HILBERT EXPANSION OF THE BOLTZMANN EQUATION IN THE INCOMPRESSIBLE EULER LEVEL IN A CHANNEL FEIMIN HUANG WEIQIANG WANG YONG WANG AND FENG XIAO

2025-04-29 0 0 794.88KB 53 页 10玖币

侵权投诉

HILBERT EXPANSION OF THE BOLTZMANN EQUATION IN THE

INCOMPRESSIBLE EULER LEVEL IN A CHANNEL

FEIMIN HUANG, WEIQIANG WANG, YONG WANG, AND FENG XIAO

Abstract. The study of hydrodynamic limit of the Boltzmann equation with physical boundary

is a challenging problem due to appearance of the viscous and Knudsen boundary layers. In this

paper, the hydrodynamic limit from the Boltzmann equation with specular reﬂection boundary

condition to the incompressible Euler in a channel is investigated. Based on the multi-scaled

Hilbert expansion, the equations with boundary conditions and compatibility conditions for inte-

rior solutions, viscous and Knudsen boundary layers are derived under diﬀerent scaling, respec-

tively. Then some uniform estimates for the interior solutions, viscous and Knudsen boundary

layers are established. With the help of L2−L∞framework and the uniform estimates obtained

above, the solutions to the Boltzmann equation are constructed by the truncated Hilbert expan-

sion with multi-scales, and hence the hydrodynamic limit in the incompressible Euler level is

justiﬁed.

Contents

1. Introduction and main results 2

1.1. Introduction 2

1.2. Asymptotic expansion 3

1.3. Hilbert expansion 9

2. Reformulation of Hilbert expansions and boundary conditions 13

2.1. Interior expansion 13

2.2. Viscous boundary layer 15

2.3. Knudsen boundary layer 18

3. Boundary conditions and criteria of uniqueness for pressure terms 20

3.1. Boundary conditions 20

3.2. Criteria of uniqueness for pressure terms 23

4. Existence of solutions for linear systems 27

4.1. Existence of solutions for a kind of linear Euler equations 27

4.2. Existence of solutions for a kind of linear parabolic system 30

5. The solutions of expansions 31

6. L2−L∞method: proof of Theorem 1.1 36

6.1. L2-energy estimate for remainder 36

6.2. Weighted L∞-estimate for remainder 40

6.3. Proof of Theorem 1.1 46

Appendix A. Proof of Lemma 2.2 46

Appendix B. Proof of Proposition 2.3 47

References 51

Date: September 11, 2023.

2010 Mathematics Subject Classiﬁcation. 35Q20, 76A02, 35Q31.

Key words and phrases. Boltzmann equation, incompressible Euler equations, hydrodynamic limit, Hilbert ex-

pansion, specular reﬂection boundary condition, viscous boundary layer, Knudsen boundary layer.

arXiv:2210.04616v2 [math.AP] 8 Sep 2023

2 F. M. HUANG, W. Q. WANG, Y. WANG, AND F. XIAO

1. Introduction and main results

1.1. Introduction. In this paper, we consider the scaled Boltzmann equation

St∂tF+v· ∇xF=1

Q(F,F),(1.1)

where F(t, x, v)≥0 is the density distribution function for the gas particles with position x=

(x1, x2, x3)∈T2×(0,1), where Tis the periodic interval [0,1], and velocity v= (v1, v2, v3)∈R3

at time t > 0, and Kn>0 is Knudsen number, which is proportional to the mean free path, St

is Strouhal number. The Boltzmann collision term Q(F1, F2) is deﬁned in the following bilinear

form

Q(F1, F2)≡ZR3ZS2

B(v−u, θ)F1(u′)F2(v′)dωdu −ZR3ZS2

B(v−u, θ)F1(u)F2(v)dωdu (1.2)

where the relationship between the post-collision velocity (v′, u′) of two particles with the pre-

collision velocity (v, u) is given by

u′=u+ [(v−u)·ω]ω, v′=v−[(v−u)·ω]ω,

for ω∈S2, which can be determined by the conservation laws of momentum and energy

u′+v′=u+v, |u′|2+|v′|2=|u|2+|v|2.

