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RECOVERING NAVIER-STOKES EQUATIONS FROM ASYMPTOTIC LIMITS OF THE BOLTZMANN GAS MIXTURE EQUATION
Carlo Bianca, Christian Dogbe
To cite this version:
Carlo Bianca, Christian Dogbe. RECOVERING NAVIER-STOKES EQUATIONS FROM ASYMP-
TOTIC LIMITS OF THE BOLTZMANN GAS MIXTURE EQUATION. Communications in Theo-
retical Physics, Chinese Physical Society, 2016. �hal-02151781�
LIMITS OF THE BOLTZMANN GAS MIXTURE EQUATION
CARLO BIANCA AND CHRISTIAN DOGBE
Abstract. This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles.
Specifically the hydrodynamics limit is performed by employing different time and space scalings. The paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) are solution of the acoustic equation, the linear incompressible Euler equa- tion and the incompressible Navier-Stokes equation. The derivation is formally tackled by the recent moment method proposed by Bardos et al. [3] and the results generalize the analysis performed in [4].
Keywords: Gas mixtures; Hydrodynamic limit, Asymptotic limit, Fluid equation, Conservation law.
AMS Subject Classification (2010): 82C40, 76P05, 35Q35
1. Introduction
The mathematical derivation of the fluid mechanics equations from the dynamics de- scribed at the mesoscopic scale is a hard challenge considering that the hydrodynamic limit techniques require assumptions and tools of nonlinear analysis. The most famous method for deriving the fluid dynamic description from kinetic equation is based on the definition of two different scalings (parabolic/diffusive and hyperbolic) and subsequently on the so-called small Knudsen number ε (proportional to the mean free path) limit, which corresponds to enhance the kinetic collision operator, responsible for the trend to the thermodynami- cal equilibrium, at a factor 1/ε. Specifically, in the diffusive scaling, a time scaling of the same order than the scaling of the collision operator is considered; the asymptotic analysis leads to parabolic equations (see, among others, papers [11], [21] and the references cited therein) and specifically to the incompressible Navier-Stokes equations. The hyperbolic scal- ing leads to conservation laws (or balance laws if source terms appear) [8]. The derivation of the macroscopic equations is based on the moment method which, differently from the the Hilbert and Chapman-Enskog expansion, consists in passing to the limit when the Knudsen number vanishes in the local conservation laws of mass, momentum and energy. It is worth stressing that, to the best of our knowledge, the moment method dates back to Bardos et al. in [3].
In this context, it has been observed (see [22]) that if the Knudsen number Kn and the Mach number go to zero with the same speed in , the Reynolds number Re (which is the ratio between the transport and viscosity effects) remains finite. Accordingly, the following interpretation of the scaling can be considered: by scaling space and time as −1 and −2 and the velocity field as , the incompressible Navier-Stokes equations are unchanged. In particular in [3], under suitable technical assumptions, it has been proved that the incompressible Navier-Stokes equations are asymptotic limit of the Boltzmann equation, see also [19].
This paper is concerned with the derivation of macroscopic fluid dynamics from the meso- scopic dynamics of a gas-particle system composed of two subsystemsLand Hof particles that collide elastically (binary gaseous mixture) and having massesmLandmH. The related distribution functionsFL= FL(t, x, v) and FH =FH(t, x, v), where t >0 is the time vari- able,x∈R3 is the space variable, andv∈R3the velocity variable, are thus solutions of the following Boltzmann equations system:
∂tFL+v·∇xFL=Q(FL, FL) +Q(FH, FL) (1.1a)
∂tFH+v·∇xFH=Q(FL, FH) +Q(FH, FH) (1.1b) whereQ(Fα, Fα) andQ(F`, F`), for {α, `} ∈ {L, H}, are the collisional integral operators which model the collisions among particles of the same subsystem (self-collisions), while Q(Fα, F`) is the collisional term which model the collisions among the particles of the two subsystemsαand`(cross-collisions) and it reads:
Q(Fα, F`)(v) = Z
R3
Z
S2
[Fα(v∗0)F`(v0)−Fα(v∗)F`(v)]B`α(ω·v,|v|)dv∗dω, (1.2a) F(v∗0) =F(t, x, v0∗), F(v0) =F(t, x, v0), F(v∗) =F(t, x, v∗), F(v) =F(t, x, v), (1.2b) where v, v∗ are the pre-collisional velocity, v0, v∗0 are the post-collisional velocity, where S2 is the 2d sphere in R3, ω is an arbitrary unit vector, dω is the surface measure on it, v := v∗−v is the relative velocity, dω is the solid angle element in the direction of ω, B`α is the nonnegative differential-scattering cross section [9]. In particular in the present paper, following Chapman and Cowling [10], we consider asB`α the collision kernel that corresponds to the hard-sphere gas case (cross-section):
B`α= 1 2√
2π
dα+d` 2
2
|(v∗−v)·ω| ω∈S2, (1.3) wheredαandd` are the atomic diameters of the particles of the two subsystemsαand`.
