of the Boltzmann Equation, I. The case M

Physics

Nonlinear Stability of Boundary Layers

of the Boltzmann Equation, I. The case M

< − ¹

Seiji Ukai¹, Tong Yang², Shih-Hsien Yu²

1 Department of Applied Mathematics, Yokohama National University, Yokohama, Japan 2 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, P.R. China Received: 4 March 2003 / Accepted: 3 July 2003

Published online: 11 November 2003 – © Springer-Verlag 2003

Abstract: This is a continuation of the paper [15] on nonlinear boundary layers of the Boltzmann equation where the existence is established and shown to be strongly depen- dent on the Mach numberM^∞of the Maxwellian state at far field. In this paper, when M^∞ < −1, we will show that the linearized operator has the exponential decay in time property and therefore a bootstrapping argument yields nonlinear stability of the boundary layers.

1. Introduction and Main Result

The nonlinear Miln´e problem can be stated as follows. Consider the 3-dimensional half- space D = {(x, y, z) ∈ R³|x > 0}, in which the mass densityF of gas particles is assumed constant on each plane parallel to the boundary∂D = {x =0}although the particle motion is 3-dimensional. That is,F is assumed to be a function of positionx (but not ofy, z) and particle velocityξ = (ξ1, ξ2, ξ3) ∈ R³. Here,ξ1 stands for the velocity component along thex-axis. Then,F is governed by the stationary Boltzmann equation





ξ₁F_x= Q(F, F ), x >0, ξ∈R³, F|x=0= F_b(ξ ), ξ₁>0, (ξ2, ξ₃)∈R²,

F →M_∞(ξ ) (x→ ∞), ξ ∈R³,

(1.1)

where

M_∞(ξ )=M[ρ_∞, u_∞, T_∞](ξ )= ρ_∞ (4π T_∞)^3/2exp

−|ξ−u_∞|² 2T_∞

, (1.2)

is a Maxwellian with constantsρ_∞>0, u_∞=(u_∞,1, u_∞,2, u_∞,3)∈R³, andT_∞>0 which are the macroscopic components in the particle distributionF. By a shift of the

variableξin the direction orthogonal to thex-axis, we can assume without loss of gen- erality thatu_∞,2 = u_∞,3 = 0, and then, the sound speed and Mach number of this equilibrium state are given by

c_∞= 5

3T_∞, M^∞=u_∞,1

c_∞ , (1.3)

respectively, see [4]. Here,Q, the collision operator, is a bilinear integral operator Q(F, G)= 1

2

R³×S²

F (ξ)G(ξ_∗)+F (ξ_∗)G(ξ)−F (ξ )G(ξ_∗)−F (ξ_∗)G(ξ )

×q(ξ −ξ_∗, ω) dξ_∗dω, (1.4)

with

ξ=ξ−[(ξ−ξ_∗)·ω]ω, ξ_∗ =ξ_∗+[(ξ−ξ_∗)·ω]ω, (1.5) where “·” is the inner product ofR³. We restrict ourselves to the hard sphere gas for which the collision kernelqis given by

q(ζ, ω)=σ0|ζ ·ω|, whereσ0is the surface area of the hard sphere.

The existence of stationary solutions, called boundary layer solutions, to the problem (1.1) is studied recently in [15]. The result there shows that the existence of boundary layer solutions depends on the Mach numberM^∞atx = ∞. WhenM^∞ =0,±1, a solvability condition is given implicitly so that the co-dimensions of the manifold for boundary dataF_b(ξ )is obtained. In the simplest case, i.e.,M^∞<−1, there is no extra solvability condition because all the information at infinity goes into the layer, which means that as long as the boundary dataF_bis close to the Maxwellian atx = ∞under some suitable norm, the boundary layer solution always exists. As the first step, to study the stability of the boundary layer solutions obtained in [15], we will study the case when M^∞<−1. The main reason why this case is easiest is that the linearized problem has exponential decay phenomena. And this decay estimate is easier to be handled in the bootstrapping argument for nonlinear stability. For the other case, the decay rate should be algebraic as for the Cauchy problem so that it is more difficult and will be pursued by authors in the future.

