Fluid Approximations from the Boltzmann Equation

for Domains with Boundary

C. David LevermoreDepartment of Mathematics and

Institute for Physical Science and TechnologyUniversity of Maryland, College Park

lvrmr@math.umd.edu

presented 11 November 2011 at the ICERM Workshop:Boltzmann Models in Kinetic Theory, 7-11 November 2011

Institute for Computational and Experimental Research in MathematicsBrown University, Providence, RI

Introduction

We study some fluid approximations derived from the Boltzmann equationover a smooth bounded spatial domain Ω ⊂ RD. Our focus will be onboundary conditions.

1. We establish the acoustic limit starting from DiPerna-Lions solutions.(Jiang-L-Masmoudi, 2010)

2. We present linearized Navier-Stokes approximations derived formallyfrom the linearized Boltzmann equation.

Acoustic System

After a suitable choice of units and Galilean frame, the acoustic systemgoverns the fluctuations in mass density ρ(x, t), bulk velocity u(x, t), andtemperature θ(x, t) over Ω × R+ by the initial-value problem

∂tρ + ∇x· u = 0 , ρ(x,0) = ρin(x) ,

∂tu + ∇x(ρ + θ) = 0 , u(x,0) = uin(x) ,D2 ∂tθ + ∇x· u = 0 , θ(x,0) = θin(x) ,

(1)

subject to the impermeable boundary condition

u ·n = 0 , on ∂Ω , (2)

where n(x) is the unit outward normal at x ∈ ∂Ω. This is one of thesimplest fluid dynamical systems, being essentially the wave equation.

The acoustic system can be derived from the Boltzmann equation for den-sities F(v, x, t) over RD × Ω × R+ that are near the global Maxwellian

M(v) = (2π)−D2 exp

(

− 12|v|

2)

. (3)

We consider families of densities in the form Fǫ(v, x, t) = M(v)Gǫ(v, x, t)where the Gǫ(v, x, t) are governed over RD × Ω × R+ by the scaledBoltzmann initial-value problem

∂tGǫ + v ·∇xGǫ =1

ǫQ(Gǫ, Gǫ) , Gǫ(v, x,0) = Gin

ǫ (v, x) . (4)

Here ǫ is the Knudsen number while Q(Gǫ, Gǫ) is given by

Q(Gǫ, Gǫ) =

∫∫

SD−1×RD

(

G′ǫ1G′

ǫ−Gǫ1Gǫ

)

b(ω, v1−v) dω M1dv1 , (5)

where b(ω, v1 − v) > 0 a.e. while Gǫ1, G′ǫ, and G′

ǫ1 denote Gǫ( · , x, t)evaluated at v1, v′ = v + ωω · (v1 − v), and v′1 = v − ωω · (v1 − v)respectively.

We impose a Maxwell reflection boundary condition on ∂Ω of the form

1Σ+Gǫ R = (1 − α) 1Σ+

Gǫ + α 1Σ+

√2π

⟨

1Σ+v ·n Gǫ

⟩

. (6)

Here (GǫR)(v, x, t) = Gǫ(R(x)v, x, t) where R(x) = I−2n(x)n(x)T

is the specular reflection matrix, α ∈ [0,1] is the accommodation coeffi-cient, 1Σ+

is the indicator function of the so-called outgoing boundary set

Σ+ =

(v, x) ∈ RD × ∂Ω : v ·n(x) > 0

, (7)

and 〈 · 〉 denotes the average

〈ξ〉 =∫

RDξ(v)M(v) dv . (8)

Because√

2π⟨

1Σ+v ·n

⟩

= 1, it seen from (6) that on ∂Ω the flux is

〈v ·nGǫ〉 =⟨

1Σ+v ·n

(

Gǫ − Gǫ R)⟩

= α

⟨

1Σ+v ·n

(

Gǫ −√

2π⟨

1Σ+v ·nGǫ

⟩

)⟩

= 0 .(9)

Formal Derivation

Fluid regimes are those in which ǫ is small. The acoustic system can be de-rived formally from the scaled Boltzmann equation for families Gǫ(v, x, t)

that are scaled so that

Gǫ = 1 + δǫgǫ , Ginǫ = 1 + δǫg

inǫ , (10)

where

δǫ → 0 as ǫ → 0 , (11)

and the fluctuations gǫ and ginǫ converge in the sense of distributions to

g ∈ L∞(dt;L2(Mdv dx)) and gin ∈ L2(Mdv dx) respectively as ǫ → 0.

