ∈Cc∞((0,T)×Ω;ℝ3),divxφ=0.
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 areflexive space Lq, 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):
(81)φ(t,x)=ψ(t)ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx],ψ∈𝒟(0,T).
We will write the moment equation (2) with simple computations again:
(82)1ϵ2∫0T‍∫Ω‍ψpϵ(ϱϵ)b(ϱϵ)dxdt=1ϵ2|Ω|∫0T‍∫Ω‍ψpϵ(ϱϵ)dx(∫Ω‍b(ϱϵ)dx)dt-1ϵ2∫0T‍∫Ω‍ψϱϵ∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍ψΠϵ·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt+Jϵ,
where
(83)Πϵ∶=-νϵdivx(∇dϵ⊗∇dϵ-(12|∇dϵ|2+F(dϵ))𝕀3),
and Jϵ is defined by
(84)Jϵ=∫0T‍∫Ω‍ψμϵ∇xuϵ:∇ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍ψϱϵuϵ⊗uϵ:∇ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt-∫0T‍∫Ω‍∂tψϱϵuϵ·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dxdt+∫0T‍∫Ω‍ψϱϵuϵ·ℬ[divx(b(ϱϵ)uϵ)]dxdt+∫0T‍ψ∫Ω‍ϱϵuϵ·ℬ[1|Ω|∫Ω‍(ϱϵb′(ϱϵ)-b(ϱϵ))divxuϵ-1|Ω|∫Ω‍(b(ϱϵ)-b′(ϱϵ)ϱϵ)divxuϵdx]dxdt.
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
(85)|b(ϱ)|+|ϱb′(ϱ)|≤Cϱγ,
with a certain γ∈(0,1). In virtue of (57), it is easy to see
(86)esssupt∈(0,T)∫Ω‍b(ϱϵ)dx≤Cϵ2,
and so we get
(87)∥b(ϱϵ)∥Lq(Ω)q≤∥[ϱϵ]resγq∥L1(Ω)≤∥[ϱϵlogϱϵ]resγq∥L1(Ω)≤Cϵ2,
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)
(88)1ϵ2∫Ω‍ϱϵ∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx=1ϵ∫Ω‍[ϱϵ-ϱ~ϵ]ess∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx+1ϵ∫Ω‍[ϱϵ-ϱ~ϵ]res∇G·ℬ[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx-p0ϵ2∫Ω‍ϱ~[b(ϱϵ)-1|Ω|∫Ω‍b(ϱϵ)dx]dx,
where the last integral is uniformly bounded due to (86). Following estimates (53), (56), and (87) together with the Lp-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 Lp-estimates for ℬ yield that the third term of the right hand side of (82) is uniformly bounded. Consequently, we deduce that
(89)∫0T‍∫Ω‍[pϵ(ϱϵ)]resϱϵνdxdt≤Cϵ2,
for a certain ν∈(0,1). We first split the integration of (78) into two parts to show (78):
(90)1ϵ2∫0T‍∫Ω‍|p0ϱϵ-pϵ(ϱϵ)|dxdt=1ϵ2∫0T‍∫ϱϵ≤M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt+1ϵ2∫0T‍∫ϱϵ>M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt.
For the first one of the right hand side, we get
(91)1ϵ2∫0T‍∫ϱϵ≤M‍|p0ϱϵ-pϵ(ϱϵ)|dxdt≤ϵα-2M22sup0≤r≤MP′′(r)|Ω|,
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 L1/ϱ~2 by the Hilbert space with the inner product
(92)〈v,w〉1/ϱ~=∫Ω‍v·wdxϱ~
and D1,2 by the completion of Cc∞(Ω¯) with respect to the norm ∥∇xφ∥L2(Ω).Theorem 4.