For the boundary layer problem, there are a lot of results on the linear existence, stability and the numerical computation, cf. [1, 2, 5–8, 12–14]. Since we will discuss the stability problem in this paper, we will not present their works in details.

The main result in this paper can be stated as follows. LetF¯ = ¯F (x, ξ )be the stationary solution to the problem (1.1). Consider the initial boundary value problem,









F_t+ξ₁F_x = Q(F, F ), t >0, x >0, ξ∈R³, F|t=0= F₀(x, ξ ), x >0, ξ∈R³,

F|x=0= F_b(ξ ), t >0, ξ1>0, (ξ2, ξ₃)∈R², F →M_∞(ξ ) (x→ ∞), t >0, ξ∈R³.

(1.6)

Theorem 1.1. WhenM^∞<−1, under the assumption that

|Fb(ξ )−M_∞(ξ )| ≤0Wβ(ξ ), ξ ∈R₊³, β >5/2,

with the weight functionWβ(ξ )defined in (2.1) and0being a sufficiently small positive constant, there exists a boundary layer solution F (x, ξ )¯ to (1.1) proved in [15]. For (1.6), when [[F0(x, ξ )− ¯F (x, ξ )]]< 1withβ > 5/2, where1>0 is a sufficiently small constant and the norm [[·]] is defined in (2.28), there exists a unique solution F (t, x, ξ ) to the problem (1.6) which decays exponentially in time to the stationary solutionF (x, ξ ). In other words, the boundary layer solution in this case is nonlinearly¯ stable.

Remark 1.2. We prove the global existence in the setting of the contraction mapping principle associated to the reduced problem (2.7) related to the quantityF − ¯F, in the space endowed with the norm (2.30). Hence, the asymptotic stability is a straightforward consequence of it. As for the existence, the method in [11] may work for (1.1).

The proof of our theorem is given in the following section. We will first consider two semigroups associated with two linearized problems of (1.6) and show that they both have exponential decay property. Then by applying the bootstrapping argument and the smallness of the strength of the boundary layer, we will have the nonlinear stability result stated in Theorem 1.1. In the following,cis used to denote a generic positive constant.

2. Stability Analysis

The stability problem to (1.6) can be discussed in two steps. The first step is to consider the corresponding linearized problem by the energy method forL²_x,ξand then the boot- strapping argument forL^∞_x,ξ. The exponential decay in time estimate obtained in the first step can be used in the second step for nonlinear stability by using Grad’s estimate on the nonlinear Boltzmann collision term to obtain an a priori estimate on the solution for the application of the fixed point theorem.

In the following, we will use the following weighted function:

W_β(ξ )=(1+ |ξ|)^−β

M[1, u_∞, T_∞](ξ )

1/2

, (2.1)

withβ∈R.

First, we shall look for the solution of (1.6) in the form

F (t, x, ξ )=M_∞(ξ )+W0(ξ )f (t, x, ξ ), (2.2) whereW0is the weight of (2.1) withβ =0. Then, the problem (1.6) reduces to









ft +ξ1fx−Lf = (f ), t >0, x >0, ξ∈R³, f|t=0 = f0(x, ξ ), x >0, ξ ∈R³,

f|x=0 = a0(ξ ), t >0, ξ1>0, (ξ2, ξ3)∈R², f →0(x→ ∞), t >0, ξ∈R³,

(2.3)

where

a₀=W₀⁻¹

F_b−M_∞ ,

and

Lf =W₀⁻¹

Q(M_∞, W₀f )+Q(W₀f, M_∞)

, (f )=(f, f ), (2.4) with

(f, g)=W₀⁻¹Q(W₀f, W₀g).

The operatorL is linear while the remainder is quadratic, both acting only on the variableξ. The following properties (and nothing else) from them will be used in the sequel. SetL^p_ξ =L^p(R³_ξ)andL^∞_ξ,β=L^∞(R³_ξ, W_β(ξ )dξ ).

Proposition 2.1. For the hard sphere model, the following holds with some positive constantsν₀,ν₁,k₀,k₁,k₂depending only onρ_∞, u_∞, T_∞.