One finds that g has the infinitesimal Maxwellian form

g = ρ + v ·u +(

12|v|

2 − D2

)

θ , (12)

where (ρ, u, θ) ∈ L∞(dt;L2(dx;R × RD× R)) solve the acoustic sys-tem (1) and boundary condition (2) with initial data given by

ρin = 〈gin〉 , uin = 〈v gin〉 , θin =⟨(

1D|v|2 − 1

)

gin⟩

. (13)

The boundary condition (2) is obtained by passing to the limit in the bound-ary flux relation (9) to see

0 = 〈v ·n gǫ〉 → 〈v ·n g〉 ,

We thereby find that 〈v ·n g〉 = 0, and finally by using the infinitesimalMaxwellian form (12) get the impermeable boundary condition (2),

u ·n = 0 .

The program initiated with Claude Bardos and Francois Golse in 1989seeks to justify fluid dynamical limits for Boltzmann equations in the set-ting of DiPerna-Lions renormalized solutions, which are the only temporallyglobal, large data solutions available.

The main obstruction to carrying out this program is that DiPerna-Lionssolutions are not known to satisfy many properties that one formally ex-pects for solutions of the Boltzmann equation. For example, they are notknown to satisfy the formally expected local conservations laws of momen-tum and energy. Moreover, their regularity is poor. The justification of fluiddynamical limits in this setting is therefore not easy.

The acoustic limit was first established in this setting by Bardos-Golse-L(2000) over a periodic domain. There idea introduced there was to passto the limit in approximate local conservations laws which are satified byDiPerna-Lions solutions. One then shows that the so-called conservationdefects vanish as the Knudsen number ǫ vanishes, thereby establishingthe local conservation laws in the limit. This was done using only relativeentropy estimates, which restricted the result to collision kernels that arebounded and to fluctuations scaled so that

δǫ → 0 andδǫ

ǫ| log(δǫ)| → 0 as ǫ → 0 , (14)

which is far from the formally expected optimal scaling (11), δǫ → 0.

In Golse-L (2002) the local conservation defects were removed using newdissipation rate estimates. This allowed the treatment of collision kernelsthat for some Cb < ∞ and β ∈ [0,1) satisfied

∫

SD−1b(ω, v1 − v) dω ≤ Cb

(

1 + |v1 − v|2)β

, (15)

and of fluctuations scaled so that

δǫ → 0 andδǫ

ǫ1/2| log(δǫ)|β/2 → 0 as ǫ → 0 . (16)

The above class of collision kernels includes all classical kernels that arederived from Maxwell or hard potentials and that satisfy a weak small de-flection cutoff. The scaling given by (16) is much less restrictive than thatgiven by (14), but is far from the formally optimal scaling (11). Finally, onlyperiodic domains are treated.

Here we improve the result of Golse-L (2002) in three ways. First, we applyestimates from L-Masmoudi (2010) to treat a broader class of collision ker-nels that includes those derived from soft potentials. Second, we improvethe scaling of the fluctuations to δǫ = O(ǫ1/2). Finally, we treat domainswith a boundary and use new estimates to derive the boundary condition(2) in the limit.

The L1 velocity averaging theory of Golse and Saint-Raymond (2002) isused through the nonlinear compactness estimate of L-Masmoudi (2010)to improve the scaling of the fluctuations to δǫ = O(ǫ1/2). Without it wewould only be able to improve the scaling to δǫ = o(ǫ1/2). This is thefirst time the L1 averaging theory has played any role in an acoustic limittheorem, albeit for a modest improvement. We remark that velocity aver-aging theory plays no role in establishing the Stokes limit with its formallyexpected optimal scaling of δǫ = o(ǫ).

We treat domains with boundary in the setting of Mischler (2002/2010),who extended DiPerna-Lions theory to bounded domains with a Maxwellreflection boundary condition. He showed that these boundary conditionsare satisfied in a renormalized sense. This means we cannot deduce that〈v ·n gǫ〉 → 0 as ǫ → 0 to derive the boundary condition (2), as we didformally.

