Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit

Abstract

Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative $L^{2}$–$L^{\infty }$ approach with new $L^{{6}}$ estimates for the hydrodynamic part $\mathbf {P}f$ of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system.

Appendices

Appendix A: Extensions and Compactness

Extension

Proof of Lemma 3.6

Step 1. In the sense of distributions on $[0, \infty )\times \Omega \times \mathbb {R}^{3}$,

$$\begin{aligned}&\varepsilon \partial _{t} f_{\delta }+ v\cdot \nabla _{x} f_{\delta } + \varepsilon ^{2} \Phi \cdot \nabla _{v} f_{\delta } \nonumber \\&\quad = \left[ 1-\chi \left( \frac{n(x) \cdot v}{\delta }\right) \chi \left( \frac{ \xi (x) }{ \delta }\right) \right] \chi (\delta |v|) \nonumber \\&\qquad \times \,\left[ \mathbf {1}_{t \in [0,\infty )} g + \mathbf {1} _{t \in ( - \infty , 0 ]} \chi (t) \left\{ \varepsilon \frac{\chi ^{\prime } (t)}{ \chi (t)} + v \cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \right\} f_{0} (x,v)\right] \nonumber \\&\qquad + \big [\mathbf {1}_{t \in [0,\infty )} f + \mathbf {1}_{t \in (- \infty , 0 ]} \chi (t) f_{0} (x,v) \big ] \{v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v}\}\nonumber \\&\qquad \times \, \left( \left[ 1-\chi (\frac{n(x) \cdot v}{\delta }) \chi \left( \frac{\xi (x)}{\delta }\right) \right] \chi (\delta |v|)\right) . \end{aligned}$$

(A.1.1)

Note that,

$$\begin{aligned}&\Big |\{v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v}\} \big [1-\chi \big (\frac{ n(x) \cdot v}{\delta }\big ) \chi \big ( \frac{ \xi (x )}{\delta }\big ) \big ] \chi (\delta |v|)\Big | \nonumber \\&\quad =\Big | - \frac{1}{\delta } \{v\cdot \nabla _{x} n(x) \cdot v + \varepsilon ^{2} \Phi \cdot n(x) \} \chi ^{\prime } \big (\frac{n(x) \cdot v}{\delta } \big ) \chi \big ( \frac{ \xi (x )}{\delta }\big ) \chi (\delta |v|) \nonumber \\&\qquad - \ \frac{1}{\delta } v\cdot \nabla _{x} \xi (x) \chi ^{\prime } \big ( \frac{ \xi (x)}{\delta }\big ) \chi \big (\frac{n(x) \cdot v}{\delta }\big ) \chi (\delta |v|) \nonumber \\&\qquad +\, \varepsilon ^{2}\delta \Phi \cdot \frac{v}{|v|} \chi ^{\prime } (\delta |v|)\Big [1-\chi \Big (\frac{ n(x) \cdot v}{\delta }\Big ) \chi \big ( \frac{\xi (x)}{\delta }\big ) \Big ] \Big | \nonumber \\&\quad \le \frac{4}{\delta }( |v|^{2}\Vert \xi \Vert _{C^2} + \varepsilon ^{2}\Vert \Phi \Vert _\infty ) \chi (\delta |v|) + \frac{C_{\Omega }}{\delta } |v|\chi (\delta |v|) + \varepsilon ^{2} \delta \Vert \Phi \Vert _\infty \mathbf {1}_{|v| \le {2}{\delta }^{-1}} \nonumber \\&\quad \lesssim {\delta ^{-3}} \mathbf {1}_{|v| \le 2 \delta ^{-1}}. \end{aligned}$$

(A.1.2)

This proves the second line of (3.20). Since $\big [1-\chi (\frac{ n(x) \cdot v}{\delta }) \chi ( \frac{\xi (x)}{\delta }) \big ] \chi (\delta |v|) \le 1$, we prove the first line of (3.20) directly.

Step 2. We claim that if $0 \le \xi (x) \le {\tilde{C}} \delta ^{4}, \ |n(x) \cdot v|> \delta$ and $|v| \le \frac{1}{\delta }$ then either $\xi ({\tilde{x}}_{\mathbf {f}} (x,v))={\tilde{C}}\delta ^{4}$ or $\xi ( {\tilde{x}}_{\mathbf {b}}(x,v)) ={\tilde{C}}\delta ^{4}$.

To show this, if $v \cdot n(x) \ge \delta$, we take $s>0$, while if $v\cdot n(x) \le - \delta$ then we take $s<0$. From (2.8),

$$\begin{aligned}&\xi (X(s; 0, x, v) ) = \xi (x) + \int _{0}^{s} V( \tau ; 0, x, v) \cdot \nabla _{x} \xi ( X( \tau ; 0, x, v) ) \mathrm {d}\tau \\&\quad =\xi (x) + \int ^{s}_{0} \{ v+ O(1)\varepsilon ^{2} \Vert \Phi \Vert _{\infty } \tau \} \cdot \{ \nabla _{x} \xi (x) + O(1)\Vert \xi \Vert _{C^{2}} (|v| + \varepsilon ^{2} \Vert \Phi \Vert _{\infty }\tau ) \tau \} \\&\quad =\xi (x) + v \cdot \nabla _{x} \xi (x) s +O(1) \Vert \xi \Vert _{C^{2}} \big \{ |v|^{2} s^{2} + \varepsilon ^{2} \Vert \Phi \Vert _{\infty } s ^{2} \\&\qquad +\, \varepsilon ^{2} \Vert \Phi \Vert _{\infty }|v|s^{3} + \varepsilon ^{4} \Vert \Phi \Vert _{\infty }^{2} s^{4} \big \}. \end{aligned}$$

From $\xi (x)\ge 0$,

$$\begin{aligned} \xi (X(s; 0, x, v) )\ge & {} \delta |s| \Big \{ 1 - \frac{ \Vert \xi \Vert _{C^{2}} }{ \delta }\Big [ |v|^{2} |s| + \varepsilon ^{2} \Vert \Phi \Vert _{\infty } |s| + \varepsilon ^{2} \Vert \Phi \Vert _{\infty }|v\Vert s|^{2} \nonumber \\&\quad +\, \varepsilon ^{4} \Vert \Phi \Vert _{\infty }^{2} |s|^{3} \Big ] \Big \} \nonumber \\\ge & {} \delta |s| \Big \{ 1 - \frac{ \Vert \xi \Vert _{C^{2}} }{\delta }\Big [\frac{1}{ \delta ^{2}} |s| + \varepsilon ^{2} \Vert \Phi \Vert _{\infty } | s| + \varepsilon ^{2} \Vert \Phi \Vert _{\infty } \frac{1}{\delta }|s|^{2}\nonumber \\&\quad +\, \varepsilon ^{4} \Vert \Phi \Vert _{\infty }^{2} |s|^{3} \Big ] \Big \} \nonumber \\\ge & {} \delta |s| \Big \{ 1- \Big [ \frac{1}{4} + \frac{ \varepsilon ^{2} \delta ^{2} \Vert \Phi \Vert _{\infty }}{4 } + \frac{\varepsilon ^{2} \delta ^{4} \Vert \Phi \Vert _{\infty }}{16} \nonumber \\&\quad +\, \frac{\varepsilon ^{4}\delta ^{8} \Vert \Phi \Vert _{\infty }^{2}}{64} \Big ] \Big \} \ \ge \ \frac{\delta |s| }{2}, \end{aligned}$$

(A.1.3)

for $0 \le |s| \le \frac{\delta ^{3}}{ 4(1+ \Vert \xi \Vert _{C^{2}})}$ and $0< \varepsilon \ll 1$. Therefore

$$\begin{aligned} \xi (Y(s;0,x,v)) = \xi \left( X\left( \frac{s}{\varepsilon }; 0,x,v \right) \right) \ge \delta \frac{|s|}{2\varepsilon }, \end{aligned}$$

for all $0\le |s| \le \frac{ \varepsilon \delta ^{3}}{ 4(1+ \Vert \xi \Vert _{C^{2}})}$ with $0< \varepsilon \ll 1$. Especially with $\varepsilon s_{*} = +\frac{ \varepsilon \delta ^{3}}{4(1+ \Vert \xi \Vert _{C^{2}})}$ for $n(x) \cdot v >\delta$ and $\varepsilon s_{*} = -\frac{\varepsilon \delta ^{3} }{4(1+ \Vert \xi \Vert _{C^{2}})}$ for $n(x) \cdot v <\delta$,

$$\begin{aligned} \xi (Y(s_{*};0,x,v)) > {\tilde{C}} \delta ^{4}. \end{aligned}$$

Therefore, by the intermediate value theorem, we prove our claim.

