SLIDE 1 Fluid Approximations from the Boltzmann Equation for Domains with Boundary
Department of Mathematics and Institute for Physical Science and Technology University 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 Mathematics Brown University, Providence, RI
SLIDE 2 Introduction We study some fluid approximations derived from the Boltzmann equation
- ver a smooth bounded spatial domain Ω ⊂ RD. Our focus will be on
boundary conditions.
- 1. We establish the acoustic limit starting from DiPerna-Lions solutions.
(Jiang-L-Masmoudi, 2010)
- 2. We present linearized Navier-Stokes approximations derived formally
from the linearized Boltzmann equation.
SLIDE 3 Acoustic System After a suitable choice of units and Galilean frame, the acoustic system governs the fluctuations in mass density ρ(x, t), bulk velocity u(x, t), and temperature 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) ,
D 2 ∂tθ + ∇ x· u = 0 ,
θ(x, 0) = θin(x) , (1) subject to the impermeable boundary condition u · n = 0 ,
(2) where n(x) is the unit outward normal at x ∈ ∂Ω. This is one of the simplest fluid dynamical systems, being essentially the wave equation.
SLIDE 4 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)−D
2 exp
2|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 scaled Boltzmann initial-value problem ∂tGǫ + v · ∇
xGǫ = 1
ǫ Q(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) =
×RD
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.
SLIDE 5 We impose a Maxwell reflection boundary condition on ∂Ω of the form
1Σ+Gǫ ◦ R = (1 − α) 1Σ+ Gǫ + α 1Σ+
√ 2π
(6) Here (Gǫ◦R)(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 =
(8) Because √ 2π
- 1Σ+ v · n
- = 1, it seen from (6) that on ∂Ω the flux is
v · n Gǫ =
- 1Σ+ v · n
- Gǫ − Gǫ ◦ R
- = α
- 1Σ+ v · n
- Gǫ −
√ 2π
(9)
SLIDE 6
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 , Gin
ǫ = 1 + δǫgin ǫ ,
(10) where δǫ → 0 as ǫ → 0 , (11) and the fluctuations gǫ and gin
ǫ
converge in the sense of distributions to g ∈ L∞(dt; L2(Mdv dx)) and gin ∈ L2(Mdv dx) respectively as ǫ → 0.
SLIDE 7 One finds that g has the infinitesimal Maxwellian form g = ρ + v · u +
1
2|v|2 − D 2
(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 =
1
D|v|2 − 1
. (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 infinitesimal Maxwellian form (12) get the impermeable boundary condition (2), u · n = 0 .
SLIDE 8
The program initiated with Claude Bardos and Francois Golse in 1989 seeks to justify fluid dynamical limits for Boltzmann equations in the set- ting of DiPerna-Lions renormalized solutions, which are the only temporally global, large data solutions available. The main obstruction to carrying out this program is that DiPerna-Lions solutions are not known to satisfy many properties that one formally ex- pects for solutions of the Boltzmann equation. For example, they are not known to satisfy the formally expected local conservations laws of momen- tum and energy. Moreover, their regularity is poor. The justification of fluid dynamical limits in this setting is therefore not easy.
SLIDE 9
The acoustic limit was first established in this setting by Bardos-Golse-L (2000) over a periodic domain. There idea introduced there was to pass to the limit in approximate local conservations laws which are satified by DiPerna-Lions solutions. One then shows that the so-called conservation defects vanish as the Knudsen number ǫ vanishes, thereby establishing the local conservation laws in the limit. This was done using only relative entropy estimates, which restricted the result to collision kernels that are bounded and to fluctuations scaled so that δǫ → 0 and δǫ ǫ | log(δǫ)| → 0 as ǫ → 0 , (14) which is far from the formally expected optimal scaling (11), δǫ → 0.
SLIDE 10 In Golse-L (2002) the local conservation defects were removed using new dissipation rate estimates. This allowed the treatment of collision kernels that for some Cb < ∞ and β ∈ [0, 1) satisfied
- SD−1 b(ω, v1 − v) dω ≤ Cb
- 1 + |v1 − v|2β ,
(15) and of fluctuations scaled so that δǫ → 0 and δǫ ǫ1/2| log(δǫ)|β/2 → 0 as ǫ → 0 . (16) The above class of collision kernels includes all classical kernels that are derived from Maxwell or hard potentials and that satisfy a weak small de- flection cutoff. The scaling given by (16) is much less restrictive than that given by (14), but is far from the formally optimal scaling (11). Finally, only periodic domains are treated.