Forv∈L1/ϱ~2(Ω:ℝ3), a vector function v:Ω→ℝ3 is written as
(93)v=Pϱ~[v]+Qϱ~[v],
where
(94)Qϱ~[v]=ϱ~∇xΨ,
and Ψ is uniquely determined as the following Neumann problem:
(95)divx(ϱ~∇xΨ)=divxvinΩ,ϱ~∇xΨ·n|∂Ω=0,∫Ω‍Ψdx=0,
where n is the outward unit normal to ∂Ω.We now write(96)ϱϵuϵ⊗uϵ=Pϱ~[ϱϵuϵ]⊗uϵ+Qϱ~[ϱϵuϵ]⊗Pϱ~[uϵ]+Qϱ~[ϱϵuϵ]⊗Qϱ~[uϵ].
Let us first show
(97)Pϱ~[ϱϵuϵ]→Pϱ~[ϱ~U]=ϱ~UinL1((0,T)×Ω;ℝ3).
To do this, we adapt the following test function to the moment equation (22):
(98)φ(t,x)=φ(t)ϱ~Pϱ~[ϱ~Ψ],Ψ∈Cc∞(Ω¯;ℝ3),Ψ·n|∂Ω=0,ψ∈Cc∞(0,T).
Taking into account the uniform estimates obtained in the previous sections, it follows that
(99)t∈[0,T]↦∫Ω‍Pϱ~[ϱϵuϵ]·Ψdx
precompact in C[0,T] and so the uniform estimates (45) and (64) with using the Sobolev embedding L5/4(Ω)↪[W1,2(Ω)]* imply that
(100)Pϱ~[ϱϵuϵ]→Pϱ~[ϱ~U]=ϱ~UinCweak([0,T];L5/4(Ω;ℝ3)).
Notice that
(101)∫0T‍∫Ω‍Pϱ~[ϱϵuϵ]·Pϱ~[ϱ~uϵ]dxϱ~dt=∫0T‍∫Ω‍Pϱ~[ϱϵuϵ]·uϵdxdt→∫0T‍∫Ω‍Pϱ~[ϱ~U]·Udxdt=∫0T‍∫Ω‍ϱ~2|U|2dxϱ~dt,
where we have here used (100). In virtue of (64) and (101), one gets
(102)Pϱ~[ϱϵuϵ]→ϱ~UinL2((0,T)×Ω;ℝ3).
Since the Helmholtz projections
(103)v→Pϱ~[v],v→Qϱ~[v]
map continuously the spaces Lp(Ω;ℝ3) and W1,p(Ω;ℝ3) into itself for any 1<p, it is easily seen that
(104)Qϱ~[ϱϵuϵ]→0weakly-(*)inL∞(0,T;L5/4(Ω;ℝ3)).
In order to show (79), we should prove
(105)limϵ→0∫0T‍∫Ω‍Qϱ~[ϱϵuϵ]⊗Qϱ~[ϱϵuϵ]:∇x(φϱ~)dxϱ~dt=0,
for any
(106)φ∈Cc∞((0,T)×Ω¯;ℝ3),divxφ=0,φ·n|∂Ω=0.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 ofQ[ϱϵuϵ]. To begin with, we rewrite (1) and (2):
(107)ϵ∂tXϵ+1ϱ~divxVϵ=0,ϵ∂tVϵ+p0ϱ~∇xXϵ=ϵ(divxGϵ1+∇Gϵ2),
where the previous estimates provide
(108)Gϵ1isboundedinLp((0,T)×Ω)3×3Gϵ2isboundedinLp((0,T)×Ω)3,
for a certain p>1 and
(109)Xϵ=ϱϵ-ϱ~ϵϱ~,Vϵ=ϱϵuϵ(110)Gϵ1=μϵ∇uϵ-(ϱϵuϵ⊗uϵ)-νϵ(∇dϵ⊗∇dϵ)(111)Gϵ2=12|∇dϵ|2+F(dϵ)-(Hϵ(ϱϵ)-Hϵ(ϱ~ϵ)-H′(ϱ~ϵ)(ϱϵ-ϱ~ϵ)).
We now use the method of spectral analysis of the wave operator. Let us consider the eigenvalue problem:
(112)ϱ~∇x(Aϱ~)=λW,p0divxW=λA,inΩ,
with the boundary condition
(113)W·n|∂Ω=0.