Masmoudi and Saint-Raymond (2003) derived boundary conditions in theStokes limit. However neither these estimates nor their recent extension tothe Navier-Stokes limit by Jiang-Masmoudi can handle the acoustic limit.Rather, we develop new boundary a priori estimates to obtain a weak formof the boundary condition (2) in this limit. In doing so, we treat a broaderclass of collision kernels than was treated earlier.

We remark that establishing the acoustic limit with its formally expectedoptimal scaling of the fluctuation size, δǫ → 0, is still open. This gap mustbe bridged before one can hope to fully establish the compressible Eulerlimit starting from DiPerna-Lions solutions to the Boltzmann equation.

In contrast, optimal scaling can be obtained within the framework of clas-sical solutions by using the nonlinear energy method developed by Guo.This has been done recently by Guo-Jang-Jiang (2009).

Framework

Let Ω ⊂ RD be a bounded domain with smooth boundary ∂Ω. Let n(x)

denote the outward unit normal vector at x ∈ ∂Ω and dσx denote theLebesgue measure on ∂Ω. The phase space domain associated with Ω

is O = RD × Ω, which has boundary ∂O = RD × ∂Ω. Let Σ+ and Σ−denote the outgoing and incoming subsets of ∂O defined by

Σ± = (v, x) ∈ ∂O : ±v ·n(x) > 0 .

The global Maxwellian M(v) given by (3) corresponds to the spatially ho-mogeneous fluid state with density and temperature equal to 1 and bulkvelocity equal to 0. The boundary condition (6) corresponds to a wall tem-perature of 1, so that M(v) is the unique equilibrium of the fluid. Associ-ated with the initial data Gin

ǫ we have the normalization∫

Ω〈Gin

ǫ 〉dx = 1 . (17)

Assumptions on the Collision Kernel

The kernel b(ω, v1−v) associated with the collision operator (5) is positivealmost everywhere.

The Galilean invariance of the collisional physics implies that b has theclassical form

b(ω, v1 − v) = |v1 − v|Σ(|ω ·n|, |v1 − v|) , (18)

where n = (v1−v)/|v1−v| and Σ is the specific differential cross-section.

We make five additional technical assumptions regarding b that are adoptedfrom L-Masmoudi (2010).

Our first technical assumption is that the collision kernel b satisfies therequirements of the DiPerna-Lions theory. That theory requires that b belocally integrable with respect to dω M1dv1 Mdv, and that it satisfies

lim|v|→∞

1

1 + |v|2∫

Kb(v1−v) dv1 = 0 for every compact K ⊂ RD , (19)

where b is defined by

b(v1 − v) ≡∫

SD−1b(ω, v1 − v) dω . (20)

Galilean symmetry (18) implies that b is a function of |v1 − v| only.

Our second technical assumption regarding b is that the attenuation coef-ficient a, which is defined by

a(v) ≡∫

RDb(v1 − v)M1dv1 , (21)

is bounded below as

Ca

(

1 + |v|2)βa ≤ a(v) for some Ca > 0 and βa ∈ R . (22)

Galilean symmetry (18) implies that a is a function of |v| only.

Our third technical assumption regarding b is that there exists s ∈ (1,∞]

and Cb ∈ (0,∞) such that

(

∫

RD

∣

∣

∣

∣

∣

b(v1 − v)

a(v1) a(v)

∣

∣

∣

∣

∣

s

a(v1)M1dv1

)1s

≤ Cb . (23)

Because this bound is uniform in v, we may take Cb to be the supremumover v of the left-hand side of (23).

Our fourth technical assumption regarding b is that the operator

K+ : L2(aMdv) → L2(aMdv) is compact , (24)

where

K+g =1

2a

∫∫

SD−1×RD

(

g′ + g′1)

b(ω, v1 − v) dω M1dv1 .

We remark that K+ : L2(aMdv) → L2(aMdv) is always bounded with‖K+‖ ≤ 1.