Step 3. We define $f_{E}(t,x,v)$ for $(x,v) \in [ \mathbb { R}^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3}$:

$$\begin{aligned} \begin{aligned} f_{E}(t,x,v)&\ := \ f_{\delta }(t- \varepsilon t_{\mathbf {b}}^{*}(x,v),x_{\mathbf {b} }^{*}(x,v), v_{\mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi \big ( {t_{\mathbf {b}}^{*} (x,v)} \big ) \\&\qquad \text {if} \ \ x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega , \\&\ := \ f_{\delta }(t+ \varepsilon t_{\mathbf {f}}^{*}(x,v),x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi \big ( {t_{\mathbf {f}}^{*} (x,v)} \big ) \\&\qquad \text {if} \ \ x_{\mathbf { f}}^{*}(x,v) \in \partial \Omega , \\&\ := \ 0 \qquad \ \ \ \ \ \text {if} \ \ x_{\mathbf {b}}^{*}(x,v) \notin \partial \Omega \ \text {and} \ x_{\mathbf {f}}^{*}(x,v) \notin \partial \Omega . \end{aligned} \end{aligned}$$

(A.1.4)

We check that $f_{E}$ is well-defined. It suffices to prove the following:

$$\begin{aligned} \begin{aligned}&\text {If} \ \ x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega \text { and } x_{ \mathbf {f}}^{*}(x,v) \in \partial \Omega \\&\ \text {then} \ \ f_{\delta }( t- \varepsilon t_{\mathbf {b}}^{*} (x,v), x_{\mathbf {b} }^{*}(x,v), v_{\mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) = 0 \\&\quad = f_{\delta }( t+ \varepsilon t_{\mathbf {f}}^{*} (x,v), x_{ \mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{ {\tilde{C}} \delta ^{4}} \big ). \end{aligned} \end{aligned}$$

If $|n(x_{\mathbf {b}}^{*}(x,v)) \cdot v_{\mathbf {b}}^{*}(x,v)| \le \delta$ or $|v_{\mathbf {b}}^{*}(x,v)| \ge \frac{1}{\delta }$ then $f_{\delta }( t- \varepsilon t^{*}_{\mathbf {b}}(x,v), x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v))=0$ due to (3.12). If $n(x_{\mathbf {b}}^{*}(x,v)) \cdot v_{ \mathbf {b}}^{*}(x,v) > \delta$ and $|v_{\mathbf {b}}^{*}(x,v)|\le \frac{1}{ \delta }$ then, due to Step 2, $\xi (x_{\mathbf {f}}^{*}(x,v)) = \xi ( x_{\mathbf {f}}^{*}( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v) ) ) = {\tilde{C}}\delta ^{4}$ so that $x_{\mathbf {f}}^{*}(x,v) \notin \partial \Omega$.

On the other hand, if $|n(x_{\mathbf {f}}^{*}(x,v)) \cdot v_{\mathbf {f} }^{*}(x,v)| \le \delta$ or $|v_{\mathbf {f}}^{*}(x,v)|\ge \frac{1}{\delta }$ then $f_{\delta }( t+ \varepsilon t^{*}_{\mathbf {f}}(x,v),x_{\mathbf {f}}^{*}(x,v), v_{ \mathbf {f}}^{*}(x,v))=0$ due to (3.12). If $n(x_{\mathbf {f} }^{*}(x,v)) \cdot v_{\mathbf {f}}^{*}(x,v) <-\delta$ and $|v_{\mathbf {f} }^{*}(x,v)|\le \frac{1}{\delta }$ then, due to Step 2, $\xi (x_{ \mathbf {b}}^{*}(x,v)) = \xi ( x_{\mathbf {b}}^{*}( x_{\mathbf {f}}^{*}(x,v), v_{ \mathbf {f}}^{*}(x,v) ) ) = {\tilde{C}}\delta ^{4}$ so that $x_{\mathbf {b} }^{*}(x,v) \notin \partial \Omega$.

Note that

$$\begin{aligned} f_{E}(t,x,v) = f_{\delta } (t,x,v) \ \ \ \text {for all } x \in \partial \Omega . \end{aligned}$$

(A.1.5)

If $x \in \partial \Omega$ and $n(x) \cdot v >\delta$ then $(x_{\mathbf {b} }^{*}(x,v), v_{\mathbf {b}}^{*}(x,v)) = (x,v)$. From the definition (A.1.4), for those (x, v), we have $f_{E}(t,x,v) = f_{\delta } (t,x,v)$. If $x \in \partial \Omega$ and $n(x) \cdot v< -\delta$ then $(x_{\mathbf {f} }^{*}(x,v), v_{\mathbf {f}}^{*}(x.v)) = (x,v)$. From the definition (A.1.4), we conclude (A.1.5) again. Otherwise, if $-\delta< n(x) \cdot v< \delta$ then $f_{E}|_{\partial \Omega } \equiv 0 \equiv f_{\delta }|_{\partial \Omega }$.

Step 4. We claim that $f_{E}(x,v) \in L^{2}([\mathbb {R} ^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3})$.

From the definition of (A.1.4), we have $f_{E}(x,v) \equiv 0$ if $x_{ \mathbf {b}}^{*} (x,v) \notin \partial \Omega$ and $x_{\mathbf {f}}^{*} (x,v) \notin \partial \Omega$. Therefore, from (A.1.4),

$$\begin{aligned}&\int _{-\infty }^{\infty } \iint _{[ \mathbb {R}^{3 } \backslash \Omega ]\times \mathbb {R}^{3}} |f_{E} (t,x,v)|^{2} \mathrm {d} x \mathrm {d} v \mathrm {d} t\nonumber \\&\quad =\int _{-\infty }^{\infty } \iint _{ [ \mathbb {R}^{3 } \backslash \Omega ] \times \mathbb {R}^{3}} \mathbf {1}_{x_{\mathbf {b}}^{*} (x,v) \in \partial \Omega } |f_{E} |^{2} + \int _{-\infty }^{\infty } \iint _{ [ \mathbb {R} ^{3 } \backslash \Omega ] \times \mathbb {R}^{3}} \mathbf {1}_{x_{\mathbf {f} }^{*} \in \partial \Omega } |f_{E} |^{2} \nonumber \\&\quad = \int _{-\infty }^{\infty } \iint _{ [ \mathbb {R}^{3 } \backslash \Omega ]\times \mathbb {R}^{3}} \mathbf {1}_{x_{\mathbf {b}}^{*} (x,v) \in \partial \Omega } |f_{\delta } ( t- \varepsilon t^{*}_{\mathbf {b}}, x_{\mathbf {b}}^{*} , v_{\mathbf {b}}^{*} )|^{2} \nonumber \\&\qquad \times \,\big | \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \big |^{2} |\chi ( {t_{\mathbf {b}}^{*} } ) |^{2}\mathrm {d} x \mathrm {d} v \mathrm {d} t \end{aligned}$$

(A.1.6)

$$\begin{aligned}&\qquad + \int _{-\infty }^{\infty } \iint _{ [ \mathbb {R}^{3 } \backslash \Omega ] \times \mathbb {R}^{3}} \mathbf {1}_{x_{\mathbf {f}}^{*} (x,v) \in \partial \Omega } |f_{\delta } ( t+ \varepsilon t^{*}_{\mathbf {f}}, x_{\mathbf {f}}^{*}, v_{\mathbf {f}}^{*} )|^{2} \nonumber \\&\qquad \times \,\big | \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \big |^{2} |\chi ( {t_{\mathbf {f}}^{*} } ) |^{2} \mathrm {d} x \mathrm {d} v \mathrm {d} t, \end{aligned}$$

(A.1.7)

where $(t^{*}_{\mathbf {b}},x^{*}_{\mathbf {b}},v^{*}_{\mathbf {b}} )$ and $(t^{*}_{\mathbf {f}},x^{*}_{\mathbf {f}},v^{*}_{\mathbf {f}} )$ are evaluated at (x, v).

