From the analysis performed during the Step 1, there is $h^n$ unique solution to Equation (6.17) associated with $H_i^n$ for $i = 1, 2, \mathfrak{H}^n$, and $h_0^n$. Moreover, there holds the representation formula (6.18), the weak formulation (6.19), and the energy estimate (6.20).

In particular, defining the trace using the representation formula (6.18) as detailed in [55, Definition 8.1], we deduce from (6.20), the linearity of Equation (6.17), and the convergences from (6.21) that $h^n$ and $\gamma h^n$ are Cauchy sequences in $L_{\omega_1}^{\infty}(\mathcal{U}^\varepsilon)$ and $L_{\omega_1}^{\infty}(\Gamma^\varepsilon)$ respectively. Therefore, there are functions $h \in L_{\omega_1}^{\infty}(\mathcal{U}^\varepsilon)$ and $\gamma h \in L_{\omega_1}^{\infty}(\Gamma^\varepsilon)$ such that, as $n \to \infty$, there holds

$$h^n \to h \quad \text{strongly in } L_{\omega_1}^{\infty}(\mathcal{O}^\varepsilon) \quad \text{and} \quad \gamma h^n \to \gamma h \quad \text{strongly in } L_{\omega_1}^{\infty}(\Gamma^\varepsilon).$$

Moreover, passing to the limit in the weak formulation of Equation (6.17) associated with $H_i^n$ for $i = 1, 2, \mathfrak{H}^n$, and $h_0^n$, we deduce that $h$, with the trace function $\gamma h$, is a weak solution of Equation (6.17) associated with $H_i$ for $i = 1, 2, \mathfrak{H}$, and $h_0$, satisfying the energy estimate

$$(6.22) \quad \|h\|_{L_{\omega_1}^{\infty}(\mathcal{U}^{\varepsilon})} + \|\gamma h\|_{L_{\omega_1}^{\infty}(\Gamma^{\varepsilon})} \le \|h_0\|_{L_{\omega_1}^{\infty}(\mathcal{O}^{\varepsilon})} + \|\mathfrak{H}\|_{L_{\omega_1}^{\infty}(\Gamma^{\varepsilon})} \\ \qquad + \|H_1\|_{L_{\omega_1}^{\infty}(\mathcal{U}^{\varepsilon})} + 2 \frac{\nu_1}{\nu_0} \|H_2\|_{L_{\omega_1^{\nu-1}}^{\infty}(\mathcal{U}^{\varepsilon})}.$$

In particular, (6.22) and the linearity of Equation (6.17) imply the uniqueness of this solution.

Furthermore, we may pass to the limit in (6.18) a.e in $(x, v) \in \mathcal{O}^\varepsilon$ such that $h$ satisfies the Duhamel formulation (6.18).

**Step 3.** We now set $\psi^0 = 0$, $\psi_0 \in L_{\omega_1}^{\infty}(\mathcal{O}^\varepsilon)$, $\alpha \in (0, 1)$, we recall that $G \in L_{\omega_1^{\nu-1}}^{\infty}(\mathcal{U}^\varepsilon)$ and we consider the recurrent sequence of solutions given by the following evolution equation

$$(6.23) \quad \left\{ \begin{array}{lll} \partial_t \psi^{k+1} & = & \mathcal{F} \psi^{k+1} + K \psi^k + G \quad \text{in } \mathcal{U}^\varepsilon \\ \gamma_- \psi^{k+1} & = & \alpha \mathcal{R} \gamma_+ \psi^k \quad \text{on } \Gamma_-^\varepsilon \\ \psi_{t=0}^{k+1} & = & \psi_0 \quad \text{in } \mathcal{O}^\varepsilon. \end{array} \right.$$

Indeed, we remark that if we assume that, for a certain $k$, $\psi^k \in L_{\omega_1}^{\infty}(\mathcal{U}^\varepsilon)$, then (K1) implies that $K\psi^k \in L_{\omega_1}^{\infty}(\mathcal{U}^\varepsilon)$. Therefore Step 2 implies the existence of $\psi^{k+1} \in L_{\omega_1}^{\infty}(\mathcal{U}^\varepsilon)$, with a trace $\gamma\psi^{k+1} \in L_{\omega_1}^{\infty}(\Gamma^\varepsilon)$, unique weak solution of Equation (6.23) in the sense provided by the Step 2.