Our fifth technical assumption regarding b is that for every δ > 0 thereexists Cδ such that b satisfies

b(v1 − v)

1 + δb(v1 − v)

1 + |v1 − v|2

≤ Cδ

(

1+a(v1))(

1+a(v))

for every v1, v ∈ RD .

(25)

The above assumptions are satisfied by all the classical collision kernelswith a weak small deflection cutoff that derive from a repulsive intermolecu-lar potential of the form c/rk with k > 2D−1

D+1. This includes all the classicalcollision kernels to which the DiPerna-Lions theory applies. Kernels thatsatisfy (15) clearly satisfy (19). If they moreover satisfy (22) with βa = β

then they also satisfy (23) and (25).

Because the kernel b satisfies (19), it can be normalized so that∫∫

SD−1×RD×RDb(ω, v1 − v) dω M1 dv1 M dv = 1 .

Because dµ = b(ω, v1− v) dω M1dv1 Mdv is a positive unit measure onSD−1× RD× RD, we denote by

⟨⟨

Ξ⟩⟩

the average over this measure ofany integrable function Ξ = Ξ(ω, v1, v)

⟨⟨

Ξ⟩⟩

=

∫∫∫

SD−1×RD×RDΞ(ω, v1, v) dµ . (26)

DiPerna-Lions-Mischler Theory

We will work in the framework of DiPerna-Lions solutions to the scaledBoltzmann equation on the phase space O = RD × Ω

∂tGǫ + v ·∇xGǫ =1

ǫQ(Gǫ, Gǫ) on O × R+ ,

Gǫ(v, x,0) = Ginǫ (v, x) on O ,

(27)

with the Maxwell reflection boundary condition (6) which can be expressedas

γ−Gǫ = (1 − α)L(γ+Gǫ) + α〈γ+Gǫ〉∂Ω on Σ− × R+ , (28)

where γ±Gǫ denote the traces of Gǫ on the outgoing and incoming setsΣ±.

Here the local reflection operator L is defined to act on any |v ·n|Mdv dσ-measurable function φ over ∂O by

Lφ(v, x) = φ(R(x)v, x) for almost every (v, x) ∈ ∂O ,

where R(x)v = v−2v ·n(x)n(x) is the specular reflection of v, while thediffuse reflection operator is defined as

〈φ〉∂Ω =√

2π∫

v·n(x)>0φ(v, x) v ·n(x)Mdv .

DiPerna-Lions theory requires that both the equation and boundary condi-tions in (27) should be understood in the renormalized sense, see (44) and(48). These solutions were initially constructed by DiPerna and Lions overthe whole space RD for any initial data satisfying natural physical bounds.For bounded domain case, Mischler recently developed a theory to treatthe Maxwell reflection boundary condition.

The DiPerna-Lions theory does not yield solutions that are known to solvethe Boltzmann equation in the usual sense of weak solutions. Rather, itgives the existence of a global weak solution to a class of formally equiva-lent initial value problems that are obtained by multiplying (27) by Γ′(Gǫ),where Γ′ is the derivative of an admissible function Γ:

(∂t + v ·∇x)Γ(Gǫ) =1

ǫΓ′(Gǫ)Q(Gǫ, Gǫ) on O × R+ . (29)

Here a function Γ : [0,∞) → R is called admissible if it is continuouslydifferentiable and for some CΓ < ∞ its derivative satisfies

|Γ′(Z)| ≤ CΓ√1 + Z

for every Z ∈ [0,∞) .

The solutions are nonnegative and lie in C([0,∞); w-L1(Mdv dx)), wherethe prefix “w-” on a space indicates that the space is endowed with its weaktopology.

Mischler (2010) extended DiPerna-Lions theory to domains with a bound-ary on which the Maxwell reflection boundary condition (28) is imposed.This required the proof of a so-called trace theorem that shows that therestriction of Gǫ to ∂O × R+, denoted γGǫ, makes sense. In particular,Mischler showed that γGǫ lies in the set of all |v ·n|Mdv dσ dt-measurablefunctions over ∂O × R+ that are finite almost everywhere, which we de-note L0(|v ·n|Mdv dσ dt). He then defines γ±Gǫ = 1Σ±γGǫ. He provesthe following.