By (2.11),

$$\begin{aligned} (A.1.6)\le & {} \int ^{\infty }_{- \infty } \mathrm {d} t \int _{\partial \Omega } \int _{n(x) \cdot v>0} \int _{0}^{ \min \{ t_{\mathbf {f} }^{*}(x,v), 1 \}}\mathrm {d} S_{x} \mathrm {d} v \mathrm {d} s \{|n(x) \cdot v| + O(\varepsilon )(1+ |v|)s \} \\&\times \big | f_{\delta } \big ( t- \varepsilon s, x_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v)),\\&\times v_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v)) \big )\big |^{2} \\\le & {} \int ^{\infty }_{-\infty }\mathrm {d} t \int _{\partial \Omega } \int _{n(x) \cdot v>0} \int _{0}^{ 1} \big | f_{\delta } (t,x,v ) \big |^{2} \{|n(x) \cdot v| + O(\varepsilon )(1+ |v|)s \} \mathrm {d} s \mathrm {d} v \mathrm {d} S_{x} \\\lesssim & {} \int _{-\infty }^{\infty } \mathrm {d} t \int _{\partial \Omega } \int _{n(x) \cdot v >0} \big | f_{\delta } (t,x,v ) \big |^{2} |n(x) \cdot v| \mathrm {d} v \mathrm {d} S_{x} \lesssim \Vert f_{\delta } \Vert _{L^{2} (\mathbb {R} \times \partial \Omega \times \mathbb {R}^{3})}^{2}, \end{aligned}$$

where we have used the fact, from (3.11), $O(\varepsilon )(1+|v|) |s| \le O(\varepsilon ) (1+ \frac{1}{\delta })\lesssim \delta \lesssim |n(x) \cdot v|$ for $(x,v) \in \text {supp} ( f_{\delta } )$, and, for $n(x) \cdot v>0, x\in \partial \Omega$, and $0 \le s \le t_{\mathbf {f}}^{*}(x,v)$,

$$\begin{aligned}&(x_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v)), v_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v)) )\\&\quad = (x_{\mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*} (x,v) ) = (x,v), \end{aligned}$$

and $t_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v)) =s$ and the change of variables $t- \varepsilon s \mapsto t$. Similarly we can show (A.1.7) $\lesssim \Vert f_{\delta } \Vert _{L^{2} (\mathbb {R} \times \partial \Omega \times \mathbb {R}^{3})}^{2}$.

Step 5. We claim that, in the sense of distributions on $\mathbb {R} \times [\Omega _{{\tilde{C}} \delta ^{4}}\backslash \bar{\Omega }] \times \mathbb {R}^{3}$,

$$\begin{aligned}&\varepsilon \partial _{t } f_{E}+ v\cdot \nabla _{x} f_{E} + \varepsilon ^{2} \Phi \cdot \nabla _{v} f_{E} \nonumber \\&\quad = \ \frac{1}{{\tilde{C}} \delta ^{4}}v \cdot \nabla _{x} \xi (x) \chi ^{\prime } \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \Big [ f_{\delta }( t- \varepsilon t_{\mathbf {b}}^{*} (x,v), x_{\mathbf {b}}^{*}(x,v), v_{ \mathbf {b}}^{*}(x,v) ) \chi (t_{\mathbf {b}}^{*} (x,v))\nonumber \\&\qquad \times \,\mathbf {1}_{x_{\mathbf {b }}^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad +\, f_{\delta }( t+ \varepsilon t_{\mathbf {f}}^{*} (x,v), x_{\mathbf {f}}^{*}(x,v), v_{ \mathbf {f}}^{*}(x,v) ) \chi (t_{\mathbf {f}}^{*} (x,v)) \mathbf {1}_{x_{\mathbf { f}}^{*}(x,v) \in \partial \Omega } \Big ] \nonumber \\&\qquad +\, f_{\delta }( t- \varepsilon t_{\mathbf {b}}^{*}(x,v), x_{\mathbf {b}}^{*} (x,v), v_{ \mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime } (t_{\mathbf {b}}^{*}(x,v))\mathbf {1}_{x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad -\, f_{\delta }( t+ \varepsilon t_{\mathbf {f}}^{*} (x,v), x_{\mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime } (t_{\mathbf {f}}^{*}(x,v))\mathbf {1}_{x_{\mathbf {f} }^{*}(x,v) \in \partial \Omega }.\nonumber \\ \end{aligned}$$

(A.1.8)

Note that

$$\begin{aligned}&[ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ]f(t- \varepsilon t_{\mathbf {b}}^{*} (x,v), x_{\mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*}(x,v)) \\&\quad = \underbrace{ [ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] (t-\varepsilon t_{\mathbf {b}}^{*} (x,v)) } \\&\qquad \times \, \partial _{t} f(t- \varepsilon t_{\mathbf {b}}^{*} (x,v), x_{\mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*} (x,v)) \\&\qquad +\, [ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] f(s, x_{ \mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*} (x,v)) |_{s= t- \varepsilon t_{\mathbf {b} }^{*} (x,v)}, \\&[ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ]f(t+ \varepsilon t_{\mathbf {f}}^{*} (x,v), x_{\mathbf {f}}^{*} (x,v), v_{\mathbf {f}}^{*} (x,v)) \\&\quad = \underbrace{ [ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] (t+\varepsilon t_{\mathbf {f}}^{*} (x,v)) } \\&\qquad \times \, \partial _{t} f(t+ \varepsilon t_{\mathbf {f}}^{*} (x,v), x_{\mathbf {f}}^{*} (x,v), v_{\mathbf {f}}^{*}(x,v)) \\&\qquad +\, [ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] f(s, x_{ \mathbf {f}}^{*} (x,v), v_{\mathbf {f}}^{*} (x,v)) |_{s= t + \varepsilon t_{\mathbf {f} }^{*} (x,v)}. \end{aligned}$$

The underbraced terms vanish because $[ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] (t- \varepsilon t_{ \mathbf {b}}^{*} (x.v)) = \frac{d}{ds} \Big |_{s=0} (t- \varepsilon t_{\mathbf {b}}^{*} (X(s;0,x,v), V(s;0,x,v) )) = \frac{d}{ds} \Big |_{s=0} (t- \varepsilon s ) = - \varepsilon$, and $[ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} ] (t+ \varepsilon t_{\mathbf {f }}^{*} (x.v)) = \frac{d}{ds} \Big |_{s=0} (t+ \varepsilon t_{\mathbf {f}}^{*} (X(s;0,x,v), V(s;0,x,v) )) = \frac{d}{ds} \Big |_{s=0} (t- \varepsilon s + \varepsilon t_{ \mathbf {f}}^{*} (x,v)) = - \varepsilon$. Moreover, in the sense of distributions on $[\Omega _{{\tilde{C}} \delta ^{4}}\backslash \bar{\Omega }] \times \mathbb {R}^{3}$,

$$\begin{aligned}&v\cdot \nabla _{x} f_{E} + \varepsilon ^{2} \Phi \cdot \nabla _{v} f_{E} \nonumber \\&\quad = \ \frac{1}{{\tilde{C}} \delta ^{4}}v \cdot \nabla _{x} \xi (x) \chi ^{\prime } \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v) ) \chi (t_{ \mathbf {b}}^{*} (x,v))\mathbf {1}_{x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad +\, f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v) ) \chi (t_{ \mathbf {f}}^{*} (x,v)) \mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } \Big ] \nonumber \\&\qquad +\, f_{\delta }(x_{\mathbf {b}}^{*} (x,v), v_{\mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime } (t_{\mathbf {b} }^{*}(x,v))\mathbf {1}_{x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad -\, f_{\delta }(x_{\mathbf {f}}^{*} (x,v), v_{\mathbf {f}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime } (t_{\mathbf {f} }^{*}(x,v))\mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega }. \end{aligned}$$