SLIDE 11
Here we improve the result of Golse-L (2002) in three ways. First, we apply estimates from L-Masmoudi (2010) to treat a broader class of collision ker- nels that includes those derived from soft potentials. Second, we improve the scaling of the fluctuations to δǫ = O(ǫ1/2). Finally, we treat domains with 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) is used through the nonlinear compactness estimate of L-Masmoudi (2010) to improve the scaling of the fluctuations to δǫ = O(ǫ1/2). Without it we would only be able to improve the scaling to δǫ = o(ǫ1/2). This is the first time the L1 averaging theory has played any role in an acoustic limit theorem, albeit for a modest improvement. We remark that velocity aver- aging theory plays no role in establishing the Stokes limit with its formally expected optimal scaling of δǫ = o(ǫ).
SLIDE 12 We treat domains with boundary in the setting of Mischler (2002/2010), who extended DiPerna-Lions theory to bounded domains with a Maxwell reflection boundary condition. He showed that these boundary conditions are 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 did formally. Masmoudi and Saint-Raymond (2003) derived boundary conditions in the Stokes limit. However neither these estimates nor their recent extension to the Navier-Stokes limit by Jiang-Masmoudi can handle the acoustic limit. Rather, we develop new boundary a priori estimates to obtain a weak form
- f the boundary condition (2) in this limit. In doing so, we treat a broader
class of collision kernels than was treated earlier.
SLIDE 13 We remark that establishing the acoustic limit with its formally expected
- ptimal scaling of the fluctuation size, δǫ → 0, is still open. This gap must
be bridged before one can hope to fully establish the compressible Euler limit 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).
SLIDE 14 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 the Lebesgue 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 bulk velocity 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
ǫ dx = 1 .
(17)
SLIDE 15
Assumptions on the Collision Kernel The kernel b(ω, v1−v) associated with the collision operator (5) is positive almost everywhere. The Galilean invariance of the collisional physics implies that b has the classical 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 adopted from L-Masmoudi (2010).
SLIDE 16 Our first technical assumption is that the collision kernel b satisfies the requirements of the DiPerna-Lions theory. That theory requires that b be locally integrable with respect to dω M1dv1 Mdv, and that it satisfies lim
|v|→∞
1 1 + |v|2
for every compact K ⊂ RD , (19) where b is defined by b(v1 − v) ≡
(20) Galilean symmetry (18) implies that b is a function of |v1 − v| only.
SLIDE 17 Our second technical assumption regarding b is that the attenuation coef- ficient a, which is defined by a(v) ≡
(21) is bounded below as Ca
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
a(v1) a(v)
a(v1) M1dv1
1
s
≤ Cb . (23) Because this bound is uniform in v, we may take Cb to be the supremum
- ver v of the left-hand side of (23).
SLIDE 18 Our fourth technical assumption regarding b is that the operator K+ : L2(aMdv) → L2(aMdv) is compact , (24) where K+˜ g = 1 2a
×RD
g′ + ˜ g′
1
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 there exists Cδ such that b satisfies b(v1 − v) 1 + δ b(v1 − v) 1 + |v1 − v|2 ≤ Cδ
- 1+a(v1)
- 1+a(v)
- for every v1, v ∈ RD .
(25)
SLIDE 19 The above assumptions are satisfied by all the classical collision kernels with 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 classical
collision kernels to which the DiPerna-Lions theory applies. Kernels that satisfy (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×RD b(ω, v1 − v) dω M1 dv1 M dv = 1 .
Because dµ = b(ω, v1 − v) dω M1dv1 Mdv is a positive unit measure on
SD−1× RD× RD, we denote by
- Ξ
- the average over this measure of
any integrable function Ξ = Ξ(ω, v1, v)
×RD×RD Ξ(ω, v1, v) dµ .
(26)
SLIDE 20 DiPerna-Lions-Mischler Theory We will work in the framework of DiPerna-Lions solutions to the scaled Boltzmann equation on the phase space O = RD × Ω ∂tGǫ + v · ∇
xGǫ = 1
ǫ Q(Gǫ, Gǫ)
O × R+ , Gǫ(v, x, 0) = Gin
ǫ (v, x)
O , (27) with the Maxwell reflection boundary condition (6) which can be expressed as γ−Gǫ = (1 − α)L(γ+Gǫ) + αγ+Gǫ
∂Ω
Σ− × R+ , (28) where γ±Gǫ denote the traces of Gǫ on the outgoing and incoming sets Σ±.