Thus, the eigenvalue problem (112) and (113) can be written into the following form:
(114)-divx[ϱ~∇x(Aϱ~)]=Λϱ~(Aϱ~),inΩ,
with
(115)∇x(Aϱ~)·n|∂Ω=0,λ2=-Λp0.
Following the eigenvalue problem (114) and (115), it is well known that there is an orthonormal basis {Aj,m}j=0,m=1∞,mj of real eigenfunctions of the weighed Lebesgue space L1/ϱ~2(Ω) corresponding to eigenvalues Λj,m such that
(116)m0=1,Λ0,1,A0,1=ϱ~,0<Λ1,1=⋯=Λ0,m1(=Λ1)<Λ2,1=⋯=Λ2,m2(=Λ2)<⋯,
where mj stands for the multiplicity of Λj. We also see that {Wj,m}j=0,m=1∞,mj is an orthonormal basis in Q[L1/ϱ~2(Ω)3] defined by
(117)W±j,m=∓ip0Λjϱ~∇xAj,mϱ~,j=1,2,...,m,m=1,...,mj.
We now take φ=ψ1(t)Aj,m and ϕ=ψ2(t)W±j,m as test functions to the system (107), and then we obtain
(118)ϵ∂t[Xϵ]j,m-p0Λjdivx[Vϵ]j,m=0,ϵ∂t[Vϵ]j,m+Λj[Xϵ]j,m=ϵ[Bϵ]j,m,
for j=1,2,..., and m=1,...,mj, where [Xϵ]j,m,  [Vϵ]j,m are defined by
(119)[Xϵ]j,m=∫Ω‍XϵAj,mdx,[Vϵ]j,m=ip0∫Ω‍Vϵ·Wj,mdx,
and the estimates in (108) show that
(120){[Bϵ]j,m}ϵ>0isboundedinL1(0,T)foranyfixj,m.
Moreover, in virtue of section 5 in [7] for a finite number of modes, we set
(121)Qϱ~,N[ϱ~Z]=-ip0∑j,0<Λj≤N‍∑m=1mj‍[ϱ~Z]j,mWj,m.We first note(122)∥Qϱ~[ϱ~uϵ]-Qϱ~,N[ϱ~uϵ]∥L1/ϱ~2(Ω;ℝ3)2≤1N∥divx(ϱ~uϵ)∥L1/ϱ~22(Ω)→0,
as N→∞ in L2(0,T;L1/ϱ~2(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 Qϱ~,N instead of Qϱ~. Observe that, thanks to (114) and (121), it is easy to see
(123)Ψϵ=1p0∑j=1N‍∑m=1mj‍[Vϵ]j,mΛj(Aj,mϱ~),-divx(ϱ~∇xΨϵ)=1p0∑j=1N‍∑m=1mj‍Λj[Vϵ]j,mAj,m.We are now ready to show (105) and so we now use (118) and (123) to rewrite the oscillation part Qϱ~[ϱϵuϵ] as
(124)limϵ→0∫0T‍∫Ω‍(Qϱ~[ϱϵuϵ]⊗Qϱ~[ϱϵuϵ]):∇x(φϱ~)dxdt=limϵ→0∫0T‍∫Ω‍(ϱ~∇xΨϵ⊗∇xΨϵ):∇x(φϱ~)dxdt=-limϵ→0∫0T‍∫Ω‍divx(ϱ~∇xΨϵ)∇xΨϵ:∇x(φϱ~)dxdt=-ϵp02limϵ→0∫0T‍∫Ω‍∑j=1N‍∑m=1mj‍Λj[Xϵ]j,m×[Aj,m]ϱ~∇xΨϵ·∂tφdxdt+ϵp02limϵ→0∫0T‍∫Ω‍∑j=1N‍∑m=1mj‍Λj[Xϵ]j,m×[Aj,m]ϱ~∂t∇xΨϵ·φdxdt
which converges to 0 as ϵ tends to 0 where we can see this proof in section 6.6 of Feireisl and Novotný [7].



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