(A.1.9)

For $\phi \in C_{c}^{\infty }( [\Omega _{{\tilde{C}} \delta ^{4}}\backslash \bar{ \Omega }] \times \mathbb {R}^{3} )$, we choose small $t>0$ such that $X(s; 0, x,v) \in \Omega _{{\tilde{C}} \delta ^{4}}\backslash \bar{\Omega }$ for all $|s| \le t$ and all $(x,v) \in \text {supp}(\phi )$. Then, from (A.1.4), for $(X(s),V(s))= (X(s;0,x,v), V(s;0,x,v))$,

$$\begin{aligned}&\frac{d}{ds}f_{E}(X(s), V(s)) \\&\quad = \frac{d}{ds} \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(X(s),V(s)), v_{\mathbf { b}}^{*} (X(s),V(s)) ) \chi (t_{\mathbf {b}}^{*} (X(s),V(s))) \mathbf {1}_{x_{ \mathbf {b}}^{*} (X(s),V(s)) \in \partial \Omega } \\&\qquad +\, f_{\delta }( x_{\mathbf {f}}^{*} (X(s),V(s)), v_{\mathbf {f}}^{*} (X(s),V(s)) ) \chi (t_{\mathbf {f}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {f} }^{*} (X(s),V(s)) \in \partial \Omega } \Big ] \\&\qquad \times \, \chi \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ) \\&\qquad +\, \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(X(s),V(s)), v_{\mathbf {b}}^{*} (X(s),V(s)) ) \chi (t_{\mathbf {b}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {b} }^{*} (X(s),V(s)) \in \partial \Omega } \\&\qquad +\, f_{\delta }( x_{\mathbf {f}}^{*} (X(s),V(s)), v_{\mathbf {f}}^{*} (X(s),V(s)) ) \chi (t_{\mathbf {f}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {f} }^{*} (X(s),V(s)) \in \partial \Omega } \Big ] \\&\qquad \times \, \frac{d}{ds}\chi \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ). \end{aligned}$$

From $(x_{\mathbf {b}}^{*}(X(s;0,x,v),V(s;0,x,v)), v_{\mathbf {b} }^{*}(X(s;0,x,v),V(s;0,x,v)))= (x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b} }^{*}(x,v))$ and $(x_{\mathbf {f}}^{*}(X(s;0,x,v),V(s;0,x,v)), v_{\mathbf {f} }^{*}(X(s;0,x,v),V(s;0,x,v)))= (x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f} }^{*}(x,v))$ and $t_{\mathbf {f}}^{*}(X(s;0,x,v),V(s;0,x,v) )=t_{\mathbf {f}}^{*}(x,v ) -s$ and $t_{\mathbf {b}}^{*}(X(s;0,x,v),V(s;0,x,v) )=t_{\mathbf {b}}^{*}(x,v )+s$,

$$\begin{aligned}&\frac{d}{ds}f_{E}(X(s), V(s)) \nonumber \\&\quad =\Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v) ) \chi ^{\prime }( t_{\mathbf {b}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {b} }^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad - f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v) ) \chi ^{\prime }( t_{\mathbf {f}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {f} }^{*}(x,v) \in \partial \Omega } \Big ] \chi \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ) \nonumber \\&\qquad +\, \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v) ) \chi (t_{\mathbf {b}}^{*}(X(s),V(s)) ) \mathbf {1}_{x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega } \nonumber \\&\qquad +\, f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v) ) \chi (t_{\mathbf {f}}^{*}(X(s),V(s)) )\mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } \Big ]\nonumber \\&\qquad \times \,\frac{1}{{\tilde{C}} \delta ^{4}}V(s ) \cdot \nabla _{x} \xi (X(s )) \chi ^{\prime } \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ). \end{aligned}$$

(A.1.10)

By the change of variables $(x,v) \mapsto (X(s; 0, x,v), V(s;0,x,v))$, for sufficiently small s,

$$\begin{aligned}&-\iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} (x,v) \{ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \}\phi (x,v) \mathrm {d} x \mathrm {d} v \nonumber \\&\quad =- \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} (X(s), V(s)) \{ V(s)\cdot \nabla _{X} + \varepsilon ^{2} \Phi \cdot \nabla _{V} \} \phi (X(s), V(s)) \mathrm {d} x \mathrm {d} v \nonumber \\&\quad = - \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} (X(s), V(s)) \frac{d}{ds} \phi (X(s), V(s)) \mathrm {d} x \mathrm {d} v. \end{aligned}$$

(A.1.11)

Since the change of variables $(x,v) \mapsto (X(s; 0, x,v), V(s;0,x,v))$ has unit Jacobian, it follows that, for s sufficiently small,

$$\begin{aligned} \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} ((X(s), V(s)) \phi (X(s), V(s)))=\iint _{[\Omega _{ {\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} (x,v) \phi (x,v), \end{aligned}$$

and hence

$$\begin{aligned} \frac{d}{ds} \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } f_{E} ((X(s), V(s)) \phi (X(s), V(s)) =0. \end{aligned}$$

Therefore we can move the s-derivative on $f_E$: By (A.1.10),

$$\begin{aligned}&(A.1.11) \\&\quad = \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } \frac{d}{ds} f_{E} (X(s), V(s)) \phi (X(s), V(s)) \mathrm {d} x \mathrm {d} v \\&\quad = \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b} }^{*}(x,v) ) \chi ^{\prime }( t_{\mathbf {b}}^{*} (X(s),V(s))) \mathbf {1}_{x_{ \mathbf {b}}^{*}(x,v) \in \partial \Omega } \\&\qquad - f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{ \mathbf {f}}^{*}(x,v) ) \chi ^{\prime }( t_{\mathbf {f}}^{*} (X(s),V(s))) \mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } \Big ] \\&\qquad \times \,\chi \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ) \phi (X(s ), V(s )) \\&\qquad + \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b} }^{*}(x,v) ) \chi (t_{\mathbf {b}}^{*}(X(s),V(s)) ) \mathbf {1}_{x_{\mathbf {b} }^{*}(x,v) \in \partial \Omega } \\&\qquad +\, f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v) ) \chi (t_{ \mathbf {f}}^{*}(X(s),V(s)) )\mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } \Big ] \\&\qquad \times \,\frac{1}{{\tilde{C}} \delta ^{4}}V(s ) \cdot \nabla _{x} \xi (X(s ))\chi ^{\prime } \big ( \frac{\xi (X(s ))}{{\tilde{C}} \delta ^{4}} \big ) \phi (X(s ), V(s )). \end{aligned}$$

From the change of variable $(X(s;0,x,v), V(s;0,x,v)) \mapsto (x,v)$,

$$\begin{aligned} (A.1.11)= & {} \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } \big [ f_{\delta }(x_{\mathbf {b}}^{*} (x,v), v_{ \mathbf {b}}^{*}(x,v)) \chi ^{\prime } (t_{\mathbf {b}}^{*}(x,v)) \mathbf {1}_{x_{ \mathbf {b}}^{*}(x,v) \in \partial \Omega } \\&- f_{\delta }(x_{\mathbf {f}}^{*} (x,v), v_{\mathbf {f}}^{*}(x,v)) \chi ^{\prime } (t_{\mathbf {f}}^{*}(x,v)) \mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } \big ] \chi \big (\frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \phi (x,v) \\&+\, \iint _{[\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }] \times \mathbb {R}^{3} } \Big [ f_{\delta }( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b} }^{*}(x,v) ) \chi (t_{\mathbf {b}}^{*}(x,v) ) \mathbf {1}_{x_{\mathbf {b} }^{*}(x,v) \in \partial \Omega } \\&+ f_{\delta }( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f} }^{*}(x,v) ) \chi (t_{\mathbf {f}}^{*}(x,v) )\mathbf {1}_{x_{\mathbf {f} }^{*}(x,v) \in \partial \Omega } \Big ]\\&\times \frac{1}{{\tilde{C}} \delta ^{4}}v \cdot \nabla _{x} \xi (x) \chi ^{\prime } \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big )\phi (x,v). \end{aligned}$$

Hence (A.1.8) is proved.

