Cercignani-Lampis boundary in the Boltzmann theory

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the local well-posedness of the Boltzmann equation in bounded domain with the Cercignani-Lampis boundary condition, which describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We prove the local-in-time well-posedness of the equation by establishing an $L^\infty$ estimate. In particular, for the $L^\infty$ bound we develop a new decomposition on the boundary term combining with repeated interaction through the characteristic. Via this method, we construct a unique steady solution of the Boltzmann equation with constraints on the wall temperature and the accommodation coefficient.


Introduction
In this paper we consider the classical Boltzmann equation, which describes the dynamics of dilute particles. Denoting F (t, x, v) the phase-space-distribution function of particles at time t, location x ∈ Ω moving with velocity v ∈ R 3 , the equation writes: (1.1) The collision operator Q describes the binary collisions between particles: In the collision process, we assume the energy and momentum are conserved. We denote the post-velocities: then they satisfy: In equation (1.2), B is called the collision kernel which is given by To describe the boundary condition for F , we denote the collection of coordinates on phase space at the boundary: And we denote n = n(x) as the outward normal vector at x ∈ Ω. We split the boundary coordinates γ into the incoming (γ − ) and the outgoing (γ + ) set: The boundary condition determines the distribution on γ − , and shows how particles back-scattered into the domain. In our model, we use the scattering kernel R(u → v; x, t): R(u → v; x, t)F (t, x, u){n(x) · u}du, on γ − . (1.5) Physically, R(u → v; x, t) represents the probability of a molecule striking in the boundary at x ∈ ∂Ω with velocity u, and to be sent back to the domain with velocity v at the same location x and time t. There are many models for it. In [3,4] Cercignani and Lampis proposed a generalized scattering kernel that encompasses pure diffusion and pure reflection molecules via two accommodation coefficients r ⊥ and r . Their model writes: (1. 6) where T w (x) is the wall temperature for x ∈ ∂Ω and I 0 (y) := π −1ˆπ 0 e y cos φ dφ .
In the formula, v ⊥ and v denote the normal and tangential components of the velocity respectively: (1.7) Similarly u ⊥ = u · n(x) and u = u − u ⊥ n(x).
There are a few properties the Cercignani-Lampis(C-L) model satisfies, including: • the reciprocity property: |n(x) · v| |n(x) · u| , (1.8) • the normalization property(see the proof in appendix) The normalization (1.9) property immediately leads to null-flux condition for F : This condition guarantees the conservation of total mass: (1.11) Remark 1. The C-L model is an extension of the following classical diffuse boundary condition. The distribution function and scattering kernel are given by: 2Tw (x) |n(x) · v|.
Other basic boundary conditions can be considered as a special case with singular R: specular reflection boundary condition: where r ⊥ = 0, r = 0.
Here we mention the Maxwell boundary condition, which is another classical model describes the intermediate reflection law. The scattering kernel is given by the convex combination of the diffuse and specular scattering kernel: R(u → v) = c 2 π(2T w (x)) 2 e − |v| 2 2Tw (x) |n(x) · v| + (1 − c)δ(u − R x v), 0 ≤ c ≤ 1.
Compared with the C-L boundary condition, the Maxwell boundary condition does not cover the combination with the bounce back boundary condition. Such combination is covered in the C-L boundary condition with r > 1. Moreover, the C-L boundary condition represents a smooth transition from the diffuse to the specular. The Maxwell boundary condition represents the convex combination of the Maxwellian and the dirac δ function.
Here we show the graphs for both boundary condition in the two dimension for comparison. We assume the particles are moving towards the boundary with velocity u = (u , u ⊥ ) = (2, −2), thus the boundary condition is given by Then the distribution function F (t, x, v)| γ− for both boundary condition can be viewed as the following graphs:  Moreover, we show the graphs for the distribution function F | γ− with C-L boundary condition with smaller accommodation coefficients.    Figure 1 represents the phenomena that half particles are specular reflected and half particles are diffusive. When we take smaller accommodation coefficient, Figure 3 and Figure 4 demonstrate that the distribution function F (t, x, v)| γ− gradually concentrate on (2,2). Moreover, the z-coordinate shows that the C-L scattering kernel indeed tends to a dirac δ function as the accommodation coefficients become smaller.
Due to the generality of the C-L model, it has been vastly used in many applications. There are other derivations of C-L model besides the original one, and we refer interested readers to [5,3,2]. Also there have been many application of this model in recent years, on the rarefied gas flow in [16,17,22,23,24]; extension to the gas surface interaction model in fluid dynamics [19,18,27]; on the linearized Boltzmann equation in [10,26,20,9]; on S-model kinetic equation in [25] etc.