Theorem. (DiPerna-Lions-Mischler Renormalized Solutions) Let b be acollision kernel that satisfies the assumptions given earlier. Fix ǫ > 0. LetGin

ǫ be any initial data in the entropy class

E(Mdv dx) =

Ginǫ ≥ 0 : H(Gin

ǫ ) < ∞

, (30)

where the relative entropy functional is given by

H(G) =

∫

Ω〈η(G)〉dx with η(G) = G log(G) − G + 1 .

Then there exists a Gǫ ≥ 0 in C([0,∞); w-L1(Mdv dx)) with γGǫ ≥ 0

in L0(|v · n|Mdv dσ dt) such that:

• Gǫ satisfies the global entropy inequality

H(Gǫ(t)) +

∫ t

0

[

1

ǫR(Gǫ(s)) +

α√2π

E(γ+Gǫ(s))

]

ds

≤ H(Ginǫ ) for every t > 0 ,

(31)

where the entropy dissipation rate functional is given by

R(G) =1

4

∫

Ω

⟨⟨

log

(

G′1G′

G1G

)

(

G′1G′ − G1G

)

⟩⟩

dx , (32)

and the so-called Darrozès-Guiraud information is given by

E(γ+G) =

∫

∂Ω

[⟨

η(γ+G)⟩

∂Ω− η

(

〈γ+G〉∂Ω

)]

dσ ; (33)

• Gǫ satisfies∫

Ω〈Γ(Gǫ(t2))Y 〉dx −

∫

Ω〈Γ(Gǫ(t1))Y 〉dx

+

∫ t2

t1

∫

∂Ω〈Γ(γGǫ)Y (v ·n)〉dσ dt −

∫ t2

t1

∫

Ω〈Γ(Gǫ) v ·∇xY 〉dxdt

=1

ǫ

∫ t2

t1

∫

Ω〈Γ′(Gǫ)Q(Gǫ, Gǫ)Y 〉dxdt ,

(34)for every admissible function Γ, every Y ∈ C1 ∩ L∞(RD × Ω), andevery [t1, t2] ⊂ [0,∞];

• Gǫ satisfies

γ−Gǫ = (1 − α)L(γ+Gǫ) + α〈γ+Gǫ〉∂Ω

almost everywhere on Σ− × R+ .(35)

Remark. Because the γGǫ is only known to exist in L0(|v · n|Mdv dσ dt)

rather than in L1loc(dt;L1(|v ·n|Mdv dσ)), we cannot conclude from the

boundary condition (35) that

〈v γGǫ〉 ·n = 0 on ∂Ω . (36)

Indeed, we cannot even conclude that the boundary mass-flux 〈v γGǫ〉 ·nis defined on ∂Ω. Moreover, in contrast to DiPerna-Lions theory over thewhole space or periodic domains, it is not asserted that Gǫ satisfies theweak form of the local mass conservation law∫

Ωχ 〈Gǫ(t2)〉dx −

∫

Ωχ 〈Gǫ(t1)〉dx −

∫ t2

t1

∫

Ω∇xχ · 〈v Gǫ〉dxdt = 0

∀χ ∈ C1(Ω) .(37)

If this were the case, it would allow a great simplification the proof of ourmain result. Rather, we will employ the boundary condition (35) inside anapproximation to (37) that has a well-defined boundary flux.

Main Result

We will consider families Gǫ of DiPerna-Lions renormalized solutions to(27) such that Gin

ǫ ≥ 0 satisfies the entropy bound

H(Ginǫ ) ≤ C inδ 2

ǫ (38)

for some C in < ∞ and δǫ > 0 that satisfies the scaling δǫ → 0 as ǫ → 0.

The value of H(G) provides a natural measure of the proximity of G to theequilibrium G = 1. We define the families gin

ǫ and gǫ of fluctuations aboutG = 1 by the relations

Ginǫ = 1 + δǫg

inǫ , Gǫ = 1 + δǫgǫ . (39)

One easily sees that H asymptotically behaves like half the square of theL2-norm of these fluctuations as ǫ → 0.