The Boltzmann collision kernel B=B(v−u, θ) in (1.2) depends only on |v−u|and θwith

cos θ= (v−u)·ω/|v−u|. Throughout the present paper, we consider the hard sphere model, i.e.,

B(v−u, θ) = |(v−u)·ω|.

From the von Karman relation, we know that the Reynolds number Reis given by

=Kn

,(1.3)

where Mais the Mach number. It was shown in Maxwell [36] and Boltzmann [5] that the Boltz-

mann equation is closely related to the ﬂuid dynamical systems for both compressible and in-

compressible ﬂows. In fact, what is called hydrodynamic limit is to derive ﬂuid systems from

Boltzmann equation by taking appropriate scaling limits of Maand Kn. For instance, formally,

by setting Ma=St=Kn→0+, one can obtain the incompressible Navier-Stokes equations with

Re= 1. One can derive the compressible Euler equations by setting Ma=St= 1 and Kn→0+.

If taking Ma=St→0+,Kn→0+ and Re→+∞, then we shall derive the incompressible Euler

equations.

In the study of hydrodynamic limit, two kinds of classical expansions are usually used. The one

is Hilbert expansion, which is proposed by Hilbert in 1912. The another is Enskog-Chapman ex-

pansion, which is independently proposed by Enskog and Chapman in 1916 and 1917, respectively.

Either Hilbert or the Chapman-Enskog expansion gives formal derivations of the compressible and

incompressible ﬂuid equations. How to justify rigorously these asymptotic expansions is a chal-

lenging problem, and is closely related to Hilbert’s sixth problem [22].

For the case of compressible Euler limit, that is, Ma=St∼O(1) and Kn→0+. When

the solution of compressible Euler equations is smooth, Caﬂisch [7] rigorously established the

compressible Euler limit from the Boltzmann equation through the truncated Hilbert expansion,

see also [30, 37, 43] and [19, 20] via a L2−L∞framework. Recently, when the boundary eﬀects are

considered, by analyzing systematically the eﬀects of viscous boundary layer and Knudsen layer,

Guo, Huang and Wang [18] justiﬁed rigorously the validity of Hilbert expansion of the Boltzmann

with specular boundary conditions in half space. On the other hand, it is well known that there

are three basic wave patterns for compressible Euler equations, that is, shock wave, rarefaction

wave and contact discontinuity, the hydrodynamic limit of Boltzmann to such wave patterns had

INCOMPRESSIBLE EULER LIMIT OF BOLTZMANN EQUATION 3

been established [23, 25, 26, 27, 47, 48] in one spatial dimensional case. We also refer to [44] for

the case of planar rarefaction wave in three spatial dimensional case.

For the case of incompressible Euler limit, that is, Ma=St→0,Kn→0+ and Re→

+∞. Under some assumptions, Bouchut, Golse and Pulvirenti [6] established the limit from

the renormalized solutions of Boltzmann equation [9] to the dissipative weak solution [31] of

incompressible Euler equations without heat transfer. Then, one of the assumptions in [6] has been

removed by Lions and Masmoudi [32, 33]. Later, all assumptions are removed by Saint-Raymond

[39, 40] through relative entropy method. Moreover, the heat transfer and specular reﬂection

boundary conditions were also studied in [40], see also [3] for the case of Maxwell reﬂection

boundary conditions. Recently, the case of complete diﬀusive boundary condition is established

by Cao, Jang and Kim [8] for analytic data when Knsatisﬁes a special relation with Ma, see also

[28]. On the other hand, when the solution of incompressible Euler equations is smooth, Masi,

Esposito and Lebowitz [34] justiﬁed the incompressible Euler limit of the Boltzmann equation on

tours by truncated Hilbert expansion, see also [46] via a L2−L∞framework.

For the hydrodynamic limit from Boltzmann equation to Navier-Stokes(-Fourier) equations,

there are also many works, such as [1, 2, 4, 13, 15, 16, 21] without physical boundary, and

[10, 11, 12, 28, 29, 35, 45] with physical boundary. We shall not go into details about the Navier-

Stokes(-Fourier) limits since we will focus on the incompressible Euler limit in present paper.