1.1. Main result. We assume that the domain is C 2 . Denote the maximum wall temperature: (1.13) Define the global Maxwellian using the maximum wall temperature: 2T M , (1.14) and weight F with it: F = √ µf , then f satisfies where the collision operator becomes: (1. 16) By the reciprocity property (1.8), the boundary condition for f becomes, for ( Thus Here we denote dσ(u, v) := R(−v → −u; x, t)du, (1.18) the probability measure in the space {(x, u), n(x) · u > 0} (well-defined due to (1.9)). Denote where the T M is defined in (1.13).
If F 0 = √ µf 0 ≥ 0 and f 0 satisfies the following estimate: Moreover, the solution F = √ µf satisfies Remark 2. In Theorem 1 the accommodation coefficient can be any number that does not correspond to the dirac δ case. Also we cover all the range for K in the collision kernel B in (1.2). We derive (1.24) and existence using the sequential argument. Assumption (1.23) is used to obtain the estimate (1.24) for the sequence solution, which is the key factor to prove the theorem.
Remark 3. There has been a lot of studies for Boltzmann equation in many aspects, the global solution [12, 11,1]; regularity estimate [14,13]; the steady solution [7,8,6]. So far we are only able to prove the local well-posedness with the C-L boundary condition. There are several obstacles to construct the global solution with the C-L boundary condition for arbitrary accommodation coefficient.
To obtain the global solution of the Boltzmann equation [12] developed the L 2 − L ∞ bootstrap and derive the time decay and continuous solution of the linearized Boltzmann equation with various boundary condition. In particular, for the diffuse boundary condition with constant wall temperature, [12] used the L 2 estimate on the boundaryˆn Here c µ is the normalization constant such that c µ √ µ|n · u|du is a probability measure. To be more specific, the diffuse boundary condition can be regarded as a projection P γ f = f | γ− . Then However, for the C-L boundary condition, such L 2 inequality does not work. We can not regard the boundary condition (1.17) as a projection because of the new probability measure dσ(u, v) in (1.18).
Another method to obtain the global solution is to use the entropy inequality. [11] used the entropy inequality and the L 1 − L ∞ bootstrap to derive the bounded solution of the linearized Boltzmann equation with periodic boundary condition. To adapt the entropy method in bounded domain, [21] used the Jensen inequality for the Darrozès-Guiraud information with Maxwell boundary condition. To be more specific, we define E as the Darrozès-Guiraud information: Since c µ µ(u)|n(x) · u|du is a probability measure then E ≥ 0 by the Jensen inequality and thus the entropy inequality follows. For the C-L boundary condition, such inequality does not work since the probability measure is given by dσ(u, v) (1.18), which is different from c µ µ(u)|n(x) · u|du. f Even though the global solution is not available for arbitrary accommodation coefficient, we are able to construct the steady and global solution when the coefficients are closed to 1. This means the we require the boundary condition to be closed to the diffuse boundary condition. We will discuss the steady solution in the following section.
1.2. Beside the local-in-time well-posedness, we can establish the stationary solution under some constraints. The steady problem is given as with F satisfying the C-L boundary condition. We use the short notation µ 0 to denote the global Maxwellian with temperature T 0 , Denote L as the standard linearized Boltzmann operator with the collision frequency ν(v) where c µ is the normalization constant.
then there exists a non-negative solution Corollary 3. For 0 < ζ < 1 4+2δ0 , set β = 0, and for ζ = 0, set β > 4 where δ 0 > 0 is in Corollary 2. There exists λ > 0 and ε 0 > 0, depending on δ 0 , such that if˜Ω then there exists a unique non-negative solution Remark 4. Different to the accommodation coefficient with almost no constraint in Theorem 1, in Corollary 2, Corollary 3 we need to restrict these two coefficients to be close to 1 in (1.29). To be more specific, we require the C-L boundary to be close to the diffuse boundary condition. In this paper we show the proof for the hard sphere case where 0 ≤ K ≤ 1. We can establish the same result for the soft potential case( −3 < K < 0 ) using the argument provided in [6].
1.3. Difficulty and proof strategy. For proving the local well-posedness we focus on establishing L ∞ estimate. In particular, for the L ∞ estimate we trace back along the characteristic until it hits the boundary or the initial datum. Thus we derive a new trajectory formula with C-L boundary condition in (1.17). Before tracing back to t = 0 there will be repeated interaction with the boundary, which creates a multiple integral due to the boundary condition (1.5). We present the formula in Lemma 1.
To understand this multiple integral we define v k , v k−1 , · · · , v 1 in Definition 1. The v i represents the integral variable at i-th interaction with the boundary. For the diffuse reflection (1.12) with constant wall temperature, the boundary condition for f = F/ √ µ is given by (1.25). Thus at the i-th interaction the boundary condition is given by If we further trace back f (v i ) in the integrand along the trajectory until the next interaction we have

Thus the integral over
The integrand for v i is symmetric for all 1 ≤ i < k and not affected by the other variables. Moreover, c µ µ(v i )|n · v i |dv i is probability measure. Thus we can apply Fubini's theorem to compute this multiple integral.