Hence, the entropy bound (38) combined with the entropy inequality (31)is consistent with these fluctuations being of order 1. Just as the relativeentropy H controls the fluctuations gǫ, the dissipation rate R given by (32)controls the scaled collision integrals defined by

qǫ =1√ǫδǫ

(

G′ǫ1G′

ǫ − Gǫ1Gǫ

)

.

Here we only state the weak acoustic limit theorem because the corre-sponding strong limit theorem is analogous to that stated in Golse-L (2002)and its proof based on the weak limit theorem and relative entropy conver-gence is essentially the same.

Theorem. (Weak Acoustic Limit) Let b be a collision kernel that satisfiesthe assumptions given earlier.

Let Ginǫ be a family in the entropy class E(Mdv dx) that satisfies the nor-

malization (17) and the entropy bound (38) for some C in < ∞ and δǫ > 0

satisfies the scaling

δǫ = O(√

ǫ)

.

Assume, moreover, that for some (ρin, uin, θin) ∈ L2(dx;R × RD× R)

the family of fluctuations ginǫ defined by (39) satisfies

(ρin, uin, θin) = limǫ→0

(

〈ginǫ 〉 , 〈v gin

ǫ 〉 ,⟨(

1D|v|2 − 1

)

ginǫ

⟩)

in the sense of distributions .(40)

Let Gǫ be any family of DiPerna-Lions-Mischler renormalized solutions tothe Boltzmann equation (27) that have Gin

ǫ as initial values.

Then, as ǫ → 0, the family of fluctuations gǫ defined by (39) satisfies

gǫ → ρ+v ·u+(12|v|

2−D2 )θ in w-L1

loc(dt; w-L1((1 + |v|2)Mdv dx)) ,

(41)where (ρ, u, θ) ∈ C([0,∞);L2(dx;R × RD× R)) is the unique solutionto the acoustic system (1) that satisfies the impermeable boundary condi-tion (2) and has initial data (ρin, uin, θin) obtain from (40). In addition, ρ

satisfies∫

Ωρdx = 0 . (42)

This result improves upon the acoustic limit result in three ways:

1. Its assumption on the collision kernel b is the same as L-Masmoudi,so it treats a broader class of cut-off kernels than in Golse-L (2002). Inparticular, it treats kernals derived from soft potentials.

2. Its scaling assumption is δǫ = O(√

ǫ), which is certainly better than thescaling assumption (16) used in Golse-L. This is still a long way from thatrequired by the formal derivation.

3. We derive a weak form of the boundary condition u ·n = 0. It is the firsttime such a boundary condition is derived for the acoustic system.

Proof of Main Theorem

In order to derive the fluid equations with boundary conditions, we needto pass to the limit in approximate local conservation laws built from therenormalized Boltzmann equation (29). We choose the renormalization

Γ(Z) =Z − 1

1 + (Z − 1)2. (43)

After dividing by δǫ, equation (29) becomes

∂tgǫ+v · ∇xgǫ =1√ǫΓ′(Gǫ)

∫∫

SD−1×RDqǫ b(ω , v1−v) dω M1dv1 , (44)

where gǫ = Γ(Gǫ)/δǫ. By introducing Nǫ = 1 + δ2ǫ g2ǫ , we can write

gǫ =gǫ

Nǫ, Γ′(Gǫ) =

2

N2ǫ− 1

Nǫ. (45)

When moment of the renormalized Boltzmann equation (44) is formallytaken with respect to any ζ ∈ span1 , v1 , · · · , vD , |v|2, one obtains

∂t〈ζ gǫ〉 + ∇x · 〈v ζ gǫ〉 =1√ǫ

⟨⟨

ζ Γ′(Gǫ) qǫ

⟩⟩

. (46)

Every DiPerna-Lions solution satisfies (46) in the sense that for every χ ∈C1(Ω) and every [t1, t2] ⊂ [0,∞) it satisfies∫

Ωχ 〈ζ gǫ(t2)〉dx −

∫

Ωχ 〈ζ gǫ(t1)〉dx +

∫ t2

t1

∫

∂Ωχ 〈v ζ γgǫ〉 ·ndσ dt

−∫ t2

t1

∫

Ω∇xχ · 〈v ζ gǫ〉dxdt =

∫ t2

t1

∫

Ωχ

1√ǫ

⟨⟨

ζ Γ′(Gǫ) qǫ

⟩⟩

dxdt .