(i)Lhas the decomposition

L= −ν(ξ )× +K,

whereν(ξ )is a positive function satisfying

ν₀≤ν(ξ )≤ν₀⁻¹(1+ |ξ|), ξ ∈R³, whereasKis an integral operator

Kh=

R³K(ξ, ξ)h(ξ)dξ with the kernel enjoying the estimate

|K(ξ, ξ)| ≤k₀(|ξ−ξ| + |ξ−ξ|⁻¹)e^{−k1|ξ−ξ|2}.

(ii)Lis non-positive self-adjoint onL²_ξ, with the estimate (Lh, h)_L2

ξ ≤ −ν₁||(1+ |ξ|)^1/2P^⊥h||²_L2 ξ

, (2.5)

whereP^⊥=I−P,P being the orthogonal projection onto the null spaceNofL.

(iii)Khas the regularizing property that it is bounded as an operator K:L^∞_ξ,β →L^∞_ξ,β+1 and K:L²_ξ →L^∞_ξ for allβ ≥0.

(iv) The bilinear operator(f, g)enjoys the estimate

||ν⁻¹(f, g)||L^∞_ξ,β≤k₃||f||L^∞_ξ,β||g||L^∞_ξ,β

for allβ.

Proof. Forρ_∞ =1,u_∞= 0, andT_∞ =1, that is, for the case of the standard Max- wellianM⁰(ξ ) =M[1,0,1](ξ ), all the statements in the above are found in, e.g. [4], pp. 197-198, except for (2.5) which is stated in [6]. Letν⁰(ξ )andK⁰(ξ, ξ)be ones corresponding to the standard MaxwellianM⁰. Their explicit formulas go back to [10, 3] (see also [4], pp. 196–197). Since

M[ρ_∞, u_∞, T_∞](ξ )=αM⁰(γ (ξ−u_∞)), forα=ρ_∞/T_∞^3/2andγ =1/T_∞^1/2, it follows from (2.4) that

ν(ξ )=c0ν⁰(γ (ξ−u_∞)), K(ξ, ξ)=c0K⁰(γ (ξ−u_∞)), γ (ξ−u_∞)), withc₀=α/γ =ρ_∞/T_∞, whence the proposition follows for the general Maxwellian.

This proposition is also valid for Grad’s cut-off hard potential [9] with due modifi- cation, particularly with(|ξ| +1)^δ (δ∈ [0,1]) in place of(|ξ| +1)in (2.5). Since the model we consider is the hard sphere (δ=1), we can letf =e^{−σ x}gin (2.3) and control by (2.5) (and byP) the termσ ξ1appearing in the deduced problem









g_t+ξ₁g_x−σ ξ₁g−Lg = e^{−σ x}(g), t >0, x >0, ξ ∈R³, g|t=0 = g₀(x, ξ ), x >0, ξ∈R³,

g|x=0 = a₀(ξ ), t >0, ξ1>0, (ξ2, ξ₃)∈R², g→0(x→ ∞), t >0, ξ ∈R³.

(2.6)

Now, denote the stationary boundary layer solution to (2.6) byg¯and let the initialg0

be a small perturbation ofg. Then the stability problem we consider can be formulated¯ as follows:









˜

g_t+ξ1g˜_x−σ ξ1g˜−Lg˜ = e^{−σ x}{ ¯Lg˜+(g)˜ }, t >0, x >0, ξ∈R³,

˜ g

t=0 = ˜g0(x, ξ ), x >0, ξ ∈R³,

˜ g

x=0 = 0, t >0, ξ1>0, (ξ2, ξ3)∈R²,

˜

g→0(x→ ∞), t >0, ξ ∈R³,

(2.7) whereg˜=g− ¯g,g˜0=g0− ¯gandL¯g˜ =2(g,¯ g).˜

LetS(t )be the solution operator (semi-group) of the linear problem





h_t +ξ₁h_x−σ ξ₁h−Lh=0, t >0, x >0, ξ ∈R³, h

x=0=0(ξ₁>0), h→0(x→ ∞), t >0, ξ∈R³, h

t=0=h0(x, ξ ), x >0, ξ∈R³.

(2.8)

Then we haveh=S(t )h0.