But for the C-L boundary condition (1.5) (1.6), the integrand is a function of both v and u, as a result the probability measure is not symmetric for v i . We are not free to apply the Fubini's theorem, which brings difficulty in bounding the trajectory formula. To be more specific, we need to compute the integral with the fixed order v k , v k−1 , · · · v 1 . We start from the integral of v k . By (1.17), the integral of v k iŝ When r ⊥ , r = 0, unlike the diffuse case, we can not decompose dσ(v k , v k−1 ) in (1.18) (1.6) into a product of a function of v k and a function of v k−1 . Thus the integral ends up with a function of v k−1 , which will be included as a part of the integral over v k−1 . This justifies that the order of the integral can not be changed. Also the integral of v i is affected by the variables v i+1 , v i+2 , · · · v k . Thus we have to compute the multiple integral with fixed order from v k to v 1 . In fact, (1.31) can be computed explicitly as e c|v k−1 | 2 ( Lemma 11,Lemma 12 ) and thus the integral for the variable v k−1 has exactly the same form as (1.31). This allows us to inductively derive an upper bound for this multiple integral. We present the induction result in Lemma 2.
With an upper bound for the trajectory formula another difficulty in the L ∞ estimate is the measure 1 {t k >0} . We need to show that this measure is small when k is large so that the L ∞ estimate follows by bounding a finite fold integral.
For this purpose [12, 1] decompose γ + into the subspace For diffuse case (1.12) the boundary condition for f is given by (1.25). We can derive that there can be only finite number of v j belong to γ δ + under the constraint that t < ∞. Meanwhile, by (1.25) the integral over γ + \γ δ + is a small magnitude number O(δ). When k( times of interaction with boundary ) is large enough one can obtain a large power of O(δ). The smallness of the measure 1 {t k >0} follows by this large power. However, for our C-L boundary condition, the integrand is given by (1.17) (1.6), which contains the term e −|v −(1−r )u | 2 in (1.18). If we apply the standard decomposition the integral over γ + \γ δ + is no longer a small number O(δ). This is because even |v | 1, |v − (1 − r )u | still depends on u . A key observation is that when |v | is large enough, if |v − (1 − r )u | < δ −1 , we can obtain |u | ≥ |v | + δ using 1 − r < 1. We take 1 − r = 1/2 as example. If |v − 1 2 u | < δ −1 , we take |v | ≥ 3δ −1 . Then we have For 1 − r = 1/2, we can choose a different number that depends on 1 − r to keep this property. Now we suppose the "bad" case |v −(1−r )u | < δ −1 happens for a large amount of times. By the discussion above, for the multiple integral with order v k , · · · , v 1 we get an extremely huge velocity |v i | with some i < k. When we compute the integral with dσ(v i , v i−1 ), once |v i−1 | is small the result is extremely small. This will provide the key factor to cancel all the other growth terms and prove the smallness of the measure 1 {t k >0} . The other one is the "good" case |v − (1 − r )u | > δ −1 . From (1.6) we can conclude the integral under this condition is a small magnitude number O(δ). Thus we can obtain some small factors to prove the smallness in both cases. Since the integrand in dσ(u, v) in (1.18) (1.6) still contains the variable u ⊥ , v ⊥ , we also need to apply the decomposition for these variables. The decomposition is similar and we skip the discussion here. But we point out that since the integrand for u ⊥ involves the first type Bessel function I 0 , we need some basic estimate to verify that the integral for u ⊥ has the same property as v , u . We put these estimates in the appendix.
Thus our new ingredient here is that we decompose the boundary term γ + into the subspace Here η is small number depends on the coefficient r to ensure |u | ≥ |v | + δ −1 when |v − (1 − r )u | < δ −1 . During computing the trajectory formula the integral involves the variable T w (x)( the wall temperature on x ∈ ∂Ω in (1.6) ). It affects the real value of the coefficient for u ( different to 1 − r ). This is the reason that we need to impose some constraint on the wall temperature, which is the condition (1.22) in Theorem 1. We present the decomposition and detail in Lemma 3 and its proof. The way to construct the stationary solution and the dynamical stability( Corollary 2 and Corollary 3 ) comes from the ideas in [7,8]. They consider the diffuse boundary condition with a small fluctuation on the wall temperature. Thus it can be regarded as a perturbation around the diffuse boundary condition with constant temperature. For our C-L boundary condition, when r ⊥ and r are close to 1, it can be also regarded as a perturbation. Thus we need to restrict the accommodation coefficient to have a small fluctuation around 1. Then we need to verify the boundary condition satisfies the property as stated in Proposition 4.1 in [7]( the condition (3.2) in this paper ). Then we can follow the standard procedure provided in [7] to prove Corollary 2 and Corollary 3.