SLIDE 21 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 the diffuse reflection operator is defined as φ∂Ω = √ 2π
- v·n(x)>0 φ(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 over the whole space RD for any initial data satisfying natural physical bounds. For bounded domain case, Mischler recently developed a theory to treat the Maxwell reflection boundary condition.
SLIDE 22 The DiPerna-Lions theory does not yield solutions that are known to solve the Boltzmann equation in the usual sense of weak solutions. Rather, it gives 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
ǫ Γ′(Gǫ)Q(Gǫ, Gǫ)
O × R+ . (29) Here a function Γ : [0, ∞) → R is called admissible if it is continuously differentiable 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)), where the prefix “w-” on a space indicates that the space is endowed with its weak topology.
SLIDE 23
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 the restriction 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-measurable functions 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 proves the following.
SLIDE 24
- Theorem. (DiPerna-Lions-Mischler Renormalized Solutions) Let b be a
collision kernel that satisfies the assumptions given earlier. Fix ǫ > 0. Let Gin
ǫ be any initial data in the entropy class
E(Mdv dx) =
ǫ ≥ 0 : H(Gin ǫ ) < ∞
(30) where the relative entropy functional is given by H(G) =
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:
SLIDE 25
- Gǫ satisfies the global entropy inequality
H(Gǫ(t)) +
t
ǫ R(Gǫ(s)) + α √ 2πE(γ+Gǫ(s))
≤ H(Gin
ǫ )
for every t > 0 , (31) where the entropy dissipation rate functional is given by R(G) = 1 4
1G′
G1G G′
1G′ − G1G
(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)
SLIDE 26
- Gǫ satisfies
- ΩΓ(Gǫ(t2)) Y dx −
- ΩΓ(Gǫ(t1)) Y dx
+
t2
t1
- ∂ΩΓ(γGǫ) Y (v · n) dσ dt −
t2
t1
xY dx dt
= 1 ǫ
t2
t1
- ΩΓ′(Gǫ) Q(Gǫ, Gǫ) Y dx dt ,
(34) for every admissible function Γ, every Y ∈ C1 ∩ L∞(RD × ¯ Ω), and every [t1, t2] ⊂ [0, ∞];
γ−Gǫ = (1 − α)L(γ+Gǫ) + αγ+Gǫ∂Ω almost everywhere on Σ− × R+ . (35)
SLIDE 27
- Remark. Because the γGǫ is only known to exist in L0(|v · n|Mdv dσ dt)
rather than in L1
loc(dt; L1(|v · n|Mdv dσ)), we cannot conclude from the
boundary condition (35) that v γGǫ · n = 0
∂Ω . (36) Indeed, we cannot even conclude that the boundary mass-flux v γGǫ · n is defined on ∂Ω. Moreover, in contrast to DiPerna-Lions theory over the whole space or periodic domains, it is not asserted that Gǫ satisfies the weak form of the local mass conservation law
- Ω χ Gǫ(t2) dx −
- Ω χ Gǫ(t1) dx −
t2
t1
xχ · v Gǫ dx dt = 0
∀χ ∈ C1(Ω) . (37) If this were the case, it would allow a great simplification the proof of our main result. Rather, we will employ the boundary condition (35) inside an approximation to (37) that has a well-defined boundary flux.
SLIDE 28
Main Result We will consider families Gǫ of DiPerna-Lions renormalized solutions to (27) such that Gin
ǫ ≥ 0 satisfies the entropy bound
H(Gin
ǫ ) ≤ Cinδ 2 ǫ
(38) for some Cin < ∞ 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 the equilibrium G = 1. We define the families gin
ǫ and gǫ of fluctuations about
G = 1 by the relations Gin
ǫ = 1 + δǫgin ǫ ,
Gǫ = 1 + δǫgǫ . (39) One easily sees that H asymptotically behaves like half the square of the L2-norm of these fluctuations as ǫ → 0.
SLIDE 29 Hence, the entropy bound (38) combined with the entropy inequality (31) is consistent with these fluctuations being of order 1. Just as the relative entropy H controls the fluctuations gǫ, the dissipation rate R given by (32) controls the scaled collision integrals defined by qǫ = 1 √ǫδǫ
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.