The purpose of present paper is to establish the hydrodynamic limit from the Boltzmann equa-

tion with specular reﬂection boundary condition to the incompressible Euler equations through

Hilbert expansion. Hereafter, let Ω := T2×(0,1) and we denote by ⃗n0= (0,0,−1) and ⃗n1=

(0,0,1) the outward normal vector of T2×[0,1] on x3= 0 and x3= 1 respectively. The phase

boundary of Ω ×R3is denoted as γ:= ∂Ω×R3, which can be split into outgoing boundary γi

incoming boundary γi

−, and grazing boundary γi

γi

+={(x, v) : x3=i, v ·⃗ni= (−1)i+1v3>0},

γi

−={(x, v) : x3=i, v ·⃗ni= (−1)i+1v3<0},

γi

0={(x, v) : x3=i, v ·⃗ni= (−1)i+1v3= 0},

with i= 0,1.

In this paper, we consider the Boltzmann equation with the specular reﬂection boundary condition:

Fε(t, x, v)|γi

−=Fε(t, xq, i, Rxv),with i= 0,1,(1.4)

where we have denoted Rxv= (v1, v2,−v3).

1.2. Asymptotic expansion. Throughout the present paper, we denote by µ(v) the normalized

global Maxwellian, i.e.,

µ(v) = 1

(2π)3

exp −|v|2

2.

It is clear that the global Maxwellian µ(v) satisﬁes the specular reﬂection boundary conditions

(1.4). Noting (1.3), we consider the case Re→+∞(i.e., the viscosity goes to zero) as Kn→0+.

On the other hand, in order to use Hilbert expansion, we need the Knudsen number Knto be an

integer power of the thickness of viscous boundary layer 1

√Re. Hence we assume that

St=Ma=εn−2,Kn=εn, n ≥3.

Then the Boltzmann equation (1.1) is rewritten as

εn−2∂tFε+v· ∇xFε=1

εnQ(Fε,Fε).(1.5)

4 F. M. HUANG, W. Q. WANG, Y. WANG, AND F. XIAO

1.2.1. Interior expansion: We deﬁne the interior expansion as

Fε(t, x, v)∼µ+Ma

∞

i=0

εiFi(t, x, v) = µ+εn−2∞

i=0

εiFi(t, x, v).(1.6)

Substituting (1.6) into (1.5), we get

∞

i=0

εn−2+i∂tFi+∞

i=0

εiv· ∇xFi

=∞

i=0

ε−n+i[Q(µ, Fi) + Q(Fi, µ)] + ∞

i,j=0

εi+j−2Q(Fi, Fj).(1.7)

Comparing the order of εin (1.7), one obtains

ε−n+i(0 ≤i≤n−3) : 0 = Q(µ, Fi) + Q(Fi, µ),

ε−2: 0 = Q(µ, Fn−2) + Q(Fn−2, µ) + Q(F0, F0),

ε−1: 0 = [Q(µ, Fn−1) + Q(Fn−1, µ)] + [Q(F0, F1) + Q(F1, F0)],

ε0:v· ∇xF0= [Q(µ, Fn) + Q(Fn, µ)] + [Q(F0, F2) + Q(F2, F0)] + Q(F1, F1),

εk(k≥1) : ∂tFk+2−n+v· ∇xFk= [Q(µ, Fk+n) + Q(Fk+n, µ)]

i+j=k+2

i,j≥0

2[Q(Fi, Fj) + Q(Fj, Fi)],

(1.8)

whereafter we use the notations Fi≡0 with 2 −n≤i≤ −1 for simplicity of presentation.

For later use, we deﬁne the linearized collision operator Lby

Lg:= −1

√µ[Q(µ, √µg) + Q(√µg, µ)],

and the nonlinear operator

Γ(g1, g2) := 1

√µQ(√µg1,√µg2).

The null space Nof Lis generated by the following standard orthogonal bases

χ0(v)≡√µ, χi(v)≡vi√µ i = 1,2,3, χ4(v)≡1

√6(|v|2−3)√µ.

We also deﬁne the collision frequency ν:

ν(v) := ZR3ZS2

B(v−u, θ)µ(u)dωdu ∼

=1 + |v|.