The momentum and energy conservation equations
mαv0+m`v∗0 =mαv+m`v∗, mα|v0|2+m`|v∗0|2=mα|v|2+m`|v∗|2, allow to derive the post-collisional velocitiesv0 andv0∗:
v0=v+ 2m`
mα+m`(ω·v)ω, v∗0 =v∗− 2mα
mα+m`(ω·v)ω. (1.4) The interested reader in the formal derivation of the above system is referred to Kogan [18], Chapman and Cowling [10], Ferziger and Kaper [13], Hirschfelder et al. [17].
The outline of the present paper is as follows. After this introduction, Section 2 provides the necessary background on the Boltzmann equations system for the binary mixture of hard-sphere gas, including the main properties of the collisional operator and the linearized Boltzmann operator. The hydrodynamic equations are derived in Section 3 which also contains our main result. Section 4 is concerned with the proof of the main result. Finally Section 5 deals with a critical analysis and future research perspectives including the possible derivation of the limiting ghost effect system.
2. Kinetic Theory of Binary Gaseous Mixture: Background
This section deals with the review of the main properties and results for the Boltzmann equations system (1.1). The aim of this section is to summarize the basic tools necessary for the derivation of the macroscopic fluid dynamics equations that will be performed in the next section.
The first step is to write the Boltzmann equations system (1.1) in compact form. Let [·,·] denote a column vector so that F(t, x, v) = [FL, FH], writing the collision operator as follows:
C(F) :=
Q(FL, FL) +Q(FH, FL) Q(FL, FH) +Q(FH, FH)
(2.1) the Boltzmann equations system (1.1) now reads:
∂tF+v·∇xF =C(F). (2.2)
Moreover, for notational convenience, we set
QL ≡Q(FL, FL), QLH≡Q(FL, FH), QHL≡Q(FH, FL), QH≡Q(FH, FH).
Bearing all above in mind, the following main properties of the collision operator Qare summarized (see, among others, papers [16], [15], [24], [2], [12] for further details):
(i) Mass conservation:
Z
R3
QLdv= 0, Z
R3
QLHdv= 0, Z
R3
QHLdv= 0, Z
R3
QHdv= 0. (2.3) (ii) Momentum conservation:
Z
R3
QLmLvdv= 0, Z
R3
(QLHmLv+QHLmHv)dv= 0, Z
R3
QHmHvdv= 0. (2.4) (iii) Energy conservation:
Z
R3
QLmL|v|2dv= 0, Z
R3
(QLHmL|v|2+QHLmH|v|2)dv= 0, Z
R3
QHmH|v|2dv= 0. (2.5) (iv) Entropy inequalities: For each functionG= (GL, GH)T ∈C0∞(R+×R3v) we have:
(C(G),logG)(L2(R3))2 =
C(G),
logGL logGH
(L2(R3))2
≤0, (2.6)
where (·,·) denotes the inner product in (L2(R3))2.
(v) Local thermal equilibria: The distribution function which maximizes the entropy (equilibrium distribution) has the form of a locally Maxwellian distribution, namely (C(G),logG) = 0 if and only if Greads:
G=
nL
mL 2πθ
3/2 exp
−mL|v−u|2 2θ
nH mH
2πθ 3/2
exp
−mH|v−u|2 2θ
, (2.7)
where uL = uH = u ∈ R3 is the mean velocity, θL = θH = θ ∈ R+ is the local temperature andnα=
Z
R3
Gαdv∈R+ is the number density of the gasα. Note that nα>0 guarantees the positivity of the distribution function andθ >0 ensures the integrability with respect tov.