For the caseM^∞<−1, theL²decay estimate for (2.8) is easy to establish. Recall that in this case, the operatorA=P ξ₁P introduced in our previous paper [15] is neg- ative definite onN, whereasLis also negative definite onN^⊥with the estimate (2.5).

Here,P andN are as in Proposition 2.1. Now for a smallσ > 0, a straightforward energy estimate gives

1 2

d

dt||h(t )||²+<|ξ₁|h⁰, h⁰>₋+ν₂||(1+ |ξ|)^1/2h(t )||²≤0,

with a constantν2>0 (sayν2=ν1/2), where||·|| = ||·||_L²

x,ξ,<·.·>₋=(·,·)_L2(ξ1>0), andh⁰=h|x=0. This implies that

d dt

e^ν2t||h(t )||² +e^ν2t

2<|ξ₁|h⁰(t ), h⁰(t ) >₋+ν₂||(1+ |ξ|)^1/2h(t )||²

≤0.

Then it follows that e^ν2t||h(t )||²+

_t

0

e^ν2t

2<|ξ1|h⁰(t ), h⁰(t ) >₋+ν2||(1+ |ξ|)^1/2h(t )||²

dt≤ ||h0||², (2.9) and

||S(t )h₀|| ≤e^−κt||h₀||, κ = ν2

2. (2.10)

As for the existence analysis, we want to prove the following estimate which is suf- ficient for the application of the fixed point theorem to get the global existence of the solution to the nonlinear problem (2.5),

||S(t )h₀||β ≤ce^−κt

||h₀||β+ ||h₀||_L²

x,ξ

, (2.11)

forβ ≥0, where|| · ||βis the norm of the spaceL^∞_x,ξ

W_β(ξ )dxdξ

=L^∞_β .

In order to prove (2.11), we first consider another simpler linear solution operator.

Letν(ξ )be as in Proposition 2.1(i) and letS₀(t )be the solution operator (semi-group)

of 





ht+ξ1hx−σ ξ1h+ν(ξ )h=0, t >0, x >0, ξ∈R³, h

x=0=0(ξ1>0), h→0(x→ ∞), t >0, ξ∈R³, ht=0=h₀(x, ξ ), x >0, ξ∈R³.

(2.12) The solution to the above linear initial boundary value problem has the following explicit expression:

h=S0(t )h0=e^{−(ν(ξ )−σ ξ1)t}χ (x−ξ1t )h0(x−ξ1t, ξ ), (2.13) whereχ (y)is the usual characteristic function fory >0. Based on this expression and with the lower boundν(ξ )≥ν₀>0, a simple calculation yields the following estimate onS₀:

||S0(t )h0||X≤ce^{−(2κ−ε)t}||h0||X, (2.14) withκ chosen to be min(ν0, ν₂)/2, for some small constant >0. Here the spaceX can be eitherL^∞_β orL²_x,ξ.

From (2.8) and (2.12), we have

















S(t )h0=S0(t )h0+_t

0S0(t−s)KS(s)h0ds

=_m−1

j=0 I_j(t )+J_m(t ) I0(t ) =S0(t )h0

Ij(t ) =t

0S0(t−s)KIj−1(s)ds=(S0K)∗Ij−1

J_m(t ) =(S₀K)∗(S₀K)∗ · · · ∗(S₀K)

m

∗h,

(2.15)

withh=S(t )h0. Here and hereafter, “∗” stands for the convolution int. By using the estimate (2.14) and the regularizing property of the compact operatorKin Proposition 2.1(iii), we have forβ≥j ≥0,

||I_j(t )||β ≤c_je^(−2κ+ε)t||h₀||β−j. (2.16) The estimate onJ_mis more complicated and can be stated in the following bootstrapping lemma.

Lemma 2.2. Forβ≥0, we have

||J_β+3(t )||β ≤ce^−κt||h₀||_L²

x,ξ.

Proof. First, again by the regularizing property ofKin Proposition 2.1(iii), we have

||J_β+3(t )||β ≤ C β!