SLIDE 30
- Theorem. (Weak Acoustic Limit) Let b be a collision kernel that satisfies
the assumptions given earlier. Let Gin
ǫ be a family in the entropy class E(Mdv dx) that satisfies the nor-
malization (17) and the entropy bound (38) for some Cin < ∞ 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 gin
ǫ defined by (39) satisfies
(ρin, uin, θin) = lim
ǫ→0
ǫ , v gin ǫ ,
1
D|v|2 − 1
ǫ
- in the sense of distributions .
(40) Let Gǫ be any family of DiPerna-Lions-Mischler renormalized solutions to the Boltzmann equation (27) that have Gin
ǫ as initial values.
SLIDE 31 Then, as ǫ → 0, the family of fluctuations gǫ defined by (39) satisfies gǫ → ρ+v · u+(1
2|v|2− D 2 )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 solution to 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
(42)
SLIDE 32 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). In particular, it treats kernals derived from soft potentials.
- 2. Its scaling assumption is δǫ = O(√ǫ), which is certainly better than the
scaling assumption (16) used in Golse-L. This is still a long way from that required by the formal derivation.
- 3. We derive a weak form of the boundary condition u · n = 0. It is the first
time such a boundary condition is derived for the acoustic system.
SLIDE 33 Proof of Main Theorem In order to derive the fluid equations with boundary conditions, we need to pass to the limit in approximate local conservation laws built from the renormalized Boltzmann equation (29). We choose the renormalization Γ(Z) = Z − 1 1 + (Z − 1)2 . (43) After dividing by δǫ, equation (29) becomes ∂t˜ gǫ+v · ∇
x˜
gǫ = 1 √ǫ Γ′(Gǫ)
×RD qǫ b(ω , v1−v) dω M1dv1 , (44)
where ˜ gǫ = Γ(Gǫ)/δǫ. By introducing Nǫ = 1 + δ2
ǫ g2 ǫ , we can write
˜ gǫ = gǫ Nǫ , Γ′(Gǫ) = 2 N2
ǫ
− 1 Nǫ . (45)
SLIDE 34 When moment of the renormalized Boltzmann equation (44) is formally taken with respect to any ζ ∈ span{1 , v1 , · · · , vD , |v|2}, one obtains ∂tζ ˜ gǫ + ∇
x· v ζ ˜
gǫ = 1 √ǫ
(46) Every DiPerna-Lions solution satisfies (46) in the sense that for every χ ∈ C1(Ω) and every [t1, t2] ⊂ [0, ∞) it satisfies
gǫ(t2) dx −
gǫ(t1) dx +
t2
t1
gǫ · n dσ dt −
t2
t1
xχ · v ζ ˜
gǫ dx dt =
t2
t1
√ǫ
(47)
SLIDE 35 Moreover, from (35) the boundary condition is understood in the renormal- ized sense: γ−˜ gǫ = (1 − α)Lγ+gǫ + αγ+gǫ∂Ω 1 + δ2
ǫ [(1 − α)Lγ+gǫ + αγ+gǫ∂Ω]2
(48) where the equality holds almost everywhere. We will pass to the limit in the weak form (47). The Main Theorem will be proved in two steps: the interior equations will be established first and the boundary condition second.
SLIDE 36 Establishing the Acoustic System The acoustic system (1) is justified in the interior of Ω by showing that the limit of (47) as ǫ → 0 is the weak form of the acoustic system whenever the test function χ vanishes on ∂Ω. We prove that the conservation defect
- n 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 case δǫ = O(√ǫ). The convergence of the density and flux terms is proved essentially the same, so we omit those arguments here. The upshot is that every converging subsequence of the family gǫ satisfies gǫ → ρ+v · u+(1
2|v|2− D 2 )θ
in w-L1
loc(dt; w-L1((1 + |v|2)Mdv dx)) ,
SLIDE 37 where (ρ, u, θ) ∈ C([0, ∞); w-L2(dx; R × RD× R)) satisfies for every [t1, t2] ⊂ [0, ∞)
- Ω χ ρ(t2) dx −
- Ω χ ρ(t1) dx −
t2
t1
xχ · u dx dt = 0
∀χ ∈ C1
0(Ω) ,
(49a)
- Ω w · u(t2) dx −
- Ω w · u(t1) dx −
t2
t1
x· w (ρ + θ) dx dt = 0
∀w ∈ C1
0(Ω; RD) ,
(49b)
D 2
2
t2
t1
xχ · u dx dt = 0
∀χ ∈ C1
0(Ω) .
(49c) This shows that the acoustic system (1) is satisfied in the interior of Ω.