For simplicity we set:
Mα≡ M[nα,u,θ](v) =nα mα
2πθ 3/2
exp
−mα|v−u|2 2θ
, α∈ {L,H} (2.8) µα(v)≡ Mα≡ M[1,0,1](v)(v) =
mα 2π
3/2 exp
−mα|v|2 2
, α∈ {L,H} (2.9)
and
µ(v) = (2π)−3/2e−|v|2/2. Therefore we have:
QL(ML,ML) +QLH(ML,MH) = 0, and QHL(MH,ML) +QH(MH,MH) = 0.
The densitynα, the mean velocityuα=uand the temperatureθα=θare such that Z
M[nα,u,θ](v)
1 mαv mα|v|2
dv=
nα mαnαu mαnα|u|2+ 3κθ
. (2.10)
The quantities mαnαuα and WMα := 12mαnα|u|2+ 32κθ are the momentum and energy densities of the Maxwellian andκis the Boltzmann constant.
We further normalize the collision operator (1.2) as follows:
Q(Fα, F`) = Z
S+×R3
|(v∗−v), ω|[Fα(v0∗)F`(v0)−Fα(v∗)F`(v)]dωdv∗. (2.11) Next, we define the standard perturbationFαwith respect toµas follows:
Fα=µ+fα√
µ, α∈ {L,H}. (2.12)
We then study the normalized vector-valued binary gaseous mixture Boltzmann equations system for the perturbation
f(t, x, v) = [fL(t, x, v), fH(t, x, v)], (2.13) which now takes the form
{∂t+v· ∇x}f+Lf = Γ(f, f), (2.14) where, for each giveng= [gL, gH], the linearized collision operator in (2.14) reads:
L= [LL,LH] := [Lg,Hg] (2.15) where
− Lg=µ−1/2
2QLL(µ, µ12gL) +QHL(µ, µ12(gL+gH)) QLH(µ, µ12(gL+gH)) + 2QHH(µ, µ12gH)
. (2.16)
Simple calculations show thatL can be decomposed in the standard way (see for instance [14]) as: Lg=ν(v)g− Kg, where K is a compact integral operator on the space of square integrable functions, and the functionν(v), called collision frequency, is a smooth function growing at large velocities as the cross section that reads:
ν(v)≡ Z
R3×S2
|v−u|µ(u)dudω.
Forg andhthe nonlinear collision operator is Γ(g, h) =µ−1/2
Q(µ12gL, µ12hL) +QHL(µ12gH, µ12hL) QLH(µ12gL, µ12hH) +QHH(µ12gH, µ12hH)
. (2.17)
LetL2M be the associated Hilbert space endowed with the usualL2inner product fL
fH
, gL
gH
M
= Z
R3
(MLfLgL+MHfHgH)dv. (2.18) The main properties ofLare summarized in the following lemma (see Proposition 2.1, pp.
635-638, [1]):
Lemma 2.1. Assuming the hard-sphere interaction for the collision kernel, we have:
1. L is the sum of a diagonal operatorf 7→νf and a compact operatorK. The domain of L is given by
D(L) ={f : k(1 +|v|)12fkL1
M<∞}.
2. L is a nonnegative self-adjoint operator in L2M:
L fL
fH
, gL
gH
M
= fL
fH
,L gL
gH
M
(2.19) 3. The kernel of L is a six-dimensional linear space:
N:= ker(L) =Span{φi, i= 0, . . . ,5}
where φ0=
1 0
, φ1= 0
1
, φi+1= mLvi
mHvi
, (i= 1,2,3), φ5=
mL|v|2 mH|v|2
.
4. Any function f ∈ D(L) can be written as f = qf +wf with qf ∈ ker(L) and w∈(ker(L))⊥ and we havehLf, fiM>δ0k(1 +|v|)12wfk2.
We will consider the basis
ψ0(v) =µ1/2(v),
ψi(v) =viµ1/2(v), (i= 1,2,3) ψ4(v) = 1
2(|v|2−3)µ1/2(v),
(2.20)
and
N0= Span{ψ0, . . . , ψ4}.
We definePas the orthogonal projectionL2(R3v) to the null spaceN. With (t, x, v) fixed, we then decompose any functiong(t, x, v) as
g(t, x, v) =Pg(t, x, v) + (I−P)g(t, x, v). (2.21) ThenPg is called the hydrodynamic part ofg and (I−P)g:=P1 the microscopic part.