Let Pgbe the L2

vprojection with respect to [χ0, ..., χ4]. It is well-known that there exists a positive

number c0>0 such that for any function g

⟨Lg, g⟩ ≥ c0∥{I−P}g∥2

ν,(1.9)

where the weighted L2-norm ∥·∥νis deﬁned as

∥g∥2

ν:= ZΩ×R3

g2(x, v)ν(v)dxdv.

INCOMPRESSIBLE EULER LIMIT OF BOLTZMANN EQUATION 5

For each i≥0, let fi:= Fi

√µ. We deﬁne the macroscopic and microscopic part of fias

fi=Pfi+ (I−P)fi={ρi+ui·v+1

2θi(|v|2−3)}√µ+ (I−P)fi,

where Pfi:= {ρi+ui·v+1

2θi|v|2−3}. It follows from (1.8) that

fi≡Pfi={ρi+v·ui+1

2θi(|v|2−3)}√µfor 0 ≤i≤n−3,

(I−P)fn−2=L−1(Γ(f0, f0)) ,

(I−P)fk+n−1=L−1n(I−P)[−∂tfk+1−n−v· ∇xfk−1]

i+j=k+1

i,j≥0

2[Γ(fi, fj) + Γ(fj, fi)]ok≥0.

(1.10)

We deﬁne the Burnett functions Aij and Bias

Aij := {vivj−δij |v|2

3}√µ, Bi:= vi

2(|v|2−5)√µ, (1.11)

where δij is the Kronecker symbol. It follows from a direct calculation that











ZR3

Fkdv =ρk,ZR3

viFkdv =uk,i,ZR3|v|2Fkdv = 3 (ρk+θk),

ZR3

vivjFkdv =δij (ρk+θk) + ⟨Aij ,(I−P)fk⟩,

ZR3

vi|v|2Fkdv = 5uk,i + 2 ⟨Bi,(I−P)fk⟩.

(1.12)

Multiplying (1.8)4by 1, v respectively, and integrating the resultant equation over R3with

respect to v, we can get

divxu0= 0,∇x(ρ0+θ0)=0.(1.13)

Thus, we may assume

ρ0+θ0=p0(t),(1.14)

where p0(t) is a function independent of x. In fact, we can prove p0(t) is a constant and p0(t)≡

(ρ0+θ0)(0) in Lemma 3.5 below due to the compatibility condition for un−2.

Moreover, taking k=n−2 in (1.8)5, and multiplying it by v, |v|2respectively, then integrating

the resultant equation over R3with respect to v, one obtains that











∂tu0+ (u0· ∇x)u0+∇x(ρn−2+θn−2−1

3|u0|2)=0,

∂tθ0+ (u0· ∇x)θ0=2

5∂tp0(t)≡0,

(1.15)

which, together with (1.13) and (1.14), yields that (ρ0, u0, θ0) satisﬁes the following incompressible

Euler system











divu0= 0, ρ0+θ0=p0(t)≡ρ0(0) + θ0(0),

∂tu0+ (u0· ∇x)u0+∇xpn−2= 0,

∂tθ0+ (u0· ∇x)θ0=2

5∂tp0(t)≡0,

(1.16)

where we have used the notation pn−2:= ρn−2+θn−2−1

3|u0|2and the fact p0(t) is a constant.

The detailed calculations of (1.16) are given in Appendix A.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

HILBERTEXPANSIONOFTHEBOLTZMANNEQUATIONINTHEINCOMPRESSIBLEEULERLEVELINACHANNELFEIMINHUANG,WEIQIANGWANG,YONGWANG,ANDFENGXIAOAbstract.ThestudyofhydrodynamiclimitoftheBoltzmannequationwithphysicalboundaryisachallengingproblemduetoappearanceoftheviscousandKnudsenboundarylayers.Inthispaper,thehydrodynamic...

展开>> 收起<<

HILBERT EXPANSION OF THE BOLTZMANN EQUATION IN THE INCOMPRESSIBLE EULER LEVEL IN A CHANNEL FEIMIN HUANG WEIQIANG WANG YONG WANG AND FENG XIAO.pdf

共53页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

HILBERT EXPANSION OF THE BOLTZMANN EQUATION IN THE INCOMPRESSIBLE EULER LEVEL IN A CHANNEL FEIMIN HUANG WEIQIANG WANG YONG WANG AND FENG XIAO

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: