\section{Introduction}

\subsection{Boltzmann equation with complete diffusive boundary condition}

In this paper, the following Boltzmann equation with Euler scaling over $(t, x, v) \in \R_+ \times \R^3_+ \times \R^3$ is considered:
\begin{equation}\label{BE}
\left\{
\begin{array}{l}
\p_t F_\eps + v \cdot \nabla_x F_\eps = \frac{1}{\eps} B (F_\eps, F_\eps) \quad \qquad {\text{on}}\  \R_+ \times \R^3_+ \times \R^3 \,,\\[1.5mm]
F_\eps (0, x, v) = F_\eps^{in} (x,v) \geq 0 \qquad \ \, \qquad {\text{on}}\  \R^3_+ \times \R^3 \,,
\end{array}
\right.
\end{equation}
where the dimensionless number $\eps>0$ is the Knudsen number which is the ratio of mean free path and macroscopic length scale, and $F_\eps (t,x,v) \geq 0$ is the number density of particles at time $t$ with velocity $v \in \R^3$ and position $x \in \R^3_+ = \{ x \in \R^3; x_3 > 0 \}$, a half-space. The collision operator $B(F_1, F_2)$ with hard sphere collision interaction is defined by
\begin{equation}\label{Collision_Operator}
  \begin{aligned}
   B (F_1, F_2) (v) = & \iint_{\R^3 \times \mathbb{S}^2} |(v-u)\cdot \omega| F_1 (u^\prime) F_2 (v^\prime)  \d \omega \d u \\
   - & \iint_{\R^3 \times \mathbb{S}^2} |(v-u)\cdot \omega| F_1 (u) F_2 (v)  \d \omega \d u\,,
  \end{aligned}
\end{equation}
where
\begin{equation*}
  \begin{aligned}
    u^\prime = u + [(v-u) \cdot \omega] \omega\,,\ v^\prime = v + [(v-u) \cdot \omega] \omega \,, \ \omega \in \mathbb{S}^2 \,.
  \end{aligned}
\end{equation*}
The outward normal vector of $\R^3_+$ is denoted by $n= (0, 0, -1)$. The symbol $\Sigma := \p \R^3_+ \times \R^3$ is the phase space boundary of $\R^3_+ \times \R^3$, which can be split as outgoing boundary $\Sigma_+$, incoming boundary $\Sigma_-$, and grazing boundary $\Sigma_0$:
\begin{equation*}
  \begin{aligned}
   & \Sigma_+ = \{ (x, v) : x \in \p \R^3_+ \,, v \cdot n = - v_3 > 0 \} \,, \\
   & \Sigma_- = \{ (x, v) : x \in \p \R^3_+ \,, v \cdot n = - v_3 < 0 \} \,, \\
   & \Sigma_0 = \{ (x, v) : x \in \p \R^3_+ \,, v \cdot n = - v_3 = 0 \} \,.
  \end{aligned}
\end{equation*}

Let $\gamma_\pm F : = \mathbbm{1}_{\Sigma_\pm} F$. The following {\em complete diffusive boundary condition} is further imposed on the scaled equation \eqref{BE}:
\begin{equation}\label{Diffusive_BC}
\gamma_- F_\eps =  M_w (t, x, v) \int_{v \cdot n >0} \gamma_+ F_\eps ( v \cdot n ) \d v \quad {\text{on}}\ \R_+ \times \Sigma_-\,,
\end{equation}
where $M_w (t, x, v)$ is the local Maxwellian distribution function corresponding to the wall (boundary) with the form
\begin{equation}\label{M_w}
\begin{aligned}
M_w (t, x, v) = \frac{\rho_w (t, x)}{[2 \pi T_w (t, x)]^{3/2}} \exp\Big\{ - \frac{|v- u_w (t, x)|^2}{2 T_w (t, x)} \Big\} \,, \ t \in \R_+ \,, \ (x, v) \in \Sigma \,.
\end{aligned}
\end{equation}
Here $u_w = (u_{w,1}, u_{w,2}, u_{w,3}) \in \R^3$ and $ T_w$ are velocity and temperature of the boundary, respectively. The $\rho_w$ is such that
\begin{equation}\label{Mw-Condition}
  \begin{aligned}
    \int_{v \cdot n < 0} |v \cdot n| M_w (t,x,v) \d v \equiv 1 \,,
  \end{aligned}
\end{equation}
which means that
\begin{equation}\label{rho_w-general}
  \begin{aligned}
    \rho_w = \sqrt{\tfrac{2 \pi}{T_w}} \Big[ \exp \Big( - \tfrac{u_{w,3}}{2 T_w} \Big) + \tfrac{u_{w, 3}}{T_w} \int_{-u_{w,3}}^{+\infty} \exp\Big( -\tfrac{z^2}{2 T_w} \Big) \d z \Big]^{-1} \,.
  \end{aligned}
\end{equation}
Once
\begin{equation}\label{uw3=0}
  \begin{aligned}
    u_{w,3} = 0
  \end{aligned}
\end{equation}
is assumed, which denotes the fixed boundary wall, the $\rho_w$ in \eqref{rho_w-general} reads $\rho_w = \sqrt{\tfrac{2 \pi}{T_w}}$. In other words, under the assumption \eqref{uw3=0} for fixed boundary wall,
\begin{equation}\label{M_w2}
  \begin{aligned}
    M_w = \sqrt{\frac{2 \pi}{T_w }} \frac{1}{(2 \pi T_w)^{3/2}} \exp\Big( - \frac{|v- u_w |^2}{2 T_w } \Big) \ \textrm{ on } \R_+ \times \Sigma \,,
  \end{aligned}
\end{equation}
which is consistent with the form in Sone's book \cite{Sone-2007-Book}. We remark that the condition \eqref{Diffusive_BC} is a special case of the so-called Maxwell reflection condition:
\begin{equation}\label{MBC}
\gamma_- F_\eps = (1-\alpha_\eps) L \gamma_+ F_\eps + \alpha_\eps K \gamma_+ F_\eps \quad \textrm{on } \R_+ \times \Sigma_- \,,
\end{equation}
where the accommodation coefficient $\alpha_\eps \in [0,1]$ describes how much the molecules accommodate to the state of the wall, and the specular-reflection $L \gamma_+ F_\eps$ and the diffuse-reflection part $K \gamma_+ F_\eps$ in \eqref{MBC} are
\begin{align*}
L \gamma_+ F_\eps (t, x, v) = & F_\eps (t, x, R_x v) \,, \ R_x v = v - 2 (v \cdot n) n = (\bar{v}, - v_3) \,, \\
K \gamma_+ F_\eps (t, x, v) = &  M_w (t, x, v) \int_{v \cdot n >0} \gamma_+ F_\eps ( v \cdot n ) \d v \,,
\end{align*}
respectively. The special case $\alpha_\eps=0$ corresponds to specular reflection, while $\alpha_\eps=1$ refers to complete diffusion, which is the cased considered in this paper.

This paper addressed the limit of the scaled Boltzmann system \eqref{BE}-\eqref{Diffusive_BC} as the Knudsen number $\eps$ goes to zero. It is well-known that (see standard references for the Boltzmann equations \cite{Cercignani-1988, CIP-1994, Sone-2007-Book}, for instance), as $\eps \to 0$, the solutions $F_\eps$ of the Boltzmann equation \eqref{BE} formally converge to a local Maxwellian
\begin{equation}\label{Max-M}
  \begin{aligned}
    \M (t, x, v)
    = \frac{\rho(t, x)}{\big(2 \pi T(t, x)\big)^{\frac{3}{2}}}
    \exp\Big( - \frac{|v - \u (t, x)|^2}{2 T(t, x)} \Big) \,,
  \end{aligned}
\end{equation}
whose parameters $(\rho, \u, T)$ satisfy the compressible Euler system. In particular, for the domain with boundary, the key (also the difficulty) is to determine the boundary condition for the compressible Euler system. It was formally derived in Sone's book \cite{Sone-2007-Book} that for the Boltzmann equation with complete diffusive boundary condition \eqref{Diffusive_BC}, the limiting Euler system will be imposed the impermeable boundary condition. The goal of the this paper is to rigorously justify this limit using the method of Hilbert expansion, in the context of short time smooth solutions.

\subsection{Compressible Euler limit from Boltzmann equation with boundary}
There have been many significant progress in the limits to incompressible fluids, such as Navier-Stokes, Stokes, or even Euler. We only list some representative results. One group of results is in the framework of DiPerna-Lions renormalized solutions of the Boltzmann equation \cite{DiPerna-Lions}, i.e., the so-called BGL program, which aimed to justify the limit to Leray solutions of the incompressible Navier-Stokes equations. This was initialized by Bardos-Golse-Levermore \cite{BGL1, BGL2}, and finished by Golse and Saint-Raymond \cite{Golse-SRM-04, Golse-SRM-09}. The corresponding results in bounded domain were carried out in \cite{Masmoudi-SRM-CPAM, Jiang-Masmoudi-CPAM}. We also should mention the incompressible Euler limits of Saint-Raymond \cite{SRM-ARMA03, SRM-IHP09}. Another group of results is in the framework of classical solutions, which are based on nonlinear energy method, semi-group method or hypercoercivity (the latter two further rely on the spectral analysis of the linearized Boltzmann operator), see \cite{Bardos-Ukai1991, Briant-JDE2015, BMM-AA2019, Gallagher-Tristani, Guo-CPAM06, Jang-Kim,JLM-CPDE, Jiang-Xu-Zhao,Levermore-Masmoudi-ARMA}.

Comparing to the incompressible limits listed above, the results to the compressible Euler system from the Boltzmann equation are much limited. This is mainly due to our still very poor understanding of the well-posedness of compressible Euler system for which we do not even know how to define weak solutions (at least for multiple spatial dimensions). The only available global-in-time solution is the calibrated BV solution of Glimm in 1965 \cite{Glimm-1965}. However, his solution was on 1-D. Regarding the higher dimensions, there are local-in-time classical solutions (see standard textbook \cite{Majda-1984}). The compressible Euler limit from the Boltzmann equations dates back to Japanese school for analytical data and Caflisch's Hilbert expansion approach which are based on the local well-posedness of compressible Euler system, see \cite{Caflish-1980-CPAM, Nishida1978}. Later improvement of Caflisch's result employing the recent progress on the $L^2\mbox{-}L^\infty$ estimate was also used to help justifying the linearized acoustic limit \cite{Guo-Jang-Jiang-2009-KRM, Guo-Jang-Jiang-2010-CPAM}. We should also mention the compressible Euler limit in the context of 1-D Riemann problem away from the initial time in \cite{HWWY-SIMA2013}.

In the domain with boundary, the situation is much more complicated. As formally (and numerically) analyzed by Japanese school (summarized in Sone's books \cite{Sone-2002book, Sone-2007-Book}), to derive the equations of compressible Euler system from the Boltzmann equations with Maxwell reflection condition using Hilbert type expansion, in most cases, there needed two coupled boundary layers: viscous and kinetic layers. The formal is also called {\em Prandtl layer}, the latter is {\em Knudsen layer} in the physics literatures. The reason that these two types of boundary layers are needed can be explained as follows: as well-known, the leading order in {\em interior} is compressible Euler system whose natural boundary condition is impermeable condition $\u\!\cdot\!n=0$. However, the local Maxwellian governed by  Euler system with this condition does not satisfy the Maxwell reflection boundary condition, except the specular reflection case (i.e., $\alpha_\eps=0$). So, kinetic layers with thickness $\eps$ are needed. In these layers, each term satisfies the linear kinetic boundary layer equations which requires {\em four} solvability conditions (by theorem of Golse-Perthame-Sulem \cite{Golse-Perthame-Sulem-1988-ARMA}), but the number of boundary conditions for compressible Euler system is {\em one}. This mismatch indicates there should be another layer with thickness $\sqrt{\eps}$: Prandtl layer. This name comes from the fact that in this layer, each term satisfies the famous Prandtl equations of compressible type. Specifically, if $\alpha_\eps=O(1)$ the leading term satisfies the {\em nonlinear} compressible Prandtl equations, the higher order terms satisfied the {\em linearized} compressible Prandtl equations. If $\alpha_\eps=O(\eps^\beta)$ with $\beta>0$, the boundary layers are weak, thus all the boundary terms appear in higher order. Thus, nonlinear Prandtl equation does not appear. Recently, there are two results in this direction for rigorous proof in the context of smooth solutions, of course, in short time interval: for specular reflection $\alpha_\eps=0$, by Guo-Huang-Wang in \cite{GHW-2020}, and for small but non-zero accommodation coeffieicnt $\alpha_\eps= O(\sqrt{\eps})$, by the authors of the current paper, in \cite{JLT-arXiv-2021}.

In the current paper, we treat the case of complete diffusion: $\alpha_\eps = 1$. The special feature of this paper is: this case is different with specular reflection \cite{GHW-2020} and the case of $0< \alpha_\eps < 1$ (for example \cite{JLT-arXiv-2021}) in which both Knudsen and Prandtl layers are used. For the case $\alpha_\eps=1$, there is no Prandtl layer, only Knudsen layers exist. The reason for this key feature is: for the kinetic boundary layer equation with complete diffusive boundary condition, only {\em one} solvability condition is needed. This provides one boundary condition for the corresponding fluid system, which exactly makes the system well-posed. So no viscous boundary layer is needed any more. In this sense, analytically, the current paper is even easier. More detailed explanations will be given in subsections \ref{Subsec:1.4} and \ref{Subsec:1.5}.


\subsection{Notations}

Throughout this paper, the notation $\bar{U} = (U_1, U_2)$ is used for any vector $U = (U_1, U_2, U_3) \in \R^3$. Moreover, for simplicity of presentations, the notation $V^0 : = V |_{x_3 = 0}$ is employed for any symbol $V = V(x)$, which may be a function, vector or operator.

For the functional spaces,  $H^s$ denotes the Sobolev space $W^{s,2} (\R^3_+)$ with norm $\| \cdot \|_{H^s}$,  $\| \cdot  \|_2$ and $\| \cdot \|_\infty$ are the $L^2$-norm and $L^\infty$-norm in both $(x,v) \in \R^3_+ \times \R^3$ variables, and $\l \cdot, \cdot \r$ is the $L^2$-inner product over $(x,v) \in \R^3_+ \times \R^3$. In particular, the symbol $\l \cdot, \cdot \r_v$ with subscript $v \in \R^3$ means the $L^2$-inner product in $v \in \R^3$.

In order to quantitatively describe the linear hyperbolic system \eqref{Linear-Hyperbolic-Syst} in the half-space $\R^3_+$, let
\begin{equation*}
\p^\alpha_{t, \bar{x}} = \p_t^{\alpha_0} \p_{x_1}^{\alpha_1} \p_{x_2}^{\alpha_2} \,,
\end{equation*}
where $\alpha = (\alpha_0, \alpha_1, \alpha_2) \in \mathbb{N}^3$, and let
\begin{equation}\label{Notation_H_k}
\begin{aligned}
& \|f(t)\|^2_{\mathcal{H}^k (\R^3_+)} = \sum_{|\alpha|+i \le k} \|\p_{t, \bar{x}}^\alpha \p^i_{x_3} f(t)\|^2_{L^2(\R^3_+)}\,, \\
& \|g(t)\|^2_{\mathcal{H}^k(\R^2)} = \sum_{|\alpha|\le k} \|\p_{t, \bar{x}}^\alpha g(t)\|^2_{L^2(\R^2)}
\end{aligned}
\end{equation}
for functions $f (t) = f(t, \bar{x}, x_3)$ and $g(t) = g (t, \bar{x})$. Here $|\alpha| = \alpha_0 + \alpha_1 + \alpha_2$. Remark that, for a function $f = f(\bar{x}, x_3)$ independent of the variable $t$, $\| f \|_{\mathcal{H}^k (\R^3_+)}$ is equivalent to the standard Sobolev norm $\| f \|_{H^k (\R^3_+)}$.

For any function $G = G(t, \bar{x}, y, v)$, with $(t, \bar{x}, y, v) \in \R_+ \times \R^2 \times \overline{\R}_+ \times \R^3$,  the Taylor expansion at $y = 0$ is
\begin{equation}
G = G^0 + \sum_{1 \leq l \leq N} \tfrac{y^l}{l !} G^{(l)} + \tfrac{y^{N+1}}{(N+1) !} \widetilde{G}^{(N+1)} \,,
\end{equation}
where the symbols
\begin{equation*}
\begin{aligned}
G^{(l)} = (\p_y^l G ) (t, \bar{x}, 0, v) \,, \ \widetilde{G}^{(N+1)} = ( \p_y^{N+1} G ) (t, \bar{x}, \eta, v) \ \textrm{for some } \eta \in (0, y).
\end{aligned}
\end{equation*}

Via the local Maxwellian $\M (t,x,v)$, the local linearized collision operator $\L$ is defined by
\begin{equation*}
\begin{aligned}
\L g = - \tfrac{1}{\sqrt{\M}} \Big\{ B (\M, \sqrt{\M} g) + B (\sqrt{\M} g, \M) \Big\} \,,
\end{aligned}
\end{equation*}
whose null space $\mathcal{N}$ is spanned by (see \cite{Caflish-1980-CPAM}, for instance)
\begin{equation*}
\begin{aligned}
\tfrac{1}{\sqrt{\rho}} \sqrt{\M} \,, \ \tfrac{v_i  - \u_i}{\sqrt{\rho T}} \sqrt{\M} \ (i=1,2,3) \,, \ \tfrac{1}{\sqrt{6 \rho}} \left\{ \tfrac{|v - \u|^2}{T} - 3 \right\} \sqrt{\M} \,.
\end{aligned}
\end{equation*}
The collision frequency $\nu (v) : = \nu (\M) (v)$ is given by
\begin{equation*}
\begin{aligned}
\nu (\M) (v) = \iint_{\R^3 \times \mathbb{S}^2} | (v - v') \cdot \omega| \M (v') \d v' \d \omega \,.
\end{aligned}
\end{equation*}
Note that
\begin{equation}\label{nu-phi}
\begin{aligned}
\nu (v) \thicksim \rho \l v \r \,,
\end{aligned}
\end{equation}
where $\l v \r = \sqrt{1 + |v|^2}$. The following weighted $L^2$-norm is introduced
\begin{equation*}
\begin{aligned}
\| g \|^2_\nu = \iint_{\R^3_+ \times \R^3} |g (x,v)|^2 \nu (v) \d x \d v \,.
\end{aligned}
\end{equation*}
Let $\mathcal{P} g$ be the $L^2_v$ projection with respect to $\mathcal{N}$. Then it is well-known (see for example \cite{Caflish-1980-CPAM}) that there exists a positive number $c_0 > 0$ such that
\begin{equation}\label{Hypocoercivity}
\begin{aligned}
\l \L g, g \r \geq c_0 \| (\mathcal{I} - \mathcal{P}) g \|^2_\nu \,,
\end{aligned}
\end{equation}
provided that $\rho$ and $T$ have positive lower and upper bounds.

\subsection{Main results}\label{Subsec:1.4}

As inspired in Caflish's work \cite{Caflish-1980-CPAM}, the Hilbert expansion approach is employed in current work. Based on the formal analyses in Section \ref{Sec:FormalAnaly} below, the compete diffusive boundary condition \eqref{Diffusive_BC} will not generate the viscous boundary layers. Moreover, as shown in Sone's book \cite{Sone-2007-Book}, the leading Knudsen layer vanishes, which, nevertheless, formally subjects to a nonlinear Knudsen layer equation. In other words, the so-called nonlinear Knudsen layer equation does NOT exist. Furthermore, the thickness of Knudsen boundary layers is $\eps$. Consequently, the solutions $F_\eps (t,x,v)$ to the scaled Boltzmann equation \eqref{BE} are therefore expanded by order $\eps$. More precisely, the ansatz of the asymptotic expansed form is
\begin{equation}\label{Hilbert_Expansion1}
  \begin{aligned}
    F_\eps (t, x, v) = \M (t, x, v) + \sum_{k=1}^2 \eps^k \big\{ F_k (t, x, v) + F^{bb}_k (t, \bar{x}, \tfrac{x_3}{\eps}, v) \big\} + \eps^2 F_{R, \eps} (t, x, v) \ge 0 \,,
  \end{aligned}
\end{equation}
where the local Maxwellian $\M (t,x,v)$ in \eqref{Max-M} for interior is governed by the compressible Euler equations
\begin{equation}\label{Compressible_Euler_Sys}
  \left\{
    \begin{aligned}
      &\p_t \rho + \div_x (\rho \u) = 0\,,\\
      &\p_t (\rho \u) + \div_x (\rho \u \otimes \u) + \nabla p = 0 \,, \\
      &\p_t \big[\rho \big( \tfrac{3}{2} T + \tfrac{1}{2} |\u|^2 \big) \big] + \div_x \big[ \rho \u \big( \tfrac{3}{2} T + \tfrac{1}{2} |\u|^2 \big) \big] + \div_x (p \u) = 0 \,, \\
      & p = \rho T
    \end{aligned}
  \right.
\end{equation}
over $(t, x) \in \R_+ \times \R^3_+$ with the impermeable boundary condition
\begin{equation}\label{BC-CEuler}
  \begin{aligned}
    \u \cdot n |_{x_3 =0} =  -\u_3 |_{x_3 = 0} = - \u_3^0 = 0 \,.
  \end{aligned}
\end{equation}
One further imposes the initial data (this can be realized by setting special form of the initial data of the Boltzmann equation \eqref{BE}):
\begin{equation}\label{IC-CEuler}
\begin{aligned}
(\rho, \u, T) (0, x) = (\rho^{in}, \u^{in}, T^{in}) (x)
\end{aligned}
\end{equation}
with compatibility condition $\u^{in} \cdot n |_{x_3} = 0$. One then can define an initial local Maxwellian
\begin{equation*}
\begin{aligned}
  \M^{in} (x,v) : = \frac{\rho^{in}(x)}{\big(2 \pi T^{in}(x)\big)^{\frac{3}{2}}}
  \exp\Big( - \frac{|v - \u^{in} (x)|^2}{2 T^{in}(x)} \Big) \,.
\end{aligned}
\end{equation*}
Denote by
\begin{equation*}
  \begin{aligned}
    \M^{in, 0} : = \M^{in} |_{x_3 = 0} \,.
  \end{aligned}
\end{equation*}
Remark that the boundary value \eqref{BC-CEuler} can eventually be derived from the solvability of nonlinear Knudsen boundary layer equations in half-space.

From \cite{Schochet-1986-CMP} or \cite{Chen-FMC-2007}, the following proposition holds.

\begin{proposition}\label{Proposition_Compressible_Euler}
	Let $s_0 \ge 3$. Assume $(\rho^{in}, \u^{in}, T^{in}) \in H^{s_0} (\R^3_+)$ satisfies $
	0 < \rho_\# \le \rho^{in} (x) \le \rho^\# \,,\quad
	0 < T_\# \le T^{in} (x) \le T^\#
	$ for some constants $\rho_\#$, $\rho^\#$, $T_\#$ and $T^\#$. Then there is a $\tau >0$ such that the compressible Euler system \eqref{Compressible_Euler_Sys}-\eqref{BC-CEuler}-\eqref{IC-CEuler} admits a unique solution
	$$
	(\rho, \u, T) \in C ( [0,\tau]; H^{s_0} (\R^3_+) ) \cap C^1 ( [0,\tau]; H^{s_0-1} (\R^3_+) )
	$$
	such that $ 0 < \tfrac{1}{2} \rho_\# \le \rho(t, x) \le 2 \rho^\#\,, \quad
	0 < \tfrac{1}{2} T_\# \le T(t, x) \le 2 T^\#
	$ hold for any $(t, x) \in [0, \tau] \times \R^3_+$. Moreover, the following estimate holds:
	\begin{align}\label{Compressible_Euler_Bound}
	\| (\rho, \u, T) \|_{C ( [0,\tau]; H^{s_0} (\R^3_+) ) \cap C^1 ( [0,\tau]; H^{s_0-1} (\R^3_+) )} \le C_0 \,.
	\end{align}
	Here the constants $\tau$, $C_0 >0$ depend only on the $H^{s_0}$-norm of $(\rho^{in}, \u^{in}, T^{in})$.
\end{proposition}

Denote by
\begin{equation}
  \begin{aligned}
    f_i : = \frac{F_i}{\sqrt{\M}} \ (i \geq 1) \,, \ f^{bb}_j : = \frac{F^{bb}_j}{\sqrt{\M^0}} \ (j \geq 0) \,.
  \end{aligned}
\end{equation}
In \eqref{Hilbert_Expansion1}, the higher order interior expanded terms $F_1$ and $F_2$ satisfy
\begin{equation}\label{Relt1}
  \begin{aligned}
    f_1 & = \P f_1 + (\I - \P) f_1 = \big[ \tfrac{\rho_1}{\rho} + u_1 \cdot \tfrac{v - \u}{T} + \tfrac{\theta_1}{6 T} ( \tfrac{|v - \u|^2}{T} - 3 ) \big] \sqrt{\M} + (\I - \P) f_1 \,, \\
    f_2 & = (\I - \P) f_2 \,,
  \end{aligned}
\end{equation}
where the kinetic parts read
\begin{equation}\label{KinPart-12}
  \begin{aligned}
    (\I - \P) f_1 = & - 2 \L^{-1} \big( (\p_t + v \cdot \nabla_x) \sqrt{\M} \big) \,, \\
    (\I - \P) f_2 = & \mathcal{G} (F_1) : = \L^{-1} \Big( - \tfrac{(\p_t + v \cdot \nabla_x) F_1 + B (F_1, F_1)}{\sqrt{\M}} \Big) \,,
  \end{aligned}
\end{equation}
and fluid variables $(\rho_1, u_1, \theta_1) \in \R \times \R^3 \times \R$ of $f_1$ obey the following linear hyperbolic system
\begin{equation}\label{Linear-Hyperbolic-Syst}
  \left\{
    \begin{aligned}
      & \p_t \rho_1 + \div_x (\rho u_1 + \rho_1 \u) = 0 \,, \\
      & \rho \big( \p_t u_1 + u_1 \cdot \nabla_x \u + \u \cdot \nabla_x u_1 \big) - \tfrac{\nabla_x (\rho T)}{\rho} \rho_1 + \nabla_x \big( \tfrac{\rho \theta_1 + 3 T \rho_1}{3} \big) = \mathcal{F}^\bot_u (f_1) \,, \\
      & \rho \big( \p_t \theta_1 + \u \cdot \nabla_x \theta_1 + \tfrac{2}{3} \big( \theta_1 \div_x \u + 3 T \div_x u_1 \big) + 3 u_1 \cdot \nabla_x T \big) = \mathcal{F}^\bot_\theta (f_1)
    \end{aligned}
  \right.
\end{equation}
with boundary condition
\begin{equation}\label{BC-LinearHypo}
  \begin{aligned}
    u^0_1 \cdot n = & \mathcal{J} (\M^0) \\
    : = & T^0 (\Psi_1 + 5 T^0 \Theta_1) (t, \bar{x}, 0) - \tfrac{1}{\rho^0} \l (v \cdot n) \sqrt{\M^0} , (\mathcal{I} - \mathcal{P}^0) f_1 (t, \bar{x}, 0, v)  \r_v \,,
  \end{aligned}
\end{equation}
where $(\rho, \u, T)$ is the smooth solution of compressible Euler equations \eqref{Compressible_Euler_Sys}, $\Psi_1$, $\Theta_1$ are defined in \eqref{f_bb_k1_Coef} below, the source terms $\mathcal{F}^\bot_u (f_1 ) \in \R^3$ and $\mathcal{F}^\bot_\theta (f_1) \in \R$ are defined as
\begin{equation}\label{Relt2}
  \begin{aligned}
    & \mathcal{F}^\bot_{u, i} (f_1) = - \sum_{j=1}^3 \p_{x_j} \l T \A_{ij}, (\I - \P) f_1 \r_v \ (i = 1,2,3) \,, \\
    & \mathcal{F}^\bot_\theta (f_1) = - \div_x \big( 2 T^{\frac{3}{2}} \l \B, (\I - \P) f_1 \r_v + 2 T \u \cdot \l \A , (\I - \P) f_1 \r_v \big) - 2 \u \cdot \mathcal{F}^\bot_u (f_1)\,.
\end{aligned}
\end{equation}
Here $\A \in \R^{3 \times 3}$ and $\B \in \R^3$ are the Burnett functions with entries
\begin{equation}
  \begin{aligned}
    \A_{ij} = & \Big\{ \tfrac{(v_i - \u_i) (v_j - \u_j)}{T} - \delta_{ij} \tfrac{|v - \u|^2}{3 T} \Big\} \sqrt{\M} \quad (1 \leq i, j \leq 3) \,, \\
    \B_i = & \tfrac{v_i - \u_i}{2 \sqrt{T}} \Big( \tfrac{|v - \u|^2}{T} - 5 \Big) \sqrt{\M} \quad (1 \leq i \leq 3) \,.
  \end{aligned}
\end{equation}
They are both in $\mathcal{N}^\perp$ and perpendicular to each other. Remark that the boundary condition \eqref{BC-LinearHypo} is derived from the solvability of the higher order Knudsen layer equation $F^{bb}_1$, see Section \ref{Sec:FormalAnaly} below. Finally, the initial data of \eqref{Linear-Hyperbolic-Syst} are imposed on
\begin{equation}\label{IC-Linear-Hyperbolic}
  \begin{aligned}
    (\rho_1, u_1, \theta_1) (0, x) = (\rho_1^{in}, u_1^{in}, \theta_1^{in}) (x) \in \R \times \R^3 \times \R
  \end{aligned}
\end{equation}
with compatibility $u_1^{in} \cdot n |_{x_3 = 0} = \mathcal{J} (\M^{in, 0})$, where the symbol $\mathcal{J}$ is given in \eqref{BC-LinearHypo}.

The Knudsen layers $f^{bb}_k (t, \bar{x}, \xi, v)$ with normal coordinate $\xi = \tfrac{x_3}{\eps}$ can be decomposed by
\begin{equation}\label{Relt3}
  \begin{aligned}
    f^{bb}_k = f^{bb}_{k,1} + f^{bb}_{k,2} \ (k =1,2) \,,
  \end{aligned}
\end{equation}
where
\begin{equation}\label{Relt4}
  \begin{aligned}
    f^{bb}_{k,1} = \big\{ \Psi_k v_3 + \Phi_{k,1} v_3 (v_1 - \u^0_1) + \Phi_{k,2} v_3 (v_2 - \u^0_2) + \Phi_{k,3} + \Theta_k v_3 |v- \u^0|^2 \big\} \sqrt{\M^0} \,,
  \end{aligned}
\end{equation}
and $f^{bb}_{k,2}$ $(k=1,2)$ subject to the Knudsen boundary layer equation
\begin{equation}\label{f_bb_k_2}
  \left\{
    \begin{aligned}
      & v_3 \p_\xi f^{bb}_{k, 2} + \mathcal{L}^0 f^{bb}_{k, 2} = S^{bb}_{k, 2} - \big( v_3 \p_\xi f^{bb}_{k,1} - S^{bb}_{k, 1} + \mathcal{L}^0 f^{bb}_{k,1} \big) \in (\mathcal{N}^0)^\perp \,, \\
      & f^{bb}_{k,2} (t, \bar{x}, 0, v) |_{v_3 >0} = \mathcal{D}_w f^{bb}_{k, 2} (t, \bar{x}, 0, v) + \mathbbm{f}_k (t, \bar{x}, v) \,, \\
      & \lim_{\xi \to +\infty} f^{bb}_{k,2} (t, \bar{x}, \xi, v) =0 \,.
    \end{aligned}
  \right.
\end{equation}
Here $\Psi_k$, $\Phi_{k,1}$, $\Phi_{k,2}$, $\Phi_{k,3}$, $\Theta_k$ are given in \eqref{f_bb_k1_Coef} below, which depend on the quantity $S^{bb}_{k,1} \in \mathcal{N}^0$ (see \eqref{S_bb_k+1_1+2} below). $S^{bb}_{k,2}$ is defined in \eqref{S_bb_k+1_1+2} below with $N=2$. The diffusive boundary operator $\mathcal{D}_w$ is defined by
\begin{equation}\label{Dw}
  \begin{aligned}
    \mathcal{D}_w f = \tfrac{M_w}{\sqrt{\M^0}} \int_{v' \cdot n >0} (v' \cdot n) f \sqrt{\M^0} \d v' \,,
  \end{aligned}
\end{equation}
where $M_w$ is the wall Maxwellian distribution. The boundary source term $\mathbbm{f}_k (t, \bar{x}, v)$ is
\begin{equation}
  \begin{aligned}
    \mathbbm{f}_k (t, \bar{x}, v) = - ( \I - \mathcal{D}_w ) ( f_k + f^{bb}_{k,1} ) (t, \bar{x}, 0, v) \,.
  \end{aligned}
\end{equation}

The remainder $F_{R, \eps} (t,x,v)$ obeys
\begin{equation}\label{Remainder_Eq}
  \begin{aligned}
    & \p_t F_{R, \eps} + v \cdot \nabla_x F_{R, \eps} - \tfrac{1}{\eps} \big[ B(F_{R, \eps}, \M) + B(\M, F_{R, \eps}) \big] \\
    = & \sum_{i=1}^2 \eps^{i-1} \big[ B(F_{R, \eps}, F_i + F^{bb}_i) + B(F_i + F^{bb}_i, F_{R, \eps}) \big] \\
    & + \eps B(F_{R, \eps}, F_{R, \eps}) + R_\eps + R^{bb}_\eps
  \end{aligned}
\end{equation}
with complete diffusive boundary condition
\begin{equation}\label{Remainder_BC}
  \gamma_- F_{R, \eps} = M_w \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \gamma_+ F_{R, \eps} \d v^\prime\,\  {\text{on}}\ \Sigma_-\,,
\end{equation}
where
\begin{equation}\label{R_eps}
  R_\eps = - ( \p_t + v \cdot \nabla_x ) F_2 + B(F_1, F_2) + B(F_2, F_1) \,,
\end{equation}
and
\begin{align}\label{R_eps_bb}
  \no R^{bb}_\eps = & - (\p_t + \bar{v} \cdot \nabla_{\bar{x}}) F^{bb}_2 + \sum_{\substack{i+j\ge 3\,,\\1 \le i, j\le 2}} \eps^{i+j-3} \big[ B(F^{bb}_i, F^{bb}_j) + B(F^0_i, F^{bb}_j) + B(F^{bb}_j, F^0_i) \big] \\
  \no & + \sum_{\substack{l+j\ge 3\,,\\1 \le l, j\le 2}} \eps^{l + j-3} \tfrac{\xi^l}{l !} \big[ B(\M^{(l)}, F^{bb}_j) + B(F^{bb}_j, \M^{(l)})  \big] \\
  \no & + \sum_{\substack{l+i+j \ge 3\,,\\1 \le l, i\le 2\,,\\ 0 \le j \le 2}} \eps^{l +i +j -3} \tfrac{\xi^l}{l !} \big[ B(F^{(l)}_i, F^{bb}_j) + B(F^{bb}_j, F^{(l)}_i)  \big] \\
  & + \sum_{j=0}^2 \eps^j \tfrac{\xi^3}{3!} \big[ B(\widetilde{\M}^{(3)}, F^{bb}_j) + B(F^{bb}_j, \widetilde{\M}^{(3)}) \big] \\
  \no & + \sum_{\substack{1 \le i \le 2\,,\\0 \le j \le 2}} \eps^{i+j} \tfrac{\xi^3}{3 !} \big[ B(\widetilde{F}^{(3)}_i, F^{bb}_j) + B(F^{bb}_j, \widetilde{F}^{(3)}_i) \big]\,.
\end{align}
Furthermore, the following initial data are imposed on the remainder equation \eqref{Remainder_Eq}:
\begin{equation}\label{IC-Remainder-F}
  \begin{aligned}
    F_{R, \eps} (0, x, v) = F_{R, \eps}^{in} (x,v) \,,
  \end{aligned}
\end{equation}
which satisfies the compatibility condition
\begin{equation*}
  \begin{aligned}
    \gamma_- F_{R, \eps}^{in} = M_w \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \gamma_+ F_{R, \eps}^{in} \d v^\prime
  \end{aligned}
\end{equation*}
on $\Sigma_-$.

For the remainder $F_{R, \eps}$, let
\begin{equation}\label{f_Reps-h_Reps}
  \begin{aligned}
    f_{R, \eps} = \tfrac{F_{R, \eps}}{\sqrt{\M}} \,, \quad h_{R, \eps}^\ell = \l v \r^\ell \tfrac{F_{R,\eps}}{\sqrt{\M_M}}
  \end{aligned}
\end{equation}
for $\ell \geq 7$, where the global Maxwellian $\M_M = \M_M (v)$ is introduced by \cite{Caflish-1980-CPAM}
\begin{align}\label{Global_Maxwellian_M}
  \M_M = \tfrac{1}{(2\pi T_M)^{\frac{3}{2}}} \exp \Big\{ - \tfrac{|v|^2}{2 T_M} \Big\} \,.
\end{align}
Here the constant $T_M$ satisfies
\begin{align} \label{T_M}
  T_M < \max_{t \in [0, \tau], x \in \R^3_+} T (t, x) < 2 T_M \,.
\end{align}
Then there exists constants $C_1$, $C_2$ such that for some $\frac{1}{2} < z <1$ and for each $(t, x, v) \in [0, \tau] \times \R^3_+ \times \R^3$, the following inequality holds:
\begin{align}\label{M-Bound}
  C_1 \M_M \le \M \le C_2 ( \M_M )^z\,.
\end{align}

One then introduces some fixed functions $F^{in}_k (x,v)$ and $F^{bb, in}_k (\bar{x}, \xi, v)$ ($k = 1,2$) constructed by the initial data $(\rho^{in}, \u^{in}, T^{in})$ and $(\rho_1^{in}, u_1^{in}, \theta_1^{in})$ mentioned above. Specifically, the construction of $F^{in}_k (x,v)$ and $F^{bb, in}_k (\bar{x}, \xi, v)$ is almost the same as that of $F_k (t,x,v)$ and $F^{bb}_k (t,\bar{x}, \xi, v)$ above. It is just to replace $(\rho, \u, T)$ and $(\rho_1, u_1, \theta_1)$ with $(\rho^{in}, \u^{in}, T^{in})$ and $(\rho_1^{in}, u_1^{in}, \theta_1^{in})$, respectively. Then the initial data $F^{in}_\eps (x,v)$ of \eqref{BE} is assumed to be well-prepared:
\begin{equation}\label{IC-wellprepared}
  \begin{aligned}
    F^{in}_\eps (x,v) = \M^{in} (x, v) + \sum_{k=1}^2 \eps^k \big\{ F_k^{in} (x, v) + F^{bb, in}_k (\bar{x}, \tfrac{x_3}{\eps}, v) \big\} + \eps^2 F_{R, \eps}^{in} (x, v) \ge 0 \,,
  \end{aligned}
\end{equation}
Furthermore, the wall velocity $u_w$ and temperature $T_3$ are assumed to match the interior fluid velocity $\u$ and temperature $T$ constrained to the boundary $\p \R^3_+$, i.e.,\footnote{ In fact, one can also assume $(u_w, T_w)$ to be a perturbation around $(\u^0, T^0)$, in which all the processes of proof are the same except that one should deal with the additional small perturbed quantities.}
\begin{equation}\label{uw-Tw-match}
  \begin{aligned}
    u_w = \u^0 \,, \ T_w = T^0  \,.
  \end{aligned}
\end{equation}

Now the main theorem is stated as follows.

\begin{theorem}\label{Main-Thm}
	Consider the wall Maxwellian $M_w$ given in \eqref{M_w2} and \eqref{uw-Tw-match}. Assume $\ell \geq 7$ and the integers $s_1 \geq 3$, $s_0 = s_1 +3$. Let $(\rho^{in}, \u^{in}, T^{in})$ satisfies the assumptions in Proposition \ref{Proposition_Compressible_Euler}, and let
	\begin{equation*}
	  \begin{aligned}
	    \mathcal{E}_1^{in} : = \| (\rho_1^{in}, u_1^{in}, \theta_1^{in}) \|_{\mathcal{H}^{s_1} (\R^3_+) } < \infty \,.
	  \end{aligned}
	\end{equation*}
	Correspondingly,
	\begin{itemize}
		\item $(\rho, \u, T) (t,x)$ is the solution to the compressible Euler system \eqref{Compressible_Euler_Sys}-\eqref{BC-CEuler}-\eqref{IC-CEuler} over time interval $t \in [0, \tau]$ constructed in Proposition \ref{Proposition_Compressible_Euler}, which governs the local Maxwellian $\M (t,x,v)$ in \eqref{Max-M};
		
		\item $F_k (t, x, v)$ and $F^{bb}_k (t, \bar{x}, \tfrac{x_3}{\eps}, v)$ $(k = 1,2)$ are constructed in Proposition \ref{Prop-HigherOrd} below.
	\end{itemize}
    Then there is a small $\eps_0 > 0$ such that if
    \begin{equation*}
      \begin{aligned}
        \mathcal{E}_R^{in} : = \sup_{\eps \in (0, \eps_0)} \Big( \| \tfrac{F_{R,\eps}^{in}}{\sqrt{\M^{in}}} \|_2 + \sqrt{\eps}^3 \| \l v \r^\ell \tfrac{F_{R,\eps}^{in}}{\sqrt{\M^{in}}} \|_\infty \Big) < \infty \,,
      \end{aligned}
    \end{equation*}
    then the scaled Boltzmann equation \eqref{BE} with complete diffusive boundary condition \eqref{Diffusive_BC} and well-prepared initial data \eqref{IC-wellprepared} admits a unique solution $F_\eps (t,x,v)$ for $\eps \in (0, \eps_0)$ over the time interval $t \in [0, \tau]$ with the expanded form \eqref{Hilbert_Expansion1}, i.e.,
    \begin{equation*}
      \begin{aligned}
        F_\eps (t, x, v) = \M (t, x, v) + \sum_{k=1}^2 \eps^k \big\{ F_k (t, x, v) + F^{bb}_k (t, \bar{x}, \tfrac{x_3}{\eps}, v) \big\} + \eps^2 F_{R, \eps} (t, x, v) \ge 0 \,,
      \end{aligned}
    \end{equation*}
    where the remainder $F_{R,\eps} (t,x,v)$ satisfies the uniform bound
    \begin{equation*}
      \begin{aligned}
        \sup_{t \in [0, \tau]} \Big\{ \| \tfrac{F_{R,\eps}}{\sqrt{\M}} (t) \|_2 + \sqrt{\eps}^3 \| \l v \r^\ell \tfrac{F_{R,\eps}}{\sqrt{\M_M}} (t) \|_\infty \Big\} \leq C( \tau, \mathcal{E}_0^{in}, \mathcal{E}_1^{in}, \mathcal{E}_R^{in} ) < \infty \,,
      \end{aligned}
    \end{equation*}
    where $\mathcal{E}_0^{in} : = \| (\rho^{in}, \u^{in}, T^{in}) \|_{H^{s_0} (\R^3_+)}$.
\end{theorem}

\begin{remark}
	Together with Proposition \ref{Prop-HigherOrd} below, Theorem \ref{Main-Thm} shows that as $\eps \to 0$,
	\begin{equation*}
	  \begin{aligned}
	    \sup_{t \in [0, \tau]} \Big\{ \| (\tfrac{F_\eps - \M}{\sqrt{\M}}) (t) \|_2 + \| \l v \r^\ell ( \tfrac{F_\eps - \M}{\sqrt{\M_M}} ) (t) \|_\infty \Big\} \leq C \sqrt{\eps} \to 0 \,.
	  \end{aligned}
	\end{equation*}
	Therefore, while imposed on the well-prepared initial data, one has justified the hydrodynamic limit from the Boltzmann equation with complete diffusive boundary condition to the compressible Euler system with impermeable boundary condition for the half-space problem.
\end{remark}

\subsection{Sketch of ideas}\label{Subsec:1.5}

As indicated in Sone's book \cite{Sone-2007-Book}, there is no leading Knudsen layer for the Boltzmann equation with complete diffusive boundary condition in half-space. Actually, even if the leading Knudsen layer is supposed to exist, the one formally obeys a nonlinear equation. However, it can be proved that the solution must be zero\footnote{This is the reason why the nonlinear Knudsen boundary layer problem under diffusive reflection boundary condition in half-space is not needed. We give an explanation in Appendix \ref{Appendix} that even if we ``pretend" to assume the $O(1)$ Knudsen layer exists, the corresponding nonlinear kinetic boundary layer equation would have only zero solution.}. So, the Knudsen layers are initially expanded from the next higher order, which satisfy the linear equations with diffusive boundary condition. The thickness of expansions is $\eps$, as the thickness of the Knudsen is $\eps$. Moreover, there only requires one solvability condition for the linear Knudsen layer equations with diffusive boundary condition, see \cite{Coron-Golse-Sulem-1988-CPAM}. The single solvability condition will solve one boundary condition such that the interior expansion is solvable. Thus, the Prandtl layers such as in the Maxwell reflection boundary case (see \cite{JLT-arXiv-2021}) or specular reflection boundary case (see \cite{GHW-2020}) are not required. Therefore, the ansatz of the expanded form \eqref{Hilbert_Expansion1} is done.

For the truncation in \eqref{Hilbert_Expansion1} with remainder $\eps^2 F_{R,\eps}$, it is actually optimal in $L^2$-$L^\infty$ framework for Boltzmann equation. In the Hilbert expansion approach, smaller the number of expanded terms is in the expansion, closer the structure of the solution is to the general solution. No matter how many terms are expanded, the remainder equations will contain a common linear structure
\begin{equation*}
  \begin{aligned}
    \p_t f_{R,\eps} + v \cdot \nabla_x f_{R,\eps} + \tfrac{1}{\eps} \L f_{R,\eps} = \textrm{ some other terms}.
  \end{aligned}
\end{equation*}
In $L^2$ estimates, the hypocoercivity of the operator $\L$ in \eqref{Hypocoercivity} leads to a kinetic dissipative rate $\tfrac{c_0}{\eps} \| (\I - \P) f_{R,\eps} \|^2_\nu$. However, there is no any such structure in $L^\infty$ estimates. These produce a disparity of order $\sqrt{\eps}^3$ between $L^2$ estimates and $L^\infty$ estimates. If one expands two terms, Theorem \ref{Main-Thm} shows $\sup_{t \in [0, \tau]} \Big\{ \| (\tfrac{F_\eps - \M}{\sqrt{\M}}) (t) \|_2 + \| \l v \r^\ell ( \tfrac{F_\eps - \M}{\sqrt{\M_M}} ) (t) \|_\infty \Big\} \leq C \sqrt{\eps} \to 0$. But if the less terms are expanded, the corresponding convergence fails. For more details on the optimality, readers can be found in \cite{JLT-arXiv-2021}.

In $L^2$ estimates for remainder equation, the key point is to deal with the boundary integral $- \tfrac{1}{2} \iint_\Sigma v_3 |f_{R, \eps} (t,\bar{x}, 0,v)|^2 \d v \d \bar{x}$. Thanks to the conditions \eqref{Mw-Condition} and \eqref{uw-Tw-match} satisfied by the wall Maxwellian $M_w (t, \bar{x}, v)$, namely, $\int_{v \cdot n < 0} |v \cdot n| M_w \d v = 1$ and $(u_w, T_w) = (\u^0, T^0)$, the boundary integral plays a role of boundary dissipative rate
\begin{equation*}
  \begin{aligned}
    \tfrac{1}{2} \iint_{\Sigma_+} |v \cdot n| |(\I - \mathcal{D}_w) f_{R,\eps} (t, \bar{x}, 0, v)|^2 \d v \d \bar{x} \,.
  \end{aligned}
\end{equation*}

In $L^\infty$ estimates for remainder equation, the goal is to derive the pointwise bounds for $h_{R,\eps}^\ell (t,x,v)$ on $(t,x,v) \in [0, \tau] \times \R^3_+ \times \R^3$ along the trajectories over $s \in [0,t]$. The set $ [0, \tau] \times \R^3_+ \times \R^3$ is split into two parts: $\mathscr{U}$ and $\mathscr{V}$, see \eqref{UV} below. In fact, as time evolves, the state of the set $\mathscr{U}$ is almost all transformed into the state of the set $\mathscr{V}$ through the boundary effect. For fixed $(t,x,v) \in \mathscr{U}$, the trajectory on $[0, t]$ does not hit the boundary $\p \R^3_+$. The trajectory on $[0,t]$ is a straight line segment. Following the almost same arguments in \cite{Guo-Jang-Jiang-2010-CPAM}, the required estimates on $\mathscr{U}$ can be obtained. For fixed $(t,x,v) \in \mathscr{V}$, the trajectory will hit the boundary $\p \R^3_+$ at time $t_1 = t - t_b (x,v) > 0$. The way is to integrate the $h_{R,\eps}^\ell$ equation along the trajectory on $[t_1, t]$, still a straight line segment. In this case, one should focus on the term $\exp\Big\{ -\tfrac{1}{\eps} \int_{t_1}^t \nu (\phi) \d \phi \Big\} h^\ell_{R, \eps} (t_1, x_b (x,v), v)$, where $(x_b (x,v), v) \in \Sigma_-$. Together with the diffusive boundary condition, $h^\ell_{R, \eps} (t_1, x_b (x,v), v)$ converts to the integral form $M_w \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \tfrac{\sqrt{\M_M}}{\l v^\prime \r^\ell} h^\ell_{R, \eps} (t_1, x_b (x,v), v^\prime) \d v^\prime$. Observe that $(t_1, x_b (x,v), v^\prime) \in \mathscr{U}$. Therefore the estimates on $\mathscr{U}$ can be applied to control this quantity. Collecting the estimates on $\mathscr{U}$ and $\mathscr{V}$, one finishes the $L^\infty$ estimate.


\subsection{Organizations of current paper}

The next section focuses on the formal analysis, in which the impermeable boundary condition of compressible Euler system and the boundary conditions of the fluid variables of higher orders are derived. The corresponding truncation is also done. In Section \ref{Sec:F1F2}, the expanded terms $F_k$ and $F^{bb}_k$ ($k=1,2$) are controlled in some sufficiently smooth norms, which is such that the coefficients consisted of $F_k$ and $F^{bb}_k$ ($k=1,2$) appeared in the remainder equation can be bounded by some harmless constant $C > 0$. In Section \ref{Sec:Remainder}, the bounds of remainder $F_{R,\eps}$ (uniform in small Knudsen number $\eps$) are obtained in the $L^2$-$L^\infty$ framework.



\section{Formal Analysis}\label{Sec:FormalAnaly}

In this section, the goal is to derive the formal analysis in the Hilbert expansion framework. Notice that the thickness of the Knudsen layer is $\eps$. As shown in Sone's book \cite{Sone-2007-Book}, there are not leading Knudsen boundary layer, which means the layers start from the oder of $\eps$. Furthermore, there is only one solvability condition for the linear Knudsen boundary layer equations with diffusive boundary condition, see \cite{Coron-Golse-Sulem-1988-CPAM}. In this situation, the viscous boundary layers do NOT exist. As a result, the asymptotic expanded form will be
\begin{equation}\label{Exp1}
  \begin{aligned}
    F_\eps (t, x, v) = F_0 (t,x,v) + \sum_{k \geq 1} \eps^k \big[ F_k (t,x,v) + F^{bb}_k (t, \bar{x}, \xi, v) \big]
  \end{aligned}
\end{equation}
with normal coordinate
\begin{equation*}
  \begin{aligned}
   \xi = \tfrac{x_3}{\eps} \,,
  \end{aligned}
\end{equation*}
where $F_k$ and $F^{bb}_k$ are interior and Knudsen boundary layers expansions, respectively.

\subsection{Interior expansion}

From plugging the interior expansion $ \sum_{k \ge 0} \eps^k F_k (t, x, v)$ into \eqref{BE} and collecting the same orders,
\begin{equation}\label{Interior_Ord_Ana}
  \begin{aligned}
   \eps^{-1}: \qquad & 0 = B(F_0, F_0) \,,\\
   \eps^{0}: \qquad & (\p_t + v \cdot \nabla_x) F_0 = B(F_0, F_1) + B(F_1, F_0)\,,\\
   \eps^{1}: \qquad & (\p_t + v \cdot \nabla_x) F_1 = B(F_0, F_2) + B(F_2, F_0) + B(F_1, F_1)\,,\\
   ......&\\
   \eps^k: \qquad &  (\p_t + v \cdot \nabla_x) F_k = B(F_0, F_{k+1}) + B(F_{k+1}, F_0) + \sum_{\substack{i+j = k+1\,,\\i,j \ge 1}} B(F_i, F_j)\,.
  \end{aligned}
\end{equation}
Together the well-known $H$-theorem, the order of $\eps^{-1}$ in \eqref{Interior_Ord_Ana} implies that $F_0 (t,x,v)$ must be a local Maxwellian, hence,
\begin{equation}\label{Local_Maxwellian}
   F_0 (t, x, v) = \M (t, x, v) \,.
\end{equation}
Then the order of $\eps^0$ in \eqref{Interior_Ord_Ana} implies that $(\rho, \u, T)$ satisfy the compressible Euler equations \eqref{Compressible_Euler_Sys}, and the kinetic part of $F_1$ subjects to \eqref{KinPart-12}. For $k \geq 1$, it is easy to derive from projecting, respectively, the order of $\eps^k$ in \eqref{Interior_Ord_Ana} into fluid and kinetic parts that the corresponding fluid variables $(\rho_k, u_k, \theta_k)$ obey the linear hyperbolic system
\begin{equation}\label{Linear-Hyperbolic-Syst-k}
  \left\{
    \begin{aligned}
      & \p_t \rho_k + \div_x (\rho u_k + \rho_k \u) = 0 \,, \\
      & \rho \big( \p_t u_k + u_k \cdot \nabla_x \u + \u \cdot \nabla_x u_k \big) - \tfrac{\nabla_x (\rho T)}{\rho} \rho_k + \nabla_x \big( \tfrac{\rho \theta_k + 3 T \rho_k}{3}\big) = \mathcal{F}^\bot_u (\tfrac{F_k}{\sqrt{\M}}) \,, \\
      & \rho \big( \p_t \theta_k + \u \cdot \nabla_x \theta_k + \tfrac{2}{3} \big( \theta_k \div_x \u + 3 T \div_x u_k \big) + 3 u_k \cdot \nabla_x T \big) = \mathcal{F}^\bot_\theta (\tfrac{F_k}{\sqrt{\M}})\,,
    \end{aligned}
  \right.
\end{equation}
and the kinetic parts satisfy
\begin{equation}\label{Ker-Arth-Inte}
(\mathcal{I} - \mathcal{P} ) \big(\tfrac{F_k}{\sqrt{\M}}\big) = \mathcal{L}^{-1} \Big( - \tfrac{(\p_t + v \cdot \nabla_x ) F_{k-1} - \sum_{\substack{i+j = k\,,\\ i, j \ge 1}} B (F_i, F_j) }{\sqrt{\M}} \Big) \,,
\end{equation}
where
\begin{equation*}
  \begin{aligned}
    \mathcal{P} ( \tfrac{F_k}{\sqrt{\M}} ) = \Big\{ \tfrac{\rho_k}{\rho} + u_k \cdot \tfrac{v - \u}{T} + \tfrac{\theta_k}{6 T} ( \tfrac{|v - \u|^2}{T} - 3 ) \Big\} \sqrt{\M} \,,
  \end{aligned}
\end{equation*}
$(\rho, \u, T)$ is the smooth solution of compressible Euler equations \eqref{Compressible_Euler_Sys}, the source terms $\mathcal{F}^\bot_u (\tfrac{F_k}{\sqrt{\M}})$ and $\mathcal{F}^\bot_\theta (\tfrac{F_k}{\sqrt{\M}})$ are defined as
\begin{equation}
  \begin{aligned}
    & \mathcal{F}^\bot_{u, i} (\tfrac{F_k}{\sqrt{\M}}) = - \sum_{j=1}^3 \p_{x_j} \l T \A_{ij}, \tfrac{F_k}{\sqrt{\M}} \r_v \ (i = 1,2,3) \,, \\
    & \mathcal{F}^\bot_\theta (\tfrac{F_k}{\sqrt{\M}}) = - \div_x \big( 2 T^{\frac{3}{2}} \l \B, \tfrac{F_k}{\sqrt{\M}} \r_v + 2 T \u \cdot \l \A, \tfrac{F_k}{\sqrt{\M}} \r_v \big) - 2 \u \cdot \mathcal{F}^\bot_u (\tfrac{F_k}{\sqrt{\M}})\,.
  \end{aligned}
\end{equation}

\subsection{Impermeable boundary condition $\u^0 \cdot n = 0$}\label{Subsec:2.2}

From substituting the expansion \eqref{Exp1} into the complete diffusive boundary condition \eqref{Diffusive_BC}, the leading order reads
\begin{equation*}
  \begin{aligned}
    \gamma_- \M = M_w \int_{v' \cdot n > 0} (v' \cdot n) \gamma_+ \M \d v' \ \textrm{ on } \Sigma_- \,.
  \end{aligned}
\end{equation*}
Integrating the above equality on $\{ v \cdot n < 0 \}$ with respect to the measure $(v \cdot n) \d v$,
\begin{equation*}
  \begin{aligned}
    \int_{v \cdot n < 0} (v \cdot n) \M^0 \d v = & \int_{v \cdot n < 0} (v \cdot n) M_w \int_{v^\prime \cdot n >0} (v^\prime \cdot n ) \M^0 \d v^\prime \d v \\
    = & \int_{v \cdot n < 0} (v \cdot n) M_w \d v \int_{v^\prime \cdot n >0} (v^\prime \cdot n ) \M^0 \d v^\prime \\
    = & - \int_{v^\prime \cdot n >0} (v^\prime \cdot n ) \M^0 \d v^\prime \,,
  \end{aligned}
\end{equation*}
where the condition \eqref{Mw-Condition} has been utilized. Consequently,
\begin{equation*}
  \begin{aligned}
    \int_{\R^3} (v \cdot n) \M^0 \d v = 0 \,.
  \end{aligned}
\end{equation*}
Observing $\int_{\R^3} [(v - \u^0) \cdot n] \M^0 \d v = 0$, one therefore has
\begin{equation*}
  \begin{aligned}
    0 = \u^0 \cdot n \int_{\R^3} \M^0 \d v = \rho^0 \u^0 \cdot n \,,
  \end{aligned}
\end{equation*}
hence $\u^0 \cdot n = 0$, the boundary condition of the compressible Euler system \eqref{Compressible_Euler_Sys}.

\subsection{Knudsen boundary layers expansion}


Together with \eqref{Interior_Ord_Ana}, it is derived from plugging the expansion \eqref{Exp1} into \eqref{BE} that
\begin{equation}\label{KL_Order_Ana}
  \begin{aligned}
   \eps^0: \qquad & v_3 \cdot \p_\xi F^{bb}_1 = \big[ B(\M^0, F^{bb}_1) + B(F^{bb}_1, \M^0) \big] \,,\\
   ......&\\
   \eps^k: \qquad  & \p_t F^{bb}_k + \bar{v} \cdot \nabla_{\bar{x}} F^{bb}_k + v_3 \p_\xi F^{bb}_{k+1} = \big[ B(\M^0, F^{bb}_{k+1}) + B(F^{bb}_{k+1}, \M^0) \big] \\
    & \qquad \qquad \qquad +\sum_{\substack{i+j=k+1\,,\\ i, j \ge 1}} \big[ B(F^{bb}_i, F^{bb}_j) + B(F^0_i, F^{bb}_j) + B(F^{bb}_j, F^0_i) \big] \\
    & \qquad \qquad \qquad + \sum_{\substack{l+j=k+1\,,\\ 1 \le l \le N, j \ge 1}} \tfrac{\xi^l}{l !} \big[ B(\M^{(l)}, F^{bb}_j) + B(F^{bb}_j, \M^{(l)}) \big] \\
    & \qquad \qquad \qquad + \sum_{\substack{l+i+j=k+1\,,\\ 1 \le l \le N, i, j \ge 1}} \tfrac{\xi^l}{l !} \big[ B(F_i^{(l)}, F^{bb}_j) + B(F^{bb}_j, F_i^{(l)}) \big] \quad {\text{for}}\ k \ge 1\,.
  \end{aligned}
\end{equation}
Here the Taylor expansions of $\M$ and $F_k$ at $x_3=0$ are used:
\begin{equation*}
  \begin{aligned}
   \M = & \M^0 + \sum_{1 \le l \le N} \tfrac{\xi^l}{l !} \M^{(l)} + \tfrac{\xi^{N+1}}{(N+1)!} \widetilde{\M}^{(N+1)}\,,\\
   F_i = & F_i^0 + \sum_{1 \le l \le N} \tfrac{\xi^l}{l !} F_i^{(l)} + \tfrac{\xi^{N+1}}{(N+1)!} \widetilde{F}_i^{(N+1)}\,,
  \end{aligned}
\end{equation*}
where $N=2$ in the truncated expansion form \eqref{Hilbert_Expansion1}. Furthermore, by plugging \eqref{Exp1} into the complete diffusive boundary condition \eqref{Diffusive_BC},
\begin{equation}\label{Diffusive_BC_Order_Ana}
  \begin{aligned}
    \gamma_- (F_k + F^{bb}_k) = M_w \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \gamma_+ (F_k + F^{bb}_k) \d v^\prime \quad {\text{on }} \Sigma_- \,.
  \end{aligned}
\end{equation}
for $k \geq 1$.

Let $f_k = \frac{F_k}{\sqrt{\M}}$ and $f^{bb}_k = \frac{F^{bb}_k}{\sqrt{\M^0}}$. \eqref{KL_Order_Ana} and \eqref{Diffusive_BC_Order_Ana} imply
\begin{equation}\label{Kudesen_Layer_Eq}
 \left\{
  \begin{aligned}
   &v_3 \p_{\xi} f^{bb}_k + \mathcal{L}^0 f^{bb}_k = \mathcal{S}^{bb}_k \,,\\
   &(\mathcal{I} - \mathcal{D}_w) f^{bb}_k |_{v_3 >0}= - (\mathcal{I} - \mathcal{D}_w) f_k |_{v_3 >0} \,,\\
   &\lim_{\xi \to +\infty} f^{bb}_k (t, \bar{x}, \xi, v) = 0\,,
  \end{aligned}
  \right.
\end{equation}
where the diffusive boundary operator $\mathcal{D}_w$ is given in \eqref{Dw}, and $S^{bb}_k = S^{bb}_{k,1} + S^{bb}_{k,2}$ with
\begin{equation}\label{S_bb_k+1_1+2}
 \begin{aligned}
   S^{bb}_{k,1} = & -\mathcal{P}^0 \Big\{  \tfrac{\big(\p_t + \bar{v} \cdot \nabla_{\bar{x}} \big) F^{bb}_{k-1}}{\sqrt{\M^0}} \Big\} \in \mathcal{N}^0 \,,\\
   S^{bb}_{k,2} = & - (\mathcal{I} - \mathcal{P}^0) \Big\{ \tfrac{ \big(\p_t + \bar{v} \cdot \nabla_{\bar{x}} \big) F^{bb}_{k-1}}{\sqrt{\M^0}} \Big\} \\
   + &  \tfrac{1}{\sqrt{\M^0}} \sum_{\substack{i+j =k,\\ i, j \ge 1}} \big[ B(\sqrt{\M^0} f^{bb}_i, \sqrt{\M^0} f^{bb}_j) + B(F^0_i, \sqrt{\M^0} f^{bb}_j) + B( \sqrt{\M^0} f^{bb}_j, F^0_i) \big] \\
   + & \sum_{\substack{l+j =k,\\ 1 \le l \le N, j \ge 1}} \tfrac{\xi^l}{l!} \tfrac{1}{\sqrt{\M^0}} \big[ B(\M^{(l)}, \sqrt{\M^0} f^{bb}_j) + B(\sqrt{\M^0} f^{bb}_j, \M^{(l)}) \big]  \\
   + & \sum_{\substack{l+i+j =k,\\ 1 \le l \le N, i, j\ge 1}} \tfrac{\xi^l}{l!} \tfrac{1}{\sqrt{\M^0}} \big[ B(F^{(l)}_i, \sqrt{\M^0} f^{bb}_j) + B(\sqrt{\M^0} f^{bb}_j, F^{(l)}_i) \big] \in \big( \mathcal{N}^0 \big)^\bot \,.
 \end{aligned}
\end{equation}
Here the notation $F^{bb}_0 = 0$ is employed. Observe that $S^{bb}_{1,1} = S^{bb}_{1,2} = 0$.

First, the following lemma is quoted to deal with the source term $S^{bb}_{k,1} \in \mathcal{N}^0$.

\begin{lemma}[\cite{Bardos-Caflisch-Nicolaenko-1986-CPAM}]\label{Lmm-fbb-k1}
	Assume that
	\begin{equation}\label{S_bb_k1}
	\begin{aligned}
	S^{bb}_{k,1}= \big\{ a_k + b_k \cdot (v- \u^0) + c_k |v-\u^0|^2 \big\} \sqrt{\M^0}
	\end{aligned}
	\end{equation}
	satisfy
	\begin{align*}
	\lim_{\xi \to \infty} e^{\eta \xi} |(a_k, b_k, c_k) (t, \bar{x}, \xi)| = 0
	\end{align*}
	for some positive constant $\eta > 0$. Then there exists a function
	\begin{equation*}
	f^{bb}_{k,1} = \big\{ \Psi_k v_3 + \Phi_{k,1} v_3 (v_1 - \u^0_1) + \Phi_{k,2} v_3 (v_2 - \u^0_2) + \Phi_{k,3} + \Theta_k v_3 |v- \u^0|^2 \big\} \sqrt{\M^0} 	
	\end{equation*}
	such that
	\begin{equation*}
	\begin{aligned}
	v_3 \p_\xi f^{bb}_{k,1} - S^{bb}_{k,1} \in (\mathcal{N}^0)^\bot \,,
	\end{aligned}
	\end{equation*}
	where
	\begin{equation}\label{f_bb_k1_Coef}
	\begin{array}{l}
	\Psi_k (t, \bar{x}, \xi)
	= - \int_\xi^{+\infty} \big( \tfrac{2}{T^0} a_k + 3 c_k \big) (t, \bar{x}, s) \d s\,, \\ [1.5mm]
	\Phi_{k,i} (t, \bar{x}, \xi)
	= - \int_\xi^{+\infty} \tfrac{1}{T^0} b_{k,i} (t, \bar{x}, s) \d s\,,\ i=1,2\,, \\ [1.5mm]
	\Phi_{k,3} (t, \bar{x}, \xi)
	= - \int_\xi^{+\infty} b_{k,3} (t, \bar{x}, s) \d s\,, \\ [1.5mm]
	\Theta_k (t, \bar{x}, \xi) = \tfrac{1}{5(T^0)^2} \int_\xi^{+\infty} a_k (t, \bar{x}, s) \d s\,.
	\end{array}
	\end{equation}
	Moreover, there holds
	\begin{equation*}
	\begin{aligned}
	& |v_3 \p_\xi f^{bb}_{k,1} - S^{bb}_{k,1}| \leq C |(a_k, b_k, c_k) (t, \bar{x}, \xi)| \l v \r^4 \sqrt{\M^0} \,, \\
	& |f^{bb}_{k,1} (t, \bar{x}, \xi, v)| \leq C \l v \r^3 \sqrt{\M^0} \int_\xi^\infty |(a_k, b_k, c_k)| \to 0 \textrm{ as } \xi \to \infty \,.
	\end{aligned}
	\end{equation*}
\end{lemma}

Let $f_{k, 2}^{bb} = f^{bb}_k - f^{bb}_{k, 1}$. One thereby obtains the linear Knudsen boundary layer equation \eqref{f_bb_k_2}, i.e.,
\begin{equation*}
\left\{
\begin{aligned}
& v_3 \p_\xi f^{bb}_{k, 2} + \mathcal{L}^0 f^{bb}_{k, 2} = S^{bb}_{k, 2} - \big( v_3 \p_\xi f^{bb}_{k,1} - S^{bb}_{k, 1} + \mathcal{L}^0 f^{bb}_{k,1} \big) \in (\mathcal{N}^0 )^\perp \,, \\
& f^{bb}_{k,2} (t, \bar{x}, 0, v) |_{v_3 >0} = \mathcal{D}_w f^{bb}_{k, 2} (t, \bar{x}, 0, v) + \mathbbm{f}_k (t, \bar{x}, v) \,, \\
& \lim_{\xi \to +\infty} f^{bb}_{k,2} (t, \bar{x}, \xi, v) =0 \,.
\end{aligned}
\right.
\end{equation*}
From Coron, Golse and Sulem's work \cite{Coron-Golse-Sulem-1988-CPAM}, the solvability of the Knudsen boundary layer equation \eqref{f_bb_k_2} is
\begin{equation}\label{KL_Sol_Condition}
  \begin{aligned}
    \int_{v \cdot n < 0} ( v \cdot n ) \mathbbm{f}_k (t, \bar{x}, v) \sqrt{\M^0} \d v = 0 \,.
  \end{aligned}
\end{equation}
Recalling the condition \eqref{Mw-Condition}, i.e., $\int_{v \cdot n < 0} |v \cdot n| M_w (t,x,v) \d v = 1$, one derives from \eqref{KL_Sol_Condition} that
\begin{equation*}
  \begin{aligned}
    \int_{v \cdot n < 0} (v \cdot n) ( f_k + f^{bb}_{k,1} ) \sqrt{\M^0} \d v = \int_{v \cdot n < 0} (v \cdot n) \mathcal{D}_w ( f_k + f^{bb}_{k,1} ) \sqrt{\M^0} \d v \\
    = \int_{v \cdot n < 0} (v \cdot n) \tfrac{M_w}{\sqrt{\M^0}} \int_{v' \cdot n > 0} (v' \cdot n) ( f_k + f^{bb}_{k,1} ) \sqrt{\M^0} \d v' \sqrt{\M^0} \d v \\
    = - \int_{v' \cdot n > 0} (v' \cdot n) ( f_k + f^{bb}_{k,1} ) \sqrt{\M^0} \d v' \,,
  \end{aligned}
\end{equation*}
which means that the solvability \eqref{KL_Sol_Condition} is equivalent to
\begin{equation}\label{Solva_Condition_f_bb_k_2}
  \begin{aligned}
    \int_{\R^3} (v \cdot n) ( f_k + f^{bb}_{k,1} ) (t, \bar{x}, 0 ,v) \sqrt{\M^0} \d v = 0 \,.
  \end{aligned}
\end{equation}

{\bf Claim:} {\em The solvability \eqref{Solva_Condition_f_bb_k_2} implies the boundary condition of the linear hyperbolic system \eqref{Linear-Hyperbolic-Syst}:
\begin{align}\label{High_Ord_Fluid_BC}
  u^0_k \cdot n = T^0 (\Psi_k + 5 T^0 \Theta_k) (t, \bar{x}, 0) - \tfrac{1}{\rho^0} \int_{\R^3} (v \cdot n) (\mathcal{I} - \mathcal{P}^0) f_k (t, \bar{x}, 0, v) \sqrt{\M^0} \d v \,.
\end{align}
}

Indeed, after splitting $f_k$ as
\begin{equation*}
  \begin{aligned}
   f_k = \big\{ \tfrac{\rho_k}{\rho} + u_k \cdot \tfrac{v - u}{T} + \tfrac{3 \theta_k}{6 T} \big( \tfrac{|v-u|^2}{T} -3 \big) \big\} \sqrt{\M} + (\mathcal{I} - \mathcal{P}) f_k \,,
  \end{aligned}
\end{equation*}
the integral $\int_{\R^3} (v \cdot n) f_k (t, \bar{x}, 0, v) \sqrt{\M^0} \d v$ can be calculated by
\begin{align*}
  & \int_{\R^3} (v \cdot n) f_k (t, \bar{x}, 0, v) \sqrt{\M^0} \d v \\
  = & \int_{\R^3} (v \cdot n) (\mathcal{I} - \mathcal{P}^0) f_k (t, \bar{x}, 0, v) \sqrt{\M^0} \d v \\
  & + \int_{\R^3} (v - \u^0 ) \cdot n \big\{ \tfrac{\rho_k^0}{\rho^0} + u_k^0 \cdot \tfrac{v - u^0}{T^0} + \tfrac{3 \theta_k^0}{6 T^0} \big( \tfrac{|v-u^0|^2}{T^0} -3 \big) \big\} \M^0 \d v \\
  = & \rho^0 u^0_k \cdot n + \int_{\R^3} (v \cdot n) (\mathcal{I} - \mathcal{P}^0) f_k (t, \bar{x}, 0, v) \sqrt{\M^0} \d v \,,
\end{align*}
where $\u^0 \cdot n =0$ has been used. On the other hand, by $\u^0 \cdot n = 0$ and the definition of $f^{bb}_{k,1}$ in Lemma \ref{Lmm-fbb-k1}, there holds
\begin{align*}
   & \int_{\R^3} (v \cdot n) f^{bb}_{k, 1} (t, \bar{x}, 0, v) \sqrt{\M^0} \d v \\
   = & \int_{\R^3} [(v - u^0) \cdot n] \big( \Psi_k v_3 + \sum_{i=1,2} \Phi_{k, i} v_3 (v_i - u^0_i) + \Phi_{k, 3} + \Theta_k v_3 |v-u^0|^2 \big) \M^0 \d v \\
   = & - \rho^0 T^0 (\Psi_k + 5 T^0 \Theta_k)\,.
\end{align*}
Collecting the above equalities reduces to the claim \eqref{High_Ord_Fluid_BC}.

Finally, the truncation of the Hilbert expansion \eqref{Exp1} is done as in \eqref{Hilbert_Expansion1}, whose remainder $F_{R,\eps}$ obeys the equations \eqref{Remainder_Eq} with complete diffusive boundary condition \eqref{Remainder_BC}.


\section{Uniform bounds for expanded coefficients}\label{Sec:F1F2}

In this section, the uniform bounds for the expansion terms $F_k$ and $F^{bb}_k$ ($k=1,2$) shall be derived, which will be applied to control the coefficients in the remainder equation \eqref{Remainder_Eq}. As shown in Section \ref{Sec:FormalAnaly}, $F_1$ is completely determined and the fluid part of $F_2$ is contained in the remainder term $F_{R,\eps}$. The kinetic part of $F_1$ is solved by the leading Maxwellian $\M$, who is governed by the compressible Euler system \eqref{Compressible_Euler_Sys} and is the known as in Proposition \ref{Proposition_Compressible_Euler}. The kinetic part of $F_2$ is solved by $F_1$ (see \eqref{KinPart-12} above). Consequently, once the fluid variables $(\rho_1, u_1, \theta_1)$ are solved, so are $F_k$ ($k=1,2$). Furthermore, $F^{bb}_k$ ($k=1,2$) are dominated by the linear Knudsen boundary layer equation with diffusive boundary condition, which has been solved in the $L^2(e^{\eta \xi} \d \xi ; L^2 (|v_3| \d v))$ space by Coron-Golse-Sulem \cite{Coron-Golse-Sulem-1988-CPAM}, and recently for which a pointwise estimate has been derived in \cite{Huang-Wang-arXiv-2021} by Huang-Wang.

First, as shown in Section \ref{Sec:FormalAnaly} above, the fluid variables $(\rho_1, u_1, \theta_1)$ of $F_1$ subject to the linear hyperbolic system \eqref{Linear-Hyperbolic-Syst} with boundary condition \eqref{BC-LinearHypo}. The corresponding initial conditions are given in \eqref{IC-Linear-Hyperbolic}. By \cite{GHW-2020}, the following conclusion holds:
\begin{lemma}\label{Lemma_Local_Sol_Hyperbolic_Sys}
	Assume that
	\begin{equation*}
	  \begin{aligned}
	    \mathbf{E}_1 : = & \|(\rho_1^{in}, u_1^{in}, \theta_1^{in})\|^2_{\mathcal{H}^k (\R^3_+)} \\
	    & + \sup_{t \in (0, \tau)} \big[\|(\mathcal{F}^\perp_\theta (f_1), \mathcal{F}^\perp_u (f_1))(t)\|^2_{\mathcal{H}^{k+1} (\R^3_+)} + \|\mathcal{J} (\M^0) (t)\|_{\mathcal{H}^{k+2} (\R^2)}^2 \big] < +\infty
	  \end{aligned}
	\end{equation*}
	with $k \ge 3$. Then there exists a unique smooth solution $(\rho_1, u_1, \theta_1)$ to \eqref{Linear-Hyperbolic-Syst}-\eqref{IC-Linear-Hyperbolic} with boundary condition \eqref{BC-LinearHypo} over $t \in [0, \tau]$ such that
	\begin{equation}\label{Energy_Inequ_LHS}
	  \begin{aligned}
	    \sup_{t \in [0, \tau]} {\|( \rho_1, u_1, \theta_1) (t)\|_{\mathcal{H}^k (\R^3_+)}^2} \le C (\tau, E_{k+2}) \mathbf{E}_1 \,,
	\end{aligned}
	\end{equation}
	where $E_{k+2} : = \sup_{t\in [0, \tau]} \| (\rho, \u, T) (t)\|_{H^{k+2}(\R^3_+)}$.
\end{lemma}

Second, the Knudsen boundary layers $F^{bb}_k$ will be controlled. Let
\begin{align*}
  w_{a, \beta} = \l v \r^\beta (\M^0)^{-a}
\end{align*}
with $\beta \ge 3$, $0 < a < \frac{1}{4}$. Denote by
\begin{equation*}
  \begin{aligned}
    \mathscr{S}_k : = S^{bb}_{k, 2} - \big( v_3 \p_\xi f^{bb}_{k,1} - S^{bb}_{k, 1} + \mathcal{L}^0 f^{bb}_{k,1} \big) \in (\mathcal{N}^0)^\perp \,.
  \end{aligned}
\end{equation*}
From Huang and Wang's work \cite{Huang-Wang-arXiv-2021}, the follow pointwise estimates to the linear Knudsen boundary layer equation \eqref{f_bb_k_2} hold:
\begin{lemma}\label{Lemma_Linear_Higher_Knudsen}
  Assume $\int_{v \cdot n < 0} (v \cdot n) \mathbbm{f}_k (t, \bar{x}, v) \sqrt{\M^0} \d v =0$. Let $\sigma_0 \in (0,1)$ be suitably small. For any fixed integer $m \geq 0$, if
  \begin{align}\label{LKL_E_m}
  	\mathcal{E}_m := \sup_{(t, \bar{x}) \in [0, \tau] \times \R^3_+} \sum_{|\alpha| \le m} \big( \| e^{\sigma_0 \xi} \nu^{-1} w_{a, \beta} \p_{t, \bar{x}}^\alpha \mathscr{S}_k \|_{L^\infty_{\xi, v}} + \| w_{a, \beta} \p_{t, \bar{x}}^\alpha \mathbbm{f}_k \|_{L^\infty_v} \big) < \infty \,,
  \end{align}
  then the linear Knudsen boundary layer equation \eqref{f_bb_k_2} admits a unique solution $f^{bb}_{k,2} (t, \bar{x}, \xi, v)$ satisfies
  \begin{equation}\label{Linear_Knudsen_Eq_Est_Higher}
  	\sup_{(t, \bar{x}) \in [0, \tau] \times \R^3_+} \sum_{|\alpha| \le m} \big( \| e^{\sigma \xi}  w_{a, \beta} \p_{t, \bar{x}}^\alpha f^{bb}_{k,2} \|_{L^\infty_{\xi, v}} + \| w_{a, \beta} \p_{t, \bar{x}}^\alpha f^{bb}_{k,2} |_{\xi =0} \|_{L^\infty_v} \big)
  	\le \tfrac{C}{\sigma_0 - \sigma} \mathcal{E}_m
  \end{equation}
  for some constant $C > 0$ and any $\sigma \in (0, \sigma_0)$.
\end{lemma}

Recall the relations \eqref{Relt1}, \eqref{KinPart-12}, \eqref{BC-LinearHypo}, \eqref{Relt2}, \eqref{Relt3} and \eqref{Relt4}. Together with the standard Sobolev theory, the following proposition is immediately implied by Lemma \ref{Lmm-fbb-k1}, \ref{Lemma_Local_Sol_Hyperbolic_Sys} and \ref{Lemma_Linear_Higher_Knudsen}. Similar arguments can be also found in \cite{GHW-2020}.

\begin{proposition}\label{Prop-HigherOrd}
	Let $\beta \geq 3$ and $0 < \tfrac{1}{4 z} (1 - z) < a < \tfrac{1}{4}$, where $z \in (\tfrac{1}{2}, 1)$ is given in \eqref{M-Bound}. Let integers $s_1 \geq 3$ and $s_0$ in Proposition \ref{Proposition_Compressible_Euler} be taken as $s_0 = s_1 + 3$. $(\rho, \u, T)$ is constructed in Proposition \ref{Proposition_Compressible_Euler} and generates the local Maxwellian $\M$. If $(\rho^{in}_1, u^{in}_1, \theta^{in}_1)$ in \eqref{IC-Linear-Hyperbolic} satisfy $\mathcal{E}_1^{in} = \| (\rho^{in}_1, u^{in}_1, \theta^{in}_1) \|_{\mathcal{H}^{s_1} (\R^3_+)} < \infty$, then there are solutions $F_k = \sqrt{\M} f_k$ and $F^{bb}_k = \sqrt{\M^0} f^{bb}_k$ $(k=1,2)$ constructed in Section \ref{Sec:FormalAnaly} over the time interval $t \in [0, \tau]$ with the bound
	\begin{equation}
	  \begin{aligned}
	    \sup_{t \in [0, \tau]} \Big\{ \sum_{|\alpha| + i \leq s_1} \| \l v \r^\beta \M^{-a} \p^\alpha_{t, \bar{x}} \p^i_{x_3} f_1 (t) \|_{L^2_x L^\infty_v} + \sum_{|\alpha| + i \leq s_1 - 1} \| \l v \r^\beta \M^{-a} \p^\alpha_{t, \bar{x}} \p^i_{x_3} f_2 (t) \|_{L^2_x L^\infty_v} \\
	    + \sum_{|\alpha| \leq s_1} \| e^{\frac{\xi}{2}} \l v \r^\beta (\M^0)^{-a} \p_{t, \bar{x}}^\alpha f^{bb}_1 (t) \|_{L^\infty_{\bar{x}, \xi, v}} + \sum_{|\alpha| \leq s_1 - 1} \| e^{\frac{\xi}{4}} \l v \r^\beta (\M^0)^{-a} \p_{t, \bar{x}}^\alpha f^{bb}_2 (t) \|_{L^\infty_{\bar{x}, \xi, v}} \Big\} \\
	    \leq C(\tau, \mathcal{E}^{in}_0, \mathcal{E}^{in}_1) < \infty \,,
	  \end{aligned}
	\end{equation}
	where $\mathcal{E}^{in}_0$ is mentioned in Theorem \ref{Main-Thm}.
\end{proposition}

Remark that the Knudsen boundary layers $F^{bb}_k$ ($k=1,2$) decay exponentially associated with $\xi > 0$. These suffice to dominate the $\xi$-polynomial factors appeared in the quantity $R^{bb}_\eps$ in \eqref{R_eps_bb} when deriving the uniform $L^2$-$L^\infty$ bounds for the remainder $F_{R, \eps}$.


\section{Uniform bounds for remainder $F_{R, \eps}$: Proof of Theorem \ref{Main-Thm}}\label{Sec:Remainder}


In this section, the $L^2$-$L^\infty$ arguments \cite{Guo-2010-ARMA,Guo-Jang-Jiang-2010-CPAM} will be applied to prove the main theorem. This approach is sufficient to estimate norms $\| f_{R,\eps} (t) \|_2$ and $\| h_{R,\eps}^\ell (t) \|_\infty$ for the remainder $F_{R,\eps}$. Here $f_{R,\eps}$ and $h_{R,\eps}^\ell$ are given in \eqref{f_Reps-h_Reps}, i.e.,
\begin{equation*}
  \begin{aligned}
    f_{R, \eps} = \tfrac{F_{R, \eps}}{\sqrt{\M}} \,, \quad h_{R, \eps}^\ell = \l v \r^\ell \tfrac{F_{R,\eps}}{\sqrt{\M_M}} \,.
  \end{aligned}
\end{equation*}
The proof relies on an interplay between $L^2$ and $L^\infty$ estimates for the Boltzmann equation.

Remark that the known coefficients $\M$, $F_k$, $F^{bb}_k$ ($k=1,2$) and quantities $R_\eps$, $R^{bb}_\eps$ composed of $\M$, $F_k$, $F^{bb}_k$ are involved in the remainder equation \eqref{Remainder_Eq} with complete diffusive boundary condition \eqref{Remainder_BC}. Thanks to Proposition \ref{Proposition_Compressible_Euler} and \ref{Prop-HigherOrd}, these knowns can be bounded by harmless constants $C > 0$ independent of $\eps$ in the $L^2$-$L^\infty$ arguments, see also the similar arguments in Section 6 of \cite{GHW-2020}. For simplicity, the corresponding details will be omitted.

There are two main lemmas to finish the proof of Theorem \ref{Main-Thm}.

\begin{lemma}[$L^2$ Estimates]\label{Lmm-L2}
	Under the same assumptions in Proposition \ref{Prop-HigherOrd} and $\ell \geq 7$, let $(\rho, \u, T)$ be a smooth solution to the Euler equations \eqref{Compressible_Euler_Sys} over $t \in [0, \tau]$ obtained in Proposition \ref{Proposition_Compressible_Euler}. Let $c_0 > 0$ be given in \eqref{Hypocoercivity}, and the wall Maxwellian $M_w$ be assumed in Theorem \ref{Main-Thm}. Then there are constants $\eps'_0 > 0$ and $C = (\M, F_k, F^{bb}_k; k=1,2) > 0$ such that for all $0 < \eps < \eps_0'$ and $t \in [0, \tau]$,
	\begin{equation}
	  \begin{aligned}
	    \tfrac{1}{2} \tfrac{\d}{\d t} \| f_{R,\eps} (t) \|_2^2 + \tfrac{c_0}{2 \eps} \| (\mathcal{I} - & \mathcal{P}) f_{R, \eps} (t) \|_\nu^2 +\tfrac{1}{2} \iint_{\Sigma_+} |v \cdot n| \big| (\mathcal{I} - \mathcal{D}_w) f_{R, \eps} (t) \big|^2 \d v \d \bar{x} \\
	    \le & C \sqrt{\eps} \| \sqrt{\eps}^3 h^\ell_{R, \eps} (t) \|_{\infty} \| f_{R, \eps} (t) \|_2 + C \| f_{R, \eps} (t) \|_2 \big( 1 + \|f_{R,\eps} (t) \|_2 \big)\,.
	  \end{aligned}
	\end{equation}
\end{lemma}

\begin{lemma}[$L^\infty$ Estimates]\label{Lmm-Linfty}
	Under the same assumptions in Lemma \ref{Lmm-L2}, there are constants $\eps''_0 > 0$ and $C = (\M, F_k, F^{bb}_k; k=1,2) > 0$ such that for all $0 < \eps < \eps_0''$ and $t \in [0, \tau]$,
	\begin{equation}\label{h-infty-est}
	  \sup_{s \in [0, t]} \| \sqrt{\eps}^3 h^\ell_{R, \eps} (s) \|_{\infty} \le C \big( \| \sqrt{\eps}^3 h^\ell_{R, \eps} (0) \|_{\infty} + \sup_{0 \le s \le \tau} \| f_{R, \eps} (s) \|_2 + \sqrt{\eps}^5 \big) \,.
	\end{equation}
\end{lemma}

\begin{proof}[Proof of Theorem \ref{Main-Thm}]
	From Lemma \ref{Lmm-L2} and \ref{Lmm-Linfty},
	\begin{equation*}
	  \begin{aligned}
	    \tfrac{\d}{\d t} \| f_{R,\eps} (t) \|_2 \leq C \sqrt{\eps} ( \| \sqrt{\eps}^3 h^{\ell}_{R, \eps} (0) \|_\infty + \sup_{0 \le s \le \tau} \| f_{R, \eps} (s) \|_2 + \sqrt{\eps}^5 ) + C ( 1 + \| f_{R,\eps} \|_2 )
	  \end{aligned}
	\end{equation*}
	holds for all $0 < \eps < \min \{ \eps_0', \eps_0'' \}$ and $t \in [0, \tau]$. The Gr\"onwall inequality yields
	\begin{equation*}
	  \begin{aligned}
	    \| f_{R,\eps} (t) \|_2 + 1 \leq (\| f_{R,\eps} (0) \|_2 + 1) e^{C t} + \sqrt{\eps} e^{C t} ( \| \sqrt{\eps}^3 h_{R,\eps} (0) \|_\infty + \sup_{t \in [0, \tau]} \| f_{R,\eps} (t) \|_2 + \sqrt{\eps}^5 )  \,.
	  \end{aligned}
	\end{equation*}
	Choosing $\eps_0 : = \min \{ \eps_0', \eps_0'', \tfrac{1}{4} e^{- 2 C \tau} \} > 0$, there holds
	\begin{equation*}
	\begin{aligned}
	  \sup_{t \in [0, \tau]} \| f_{R,\eps} (t) \|_2 \leq C_\tau \big( 1 + \| f_{R, \eps} (0) \|_2 + \| \sqrt{\eps}^3 h_{R,\eps}^\ell (0) \|_\infty \big)
	\end{aligned}
	\end{equation*}
	for some $C_\tau > 0$ and any $0 < \eps < \eps_0$, which, together with \eqref{h-infty-est}, finishes the proof of Theorem \ref{Main-Thm}.
\end{proof}

\subsection{$L^2$ estimates for $f_{R,\eps}$: Proof of Lemma \ref{Lmm-L2}}

From the definition of $f_{R,\eps}$ in \eqref{f_Reps-h_Reps} and the equation of $F_{R, \eps}$ in \eqref{Remainder_Eq} with diffusive boundary condition \eqref{Remainder_BC}, $f_{R,\eps}$ subjects to
\begin{equation}\label{Remainder_f}
 \begin{aligned}
   \p_t f_{R, \eps} + v \cdot \nabla_x f_{R, \eps} & + \tfrac{1}{\eps} \mathcal{L} f_{R, \eps}
   \\
   = & - \tfrac{(\p_t + \bar{x} \cdot \nabla_{\bar{x}}) \sqrt{\M}}{\sqrt{\M}} f_{R, \eps} + \eps \tfrac{1}{\sqrt{\M}} B (\sqrt{\M} f_{R, \eps}, \sqrt{\M} f_{R, \eps}) + \tfrac{1}{\sqrt{\M}} (R_\eps + R^{bb}_\eps) \\
   & + \sum_{i=1}^2 \eps^{i-1} \tfrac{1}{\sqrt{\M}} \big[ B ( F_i + F^{bb}_i , \sqrt{\M} f_{R, \eps}) + B ( \sqrt{\M} f_{R, \eps}, F_i + F^{bb}_i) \big]
 \end{aligned}
\end{equation}
with
\begin{equation}\label{Remainder_BC_f}
  \gamma_- f_{R, \eps} = \mathcal{D}_w f_{R, \eps} \ \textrm{ on } \Sigma_- \,,
\end{equation}
and initial data
\begin{equation}\label{Remainder_IC_f}
 f_{R, \eps} (0, x, v) = f^{in}_{R, \eps} (x, v) : = \tfrac{F_{R, \eps}^{in} (x,v)}{\sqrt{\M^{in}}} \,.
\end{equation}
Taking inner product with $f_{R, \eps}$ with respect to $(x, v)$ on $\R^3_+ \times \R^3$, one has
\begin{equation}\label{Remainder_L2-1}
  \begin{aligned}
    \tfrac{1}{2} \tfrac{\d}{\d t} \| f_{R,\eps} \|_2^2 +  \tfrac{1}{\eps} \l \mathcal{L} f_{R, \eps}, f_{R, \eps} \r - \tfrac{1}{2} \iint_{\Sigma} v_3 \big| f_{R, \eps} (t, \bar{x}, 0, v) \big|^2 \d v \d \bar{x} \\
    = - \iint_{\R^3_+ \times R^3} \tfrac{(\p_t + v \cdot \nabla_{x}) \sqrt{\M}}{\sqrt{\M}} |f_{R, \eps}|^2 \d v \d x + \eps \iint_{\R^3_+ \times R^3} \tfrac{1}{\sqrt{\M}} B(\sqrt{\M} f_{R, \eps}, \sqrt{\M} f_{R, \eps}) f_{R, \eps} \d v \d x \\
    + \sum_{i=1}^2 \eps^{i-1} \iint_{\R^3_+ \times R^3} \tfrac{1}{\sqrt{\M}} \big[ B (F_i + F^{bb}_i, \sqrt{\M} f_{R, \eps}) + B( \sqrt{\M} f_{R, \eps}, F_i + F^{bb}_i) \big] f_{R, \eps} \d v \d x \\
    + \iint_{\R^3_+ \times R^3} \tfrac{1}{\sqrt{\M}} (R_\eps + R^{bb}_\eps) f_{R, \eps} \d v \d x\,.
  \end{aligned}
\end{equation}


Following the arguments of \cite{Guo-Jang-Jiang-2010-CPAM}, the terms on the right-hand side of \eqref{Remainder_L2-1} can be bounded by
\begin{equation}\label{Remainder_L2_RHS}
  \begin{aligned}
    {\text{RHS\ of\ \eqref{Remainder_L2-1}}}
    \le C_\lambda \eps^2 \| h^\ell_{R, \eps} \|_{\infty} \| f_{R, \eps} \|_2 + C \| f_{R, \eps} \|_2 (1 + \| f_{R, \eps} \|_2) + \tfrac{C \lambda^2}{\eps} \| (\mathcal{I} - \mathcal{P}) f_{R, \eps} \|_{\nu}^2
  \end{aligned}
\end{equation}
for some small $\lambda > 0$ to be determined, where $C_\lambda >0$ is a constants depending on $\lambda$. Furthermore, the inequality \eqref{Hypocoercivity} reduces to
\begin{equation}\label{L2-hypercoer}
  \begin{aligned}
    \tfrac{1}{\eps} \l \mathcal{L} f_{R, \eps}, f_{R, \eps} \r \geq \tfrac{c_0}{\eps} \| (\I - \P) f_{R, \eps} \|^2_\nu \,.
  \end{aligned}
\end{equation}
It remains to deal with the boundary integral term $- \tfrac{1}{2} \iint_{\Sigma} v_3 \big| f_{R, \eps} (t, \bar{x}, 0, v) \big|^2 \d v \d \bar{x}$ in \eqref{Remainder_L2-1}. In fact, there holds
\begin{align}\label{BC-integral-f}
  - \tfrac{1}{2} \iint_{\Sigma} v_3 |f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \d \bar{x} = \tfrac{1}{2} \iint_{\Sigma_+} |v \cdot n| \big| (\mathcal{I} - \mathcal{D}_w) f_{R, \eps} \big|^2 \d v \d \bar{x}
\end{align}
under the diffusive boundary condition \eqref{Remainder_BC_f}. Indeed, by \eqref{Remainder_BC_f} and $v \cdot n = - v_3$,
\begin{align*}
  & - \tfrac{1}{2} \iint_{\Sigma} v_3 |f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \d \bar{x} \\
  = & \tfrac{1}{2} \iint_{\Sigma_+} v \cdot n |f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \d \bar{x} + \tfrac{1}{2} \iint_{\Sigma_-} v \cdot n |\mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \d \bar{x} \,.
\end{align*}
Recalling the definition of the diffusive boundary operator $\mathcal{D}_w$ in \eqref{Dw}, the integral $\int_{v \cdot n < 0} (v \cdot n) |\mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v $ can be calculated by
\begin{equation*}
  \begin{aligned}
    & \int_{v \cdot n < 0} (v \cdot n) |\mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \\
    = & \Big( \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v^\prime) \sqrt{\M^0} \d v^\prime \Big)^2 \int_{v \cdot n < 0} (v \cdot n ) \tfrac{M_w^2}{\M^0} \d v \\
    = & - \tfrac{1}{\rho_0} \sqrt{\tfrac{2 \pi}{T^0}} \Big( \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v^\prime) \sqrt{\M^0} \d v^\prime \Big)^2 \,,
  \end{aligned}
\end{equation*}
where one has used the fact $\int_{v \cdot n < 0} (v \cdot n ) \frac{M_w^2}{\M^0} \d v = \tfrac{1}{\rho_0} \sqrt{\tfrac{2 \pi}{T^0}}$ derived from the relations $\frac{M_w}{\M^0} = \frac{1}{\rho_0} \sqrt{\frac{2 \pi}{T^0} }$ and \eqref{Mw-Condition}, i.e., $\int_{v \cdot n < 0} |v \cdot n| M_w \d v = 1$.

Next, one focuses on the integral $\int_{v \cdot n > 0} (v \cdot n) |f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v$. By the decomposition $f_{R, \eps} = (\mathcal{I} - \mathcal{D}_w) f_{R, \eps} + \mathcal{D}_w f_{R, \eps}$,


\begin{equation*}
\begin{aligned}
  & \int_{v \cdot n >0} (v \cdot n) |f_{R, \eps} (t, \bar{x}, 0, v)|^2 \d v \\
  = & \int_{v \cdot n >0} (v \cdot n) \big| (\mathcal{I} - \mathcal{D}_w) f_{R, \eps} (t, \bar{x}, 0, v) \big|^2 \d v + \underbrace{\int_{v \cdot n >0} (v \cdot n) \big| \mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v) \big|^2 \d v }_{B_1} \\
  & + 2 \underbrace{ \int_{v \cdot n >0} (v \cdot n) \big( \mathcal{I} - \mathcal{D}_w \big) f_{R, \eps} (t, \bar{x}, 0, v) \cdot \mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v) \d v }_{B_2} \,.
\end{aligned}
\end{equation*}
By $\u^0 \cdot n = 0$ and $(u_w, T_w) = (\u^0, T^0)$, one has $\int_{\R^3} (v \cdot n) M_w \d v = 0$, which, combining with $\int_{v \cdot n < 0} (v \cdot n) M_w \d v = - 1$ given in \eqref{Mw-Condition}, implies that $\int_{v \cdot n > 0} (v \cdot n) M_w \d v = 1$. Together with the definition of $\mathcal{D}_w$ in \eqref{Dw} and $\frac{M_w}{\M^0} = \frac{1}{\rho_0} \sqrt{\frac{2 \pi}{T^0} }$,
\begin{align*}
  B_1 = & \int_{v \cdot n >0} (v \cdot n) \Big\{ \tfrac{M_w}{\sqrt{\M^0}} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v^\prime) \sqrt{\M^0} \d v^\prime \Big\}^2 \d v \\
  = & \tfrac{1}{\rho^0} \sqrt{\tfrac{2 \pi}{T^0}} \int_{v \cdot n >0} (v \cdot n) M_w \d v \Big( \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v^\prime) \sqrt{\M^0} \d v^\prime \Big)^2 \\
  = & \tfrac{1}{\rho^0} \sqrt{\tfrac{2 \pi}{T^0}} \Big( \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v^\prime) \sqrt{\M^0} \d v^\prime \Big)^2 \,.
\end{align*}
For $B_2$,
\begin{align*}
  B_2 = & \int_{v \cdot n >0} (v \cdot n) f_{R, \eps} (t, \bar{x}, 0, v) \mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v) \d v - \int_{v \cdot n >0} (v \cdot n) \big| \mathcal{D}_w f_{R, \eps} (t, \bar{x}, 0, v) \big|^2 \d v \\
  := & B_{21} - B_1 \,.
\end{align*}
The goal is to show that $B_{21}=B_1$, which means $B_2 = 0$. Actually, by $\frac{M_w}{\M^0} = \frac{1}{\rho_0} \sqrt{\frac{2 \pi}{T^0} }$,
\begin{align*}
B_{21} = & \int_{v \cdot n > 0} (v \cdot n) f_{R, \eps} (t, \bar{x}, 0, v) \tfrac{1}{\rho^0} \sqrt{\tfrac{2\pi}{T^0}} \sqrt{\M^0} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v' ) \sqrt{\M^0} \d v^\prime \d v \\
= & \tfrac{1}{\rho^0} \sqrt{\tfrac{2\pi}{T^0}} \Big( \int_{v^\prime \cdot n >0} (v^\prime \cdot n) f_{R, \eps} (t, \bar{x}, 0, v') \sqrt{\M^0} \d v^\prime \Big)^2 = B_1 \,.
\end{align*}
Thus $B_2 = 0$. Observing $\Sigma_+ = \R^2 \times \{ v: v \cdot n > 0 \}$ and collecting the above calculations, one concludes the claim \eqref{BC-integral-f}.

Consequently, from \eqref{Remainder_L2-1}, \eqref{Remainder_L2_RHS}, \eqref{L2-hypercoer} and \eqref{BC-integral-f},
\begin{equation}\label{Remainder_L2-2}
  \begin{aligned}
   & \tfrac{1}{2} \tfrac{\d}{\d t} \| f_{R,\eps} \|_2^2 + \tfrac{c_0}{\eps} \| (\mathcal{I} - \mathcal{P}) f_{R, \eps} \|_\nu^2 + \tfrac{1}{2} \iint_{\Sigma_+} |v \cdot n| \big| (\mathcal{I} - \mathcal{D}_w) f_{R, \eps} \big|^2 \d v \d \bar{x} \\
   \le & C_\lambda \eps^2 \| h^\ell_{R, \eps} \|_{\infty} \| f_{R, \eps} \|_2 + C \| f_{R, \eps} \|_2 \big( 1 + \|f_{R,\eps}\|_2 \big) + \tfrac{C \lambda^2}{\eps} \| (\mathcal{I} - \mathcal{P}) f_{R,\eps} \|_\nu^2
  \end{aligned}
\end{equation}
for small $\lambda > 0$ to be determined. Taking small $\lambda >0$ such that $c_0 - C \lambda^2 \ge \frac{1}{2} c_0 >0$, one then concludes the proof of Lemma \ref{Lmm-L2}.

\subsection{$L^\infty$ estimates for $h^\ell_{R, \eps}$: Proof of Lemma \ref{Lmm-Linfty}}

As in \cite{Guo-2010-ARMA}, the following backward $b$-characteristics of Boltzmann equation are required to derive the $L^\infty$ estimates for $h^\ell_{R, \eps}$. Given $(t, x, v)$, define the trajectory $[X(s), V(s)]$ as
\begin{equation*}
	\left\{
	  \begin{aligned}
	   & \tfrac{\d}{\d s} X(s) = V(s)\,,\quad
	   \tfrac{\d}{\d s} V(s) = 0\,, \\
	   & [X(t), V(t)] = [x, v] \,.
	  \end{aligned}
	\right.
\end{equation*}
The solution is given by
\begin{equation*}
  [X (s), V (s)] = [X(s; t, x, v), V(s; t, x, v)] = [x- (t -s)v, v] \,.
\end{equation*}
For any fixed $(x, v) \in \overline{\R}^3_+ \times \R^3$ with $v_3 \ne 0$, its backward exist time $t_b (x , v) \ge 0$ is defined as the last moment at which the back-time straight line $[X(s; 0, x, v), V(s; 0, x, v)]$ remains in $x \in \overline{\R}^3_+$. The backward exist time can be explicitly represented by
\begin{align*}
  t_b (x, v) = \sup\big\{ \tau_0 \ge 0: x - \tau_0 v \in \R^3_+ \big\} \,,
\end{align*}
which implies $x - t_b v \in \p \R^3_+$, i.e., $x_3 - t_b v_3 =0$. Denote by
$$
x_b (x, v) = x- t_b v \in \p \R^3_+ \,.
$$
Remark here that for the case $v \cdot n <0$, the cycle will hit the boundary for one time, see Figure \ref{tb-exist}. For the case $v \cdot n \geq 0$, the back-time trajectory is a straight line and does not intersect the boundary, see Figure \ref{tb-NOTexist}.


\begin{figure}[h]
	\centering
	\begin{tikzpicture}[node distance=2cm]
	\draw[-, very thick] node[left,scale=1]{$\p \R^3_+$}(0,0)--(3,0);
    \draw[->, thick] (2,-0.1)--(2,-0.6) node[below]{$n$};
    \draw[-,thick, dashed] (1,0)--(2,1);
    \draw[->,thick] (2,1)--(2.7,1.7) node[right]{$v$};
    \draw (2,1) node[below right, scale=1]{$x$};
    \draw (1.5, 0.5) node[above left]{$t_b (x,v)$};
    \draw[->,dashed, color=red] (2,1)--(3.8,1) node[right, color=black]{time $t$};
    \draw[->,dashed, color=red] (1,0)--(3.8,0) node[right, color=black]{time $t_1 = t - t_b (x,v)$};
    \draw (1,0) node[below]{$x_b (x,v)$};
    \draw[->, thick, color=blue] (0, 0.5)--(1,0);
    \draw (0.6,0.8) node[below]{\color{blue}$v'$};
	\end{tikzpicture}
	\caption{The case $v \cdot n < 0$: $t_b (x,v)$ exists.}\label{tb-exist}
\end{figure}

\begin{figure}[h]
	\centering
	\begin{tikzpicture}[node distance=2cm]
	\draw[-, very thick] node[above left,scale=1]{$\p \R^3_+$}(0,0.5)--(4,0.5);
	\draw[->, thick] (2,0.4)--(2,-0.1) node[below]{$n$};
	\draw[-,thick, dashed] (0.5,2.5)--(1.5,1.5);
	\draw[->,thick] (1.5,1.5)--(2,1) node[right]{$v$};
	\draw (1.5,1.5) node[below left, scale=1]{$x$};
	\draw[->,dashed, color=red] (1.5,1.5)--(4,1.5) node[right, color=black]{time $t$};
	\end{tikzpicture}
	\caption{The case $v \cdot n \geq 0$: $t_b (x,v)$ does not exists.}\label{tb-NOTexist}
\end{figure}

Given $x \in \overline{\R}^3_+$, $(x, v) \notin \Sigma_0 \cup \Sigma_-$ and $(t_0, v_0, x_0) = (t, x, v)$. The back-time cycle is defined as
\begin{equation}
\begin{aligned}
	X_{cl} (s; t, x, v) = & \mathbbm{1}_{[t_1, t_0)} (s) \big\{ x - (t-s) v \big\} + \mathbbm{1}_{(-\infty, t_1)} (s) \big\{ x - (t- t_1) v - (t_1 - s) v^\prime \big\} \,,\\
	V_{cl} (s; t, x, v) = & \mathbbm{1}_{[t_1, t_0)} (s) v + \mathbbm{1}_{(-\infty, t_1)} (s) v^\prime \,,
\end{aligned}
\end{equation}
where $t_1 = t - t_b (x, v)$, $v^\prime \in \mathcal{V} := \{v^\prime: v^\prime \cdot n >0\}$, see Figure \ref{tb-exist}.

From \eqref{f_Reps-h_Reps}, one has
$$
h^\ell_{R, \eps} = \tfrac{\sqrt{\M}}{\sqrt{\M_M}} \l v \r^\ell f_{R, \eps}\,.
$$
Then \eqref{Remainder_f}-\eqref{Remainder_BC_f} can be rewritten as
\begin{equation}\label{Remainder_h}
 \begin{aligned}
   \p_t h^\ell_{R, \eps} + v \cdot \nabla_x h^\ell_{R, \eps} + \tfrac{\nu (\M)}{\eps} h^\ell_{R, \eps} = q_{R,\eps}^\ell (t,x,v)
 \end{aligned}
\end{equation}
with the boundary condition
\begin{equation}\label{Remainder_BC_h}
  \gamma_- h^\ell_{R, \eps} = M_w  \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \tfrac{\sqrt{\M_M}}{\l v^\prime \r^\ell} h^\ell_{R, \eps} (t, x, v^\prime) \d v^\prime\ {\text{on}}\ \Sigma_-\,,
\end{equation}
where
\begin{equation}\label{q}
  \begin{aligned}
    q_{R,\eps}^\ell (t,x,v) = & \tfrac{1}{\eps} K_\ell h^\ell_{R, \eps} +\tfrac{\eps \l v \r^\ell}{\sqrt{\M_M}} B \big( \tfrac{\sqrt{\M_M}}{\l v \r^\ell} h^\ell_{R, \eps}, \tfrac{\sqrt{\M_M}}{\l v \r^\ell} h^\ell_{R, \eps} \big) + \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \big( R_\eps + R^{bb}_\eps \big) \\
    & + \sum_{i=1}^2 \eps^{i-1} \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \big\{ B \big( F_i + F^{bb}_i, \tfrac{\sqrt{\M_M}}{\l v \r^\ell} h^\ell_{R, \eps} \big) + B\big( \tfrac{\sqrt{\M_M}}{\l v \r^\ell} h^\ell_{R, \eps}, F_i + F^{bb}_i \big) \big\} \,,
  \end{aligned}
\end{equation}
and the operator $K_\ell g : = \l v \r^\ell K (\tfrac{g}{\l v \r^\ell})$ with
\begin{equation*}
  \begin{aligned}
    K h = & \int_{\R^3 \times \mathbb{S}^2} | (u -v) \cdot \omega | \sqrt{\M_M (u)} \tfrac{\M (v)}{\sqrt{\M_M (v)}} h (u) \d \omega \d u  \\
    & - \int_{\R^3 \times \mathbb{S}^2} | (u -v) \cdot \omega | \M (u^\prime) \tfrac{\sqrt{\M_M (v^\prime)}}{\sqrt{\M_M (v)}} h(v^\prime) \d \omega \d u \\
    & - \int_{\R^3 \times \mathbb{S}^2} | (u -v) \cdot \omega | \M (v^\prime) \tfrac{\sqrt{\M_M (u^\prime)}}{\sqrt{\M_M (v)}} h(u^\prime) \d \omega \d u \,.
  \end{aligned}
\end{equation*}

The goal of this subsection is to derive the pointwise estimates for $h_{R,\eps}^\ell (t,x,v)$ over $(t,x,v) \in [0, \tau] \times \R^3_+ \times \R^3$. According to the trajectories showed in Figure \ref{tb-exist} and \ref{tb-NOTexist}, the points set $[0, \tau] \times \R^3_+ \times \R^3$ can be clarified by following two parts:
\begin{equation}\label{UV}
  \begin{aligned}
    \mathscr{U} = & \{ (t,x,v) \in [0, \tau] \times \R^3_+ \times \R^3 : t \in [0, \tau], v \cdot n \geq 0 \} \\
    & \qquad \cup \{ (t,x,v) \in [0, \tau] \times \R^3_+ \times \R^3 : v \cdot n < 0 \textrm{ and } t_1 = t - t_b (x,v) \leq 0 \} \,, \\
    \mathscr{V} = & [0, \tau] \times \R^3_+ \times \R^3 - \mathscr{U} \\
    = & \{ (t,x,v) \in [0, \tau] \times \R^3_+ \times \R^3 : v \cdot n < 0 \textrm{ and } t_1 = t - t_b (x,v) > 0 \} \,.
  \end{aligned}
\end{equation}
Define a number
\begin{equation}\label{t-star}
  t_\star =
  \left\{
    \begin{aligned}
      0 \,, \quad \textrm{if } (t,x,v) \in \mathscr{U} \,, \\
      t_1 \,, \quad \textrm{if } (t,x,v) \in \mathscr{V} \,.
    \end{aligned}
  \right.
\end{equation}
Observe that the trajectory over $s \in [t_\star, t]$ is
\begin{equation}\label{Trajectory}
  \begin{aligned}
    X_{cl} (s; t, x, v) = x - (t-s) v \,, \ V_{cl} (s; t, x, v) = v \,.
  \end{aligned}
\end{equation}

From integrating the equation \eqref{Remainder_h} along trajectory $[X_{cl} (s; t,x,v), V_{cl} (s; t,x,v)]$ in \eqref{Trajectory} over $s \in [t_\star , t]$,
\begin{equation}\label{h-1}
  \begin{aligned}
    h^\ell_{R, \eps} (t, x, v) = \exp\Big\{ -\tfrac{1}{\eps} \int_{t_\star}^t \nu (\phi) \d \phi \Big\} h^\ell_{R, \eps} (t_\star, X_{cl} (t_\star), V_{cl} (t_\star)) \\
    + \int_{t_\star}^t \exp\Big\{ -\tfrac{1}{\eps} \int_{t_\star}^t \nu (\phi) \d \phi \Big\} q_{R,\eps}^\ell \big( s, X_{cl} (s), V_{cl} (s) \big) \d s \,,
  \end{aligned}
\end{equation}
where $q_{R,\eps}^\ell$ is given in \eqref{q}, $t_\star$ is defined in \eqref{t-star} and the following simplified notations are employed:
\begin{equation*}
  \begin{aligned}
    [X_{cl} (s), V_{cl} (s)] : = [X_{cl} (s; t,x,v), V_{cl} (s; t,x,v)] \,, \ \nu (\phi) : = \nu (\M) \big(\phi, X_{cl} (\phi), V(\phi)\big) \,.
  \end{aligned}
\end{equation*}
Following almost the same arguments in \cite{Guo-Jang-Jiang-2010-CPAM}, the second term in right-hand side of \eqref{h-1} can be bounded by
\begin{equation}\label{h-2}
  \begin{aligned}
    \int_{t_\star}^t \exp \Big\{ -\tfrac{1}{\eps} & \int_{t_\star}^t \nu (\phi) \d \phi \Big\} q_{R,\eps}^\ell \big( s, X_{cl} (s), V_{cl} (s) \big) \d s \\
    & \leq C (m^4 + \sqrt{\eps} + \tfrac{1}{N^0}) \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|_\infty \\
    & + C \sqrt{\eps}^3 \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|^2_\infty + \tfrac{C}{\sqrt{\eps}^3} \sup_{s \in [0, \tau]} \| f_{R,\eps} (s) \|_2 + C \eps
  \end{aligned}
\end{equation}
for sufficiently large $N_0 > 0$ and sufficiently small $m > 0$ to be determined. Here $C > 0$ is a harmless constant. It remains to control the first term in the right-hand side of \eqref{h-1}.

\vspace*{2mm}

{\bf Case 1:} $(t,x,v) \in \mathscr{U}$.

\vspace*{2mm}

Notice that $t_\star = 0$. Together with \eqref{nu-phi}, the first term in the right-hand side of \eqref{h-1} can be bounded by
\begin{equation}\label{h-3}
  \begin{aligned}
    \exp\Big\{ -\tfrac{1}{\eps} \int_0^t \nu (\phi) \d \phi \Big\} h^\ell_{R, \eps} (0, X_{cl} (0), V_{cl} (0)) \leq e^{ -\tfrac{C}{\eps} \l v \r t } \| h_{R,\eps}^\ell (0) \|_\infty \leq \| h_{R,\eps}^\ell (0) \|_\infty \,.
  \end{aligned}
\end{equation}
From \eqref{h-1}, \eqref{h-2} and \eqref{h-3}, one obtains that for any $(t,x,v) \in \mathscr{U}$
\begin{equation}\label{h-4}
  \begin{aligned}
    |h_{R,\eps}^\ell (t,x,v) | \leq & \| h_{R,\eps}^\ell (0) \|_\infty + C (m^4 + \sqrt{\eps} + \tfrac{1}{N^0}) \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|_\infty \\
    & + C \sqrt{\eps}^3 \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|^2_\infty + \tfrac{C}{\sqrt{\eps}^3} \sup_{s \in [0, \tau]} \| f_{R,\eps} (s) \|_2 + C \eps
  \end{aligned}
\end{equation}
for sufficiently large $N_0 > 0$ and sufficiently small $m > 0$ to be determined.

\vspace*{2mm}

{\bf Case 2:} $(t,x,v) \in \mathscr{V}$.

\vspace*{2mm}

Notice that $t_\star = t_1 > 0$, and $( X_{cl} (t_1), V_{cl} (t_1)) = ( x_b (x,v), v ) \in \Sigma_-$. Then the first term in the right-hand side of \eqref{h-1} is
\begin{equation*}
  \begin{aligned}
    \exp\Big\{ -\tfrac{1}{\eps} \int_{t_1}^t \nu (\phi) \d \phi \Big\} h^\ell_{R, \eps} (t_1, x_b (x,v), v) \,.
  \end{aligned}
\end{equation*}
Recalling the boundary condition \eqref{Remainder_BC_h}, one has
\begin{align*}
  h^\ell_{R, \eps} (t_1, x_b (x,v), v) = M_w \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \tfrac{\sqrt{\M_M}}{\l v^\prime \r^\ell} h^\ell_{R, \eps} (t_1, x_b (x,v), v^\prime) \d v^\prime\,.
\end{align*}
Thanks to \eqref{M-Bound} and $M_w = \tfrac{1}{\rho^0} \sqrt{\tfrac{2 \pi}{T^0}} \M^0$, one easily knows
\begin{equation*}
  \begin{aligned}
    0 < M_w \tfrac{\l v \r^\ell}{\sqrt{\M_M}} \int_{v^\prime \cdot n >0} (v^\prime \cdot n) \tfrac{\sqrt{\M_M}}{\l v^\prime \r^\ell} \d v' \leq C \,.
  \end{aligned}
\end{equation*}
Since $(t_1, x_b (x,v), v') \in \mathscr{U}$ here, $|h_{R,\eps}^\ell (t_1, x_b (x,v), v') |$ is bounded by \eqref{h-4}. Consequently, the first term in the right-hand side of \eqref{h-1} is dominated by
\begin{equation}\label{h-5}
  \begin{aligned}
    & \exp\Big\{ -\tfrac{1}{\eps} \int_{t_1}^t \nu (\phi) \d \phi \Big\} | h^\ell_{R, \eps} (t_1, x_b (x,v), v) | \\
    \leq & C \| h_{R,\eps}^\ell (0) \|_\infty + C (m^4 + \sqrt{\eps} + \tfrac{1}{N^0}) \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|_\infty \\
    & + C \sqrt{\eps}^3 \sup_{s \in [0, \tau]} \| h_{R,\eps}^\ell (s) \|^2_\infty + \tfrac{C}{\sqrt{\eps}^3} \sup_{s \in [0, \tau]} \| f_{R,\eps} (s) \|_2 + C \eps
  \end{aligned}
\end{equation}
for sufficiently large $N_0 > 0$ and sufficiently small $m > 0$ to be determined. From \eqref{h-1}, \eqref{h-2} and \eqref{h-5}, the bound also holds for all $(t,x,v) \in \mathscr{V}$.

Finally, the proof of Lemma \ref{Lmm-Linfty} can be finished by choosing sufficiently small $\eps, m > 0$ and large $N_0 > 0$.