On the other hand, following the bounds of (A.1.6) and (A.1.7) in Step 4 we prove the third line of (3.20).

Step 6. We define ${\bar{f}}(t,x,v)$ for $(t,x,v) \in \mathbb {R} \times \mathbb {R}^{3} \times \mathbb {R}^{3}$:

$$\begin{aligned} \begin{aligned} {\bar{f}}(t,x,v)&\ : = \ f_{\delta }(t,x,v) \mathbf {1}_{(x,v) \in \bar{\Omega } \times \mathbb {R}^{3}} + f_{E}(t,x,v) \mathbf {1}_{(x,v) \in [\mathbb {R}^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3}}. \end{aligned} \end{aligned}$$

(A.1.12)

For $\phi \in C^{\infty }_{c}(\mathbb {R} \times \mathbb {R}^{3} \times \mathbb { R}^{3})$, by Lemma 3.3,

$$\begin{aligned}&- \int _{- \infty }^{\infty } \mathrm {d} t \iint _{\mathbb {R}^{3} \times \mathbb {R}^{3}} {\bar{f}}(t,x,v) \{\varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \} \phi (t,x,v) \mathrm {d} x \mathrm {d} v \\&\quad =- \int _{- \infty }^{\infty } \mathrm {d} t\iint _{\Omega \times \mathbb {R} ^{3}} f_{\delta }(t,x,v) \{ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \} \phi (t,x,v) \mathrm {d} x \mathrm {d} v \\&\qquad - \int _{- \infty }^{\infty } \mathrm {d} t \iint _{ [\mathbb {R}^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3}} f_{E}(t,x,v) \{\varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \} \phi (t,x,v) \mathrm {d} x\mathrm {d} v \\&\quad = \int _{- \infty }^{\infty } \mathrm {d} t \int _{\gamma } f_{\delta } (t,x,v) \phi (t,x,v) \{n(x) \cdot v\} \mathrm {d} S_{x} \mathrm {d} v \\&\qquad + \int _{- \infty }^{\infty } \mathrm {d} t \int _{\gamma } f_{E}(t,x,v) \phi (t,x,v) \{-n(x) \cdot v\} \mathrm {d} S_{x} \mathrm {d} v \\&\qquad + \int _{- \infty }^{\infty } \mathrm {d} t \iint _{\Omega \times \mathbb {R} ^{3}} \{\varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \} f_{\delta }(t,x,v) \phi (t,x,v) \mathrm {d} x \mathrm {d} v \\&\qquad + \int _{- \infty }^{\infty } \mathrm {d} t \iint _{ [\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }]\times \mathbb {R}^{3}} \{ \varepsilon \partial _{t} + v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \} {f} _{E}(x,v) \phi (x,v) \mathrm {d} x \mathrm {d} v, \end{aligned}$$

where the contributions of $\{t= \infty \}$ and $\{t= -\infty \}$ vanish since $\phi (t) \in C_{c}^{\infty } (\mathbb {R})$.

From (A.1.5), the boundary contributions are cancelled:

$$\begin{aligned} \int _{- \infty }^{\infty } \int _{\gamma } f_{\delta } (t,x,v) \phi (t,x,v) \mathrm {d} \gamma \mathrm {d} t - \int _{- \infty }^{\infty } \int _{\gamma } f_{E}(t,x,v) \phi (t,x,v) \mathrm {d} \gamma \mathrm {d} t =0. \end{aligned}$$

Further from (A.1.1) and (A.1.8), we prove that ${\bar{f}}$ solves (3.13) in the sense of distributions on $\mathbb {R} \times \mathbb {R}^{3} \times \mathbb {R}^{3}$. $\square$

Compactness of $K\mathcal {L}^{-1}$

Proof of Lemma 2.11

From Proposition 2.10, $\mathcal {L}^{-1}$ maps $L^{2}$ to $L^{2}$ so that $\sup _{n} \Vert f^{n}\Vert _{L^{2}} < +\infty$.

Step 1. We approximate K by a compactly supported smooth $K_{N}$.

For any $N\gg 1$, by the Hölder inequality,

$$\begin{aligned}&\Vert K(\mathbf {1}_{\{|u|> N \text { or } |v|>N \text { or } |v-u|< \frac{1}{N} \}} f) \Vert _{2} \\&\quad = \Big \Vert \int _{\mathbb {R}^{3}} \mathbf {k}(v,u) \mathbf {1}_{\{|u|>N \text { or } |v|> {N} \text { or } |v-u|< \frac{1}{N} \}} f(u) \mathrm {d} u \Big \Vert _{2}\\&\quad \le \sup _{v} \sqrt{\int _{|u|>N} | \mathbf {k}(v,u)| \mathrm {d} u } \times \sup _{u} \sqrt{\int _{\mathbb {R}^{3}} | \mathbf {k}(v,u)| \mathrm {d} v }\times \Vert f \Vert _{2} \\&\qquad +\sup _{v} \sqrt{\int _{\mathbb {R}^{3}} |\mathbf {k}(v,u)| \mathrm {d} u } \times \sup _{u} \sqrt{\int _{|v|>N}| \mathbf {k}(v,u)| \mathrm {d} v } \times \Vert f \Vert _{2} \\&\qquad + \sup _{v} \sqrt{\int _{\mathbb {R}^{3}} |\mathbf {k}(v,u)| \mathrm {d} u } \times \sup _{u} \sqrt{\int _{\mathbb {R}^{3}} \frac{e^{-\theta |v-u|^{2}}}{ |v-u|} \mathbf {1}_{|v-u| < \frac{1}{N}} \mathrm {d} v}\\&\qquad \times \, \Vert f\Vert _{2} \ \lesssim \ o(1) \Vert f\Vert _{2}, \end{aligned}$$

where we have used the fact $\int _{\mathbb {R}^{3}} \mathbf {k}(v,u) \mathrm {d} u \lesssim 1, \int _{\mathbb {R}^{3}} \mathbf {k}(v,u) \mathrm {d} v \lesssim 1$, and $\int _{\mathbb {R}^3}\frac{e^{-\beta |v|^{2}}}{|v|} \mathrm {1}_{|v| < \frac{1}{N}}=o(1)$.

Note that

$$\begin{aligned} \mathbf {k}(v,u) = \mathbf {k}_{1} (v,u) - \mathbf {k}_{2} (v,u), \ \ \ \text { with} \ \ \ \mathbf {k}_{1}, \mathbf {k}_{2} >0, \end{aligned}$$

and therefore $\mathbf {k}_{i}(v,u) \mathbf {1}_{\{|u|\le N \ \& \ |v|\le N \ \& \ |v-u| \ge \frac{1}{N}\}}> \delta$ for $0< \delta \ll 1$.

We now define $K_{N} f:= \int \mathbf {k}_{N} f$ with

$$ \begin{aligned}&\mathbf {k}_{N}(v,u) = \ \mathbf {k}_{1,N} (v,u) - \mathbf {k}_{2,N} (v,u) \nonumber \\&\quad := \ \mathbf {k}_{1}(v,u)\mathbf {1}_{\{|u|\le N \ \& \ |v|\le N \ \& \ |v-u| \ge \frac{1}{N}\}} * \phi _{ \frac{1}{N}}(v)\phi _{ \frac{1}{N}}(u) \nonumber \\&\qquad -\,\mathbf {k}_{2}(v,u)\mathbf {1}_{\{|u|\le N \ \& \ |v|\le N \ \& \ |v-u| \ge \frac{1}{N}\}} * \phi _{ \frac{1}{N}}(v)\phi _{ \frac{1}{N}}(u) \nonumber \\&\quad = \ \int _{\mathbb {R}^{3}} \mathrm {d} u^{\prime } \phi _{ \frac{1}{N}}(u- u^{\prime })\int _{\mathbb {R}^{3}} \mathrm {d} v^{\prime } \phi _{ \frac{1}{N} }(v-v^{\prime }) \mathbf {k}_{1}(v,u)\mathbf {1}_{\left\{ |u|\le N \ \& \ |v|\le N \ \& \ |v-u| \le \frac{1}{N}\right\} } \nonumber \\&\qquad - \int _{\mathbb {R}^{3}} \mathrm {d} u^{\prime } \phi _{ \frac{1}{N}}(u- u^{\prime })\int _{\mathbb {R}^{3}} \mathrm {d} v^{\prime } \phi _{ \frac{1}{N} }(v-v^{\prime }) \mathbf {k}_{2}(v,u)\mathbf {1}_{\left\{ |u|\le N \ \& \ |v|\le N \ \& \ |v-u| \le \frac{1}{N}\right\} },\nonumber \\ \end{aligned}$$

(A.2.1)

where $\phi _{ \frac{1}{N}}$ is the standard mollifiers. In particular, for $i=1,2$,

$$ \begin{aligned}&\sup _{v}\int _{\mathbb {R}^{3}} \big | \mathbf {k}_{i,N}(v,u) - \mathbf {k}_{i} (v,u) \mathbf {1}_{\{|u|\le N \ \& \ |v|\le N \ \& \ |v-u| \le \frac{1}{N} \}} \big | \mathrm {d} u =o(1), \\&\sup _{u}\int _{\mathbb {R}^{3}} \big | \mathbf {k}_{i,N}(v,u) - \mathbf {k}_{i} (v,u) \mathbf {1}_{\{|u|\le N \ \& \ |v|\le N \ \& \ |v-u| \le \frac{1}{N} \}} \big | \mathrm {d} v =o(1). \end{aligned}$$

Consequently, $\Vert K_{N } f- K f\Vert _{2} \lesssim o(1) \Vert f\Vert _{2}$.

We denote

$$\begin{aligned} \bar{\mathbf {k}}_{N} (v,u) := \mathbf {k}_{1,N} (v,u) + \mathbf {k}_{2,N} (v,u), \ \ \ {\bar{K}}_{N} f (v) : = \int _{\mathbb {R}^{3}} \bar{\mathbf {k}} _{N} (v,u) f(u) \mathrm {d} u. \end{aligned}$$

Note that $\mathbf {k}_{1,N}, \mathbf {k}_{2,N} \ge 0$ and hence $\bar{ \mathbf {k}}_{N} \ge 0$ and $\bar{\mathbf {k}}_{N} \ge | \mathbf {k}_{N}|$.

Step 2.

We fix $\delta \ll 1$. Given $f^n$, we define $f_{\delta }^{n}$ as

$$\begin{aligned} f^n_{\delta }(x,v) : = \Big [1-\chi \big (\frac{n(x) \cdot v}{\delta }\big ) \chi \big (\frac{ \xi (x)}{\delta }\big ) \Big ] \chi (\delta |v|) f^(x,v), \end{aligned}$$

(A.2.2)

and extend it to the whole space $\mathbb {R}^{3} \times \mathbb {R}^{3}$. We follow the process in the proof of Lemma 3.6 and we only pinpoint the difference.

Similarly to Step 3 of the proof of Lemma 3.6, we define $f_{E}^{n}(x,v)$ for $(x,v) \in [\mathbb {R}^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3}$

$$\begin{aligned}&f_{E}^{n}(x,v) \nonumber \\&\quad := \ e^{- \lambda t_{\mathbf {b}}^{*} (x,v) + \int ^{- t_{\mathbf {b} }^{*}(x,v)}_{0} \big [\frac{ {\nu }(X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \nonumber \\&\qquad \times f_{\delta }^{n}(x_{ \mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{ {\tilde{C}} \delta ^{4}} \big ) \chi ( t_{\mathbf {b}}^{*} (x,v) ) \ \ \ \ \text { for} \ \ x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega , \nonumber \\&\quad := \ e^{ \lambda t_{\mathbf {f}}^{*} (x,v) + \int ^{ t_{\mathbf {f} }^{*}(x,v)}_{0} \big [\frac{ {\nu }(X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \nonumber \\&\qquad \times \, f_{\delta }^{n}(x_{ \mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v)) \chi \big ( \frac{\xi (x)}{ {\tilde{C}} \delta ^{4}} \big ) \chi ( t_{\mathbf {f}}^{*} (x,v) ) \ \ \ \ \text { for} \ \ x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega , \nonumber \\&\quad := 0\quad \text {for} \quad x_{\mathbf {b}}^{*}(x,v) \notin \partial \Omega \ \text {and} \ x_{\mathbf {f}}^{*}(x,v) \notin \partial \Omega , \end{aligned}$$

(A.2.3)

where $(X(\tau ), V(\tau )) = (X(\tau ; 0,x,v), V(\tau ;0,x,v))$.

Then

$$\begin{aligned} f^n_{E}(x,v) = f^n_{\delta } (x,v) \ \ \ \text {for all } x \in \partial \Omega , \end{aligned}$$

(A.2.4)

because, if $x \in \partial \Omega$ and $n(x) \cdot v>\delta$, then $t_{ \mathbf {b}}^{*}(x,v)=0$. If $x \in \partial \Omega$ and $n(x) \cdot v<\delta$ then $t_{\mathbf {f}}^{*}(x,v)=0$.

We define ${\bar{f}}^{n}(x,v)$ as

$$\begin{aligned} \begin{aligned} {\bar{f}}(x,v)&\ : = \ f_{\delta }(x,v) \mathbf {1}_{(x,v) \in \bar{\Omega } \times \mathbb {R}^{3}} + f_{E}(x,v) \mathbf {1}_{(x,v) \in [\mathbb {R}^{3} \backslash \bar{\Omega }] \times \mathbb {R}^{3}}. \end{aligned} \end{aligned}$$

(A.2.5)

Note that ${\bar{f}}^{n}$ solves

$$\begin{aligned} \lambda {\bar{f}}^{n} + v\cdot \nabla _{x} {\bar{f}} ^{n} + \frac{1}{\varepsilon } \nu \bar{ f} ^{n} +\varepsilon ^{2} \Phi \cdot \nabla _{v} {\bar{f}}^{n} - \frac{1}{2}\varepsilon ^{2} \Phi \cdot v {\bar{f}}^{n} \ = \ h^{n} \ = \ h_{1}^{n}+ h_{2}^{n} + h_{3}^{n} + h_{4}^{n}, \end{aligned}$$

where

$$\begin{aligned}&h_{1}^{n} (x,v)= \mathbf {1}_{(x,v) \in \Omega \times \mathbb {R}^{3}} \left[ 1-\chi \left( \frac{n(x) \cdot v}{\delta }\right) \chi \left( \frac{\xi (x)}{\delta }\right) \chi \left( \frac{ \xi (x)}{\delta }\right) \right] \chi (\delta |v|) g^{n}, \\&h_{2}^{n} (x,v) = \mathbf {1}_{(x,v) \in \Omega \times \mathbb {R}^{3}} f^{n} \left\{ v\cdot \nabla _{x} + \varepsilon ^{2} \Phi \cdot \nabla _{v} \right\} \\&\quad \times \Big ( \big [1-\chi \big ( \frac{n(x) \cdot v}{\delta }\big ) \chi \big ( \frac{\xi (x)}{\delta }\big ) \big ] \chi \big (\frac{ \xi (x)}{\delta }\big ) \chi (\delta |v|)\Big ), \\&h_{3}^{n} (x,v) = \mathbf {1}_{{\ (x,v) \in [\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }]\times \mathbb {R}^{3}}} \ \frac{1}{{\tilde{C}} \delta ^{4}}v \cdot \nabla _{x} \xi (x) \chi ^{\prime } \big ( \frac{\xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \nonumber \\&\quad \times \Big [ f_{\delta }^{n}( x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b} }^{*}(x,v) ) e^{- \lambda t_{\mathbf {b}}^{*} (x,v) + \int ^{- t_{\mathbf {b} }^{*}(x,v)}_{0} \big [\frac{ {\nu }(X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \mathbf {1}_{x_{ \mathbf {b}}^{*}(x,v) \in \partial \Omega } \\&\quad +\, f_{\delta }^{n}( x_{\mathbf {f}}^{*}(x,v), v_{\mathbf {f}}^{*}(x,v) ) \mathbf {1}_{x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega } e^{ \lambda t_{ \mathbf {f}}^{*} (x,v) + \int ^{ t_{\mathbf {f}}^{*}(x,v)}_{0} \big [\frac{ {\nu } (X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \Big ], \\&h_{4}^{n} (x,v) = \mathbf {1}_{{\ (x,v) \in [\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{\Omega }]\times \mathbb {R}^{3}}} f^{n}_{\delta }(x_{\mathbf {b}}^{*}(x,v), v_{\mathbf {b}}^{*}(x,v)) \\&\quad \times e^{- \lambda t_{\mathbf {b}}^{*} (x,v) + \int ^{- t_{\mathbf {b} }^{*}(x,v)}_{0} \big [\frac{ {\nu }(X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \chi \big ( \frac{ \xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime }(t_{\mathbf {b}}^{*}(x,v)) \mathbf {1}_{ x_{\mathbf {b}}^{*}(x,v) \in \partial \Omega } \\&\quad +\, \mathbf {1}_{{\ (x,v) \in [\Omega _{{\tilde{C}} \delta ^{4}} \backslash \bar{ \Omega }]\times \mathbb {R}^{3}}} f^{n}_{\delta }(x_{\mathbf {f}}^{*}(x,v), v_{ \mathbf {f}}^{*}(x,v)) \\&\quad \times e^{ \lambda t_{\mathbf {f}}^{*} (x,v) + \int ^{ t_{\mathbf {f} }^{*}(x,v)}_{0} \big [\frac{ {\nu }(X(\tau ),V(\tau ))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau )) \cdot V(\tau ) \big ] \mathrm {d} \tau } \chi \big ( \frac{ \xi (x)}{{\tilde{C}} \delta ^{4}} \big ) \chi ^{\prime }(t_{\mathbf {f}}^{*}(x,v)) \mathbf {1}_{ x_{\mathbf {f}}^{*}(x,v) \in \partial \Omega }, \end{aligned}$$

and

$$\begin{aligned}&\Vert h_{1}^{n} \Vert _{L^{2}(\mathbb {R}^{3} \times \mathbb {R}^{3})} \ \lesssim \ \Vert g^{n}\Vert _{L^{2}(\Omega \times \mathbb {R}^{3})}, \ \ \Vert h_{2}^{n} \Vert _{L^{2} (\mathbb {R}^{3} \times \mathbb {R}^{3})} \ \lesssim _{\delta } \ \Vert f^{n} \Vert _{L^{2}(\Omega \times \mathbb {R}^{3})}, \\&\Vert h_{3}^{n} \Vert _{L^{2} (\mathbb {R}^{3} \times \mathbb {R}^{3})} +\Vert h_{4}^{n} \Vert _{L^{2} (\mathbb {R}^{3} \times \mathbb {R}^{3})} \ \lesssim _{\delta } \ \Vert f_{\delta }^{n} \Vert _{L^{2} (\gamma )}. \end{aligned}$$

Step 3. For $(x,v) \in \text {supp}({\bar{f}})$, we can choose a fixed $T>0$ such that

$$\begin{aligned} X(T; 0,x,v) \notin \text {supp}({\bar{f}} ) \ \ \text {and} \ \ X(T; 0,x,v) \notin \text {supp}(h), \end{aligned}$$

(A.2.6)

so that

$$\begin{aligned} {\bar{f}} (X(T;0,x,v), V( T;0,x,v))=0. \end{aligned}$$

Directly,

$$\begin{aligned} | X( T;0,x,v) -x| = | v T+ O(\varepsilon ^{2}) \Vert \Phi \Vert _{\infty }T^{2}| \ge \delta T- O(\varepsilon ^{2}) \Vert \Phi \Vert _{\infty }T^{2}. \end{aligned}$$

We choose $T= \frac{C}{\delta }$ for large but fixed $C\gg 1$ such that $| X( T;0,x,v) -x|\ge C - O(\frac{\varepsilon ^{2}}{\delta ^{2}}) \ge \frac{C}{2} \gg 1$. This proves our claim (A.2.6).

With this choice T,

$$\begin{aligned} {\bar{f}}^{n}(x,v)= & {} - \int ^{T}_{0} h^{n}(X(s;0,x,v), V(s;0,x,v))\\&\quad e^{\lambda s + \int ^{s}_{0} \big [\frac{\nu ( V(\tau ;0,x,v))}{\varepsilon } - \frac{1}{2} \varepsilon ^{2} \Phi (X(\tau ;0,x,v) ) \big ] \mathrm {d} \tau } \mathrm {d} s. \end{aligned}$$

By the averaging lemma [23], for any given v,

$$\begin{aligned} \int \mathbf {k}_{N}(v,u){\bar{f}}^{n}(x,u) \mathrm {d} u\in H^{1/4}(\mathbb {R} ^{3}). \end{aligned}$$

Since ${\bar{f}}^{n}$’s support is bounded uniformly, by a diaganolization argument, it follows that there exists a weak limit ${\bar{f}}\in L^{2}$ of ${\bar{f}}^{n}$ such that for any rational point v

$$\begin{aligned} \int \mathbf {k}_{N}(v,u){\bar{f}}^{n}(x,u)\mathrm {d} u\rightarrow \int \mathbf { k}_{N}(v,u){\bar{f}}(x,u) \mathrm {d} u \ \ \ \text {strongly in} \ L_{x}^{2}. \end{aligned}$$

(A.2.7)

Since $\mathbf {k}_{N}(v,u)$ is smooth in v with compact support, we deduce (A.2.7) for all $v \in \mathbb {R}^{3}.$

Step 4. Finally,

$$\begin{aligned} K f^{n}= & {} K_{N} f^{n} + (Kf^{n}-K_{N}f^{n}) = {K_{N} f^{n}_{\delta }} \ + \ {\ K_{N}( f^{n}- f^{n}_{\delta }) + (Kf^{n}-K_{N}f^{n})}. \end{aligned}$$

From Step 3,

$$\begin{aligned} K_{N} {f}^{n}_{\delta } = K_{N} {\bar{f}}^{n}|_{\Omega } \rightarrow K_{N} \bar{f }|_{\Omega } \ \ \ \text {strongly in } L^{2}. \end{aligned}$$

Note that

$$\begin{aligned} 1-\Big [1-\chi \big (\frac{n(x) \cdot v}{\delta }\big ) \chi \big (\frac{ \xi (x )}{\delta }\big ) \Big ] \chi (\delta |v|) \le \mathbf {1}_{|v| \ge \frac{1}{\delta }} + \mathbf {1 }_{|v| \le \frac{1}{\delta }, \text {dist} (x, \partial \Omega )< \frac{\delta }{2}, |n(x) \cdot v|< \delta }. \end{aligned}$$

(A.2.8)

Hence, from (A.2.1),

$$\begin{aligned} \Vert K_{N} ( f^{n} - f^{n}_{\delta })\Vert _{L^{2}_{x,v}}\lesssim & {} \Vert f^{n} -f^{n}_{\delta }\Vert _{L^{2}_{x,v}} \lesssim \big \Vert \chi _{\delta }^{c} \big \Vert _{\infty } \Vert f^{n}\Vert _{L^{2}_{x,v}} \lesssim O(\delta ), \\ \Vert Kf^{n} - K_{N} f^{n} \Vert _{L^{2}_{x,v}}\lesssim & {} o(1) \Vert f^{n} \Vert _{L^{2}_{x,v}} \lesssim o(1). \end{aligned}$$

We conclude the proof by choosing $\delta \ll 1, \ 1 \ll N$ then letting $n\rightarrow \infty$. $\square$

Appendix B: List of symbols

$\Vert \cdot \Vert _p$	the $L^p(\bar{\Omega }\times \mathbb {R}^3)$ norm
$\Vert \cdot \Vert _{L^{p}}$	the $L^p(\bar{\Omega } )$ norm
$\Vert \cdot \Vert _{L^{p}_y}$	the $L^p( \{ y \in Y\} )$ norm
$( \, \cdot \,,\, \cdot \, )$	$L^2(\bar{\Omega }\times \mathbb {R}^3)$ inner product or $L^2( \mathbb {R}^3)$ inner product
$\bar{\Omega }$	$\Omega \cup \partial \Omega$
$\Vert \cdot \Vert _{\nu }$	$\Vert \nu ^{1/2}\cdot \Vert _2$
$W^{k,p}$	Sobolev space
$H^k$	a Sobolev space $W^{k,p}$ with $p=2$
$\Vert \cdot \Vert _{L^{p}L^{q}}$	a norm of $\Vert \cdot \Vert _{L^{p}(L^{q})}$ which equals $\big \Vert \Vert \cdot \Vert _{L^{q}} \big \Vert _{L^{p}}$
$\mathrm {d} \gamma$	$|n(x)\cdot v|\mathrm {d} S(x)\mathrm {d} v$ with $\mathrm {d} S(x)$ the surface measure of $\partial \Omega$
$L^p(\partial \Omega \times \mathbb {R}^3)$	$L^p$-norm on $\partial \Omega \times \mathbb {R}^3$ with a measure $\mathrm {d}\gamma$
$|f|_p^p$	$\int _{\gamma } |f(x,v)|^p \mathrm {d}\gamma$ or $\int _{\partial \Omega } |f(x)|^p \mathrm {d} S(x)$
$f_\gamma , f_\pm$	a trace of f on $\gamma$, a trace of f on $\gamma _\pm$
$|f|_{p,\pm }$	$|f \mathbf {1}_{\gamma _{\pm }}|_p$
$X \lesssim _\alpha Y$	$X \le C_\alpha Y$ where $C_\alpha$ only depends a parameter $\alpha$
$X \ll _{a} Y$	$X\le C_{a} Y$ where $C_{a}>0$ is sufficiently small
$0+$	Some positive number $\delta >0$ without specifying the size
$\vec {G}$	A force field	(1.1)
$\Phi$	A force field	(1.9)
$M_w$	A wall Maxwellian	(1.3)
$\mathcal {P}^w_\gamma$	The diffuse reflection boundary operator	(1.5)
$\mu$	The global Maxwellian	(1.8)
$T_w$	A wall temperature	(1.9)
$\vartheta _w$	An $\varepsilon$-order variation of the wall temperature around 1	(1.9)
$\Theta _w$	An extension of $\vartheta _w$ in $\Omega$	(1.10)
$\rho _w$	A function defined in (1.11)	(1.11)
$f_w$	A correction term at the boundary	(1.11)
$\mathbf {P}$	A $L^2_v$-projection on the null space of L
a, b, c,	Coefficients of $\mathbf {P}$	(1.19)
$\mathbf {I} -\mathbf {P}$	$(\mathbf {I} -\mathbf {P}) f := f -\mathbf {P}f$

$\mathfrak {M}$	Mach number
$\varphi _\varepsilon$	A correction on the boundary	(1.22)
r	$\mu ^{-1/2} \mathcal {P}^w_\gamma \big (\sqrt{\mu }\varphi _{\varepsilon } \big )-\varphi _{\varepsilon }$	(1.25)
$P_\gamma f$	$\sqrt{2\pi }\sqrt{\mu (v)} \int _{n(x)\cdot u>0} f(u) \sqrt{ \mu (u)} \{n(x) \cdot u\} \mathrm {d} u$	(1.26)
$\mathcal {Q}$	$\varepsilon ^{-1}\big [\mu ^{-1/2} \mathcal {P}^w_\gamma ( \sqrt{\mu } f) -{P}_\gamma f\big ]$	(1.27)
$f_s$	An $\varepsilon$-order correction of steady solutions substracted $f_w$	(1.32)
$L_1, A_s$	Terms defined in (1.34), (1.35)
$L^{-1}$	$L^{-1}$ is an inverse with $\mathbf {P}L^{-1} \equiv 0$
w(v)	$e^{\beta |v|^2}$ with $0< \beta \ll 1$	Theorem 1.1
$g_1, g_2$	Defined in (1.41)
${\tilde{f}}$	An unsteady perturbation around steady solutions	(1.48)
$L_\phi$	$L_\phi \psi : = - [\Gamma (\phi , \psi ) + \Gamma (\psi ,\phi )]$
$\mathcal {E}_{\lambda }[\cdot ](t)$	An energy term in $L^2$ energy estimate	(1.53)
$\mathcal {D}_{\lambda }[\cdot ](t)$	A dissipation term in $L^2$ energy estimate	(1.54)
$\xi$	A function defined in (2.3)	(2.3)
n(x)	The outward normal at $x \in \partial \Omega$	(2.5)
$t_{\mathbf {b}}, x_{\mathbf {b}}$	A backward exit time, a backward exit point	(2.9)
$t_{\mathbf {f}}, x_{\mathbf {f}}$	A forward exit time, a forward exit point	(2.10)
$t_i (t, \ldots , v_{i-1})$	i-th backward exit time	Definition 2.7
$X_\mathbf {cl}, V_\mathbf {cl}$	A stochastic trajectory	Definition 2.7
$\gamma ^\delta _\pm$	Non-grazing phase boundary	(2.15)
$\llbracket \cdot \rrbracket$	A closing norm for the steady case	(2.20)
$\mathbf {k}_\beta$	A function defined in (2.26)	(2.26)
$K_\beta$	$K_\beta g := \int _{\mathbb {R}^3} \mathbf {k}_\beta (v,u) g(u) \mathrm {d}u$
$\beta ^\prime$	A number $0< \beta ^\prime \ll \beta$	Proposition 2.6
$\mathcal {V}(x), \mathcal {V}_j$	velocity sections of $\gamma _+$	(2.28), (2.29)
$\mathrm {d}\sigma , \mathrm {d}\sigma _j$	Measures defined on the velocity sections of $\mathcal {V}(x), \mathcal {V}_j$	(2.28)
${\tilde{w}}$	$w^{-1} \mu ^{-1/2}$	(2.35)
$\mathrm {d}\Sigma _l$	a stochastic measure	(2.46)
Y, W	Scaled trajectory for the dynamic problem	(3.2)
${\tilde{t}}_\mathbf {b}, {\tilde{x}}_\mathbf {b}$	Scaled backward exit time and position	(3.4)
${\tilde{t}}_\mathbf {f}, {\tilde{x}}_\mathbf {f}$	Scaled forward exit time and position	(3.4)
$f_\delta$	An interior and non-grazing parts of f	(3.11)
$f_{la}, f_{sm}$	Functions defined in (3.22)
$\mathbf {S}_i f$	Functions defined in (3.40) and (3.40)
$[[[ \cdot ]]]$	A closing norm for the unsteady case	(3.93)
$\overline{f_{\delta }}$	Defined in (A.1.12)