(47)

Moreover, from (35) the boundary condition is understood in the renormal-ized sense:

γ−gǫ =(1 − α)Lγ+gǫ + α〈γ+gǫ〉∂Ω

1 + δ2ǫ [(1 − α)Lγ+gǫ + α〈γ+gǫ〉∂Ω]2on Σ− × R+ , (48)

where the equality holds almost everywhere. We will pass to the limit inthe weak form (47).

The Main Theorem will be proved in two steps: the interior equations willbe established first and the boundary condition second.

Establishing the Acoustic System

The acoustic system (1) is justified in the interior of Ω by showing that thelimit of (47) as ǫ → 0 is the weak form of the acoustic system wheneverthe test function χ vanishes on ∂Ω. We prove that the conservation defecton the right-hand side of (47) vanishes as ǫ → 0. The proof of the analo-gous result in Golse-L (2002) must be modified in order to include the caseδǫ = O(

√ǫ). The convergence of the density and flux terms is proved

essentially the same, so we omit those arguments here. The upshot is thatevery converging subsequence of the family gǫ satisfies

gǫ → ρ+v ·u+(12|v|

2−D2 )θ in w-L1

loc(dt; w-L1((1 + |v|2)Mdv dx)) ,

where (ρ, u, θ) ∈ C([0,∞); w-L2(dx;R × RD× R)) satisfies for every[t1, t2] ⊂ [0,∞)

∫

Ωχ ρ(t2) dx −

∫

Ωχ ρ(t1) dx −

∫ t2

t1

∫

Ω∇xχ ·udxdt = 0

∀χ ∈ C10(Ω) ,

(49a)

∫

Ωw ·u(t2) dx −

∫

Ωw ·u(t1) dx −

∫ t2

t1

∫

Ω∇x · w (ρ + θ) dxdt = 0

∀w ∈ C10(Ω;RD) ,

(49b)

D2

∫

Ωχ θ(t2) dx − D

2

∫

Ωχ θ(t1) dx −

∫ t2

t1

∫

Ω∇xχ ·udxdt = 0

∀χ ∈ C10(Ω) .

(49c)

This shows that the acoustic system (1) is satisfied in the interior of Ω.

Establishing the Boundary Condition

The more significant step is to justify the impermeable boundary condi-tion (2). Unlike to what is done for the incompressible Stokes and Navier-Stokes limits, here we do not have enough control to pass to the limit in theboundary terms in (47) for the local conservation laws of momentum andenergy. We can however do so for the local conservation law of mass —i.e. when ζ = 1. Indeed, we can extend (49a) to∫

Ωχ ρ(t2) dx−

∫

Ωχ ρ(t1) dx−

∫ t2

t1

∫

Ω∇xχ ·udxdt = 0 ∀χ ∈ C1(Ω) .

(50)We obtain 42 by setting χ = 1 and t1 = 0 above, and using the fact thatthe family Gin

ǫ satisfies the normalization (17).

Because for every χ ∈ C1(Ω) we can find a sequence χn ⊂ C10(Ω)

such that χn → χ in L2(dx), it follows from (49a) and (49c) that

D2

∫

Ωχ θ(t2) dx − D

2

∫

Ωχ θ(t1) dx

= limn→∞

D2

∫

Ωχn θ(t2) dx − lim

n→∞D2

∫

Ωχn θ(t1) dx

= limn→∞

∫

Ωχn ρ(t2) dx − lim

n→∞

∫

Ωχn ρ(t1) dx

=

∫

Ωχ ρ(t2) dx −

∫

Ωχ ρ(t1) dx .

It thereby follows from (50) that we can extend (49c) to

D2

∫

Ωχ θ(t2) dx − D

2

∫

Ωχ θ(t1) dx

−∫ t2

t1

∫

Ω∇xχ ·udxdt = 0 ∀χ ∈ C1(Ω) .