_t

0

(t−τ )^βe−(2κ−)(t−τ )||J₂||_L∞

x (L²_ξ)(τ )dτ, (2.17) where

J2(t )=(S0K)∗(S0K)∗h=S0∗ ¯J , (2.18) with

J¯=KS0K∗h= _t

0

KS0(t−s)Kh(s)ds= _t

0

J¯0(t−s, s)ds. (2.19)

We now estimateJ¯₀(t, s)as follows. Here, we need to use some integral property of the compact operatorK. By definition, we have

J¯0(t, s)=KS0(t )Kh(s)

=

R³×R³K(ξ, ξ)K(ξ, ξ)e^{−(ν(ξ)−σ ξ1)t}χ (y)h(s, y, ξ)dξdξ, (2.20) wherey=x−ξ₁t. Hence,

| ¯J0(t, s)| ≤e^{−(ν0−ε)t}

R×R³K0(ξ, ξ₁, ξ)χ (y)|h(s, y, ξ)|dξ₁dξ, (2.21) where

K₀(ξ, ξ₁, ξ)≡

R²|K(ξ, ξ)||K(ξ, ξ)|dξ₂dξ₃, withξ=(ξ₁, ξ₂, ξ₃).

Notice that the estimate of the kernelK(ξ, ξ)stated in Proposition 2.1(i) gives

R³|K(ξ, ξ)|dξ=

R³|K(ξ, ξ )|dξ≤C₀,

R²|K(ξ, ξ)|dξ₂ dξ₃ ≤C1,

where C0 and C1 are some positive constants depending only on the parameters ρ_∞, u_∞, T_∞. Thus, we have

R×R³K₀(ξ, ξ₁, ξ)dξ₁dξ =

R³×R³ |K(ξ, ξ)| |K(ξ, ξ)|dξdξ≤C₀²,

R³K0(ξ, ξ₁, ξ)dξ ≤C0

R² |K(ξ, ξ) dξ₂ dξ₂ ≤C0C1. By (2.21) and the Schwartz inequality,

J¯0(t, s)²≤e^{−2(2κ−)t}

R²×R³K0(ξ, ξ₁, ξ) dξ₁ dξ

×

R²×R³K0(ξ, ξ₁, ξ)χ (y)h(s, y, ξ)²dξ₁ dξ

≤C₀²e^{−2(2κ−)t}

R²×R³ K0(ξ, ξ₁, ξ)χ (y)h(s, y, ξ)²dξ₁ dξ. (2.22) Therefore, we have

¯J₀(t, s)²_L∞

x (L²_ξ)=sup

x>0

R³

J¯₀(t, s)²dξ

≤C₀²C₀C₁e^{−2(2κ−)t}

R×R³

χ (y)h(s, y, ξ)²dξ₁ dξ

= c

te^{−2(2k−ε)t} _∞

0

dy

R³ dξh(s, y, ξ)²

≤ c

te^{−2(2k−ε)t}e^−2ksh0²_L2 x,ξ

. (2.23)

Here, we have used theL²decay estimate (2.10). Hence (2.19) and (2.23) give ¯J (t )_L∞

x (L²_ξ)≤ _t

0

¯J₀(t−s, s)_L∞ x (L²_ξ)ds

≤c _t

0

e^{−(2κ−)(t−s)}

√t−s e^−κsh_0L²

x,ξ ds

≤c e^−kt _t

0

e^{−(κ−)(t−s)}

√t−s dsh0 ≤ce^−κth0. (2.24) This and (2.14), (2.18) give

J₂(t )_L∞

x (L²_ξ)= S₀∗ ¯J ≤ _t

0

e^{−(2κ−ε)(t−s)} ¯J (s)_L∞ x(L²_ξ)ds

≤c _t

0

e^{−(2κ−ε)(t−s)} e^−κs dsh_0L²

x,ξ

≤ce^−κth_0L²

x,ξ. (2.25)

Plugging this into (2.17) yields

||Jβ+3(t )||β ≤e^−κt c β!

_t

0

(t−τ )^βe^{−(κ−)(t−τ )}dτ||h0||_L²

x,ξ

≤ce^−κt||h0||_L²

x,ξ. (2.26)

And this completes the proof of the lemma.

This lemma and (2.16) complete the proof of theL^∞_β decay estimate (2.11).