Let’s now introduce thepseudo-inverseof the linear operatorL. The Fredholm alternative states that the equation
Lg=φ, φ∈R(L), g∈R(L) (2.22) has a solution if and only if φ∈ N⊥, in which case there is a unique solution inφ ∈N⊥ denoted byg=L−1φ. The Fredholm property ofLimplies that for every p∈(1,∞)
L:Lp(µdv)→Lp(µdv) is bounded,
and that for everyξ∈Lp(µ dv) there exists a uniqueξb∈Lp(µ dv) such that Lbξ=P1ξ, Pbξ= 0.
For every ξ∈ Lp(µdv) we define L−1ξ =ξbwhere ξbis determined above. This defines an operatorL−1 such that
L−1:Lp(µ dv)→Lp(µ dv), is bounded,
L−1L=P1 overLp(µ dv), LL−1=P1 overLp(µ dv)
and N(L−1) = N(L). The operator L−1 is the unique pseudo inverse of L with these properties.
In what follows we denote the weighted inner product ofhandgin L2µ(R3) with respect to a given Maxwellianµby:
hh, giµ≡ Z
R3
h(v)g(v)µ dv. (2.24)
2.1. Moments and conservation laws. Macroscopic quantities such as the number, mo- mentum and energy densities are defined as moments of the distribution function with respect to the velocity. Specifically, the densitynα, the mean velocityuα(mαnαuαis theα- subsystem momentum density) and the energy densityWαof the subsystemαare computed from theα-subsystem distribution functionfα as follows:
nα mαnαuα
Wα
= Z
Fα(v)
1 mαvα 1 2mα|v|2
dv. (2.25)
The quantitymαnαuαis theα-species momentum density andWα is the energy density.
Moreover the following fields can be defined:
c=v−uα (random velocity with bulk velocityu)
qiα= 1 2
Z
R3
ci|c|2Fαdv (i= 1,2), (heat flow vector)
pαij = Z
R3
cicjFαdv (i, j= 1,2), (stress tensor with componentspαij).
It is common to separate the drift motion (defined by the average velocity uα) and the random kinetic motion (defined by the velocityc) in evaluating these integrals. By definition ofuα, one has
Z
v⊗vFαmαdv=mαnαuα⊗uα+Pα where
Pα:= trace(pij) = Z
(v−uα)⊗(v−uα)mαFαdv, (i, j= 1,2), and
Z 1
2mα|v|2vFαmαdv=Wαuα+Pαuα+Qα with
Qα:=
Z mα|v−uα|2
2 (v−uα)Fαdv.
The internal energy per particleeα and the temperatureθαofα-species gas are defined as:
eα= mα 2nα
Z
R3
|v−uα|2Fαdv (2.26)
where, unless otherwise stated, the domain of integration is the whole space of v (or of the variable of integration). We define the mass density %α by %α = mαnα. We also define global quantities for the mixture: the counterparts of the mixture, i.e., the molecular number densityn, mass density %, mass average velocity v, pressure pand temperatureθ, are expressed by a proper combination of the quantities above as
n= X
α∈{L,H}
nα, %= X
α∈{L,H}
%α, E=nκθ= X
α∈{L,H}
pα+1
3|u−vα|2
.
The following proposition holds true, see [12] for the proof.
Proposition 2.2. LetFα=Fα(t, x, v)be a solution of the Boltzmann mixture system (1.1) that is locally integrable and rapidly decaying inv for each(t, x). Then the mass, momentum
and energy conservation laws hold:
∂t Z
R3
Fαvdv+∇x Z
R3
vFαdv= 0 (2.27a)
∂t Z
R3
mαvFαdv+∇x Z
R3
mαv⊗vFαdv= Z
R3
mαvQ(Fα, F`)dv (2.27b)
∂t
Z
R3
mα1
2|v|2Fαdv+∇x
Z
R3
mα1
2|v|L2vFαdv= Z
R3
mα1
2|v|2Q(Fα, F`)dv (2.27c) Bearing the previous notations for the thermodynamic fields in mind, these continuity equations are, for theL-subsystem:
∂tnL+∇x·(nLu) = 0 (2.28a)
∂t(mLnLu) +∇x
Z
v⊗vFLmLdv
=QLH (2.28b)
∂tWL+∇x·
Z mL|v|2 2 vFLdv
=QLH (2.28c)
and similarly for the subsystemH:
∂tnH+∇x·(nHu) = 0 (2.29a)
∂t(mHnHu) +∇x Z
v⊗vFHmHdv
=QHL (2.29b)
∂tWH+∇x·
Z mH|v|2 2 vFHdv
=QHL, (2.29c)
whereQα` and Qα` are the momentum and energy transfer rates toward the subsystemα from the subsystem`:
Qα`
Q`α
= Z
Qα`(Fα, F`)(v)
mαv
1 2mα|v|2
dv (2.30)
so that, because of the momentum and energy conservation properties of the unlike-collision operators (2.4) and (2.5), we have
Qα`+Q`α= 0, Qα`+Q`α= 0. (2.31)
3. Derivation of the macroscopic equations
This section is devoted to the formal derivation of the macroscopic equations for the density, momentum and temperature from the Boltzmann equations system for the binary mixture hard-sphere gas.
As already mentioned in the introduction, the main aim of this paper is the asymptotic derivation of macroscopic equations. In particular the derivation of different hydrodynamical equations depends on the time and space scales employed. Specifically we consider the following rescaled Boltzmann mixture equations system
εγ∂tFεH+ v·∇xFεH = εqQ(FεH, FεL) + 1
εQ(FεH, FεH) (3.1a) εγ+1∂tFεL+v·∇xFεL = 1
εQ(FεL, FεL) + εq+1Q(FεL, FεH), (3.1b) whereFα(t, x, v) = Fα(εtγ,x, v), q >1 is a real parameter, >0 is a real parameter that can be interpreted as the Knudsen number or the mean-free path between two intermolecular collisions, and γ >0 is the strength parameter of the scaling. It is worth noting that the time scaling, which is related to the Strouhal number (ratio of the oscillation frequency to
the bulk velocity), is introduced to suppress, whenγ6= 0, the acoustic modes varying in a faster timescale than rotational modes of the fluid.
In the hydrodynamic regime, where the Knudsen number vanishes, it is expected to obtain a continuum limit and thus reaches statistical equilibrium. Specifically the paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) of the two subsystems are solution of the acoustic equation, the linear incompressible Euler equation and the incompressible Navier-Stokes equation. The derivation is formally tackled by the recent moment method proposed by Bardos et al. [3] and the results generalize the analysis performed in [4].
It is worth stressing that we did not try to get all possible scalings. In particular, one can easily derive simplified models to those we have here [4].
It is worth stressing that for the derivation of the rescaled equations system we can think the solutions of (3.1a)-(3.1b) as scaled solutions of (1.1a)-(1.1b), that is, ifFα(t, x, v) solves the Boltzmann mixture equations (1.1a)-(1.1b) then the parabolic scaling Fα(t, x, v) = Fα(εtγ,x, v) generates a solution of the dimensionless Boltzmann mixture equations system (1.1a)-(1.1b) (see for instance [3], [20]).
In the hydrodynamic regime where the Knudsen number vanishes →0, it is clear that the right-hand side of (3.1a) would become singular. The only possibility to avoid this singularity is that
ε→0limQ(FεH, FεH) = 0. (3.2) Then, by (2.7), the limit must be a Maxwellian, namely,
F0H=M[nH, u, θ], (3.3)
for some (nH, u, θ) which can be function oftandx(similarly for the subsystemL).
If FεLand FεH solve the binary gas mixture Boltzmann equations (3.1b) and (3.1a) then FεLsatisfies the local conservation laws of mass, momentum, and energy:
εγ+1∂t
Z
R3
FεL
1 vmL
1 2|v|2mL
dv+ divx
Z
R3
FεL
v v⊗vmL v12|v|2mL
dv=εq+1mL
0 Q(FεL, FεH) Q(FεL, FεH)
(3.4) and forFεHone has:
εγ∂t
Z
R3
FεH
1 vmH
1 2|v|2mH
dv+ divx
Z
R3
FεH
v v⊗vmH v12|v|2mL
dv=εqmH
0 Q(FεH, FεL) Q(FεH, FεL)
. (3.5) Since the solutions of the asymptotic equations are not guaranteed to exist or to be regular, our proof is only formal. We assume that for eachε > 0, Fεα is a solution of (3.1b) and (3.1a) that satisfies the local conservation laws of mass, momentum, and energy, as well as the local entropy relation. Assume that
Fεα→Fα a.e as well as
Z
R3
Fεαdv→ Z
R3
Fαdv, Z
R3
vFεαdv→ Z
R3
vFαdv, Z
R3
|v|2Fεαdv→ Z
R3
|v|2Fαdv
inC(R+;D0(R3)), while Z
R3
v⊗vFεαdv→ Z
R3
v⊗vFαdv Z
R3
v|v|2Fεαdv→ Z
R3
v|v|2FαdvD0(R3)), inC(R+;D0(R3)), asε→0.