1.4.
Outline. In section 2 we conclude Theorem 1 by proving the L ∞ bound for the sequence f m as well as the existence and L ∞ stability. In section 3, we conclude Corollary 2 and Corollary 3 by using the key propositions provided in [7]. In the appendix we prove some necessary estimates.

Local well-posedness
We start with the construction of the following iteration equation, which is positive preserving as in [12,15]. Then equation is given by The equation for h m+1 reads Here (2.5) We use this section to establish the L ∞ estimate of the sequence h m+1 and derive the existence and uniqueness of the equation (1.1). The L ∞ estimate is given by the following proposition.
Here C ∞ is a constant defined in (2.134) and Remark 5. The condition (2.9) is important. The smallness of the time will be used in the proof many times. And the parameters in (2.9) guarantee that the time only depends on the temperature, accommodation and the initial condition.
The strategy to prove Proposition 4 is to express h m+1 along the characteristic using the C-L boundary condition. We present the formula in Lemma 1. We will use Lemma 2 and Lemma 3 to bound the formula.
We represent h m+1 with the stochastic cycles defined as follows.
, v be the location and velocity along the trajectory before hitting the boundary for the first time, d ds Therefore, from (2.11), we have Define the back-time cycle as Inductively, before hitting the boundary for the k-th time, define Here we set For simplicity, we denote in the following lemmas and propositions.
where H is bounded by

(2.15)
Here we use a notation Proof. For simplicity, we denoteμ we apply the fundamental theorem of calculus to get (2.20) We use an induction of k to prove (2.13). The first term of the RHS of (2.20) can be expressed by the boundary condition. For 1 ≤ k ≤ m, we rewrite the boundary condition (2.4) using (2.17) as Directly applying (2.21) with k = 1 the first term of the RHS of (2.20) is bounded by Then we apply (2.12) and (2.20) to derive Therefore, the formula (2.13) is valid for k = 2. Assume (2.13) is valid for k ≥ 2 (induction hypothesis). Now we prove that (2.13) holds for k + 1. We express the last term in (2.14) using the boundary condition. In (2.21), since 1 µ(t k ,x k ,v k−1 ) depends on v k−1 , we move this term to the integration over V k−1 in (2.13). Using the second line of (2.15), the integration over Therefore, by (2.23) the integration over V k−1 readŝ For the remaining integration in (2.21), we split the integration over V k into two terms aŝ For the first term of the RHS of (2.25), we use the similar bound of (2.12) and derive that In the first line of (2.26), , is consistent with the second line of (2.15) with l = k, s = t k . In the second line of (2.26) is consistent with the second line of (2.15) with l = k.
From the induction hypothesis( (2.13) is valid for k) and (2.24), we derive the integration over V j for j ≤ k −1 is consistent with the third line of (2.15). After taking integration´ k−1 j=1 Vj we change dΣ k k−1,m in (2.15) to dΣ k+1 k,m . Thus the contribution of (2.26) iŝ k j=1 Vj For the second term of the RHS of (2.25), we use the same estimate as (2.12) and we derive (2.28) Similar to (2.27), after taking integration over´ k−1 j=1 Vj the contribution of (2.28) iŝ k j=1 Vj (2.29) From (2.29) (2.27), the summation in the first and second lines of (2.14) extends to k. And the index of the third line of (2.14) changes from k to k + 1. For the rest terms, the index l ≤ k − 1, we haven't done any change to them. Thus their integration are over Therefore, the formula (2.14) is valid for k + 1 and we derive the lemma.
The next lemma is the key to prove the L ∞ bound for h m+1 . Below we define several notation: let r max := max(r (2 − r ), r ⊥ ), r min := min(r (2 − r ), r ⊥ ). (2.30) We inductively define: By a direct computation, for 1 ≤ i ≤ l, we have Then by the definition of (2.35) and (2.15), we have (2.38) Remark 6. We aim to bound the multiple integral in the trajectory formula in Lemma 1. Each integral in the formula involves the variable T w (x), T M , r ⊥ , r , thus we need to find the pattern of the upper bound for each fold integral. This is the reason we define these inductive notations.
Now we state the lemma.
Lemma 2. Given the formula for h m+1 in (2.12) and (2.13) of lemma 1, there exists such that when t ≤ t * , for any 0 ≤ s ≤ t l we havê where we define: Here C T M ,ξ is a constant defined in (2.49) and C is constant defined in (2.52). Moreover, for any p < p ≤ l, we havê Proof. From (1.9) and (1.18), for the first bracket of the first line in (2.15) with l + 1 ≤ j ≤ k − 1, we havê Without loss of generality we can assume k = l + 1. Thus dΦ k,l p,m = dΦ l+1,l p,m . We use an induction of p with 1 ≤ p ≤ l to prove (2.40).
When p = l, by the second line of (2.35), the integration over V l is written aŝ (2.43) (2.46) v l−1, and v l−1,⊥ are defined similarly.
First we compute the integration over V l, , the second line of (2.45). To apply (4.6) in Lemma 11, we set .
Next we compute first line of (2.45). To apply (4.9) in Lemma 12, we set Thus we can compute b b−a−ε and (a+ε)b b−a−ε using the exactly the way as (2.49) and (2.51) with replacing r (2 − r ) by r ⊥ . Hence replacing r (2 − r ) by r ⊥ and replacing v l−1, by v l−1,⊥ in (2.53), we bound the first line of (2.45) by where A l,l is defined in (2.41) and T l,l = 2ξ ξ+1 T M . Therefore, (2.40) is valid for p = l. Suppose (2.40) is valid for the p = q + 1(induction hypothesis) with q + 1 ≤ l, then l j=q+1 Vj We want to show (2.40) holds for p = q. By the hypothesis and the third line of (2.35), l j=q Vj Using the definition of A l,q+1 in (2.41), we obtain (2.57) We focus on the coefficient of |v q | 2 in (2.57), we derive .
By the Definition 1, x q+1 = x q+1 (t, x, v, v 1 , · · · , v q ), thus T w (x q+1 ) depends on v q . In order to explicitly compute the integration over V q , we need to get rid of the dependence of the T w (x q+1 ) on v q . Then we bound (2.59) In the third line of (2.59), to apply (4.6) in Lemma 11, we set Taking (2.47) for comparison, we can replace 2ξ ξ+1 T M by T l,q and replace t by C l−q t. Then we apply the replacement to (2.48) and obtain where we take t * = t * (T M , ξ, C, k) to be small enough and t ≤ t * . Also we require the t satisfy We conclude the t * only depends on the parameter in (2.39). Thus by the same computation as (2.49) we obtain where we use T l,q ≤ 2ξ ξ+1 T M from (2.33) and (2.30). C T M ,ξ is defined in (2.49). By the same computation as (2.51), we obtain Here we use T l,q ≤ 2ξ ξ+1 T M and (2.30) to obtain with C defined in (2.52). Thus by Lemma 11 with w = (1 − r )v q−1, , the third line of (2.59) is bounded by (2.60) By the same computation the second line of (2.59) is bounded by (2.62) In the proof for (2.40) we have Then by replacing q by p − 1 in the estimate (2.56) ≤ (C T M ,ξ ) 2(l−q+1) A l,q we have Keep doing this computation until integrating over V p we obtain the second inequality in (2.42).
The next result is the Lemma 3, which is the smallness of the last term of (2.14).
For the last term of (2.14), there exists where A k0−1,1 is defined in (2.41).
Remark 7. The difference between this lemma and Lemma 2 is that we have the small term ( 1 2 ) k0 . This lemma implies when k = k 0 is large enough, the measure of the last term of (2.14) is small.
We need several lemmas to prove it.
Here η i, is a constant defined in (2.78).
Here η i,⊥ is a constant defined in (2.81).
Proof. First we focus on (2.68). By (2.59) in Lemma 2, we can replace l by k − 1 and replace q by i to obtain (2.73) Under the condition (2.67), we consider the second line of (2.73) with integrating over {v i,⊥ ∈ V i,⊥ : |v i ·n(x i )| < 1−η 2(1+η) δ}. To apply (4.10) in Lemma 12, we set Under the condition |v i · n(x i )| < 1−η 2(1+η) δ, applying (4.10) in Lemma 12 and using (2.61) with q = i, l = k − 1, we bound the second line of (2.73) by Taking (2.61) for comparison, we conclude the second line of (2.73) provides one more constant term δ. The third line of (2.73) is bounded by (2.60) with q = i, l = k − 1. Therefore, we derive (2.68). Then we focus on (2.70). We consider the third line of (2.73). To apply (4.8) in Lemma 11, we set where we define Thus under the condition (2.69), applying (4.8) in Lemma 4.6 with b b−a−ε w = η i, v i−1, and using (2.60) with q = i, l = k − 1, we bound the third line of (2.73) by By the same computation in Lemma 4, we derive (2.70) because of the extra constant δ. Last we focus on (2.72). We consider the second line of (2.73) with integrating over To apply (4.10) in Lemma 13, we set (2.80) By the same computation as (2.77) where we define Thus under the condition (2.71), applying (4.13) in Lemma 13 with b b−a−ε w = η i,⊥ v i−1,⊥ and using (2.61) with q = i, l = k − 1, we bound the second line of (2.73) by Then we derive (2.70) because of the extra constant δ.
Lemma 5. For η i, and η i,⊥ defined in Lemma 4, we suppose there exists η < 1 such that Remark 8. Lemma 4 includes the cases that are controllable because of the small magnitude number δ, which is the "good" factor for us to establish the Lemma 3. This lemma discusses those "bad" cases, which are the main difficulty since they do not directly provide δ.
Proof. Under the condition (2.83) we have Thus we derive where we use |v i, | > 1+η 1−η δ −1 in the second line and 1 > η ≥ η i, in the third line. Then we obtain (2.84). Under the condition (2.85), we apply the same computation above to obtain (2.86). Lemma 6. Suppose there are n number of v j such that

87)
and also suppose the index j in these v j are i 1 < i 2 < · · · < i n , then Proof. By (2.42) in Lemma 2 with l = k − 1, p = i 1 , p = i n and using (2.70) with i = i n , we havê .
(2.89) Again by (2.42) and (2.70) with i = i n−1 we have Keep doing this computation until integrating over V i1 we derive (2.88).
Here the η satisfies the condition (2.82).

Remark 9.
In this lemma we combine the estimates and properties in Lemma 4 and Lemma 5. In the proof we will address the difficulty stated in Lemma 5 to obtain the key factor (3δ) L/2 .
Proof. By the definition (2.90) we have Here we summarize the result of Lemma 4 and Lemma 5.
We define W i,δ as the space that provides the smallness: Then we have For the subsequence {v l+1 , · · · , v l+L } in (2.91), when the number of v j ∈ W j,δ is larger than L/2, by (2.88) in Lemma 6 with n = L/2 and replacing the condition (2.87) by v j ∈ W j,δ , we obtain (2.95) We finish the discussion with the case(1),(2b),(2d). Then we focus on the case (2a),(2c).
By the same computation as (2.110), we have Then we use η qi, < 1 to obtain By (4.7) in Lemma 11 and (2.104), we apply (2.60) with q = q i to bound the third line of (2.73)( the integration over V qi, ) by Hence by the constant in (2.105) we draw a similar conclusion as (2.94): Therefore, by Lemma 6, after integrating over V q1, , V q2, , · · · , V q M , we obtain an extra constant The second case is that the number of v j ∈ {v j / ∈ W j,δ : |v j,⊥ | > 1+η 1−η δ −1 } is larger than L/4. We categorize v j,⊥ into Denote |Set4| = M 1 and the corresponding index as p 1 , p 2 , · · · , p M1 , |Set5| = M 1 and the corresponding index as q 1 , q 2 , · · · , q M1 , |Set6| = N 1 and the corresponding index as o 1 , o 2 , · · · , o N1 . Also define b j := |v q j ,⊥ | − |v q j −1,⊥ |. By the same computation as (2.102), we have We focus on the integration over v q j . Let 1 ≤ i ≤ M 1 , we consider the second line of (2.73) with i = q i and with integrating over To apply (4.12) in Lemma 11, we set By the same computation as (2.110), we have (2.108) Similar to (2.104), we have Hence by (4.12) in Lemma 13 and applying (2.61), we bound the integration over V q i ,⊥ by The integration over V q 1 ,⊥ , V q 2 ,⊥ , · · · , V q M 1 ,⊥ provides an extra constant where we set δ 1 in the last step. Then e −Lδ −1 is smaller than (3δ) L/2 in (2.95) and we concludê Finally collecting (2.95), (2.107) and (2.109) we derive the lemma.
Now we prove the Lemma 3.
Proof of Lemma 3.
Step 1 To prove (2.66) holds for the C-L boundary condition, we mainly use the decomposition (2.90) done by [1] and [14] for the diffuse boundary condition. In order to apply Lemma 7, here we consider the space V 1−η 2(1+η) δ i and ensure η satisfy the condition (2.82). In this step we mainly focus on constructing the η, which is defined in (2.120). First we consider η i, , which is defined in (2.78). In regard to (2.75) and (2.76), we take t = t (ξ, k, T M )( consistent with (2.65) ) to be small enough and set t ≤ t to obtain For any ε 1 > 0, there exists k 1 s.t when Moreover, by (2.63), there exists ε 2 s.t Then we have ε 2 = ε 2 (min{T w (x)}, T M , r , r ⊥ ).
(2.113) Thus we can bound T w (x i ) in the η i, ( defined in (2.78)) below as Thus we obtain (2.115) By (2.111), we take k = k 1 = k 1 (ε 2 , T M , r min ) (2.116) to be large enough such that ε 1 < ε 2 /4. By (2.110) and (2.115), we derive that when k = k 1 , (2.117) Here we define and we take t = t (k, T M , ε 2 , C, r ) to be small enough and t ≤ t such that 4T M C k t 1 to ensure the second inequality in (2.117). Combining (2.113) and (2.116), we conclude the t we choose only depends on the parameter in (2.65).
In consequence, when t k > 0, by (2.121) and t 1, there can be at most Step 3 In this step we combine Step 1 and Step 2 and focus on the integration over k−1 j=1 V j . By (2.121) in Step 2, we define In order to get (2.118),(2.119)< 1, we need to ensure the condition (2.111). Thus we take k = k 1 (T M , ξ, r ⊥ , r ) and only use the decomposition V j = V j \V Then we only consider the half sequence {v 1 , v 2 , · · · , v k/2 }. We derive that when t k > 0, there are at most N number of v j ∈ V 1−η 2(1+η) δ j and at In this single half sequence {v 1 , · · · , v k/2 }, in order to apply Lemma 7, we only want to consider the subsequence (2.91) with l + 1 < l + L ≤ k/2 and L ≥ 100 1+η 1−η . Thus we need to ignore those subsequence with L < 100 1+η 1−η . By (2.91), we conclude that at the end of this subsequence, it is adjacent to a v l ∈ V 1−η 2(1+η) δ l . By (2.124), we conclude There are at most N number of subsequences (2.91) with L ≤ 100 1 + η 1 − η . (2.125) We ignore these subsequences. Then we define the parameters for the remaining subsequence( with L ≥ 100 1+η 1−η ) as: in the first subsequence starting from v 1 , n := the number of these subsequences.
Similarly we can define M 2 , M 3 , · · · , M n as the number in the second, third, · · · , n-th subsequence. Recall that we only consider k/2 j=1 V j , thus we have (2.126) By (2.125), we obtain Step 4 Now we are ready to prove the lemma. By (2.124), we havê Since (2.129) holds for a single sequence, we derive where we use (2.127) in the second line. Take k = N 3 , the coefficient in (2.131) is bounded by Using (2.124), we derive 3δ = C(Ω, η)N −1/3 . Finally we bound (2.132) by where we choose δ to be small enough in the second line such that N = N (Ω, η, C T M ,ξ ) is large enough to satisfy And thus we choose k = N 3 = k 2 = k 2 (Ω, η, C T M ,ξ ) and we also require log k > 150 in the last step. Then we get (2.66). Therefore, by the condition (2.111), we choose k = k 0 = max{k 1 , k 2 }. By the definition of η (2.120) with (2.118) and (2.119), we obtain η = η(T M , C, r ⊥ , r , ε 2 ). Thus by (2.113) and (2.116), we conclude the k 0 we choose here does not depend on t and only depends on the parameter in (2.64). We derive the lemma.
Proof of Proposition 4. First we take t ∞ ≤ t .
(2.133) with t defined in (2.65). Then we let k = k 0 with k 0 defined in (2.64) so that we can apply Lemma 3 and Lemma 2. Define the constant in (2.7) as (2.134) We mainly use the formula given in Lemma 1. We consider two cases.
Case1: t 1 ≤ 0, By (2.12) and using the definition of Γ m gain (s) in (2.16) we have where u = u (u, v) and v = v (u, v) are defined by (1.3). Then we have where we choose where C ∞ is defined in (2.134 Then we focus on the second line of (2.13). Using θ = 1 4T M ξ we bound the second line of (2.13) by Now we focus on´ k 0 −1 j=1 Vj H. We compute H term by term with the formula given in (2.14). First we compute the first line of (2.14). By Lemma 2 with p = 1, for every 1 ≤ l ≤ k 0 − 1, we havê In regard to (2.141) we have Using the definition (2.33) we have T w (x 1 ) < 2ξ ξ+1 T M and T l,1 < 2ξ ξ+1 T M . Then we take t ∞ = t ∞ (T M , k 0 , ξ, C) (2.143) to be small enough and t ≤ t ∞ so that the coefficient for (2.144) Since (2.142) holds for all 1 ≤ l ≤ k 0 − 1, by (2.144) the contribution of the first line of (2.14) in (2.141) is bounded by (2.145) Then we compute the second line of (2.14). For each 1 ≤ l ≤ k 0 − 1 such that max{0, t l+1 } ≤ s ≤ t l , by (2.15), we have dΣ k0 l,m (s) = e −|v l | 2 (t l −s) dΣ k0 l,m (t l ). Therefore, we deriveˆt l max{0,t l+1 }ˆ where we apply (2.137) in the third line and we apply Lemma 2 in the last line.
In regard to (2.141), by (2.144) we obtain Since (2.146) holds for all 1 ≤ l ≤ k 0 − 1, the contribution of the second line of (2.14) in (2.141) is bounded by Last we compute the third term of (2.14). By Lemma 3 and the assumption (2.7) we obtain In regard to (2.141), by (2.144) we have Thus the contribution of the third line of (2.14) in (2.141) is bounded by By the definition of k 0 in (2.64), definition of C T M ,ξ in (2.49), definition of C in (2.52), we derive (2.9).
Then we can conclude the well-posedness.
Proof of Theorem 1. First of all we take t < t ∞ , where t ∞ is defined in (2.9) so that we can apply Proposition 4. We have sup • Existence For h m given in (2.2), we take the difference h m+1 − h m and deduce that By the same derivation as (2.12) (2.13), when t 1 ≤ 0, we have where we use h m+1 (0) = h m (0).
Then we follow the computation for (2.136) to obtain where we take N = N ( h m ∞ ) to be large and t < t ∞ = t ∞ (N ) to be small as in (2.138). When t 1 > 0, by the same derivation as (2.13), we have where H d is bounded by where we take t < t ∞ = t ∞ (k 0 ) to be small.
Thus in the case t 1 > 0 we obtain Therefore, h m is a Cauchy-sequence in L ∞ . The existence follows by taking the limit m → ∞ and the solution h = e (θ−t)|v| 2 f satisfies (2.155) This concludes the existence of f and (1.24).
• Stability Suppose there are two solutions h 1 and h 2 satisfy (2.155). Also suppose there initial condition satisfy When t 1 ≤ 0, by the same derivation as (2.137) and (2.152) we have By taking N = N ( h 1 ∞ , h 2 ∞ ) to be large as in (2.138) so that ( h 2 ∞ + h 1 ∞ )O( 1 N 2 ) 1, we derive the L ∞ stability by the Gronwall's inequality.
When t 1 > 0, the argument is exactly the same as the existence part and we conclude the L ∞ stability for all cases. The uniqueness follows immediately by setting h 1 (0) = h 2 (0).
The positivity follows from the the property that iteration equation (2.1) is positive preserving and (2.154).

Steady problem with C-L boundary condition
This section is devoted to the steady solution to the Boltzmann equation with the Cercignani-Lampis boundary condition as mention in Section 1.2.
Remark 10. The setting of the steady solution is given in Section 1.2. We remark here that in this section we no longer use notation µ. Instead we put the subscript µ 0 , δ 0 only for this section in order to avoid confusion.
To prove Corollary 2 we need the following Proposition.
Proposition 5 (Proposition 4.1 of [7]). Define a weight function scaled with parameter as and β > 4. Then the solution f to the linear Boltzmann equation For the purpose of applying Proposition 5, we focus on the boundary condition for the linearized equation f s .
√ µ 0 f s with F s satisfying the boundary condition (1.5), (1.6), the boundary condition for f s can be represented as Before proving this lemma we need the following lemma for the C-L boundary condition.
Lemma 9. In regard to the boundary condition (1.6), we have where (3.8) Moreover, for any x ∈ ∂Ω and r , r ⊥ , we havê Proof. Using the definition of R(u → v; x, t) in (1.6) we can write the LHS of (3.7) aŝ (3.10) First we compute the second line of (3.10), in order to apply Lemma 11, we set Then the second line of (3.10) equals to .
Then we compute the first line of (3.10), in order to apply Lemma 12, we set Then the first line of (3.10) is equal to Thus we conclude (3.7). Then we focus on (3.9). The LHS of (3.9) can be written aŝ (3.11) Clearly (3.11) = 1.
Proof of Lemma 8. By plugging the linearization F s = µ 0 + √ µ 0 f s into the boundary condition (1.5) and using Lemma 9 we obtain .
We can rewrite the boundary condition into f s (v) = r 1 + r 2 (f s ) − P γ f s + P γ f s . To prove the Lemma we just need to focus on r 2 (f s ) − P γ f s . By Tonelli theorem, we havê Thus we prove (3.5). Then we focus on (3.6). By the assumption in (1.29) and ζ < 1 θ(4+2δ0) , for x ∈ ∂Ω we have where we apply Lemma 9 in the last line. Then we conclude the Lemma.

And by Proposition 5 again for
Hence f is Cauchy in L ∞ and we construct our solution by taking the limit f → f s . Uniqueness follows in the standard way.
Then we focus on the dynamical stability, which is the Corollary 3. We need this Proposition.
Proof of Corollary 3. With the stationary solution for (1.26) given in Corollary 2, we set the solution to (1.1) as Then the equation for f reads We consider the following iteration sequence By taking difference f +1 − f , we deduce that with f +1 − f = 0 initially. Repeating the same argument, we obtain This implies that f +1 is a Cauchy sequence. The uniqueness is standard.
To conclude the positivity, we use another sequence in [7], We pose F = F s + √ µ 0 f , then the equation for f reads It is shown in [7] that f is a Cauchy sequence. Thus by the uniqueness of the solution we conclude the positivity of F and F s by positive preserving property of this sequence solution.
where we apply change of variable v + b a+ε−b w → v in the first step of the last line, then we obtain (4.6). Following the same derivation b thus we obtain (4.8).