SLIDE 38 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 the boundary terms in (47) for the local conservation laws of momentum and
- energy. We can however do so for the local conservation law of mass —
i.e. when ζ = 1. Indeed, we can extend (49a) to
- Ω χ ρ(t2) dx−
- Ω χ ρ(t1) dx−
t2
t1
xχ · u dx dt = 0
∀χ ∈ C1(Ω) . (50) We obtain 42 by setting χ = 1 and t1 = 0 above, and using the fact that the family Gin
ǫ satisfies the normalization (17).
SLIDE 39 Because for every χ ∈ C1(Ω) we can find a sequence {χn} ⊂ C1
0(Ω)
such that χn → χ in L2(dx), it follows from (49a) and (49c) that
D 2
2
= lim
n→∞ D 2
n→∞ D 2
= lim
n→∞
n→∞
=
- Ω χ ρ(t2) dx −
- Ω χ ρ(t1) dx .
It thereby follows from (50) that we can extend (49c) to
D 2
2
−
t2
t1
xχ · u dx dt = 0
∀χ ∈ C1(Ω) . (51)
SLIDE 40 Finally, because for every w ∈ C1(Ω; RD) such that w · n = 0 on ∂Ω we can find a sequence {wn} ⊂ C1
0(Ω; RD) such that wn → w in
L2(dx; RD) and ∇
x · wn → ∇ x · w in L2(dx), it follows from (49b) that
- Ω w · u(t2) dx −
- Ω w · u(t1) dx
= lim
n→∞
n→∞
= lim
n→∞
t2
t1
x· wn (ρ + θ) dx dt
=
t2
t1
x· w (ρ + θ) dx dt .
But this combined with (50) and (51) is the weak formulation of the acoustic system (1) with the boundary condition (2).
SLIDE 41 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 this limit must be the strong solution in C([0, ∞); L2(dx; R × RD× R)). The family of fluctuations gǫ therefore converges as asserted by (41).
- Remark. The most important difference between the acoustic limit and the
incompressible limits (Stokes and Navier-Stokes) is that the compactness
- f the renormalized traces γ˜
gǫ in the acoustic limit case is not available. The pointwise convergence δǫ˜ gǫ → 0 a.e. is also unavailable. In contrast, for the incompressible limits the entropy bounds from boundary provide a priori estimates on the quantity γǫ = γ+gǫ−1Σ+γ+gǫ∂Ω. Specifically, we have the L2 bound on 1
δǫ γ(1)
ǫ
nǫ with some renormalizer nǫ. However, in the
acoustic limit, because of the acoustic scaling, we have only the L2 bound
ǫ
nǫ which is much weaker than in the incompressible limits cases.
SLIDE 42 Linearized Boltzmann Setting Now consider the linearized Boltzmann ∂tgǫ + v · ∇
xgǫ + 1
ǫ Lgǫ = 0 , Here Lg = −2Q(1, g) as before. This is well-posed in L2(Mdvdx) over a smooth spatial domain Ω with proper boundary conditions. Let
m =
v1 · · · vD
1 2|v|2T .
The local conservation laws are then ∂tm gǫ + ∇
x· v m gǫ = 0 .
SLIDE 43 In the interior of Ω we can approximate gǫ by the Enskog expansion gǫ = mTαǫ − ǫ L−1mTv · ∇
xαǫ
+ ǫ2L−1 (∂t + v · ∇
x) L−1mTv · ∇ xαǫ + O(ǫ3) ,
where m mTαǫ = m gǫ. If we approximate αǫ by the solution of the acoustic system m mT∂tαǫ + m mTv · ∇
xαǫ = 0 ,
m mTαǫ
ǫ ,
then we expect to prove that gǫ = mTαǫ + O(ǫ) .
SLIDE 44 If we approximate αǫ by the solution of the compressible Stokes system m mT ∂tαǫ + m mTv · ∇
xαǫ = ǫ ∇ x·
xαǫ
m mTαǫ
ǫ − ǫ m v · ∇ xL−1gin ǫ ,
then we expect to prove that gǫ = mTαǫ − ǫ L−1mTv · ∇
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 insure
that the solution of a half-space problem decays is the number of incom- ing plus the number of tangential characteristic velocities of the acoustic
- system. This is generally greater than the number of conditions required
to make the acoustic system well-posed, but is equal to the number of conditions required to make the compressible Stokes system well-posed.
SLIDE 45 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!