The small Mach number (ratio of the bulk velocity to the sound speed) is realized if Fεα is close to an absolute Maxwellian. If one takes the standard Maxwellian, the distance from this absolute Maxwellian can be scaled in the unit of the Knudsen numberεas
Fεα(t, x, v) =µα+εβµ1/2α (v)gαε(t0, x, v), α∈ {L, H} (3.6) whereβ>0 is the strength parameter of the scaling. The distribution functiongα can be viewed as the microscopic response of the system to gradients of macroscopic variables.
Varying the magnitude of the two scalings, it yields different limits as ε→0. Thus, one can derive different fluid equations (and in particular incompressible models) depending on the chosen scaling.
Before going further into the analysis, a linearization of Fεαis required. We will look for the solutionFεα nearµ, that is, the solution having the form
FεH=µ+εβµ
1
2gHε, FεL=µ+εmµ
1
2gLε, (3.7)
withgαε =O(1) asε→0. Plugging the scaling (3.7) into the mixture system (3.1) to deduce the governing equation of the new unknowngαε, one finds:
γ+1∂tgL+v·∇xgL = 1
εL(gL) +εm−1Γ(gL) +µ−
1 2
β+q−m+1QLH(µ, µ−
1 2gH) +q+1QLH(µ−
1
2gL, µ) +β+q+1QLH(µ
1 2gεL, µ
1 2gHε)
(3.8) εγ∂tFεH+ v·∇xFεH = 1
εH(gH) +εβ−1Γ(gH ) +µ−
1 2
εm+q−βQHL(µ, µ
1 2gεL) + εqQLH(µ12gεH, µ) +εm+qQHL(µ12gHε, µ12gεL)
. (3.9)
Here, εγ allows us to choose the phenomenon we want to emphasis. For notational conve- nience, we set:
LLH1 (gLH) =µ−12QLH(µ, µ−12gH), LLH2 (gLH ) =µ−12QLH(µ−12gL, µ) (3.10) HHL1 (gHL ) =µ−12QHL(µ, µ12gLε) HHL2 (gHL ) =µ−12QLH(µ12gHε, µ) (3.11) ΓLH1 (gLH) =µ−
1 2QLH(µ
1 2gLε, µ
1
2gHε), ΓHL2 (gHL ) =µ−
1 2QHL(µ
1 2gεH, µ
1
2gεL). (3.12) By varyingε,βwe can formally derive the systems of the fluids dynamics.
The main result of the present paper is to perform the asymptotic limit when γ,β>0, m > 1. Specifically, the evolution of (nH, u, θ) is governed by theacoustic system for FH andIncompressible Navier-Stokes equationsforFL.
We state the main result of this article.
Theorem 3.1. Let FεL andFεH be a family of distribution solutions to the scaled mixture systems (3.1b)-(3.1a) that satisfies the local conservation laws of mass, momentum, and energy. Moreover, forα, β∈ {L,H}, assume that:
gεα= Fεα−µ ε√
µ →fα in the sense of distributions, (3.13) and
εQαβ(gαε, gβε)→0 in the sense of distributions
Z
R3
gεα√ µdv→
Z
R3
gα√ µdv,
Z
R3
vgεα√ µdv→
Z
R3
vgα√
µ inC(R+;D(R3)), Z
R3
v⊗vgεα√ µdv→
Z
R3
v⊗vgα√ µdv,
Z
R3
vgεα√ µdv→
Z
R3
vgα√
µdv inC(R+;D(R3)), Z
R3
|v|2gαε√ µdv→
Z
R3
|v|2gα√ µdv,
Z
R3
v|v|2gαε√ µdv→
Z
R3
v|v|2gα√
µdv in C(R+;D(R3)), asε→0. Then the asymptotic limitf := [fεL, fεH]T reads:
f(t, x, v) =n(t, x, v) +v·u(t, x, v) +θ(t, x, v)12(|v|2−3).