In order to estimate the nonlinear term(g)˜ and the coupling termL¯g˜in (2.7) by Proposition 2.1(iv), we also need the following lemma.

Lemma 2.3. Whenβ ≥0, for the two semigroupsS₀andS, we have

||S0∗ν(ξ )h||β(t )≤ce^−κt sup

0≤τ≤t{e^κτ||h||β(τ )},

||S∗ν(ξ )h||β(t )≤ce^−κ/2t{ sup

0≤τ≤t

(e^κ/2τ||h||β(τ ))+ sup

0≤τ≤t

(e^κ/2τ||νh||_L²

x,ξ(τ ))},

both for every functionh(t, x, ξ )with the relevant norm bounded.

Proof. First, by the special property of the semigroupS0and the linear growth rate of ν(ξ ), we have

||S0∗νh||β ≤sup

x,ξ

t 0

(1+ |ξ|^β)e^{−(ν(ξ )−σ ξ1)(t−s)}χ (x−ξ1s)ν(ξ )|h(s, x−ξ1s, ξ )|ds

≤e^−κt sup

0≤τ≤t

{e^κτ||h||β(τ )}sup

ξ

_t

0

e^{−(ν(ξ )−κ−σ ξ1)(t−s)}ν(ξ )ds

≤ce^−κt sup

0≤τ≤t

{e^κτ||h||β(τ )}.

To give the estimate forS, we use the relation betweenSandS₀, S=S0+S0∗KS.

First, write (2.11) as

||S(t )h0||β ≤ce^−κt[[h0]]β, (2.27) with

[[·]]β = || · ||β + || · ||_L²

x,ξ. (2.28)

We assumeβ ≥1 but the proof is similar for otherβ. By the regularizing property of the operatorKagain, we have

||S₀∗KS∗νh||β ≤ _t

0

e^−κ(t−s)||KS∗νh||β(s)ds

≤c _t

0

e^−κ(t−s)||S∗νh||β−1(s)ds

≤c _t

0

e^−κ(t−s) _s

0

e^{−κ(s−τ )}[[νh]]β−1(τ )dτ ds (by (2.27))

≤c sup

0≤τ≤t

{e^κ/2τ[[νh]]β−1(τ )} _t

0

e^−κ(t−s)e^−κ/2ssds

≤ce^−κ/2t sup

0≤τ≤t

{e^κ/2τ[[νh]]β−1(τ )}.

Combining this with the estimate forS0, we have

||S∗ν(ξ )h||β(t )≤ce^−κ/2t

sup

0≤τ≤t

e^κ/2τ||h||β(τ ) + sup

0≤τ≤t

e^κ/2τ[[νh]]β−1(τ ) . Recalling the linear growth ofν(ξ )and the definition (2.28) completes the proof of the lemma.

By using the estimates in the above lemmas and (2.27), we can now construct a global solution to the nonlinear problem (2.7). The definition of the semigroup implies that

˜

g=S(t )g˜₀+S∗ {e^{−σ x}(L¯g˜+(g))˜ }. (2.29) Write the right-hand side by[g]. We have˜

||[g]||˜ β ≤ ||S(t )g˜₀||β+ ||S∗ {νν⁻¹e^{−σ x}(L¯g˜+(g))˜ }||β

≤ce^−κ/2t

[[g˜0]]β+sup

τ≥0

e^κ/2τ||e^{−σ x}νν⁻¹(L˜g¯+(g))˜ ||β(τ )

+sup

τ≥0

e^κ/2τ||e^{−σ x}νν⁻¹(L˜g¯+(g))˜ ||_L²

x,ξ(τ )

≤ce^−κ/2t{[[g˜₀]]β + || ¯g||β||| ˜g||| + ||| ˜g|||²}, where

|||h||| =sup

t≥0{e^κ/2t||h||β(t )}. (2.30) In the above we have used the estimate in Proposition 2.1(iv) and the relation

||e^{−σ x}νh||²_L2 x,ξ ≤

_∞

0

e^{−2σ x}dx

R³ν²(ξ )(1+ |ξ|)^−2βdξ

||h||²β

=c||h||²β,

β > 5 2

. Consequently, we have

|||[g]|||˜ β ≤c([[g˜₀]]β+ || ¯g||β||| ˜g||| + ||| ˜g|||²), and similarly,

|||[g]˜ −[h]˜ |||β ≤c(|| ¯g||β||| ˜g− ˜h||| + ||| ˜g+ ˜h||||| ˜g− ˜h|||), with the same constantc.