(51)

Finally, because for every w ∈ C1(Ω;RD) such that w ·n = 0 on ∂Ω

we can find a sequence wn ⊂ C10(Ω;RD) such that wn → w in

L2(dx;RD) and ∇x · wn → ∇x · w in L2(dx), it follows from (49b) that

∫

Ωw ·u(t2) dx −

∫

Ωw ·u(t1) dx

= limn→∞

∫

Ωwn ·u(t2) dx − lim

n→∞

∫

Ωwn ·u(t1) dx

= limn→∞

∫ t2

t1

∫

Ω∇x · wn (ρ + θ) dxdt

=∫ t2

t1

∫

Ω∇x · w (ρ + θ) dxdt .

But this combined with (50) and (51) is the weak formulation of the acousticsystem (1) with the boundary condition (2).

This system has a unique solution in C([0,∞); w-L2(dx;R × RD× R)),so all converging sequences of the family gǫ have this same limit. So thislimit must be the strong solution in C([0,∞);L2(dx;R × RD× R)). Thefamily of fluctuations gǫ therefore converges as asserted by (41).

Remark. The most important difference between the acoustic limit and theincompressible limits (Stokes and Navier-Stokes) is that the compactnessof the renormalized traces γgǫ in the acoustic limit case is not available.The pointwise convergence δǫgǫ → 0 a.e. is also unavailable. In contrast,for the incompressible limits the entropy bounds from boundary provide apriori estimates on the quantity γǫ = γ+gǫ−1Σ+〈γ+gǫ〉∂Ω

. Specifically, we

have the L2 bound on 1δǫ

γ(1)ǫnǫ

with some renormalizer nǫ. However, in theacoustic limit, because of the acoustic scaling, we have only the L2 bound

on γ(1)ǫnǫ

which is much weaker than in the incompressible limits cases.

Linearized Boltzmann Setting

Now consider the linearized Boltzmann

∂tgǫ + v ·∇xgǫ +1

ǫLgǫ = 0 ,

Here Lg = −2Q(1, g) as before. This is well-posed in L2(Mdvdx) overa smooth spatial domain Ω with proper boundary conditions. Let

m =(

1 v1 · · · vD12|v|2

)T.

The local conservation laws are then

∂t〈m gǫ〉 + ∇x · 〈v m gǫ〉 = 0 .

In the interior of Ω we can approximate gǫ by the Enskog expansion

gǫ = mTαǫ − ǫL−1

mTv ·∇xαǫ

+ ǫ2L−1 (∂t + v ·∇x)L−1m

Tv ·∇xαǫ + O(ǫ3) ,

where 〈mmT 〉αǫ = 〈m gǫ〉.

If we approximate αǫ by the solution of the acoustic system

〈mmT 〉∂tαǫ + 〈mm

Tv〉 ·∇xαǫ = 0 ,

〈mmT 〉αǫ

∣

∣

∣

t=0= 〈m gin

ǫ 〉 ,

then we expect to prove that

gǫ = mTαǫ + O(ǫ) .

If we approximate αǫ by the solution of the compressible Stokes system

〈mmT 〉∂tαǫ + 〈mm

Tv〉 ·∇xαǫ = ǫ∇x ·[

〈v mL−1m

Tv〉 ·∇xαǫ

]

,

〈mmT 〉αǫ

∣

∣

∣

t=0= 〈m gin

ǫ 〉 − ǫ 〈m v ·∇xL−1ginǫ 〉 ,

then we expect to prove that

gǫ = mTαǫ − ǫL−1

mTv ·∇xαǫ + O(ǫ2) .

This requires a boundary layer construction through order ǫ.

The “count” for such a construction is correct, unlike for the acoustic ap-proximation. More specifically, the number of conditions needed to insurethat the solution of a half-space problem decays is the number of incom-ing plus the number of tangential characteristic velocities of the acousticsystem. This is generally greater than the number of conditions requiredto make the acoustic system well-posed, but is equal to the number ofconditions required to make the compressible Stokes system well-posed.

Open Problems in the BGL Program

1. Acoustic limit with optimal scaling, δǫ → 0.

2. Compressible Stokes approximation (linearized compressible N-S).

3. Weakly nonlinear/dissipative approximation to compressible N-S.

4. Dominant-balance incompressible approximations (Bardos-L-Ukai-Yang)

5. Bounded domains (Bardos, Jiang, Masmoudi, L, Saint-Raymond, · · · )

Thank You!