Furthermore
• If γ= 0, β>0, (u, θ) is solution of the linear compressible equation (the so-called acoustic equation) for the subsystem Hand solution of the incompressible Navier-Stokes equation for the subsystemL:
∂tnH+ divu= 0
∂tu+∇x(nH+θ) = 0
∂tθ+2
3divu= 0.
∂u˜
∂t + ˜u·∇˜u+∇p˜L=ν∆˜u
∂θ˜
∂t + ˜u·∇θ˜=κ∆˜θ.
(3.15)
• If γ>0andβ>0, thenusatisfies the divergence conditions divu= 0,
whilenL andθsatisfies
∇x(nL+θ) = 0.
• If 0< γ=β<1, then the subsystemHis solution of theIncompressible Euler equation:
∂u
∂t +u·∇u+∇pH = 0
∂θ
∂t +u·∇θ= 0,
(3.17)
and the subsystem Lis solution of the incompressible Navier-Stokes equation.
4. Proof of Theorem 3.1
In this subsection, we study the group which governs the acoustic waves and then we will pass to the limit in the equation of conservation of the momentum projected on the space of divergence-free vector fields. We apply the strategy of Bardos et al. [3] to derive the limiting equations.
• The Case γ= 0, β>0.
• For the subsystem H, we get the acoustic system from the Boltzmann equation. We write (3.9) as
ε ∂gεH
∂t +v·∇xgHε
= H(gH) +εβ+1Γ(gH) +εm+q−β+1HHL1 (gHL) +εq+1HHL2 (gHL )
+εm+q+1ΓHL2(gHL) (4.1)
By letting ε → 0 in (4.1), one finds that H(g0H) = 0. Hence g(t, x,·) takes values in Null(H), the null space ofH. BecausegHis assumed to belong toL∞(dt;L2(µdvdx)), we conclude that the limitg0Hhas the form
gH0=
nH+v·u+ v2
2 −3 2
θ
√
µ, (4.2)
wherenH, uandθare functions oft, x. In order to determine the dynamics ofnH, u, θ, we project (3.9) onto{µ1/2, vµ1/2,
v2 2 −32
µ1/2}. The inner product of (3.9) withψj gives (γ= 0):
∂t Z
R3
µ1/2gεHdv+∇x· Z
R3
vµ1/2gHε= 0 (4.3a)
∂t Z
R3
vµ1/2gεHdv+∇x· Z
R3
v⊗vµ1/2gεHdv
= Z
R3
vµ1/2
εm+q−βHHL1 (gHL) +εqHHL2 (gHL ) +εm+qΓHL2 (gHL )
dv (4.3b)
∂t
Z
R3
v2 2 −3
2
µ1/2gHεdv+∇x· Z
R3
v v2
2 −3 2
µ1/2gεHdv
= Z
R3
v2 2 −3
2
µ1/2
εm+q−βHHL1 (gHL) +εqHHL2 (gHL ) +εm+qΓHL2 (gHL )
dv. (4.3c) Next, passing to the limit in both sides of this equality, we arrive at
∂t
Z
R3
µ1/2gHdv+∇x· Z
R3
vµ1/2gH= 0 (4.4a)
∂t Z
R3
vµ1/2gHdv+∇x· Z
R3
v⊗vµ1/2gHdv= 0 (4.4b)
∂t Z
R3
v2 2 −3
2
µ1/2gHdv+∇x· Z
R3
v v2
2 −3 2
µ1/2gHdv= 0 (4.4c) SincegH=PgH, the fluid variables associated with the fluctuation of the number density gεis defined by:
nHε=hgεHi, uε=hvgHεi, θε= v2
2 −3 2
gHε
, with hζi= Z
ζ(v)µ1/2(v)dv.
Therefore, in the limit one has
∇x· Z
R3
v⊗vµ1/2gHdv = ∇x·
v⊗2
nH+v·u+ v2
2 −3 2
θ . (4.5)
Observe that by our normalization, it follows that
h1, µi= 1, h|vj|2, µi= 1, h|v|2, µi= 3 h|vj|2|vm|2, µi= 1, j6=m
h|v|4, µi= 15, h|vj|4, µi= 3, h|v|2|vj|2, µi= 5 and in view of the identity
hv⊗2i=hv⊗2 12(|v|2−3)i=h16|v|2(|v|2−3)iI=I,
substituting the explicit formula for gH in the left-hand side of the identity above leads to:
∇x· hv⊗vnHi=∇x·(nHI) =∇xnH (4.6)