The smallness assumption on [[g¯0]]]β and that on||| ¯g||| which follows from the smallness assumption on the boundary dataa0in (2.6) now assure that the nonlinear mapis a contraction map in a small ball of the Banach space defined with the norm (2.30) and therefore a unique fixed point exists. This implies, taking into account the choice of the norm (2.30), that (2.7) has a unique global in time solution converging exponentially to 0 ast → ∞in the norm (2.28). Thus Theorem 1.1 follows.

Acknowledgement. The research of the first author was supported by Grant-in Aid for Scientific Research (C) 136470207, Japan Society for the Promotion of Science (JSPS). The research of the second author was supported by the Competitive Earmarked Research Grant of Hong Kong CityU 1092/02P# 9040737.

The research of the third author was supported by the Competitive Earmarked Research Grant of Hong Kong # 9040645.

References

1. Aoki, K., Nishino, K., Sone, Y., Sugimoto, H.: Numerical analysis of steady flows of a gas condens- ing on or evaporating from its plane condensed phase on the basis of kinetic theory: Effect of gas motion along the condensed phase. Phys. Fluids A 3, 2260–2275 (1991)

2. Bardos, C., Caflish, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann equation of a hard sphere gas. Comm. Pure Appl. Math. 49, 323–352 (1986)

3. Carleman, T.: Sur La Th´eorie de l’´Equation Int´egrodiff´erentielle de Boltzmann. Acta Mathematica 60, 91–142 (1932)

4. Cercignani, C., Illner, R., Purvelenti, M.: The Mathematical Theory of Dilute Gases. Berlin: Springer- Verlag, 1994

5. Cercignani, C.: Half-space problem in the kinetic theory of gases. In: E. Kr¨oner, K. Kirchg¨assner, (eds.), Trends in Applications of Pure Mathematics to Mechanics, Berlin: Springer-Verlag, 1986, pp. 35–50

6. Coron, F., Golse, F., Sulem, C.: A classification of well-posed kinetic layer problems. Commun.

Pure Appl. Math. 41, 409–435 (1988)

7. Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rat. Mech. Anal. 103(1), 81–96 (1988)

8. Golse. F., Poupaud, F.: Stationary solutions of the linearized Boltzmann equation in a half-space.

Math. Methods Appl. Sci. 11, 483–502 (1989)

9. Grad, H.: Asymptotic Theory of the Boltzmann Equation. In: Rarefied Gas Dynamics, J.A. Laur- mann, (ed.), Vol 1, 26, New York: Academic Press, 1963, pp. 26–59

10. Hilbert, D.: Grundz¨uge einer Allgemeinen Theorie der Linearen Integralgleichungen. (German) New York, N.Y.: Chelsea Publishing Company, 1953, pp. xxvi+282

11. Lions, P.-L.: Conditions at infinity for Boltzmann’s equation. Commun. Partial Diff. Eqs. 19, 335–

367 (1994)

12. Sone, Y.: Kinetic Theory of Evaporation and Condensation-Linear and Nonlinear Problems. J. Phys.

Soc. Japan 45(1), (1978)

13. Sone, Y.: Kinetic Theory and Fluid Dynamics. Berlin: Birkh¨auser, 2002

14. Ukai, S.: On the half-space problem for the discrete velocity model of the Boltzmann equation. In:

Advances in Nonlinear Partial Differential Equations and Stochastic, Kawashima, T. Yangisawa, (eds.), Series on Advances in Mathematics for Applied Sciences, Vol. 48, Singapore–New York:

World Scientific, 1998, pp. 160–174

15. Ukai, S., Yang, T., Yu, S.-H.: Nonlinear Boundary Layers of the Boltzmann Equation: I, Existence.

Commun. Math. Phys. 236, 373–393 (2003) Communicated by H.-T. Yau