We take now $A > 0$ to be fixed later and, inspired by the recent series of papers [15, 16, 12], we define the modified weight functions

$$\omega_1^A = \omega_1^A(v) := \mathcal{M}^{-1}\xi_A(v) + (1 - \xi_A(v))\omega_1,$$

where $\xi_A(v) := \xi(|v|/A)$, for a function $\xi \in C^2(\mathbb{R}_+, \mathbb{R})$, such that $\mathbf{1}_{[0,1]} \le \xi \le \mathbf{1}_{[0,2]}$. Moreover, we remark that there is a constant $c_A > 0$ such that

$$(6.24) \qquad c_A^{-1} \omega_1 \le \omega_1^A \le c_A \omega.$$

Using now the representation formula given by the Step 2 for the solutions of Equation (6.23), multiplying by $\omega_1^A$, using (K1) together with (6.24), and arguing as during the proof of (K2) with $\nu_2 = \nu_0/2$, we deduce that there is a constant $C_0 > 0$ satisfying

$$(6.25) \quad |\psi^{k+1}(t, x, v)| \omega_1^A = e^{-\nu_0 t} \|\psi_0\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + t C_0 \|\psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} \\ \qquad + C_0 \|G\|_{L_{\omega_1^{\nu-1}}^{\infty}(\mathcal{O}^\varepsilon)} + \alpha \omega_1^A(v) |\mathcal{R}\gamma_+\psi^k(t_1, x_1, v)|.$$

To bound the boundary term from (6.25) we remark that

$$\begin{aligned} |\mathcal{R}\gamma_+\psi^k(t_1, x_1, v)| & \le (1 - \iota^\varepsilon)\|\psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + \iota^\varepsilon \omega_1^A \mathcal{M} \left|\int_{\mathbb{R}^3} \gamma_+\psi^k(t_1, x_1, u) (n_x \cdot u)_+ \, du \right| \\ & \le (1 - \iota^\varepsilon)\|\psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + \iota^\varepsilon \|\psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} \omega_1^A \mathcal{M} \int_{\mathbb{R}^3} (n_x \cdot u)_+ \omega_1^A(u)^{-1} \, du. \end{aligned}$$
On the one hand, we observe that from the very definition of $\omega_A$ there holds

$$ \omega_1^A \mathcal{M} = \xi_A + (1 - \xi_A)\omega_1 \mathcal{M} \le 1. $$

On the other hand, we have that

$$ 1 \le \int_{\mathbb{R}^3} (n_x \cdot u)_+ \omega_1^A(u)^{-1} du = \int_{\mathbb{R}^3} \mathcal{M}(u)(n_x \cdot u)_+ (\xi_A + (1 - \xi_A)\omega \mathcal{M})^{-1} du := \Theta_A \underset{A \to \infty}{\longrightarrow} 1, $$

where we have used the fact that $\xi_A + (1 - \xi_A)\omega \mathcal{M} \to 1$ as $A \to \infty$ and $\tilde{\mathcal{M}} = 1$.

Using then that $\alpha \in (0, 1)$ and $\Theta_A \to 1$ as $A \to \infty$, we can choose $A > 0$ large enough such that $\Theta_A < 1/\alpha$, this implies that

$$ \alpha (1 - \iota^\varepsilon + \iota^\varepsilon \Theta_A) = \alpha (1 + \iota^\varepsilon (\Theta_A - 1)) \le \alpha \Theta_A < 1, $$

where we have used the fact that $\Theta_A \ge 1$ for every $A > 0$ and $\iota^\varepsilon(x) \le 1$ for almost every $x \in \partial \Omega^\varepsilon$. Using then the linearity of Equation (6.23) and the previous analysis we have that

$$ \|\psi^{k+1} - \psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + \|\gamma\psi^{k+1} - \gamma\psi^k\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} \le (t C_0 + \alpha \Theta_A) \|\psi^k - \psi^{k-1}\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)}, $$

These informations imply that, for a small time $T_0 > 0$, $\psi^k$ and $\gamma\psi^k$ are Cauchy sequences in the the Banach spaces $L_{\omega_1^A}^{\infty}(\mathcal{U}_{T_0}^\varepsilon)$ and $L_{\omega_1^A}^{\infty}(\Gamma_{T_0}^\varepsilon)$. Therefore there are functions $\psi \in L_{\omega_1^A}^{\infty}(\mathcal{U}_{T_0}^\varepsilon)$ and $\gamma\psi \in L_{\omega_1^A}^{\infty}(\Gamma_{T_0}^\varepsilon)$ such that, as $k \to \infty$, there holds

$$ \psi^k \to \psi \text{ strongly in } L_{\omega_1^A}^{\infty}(\mathcal{U}_{T_0}^\varepsilon) \quad \text{and} \quad \gamma\psi^k \to \gamma\psi \text{ strongly in } L_{\omega_1^A}^{\infty}(\Gamma_{T_0}^\varepsilon). $$

Moreover, using (6.24) and passing to the limit in the weak formulation associated with Equation (6.23) we deduce that $\psi$ solves the evolution equation

$$ (6.26) \qquad \left\{ \begin{array}{lll} \partial_t \psi = \mathcal{T} \psi + K\psi + G & \text{in } \mathcal{U}^\varepsilon \\ \gamma_- \psi = \alpha \mathcal{R} \gamma_+ \psi & \text{on } \Gamma_-^\varepsilon \\ \psi_{t=0} = \psi_0 & \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$

in the time interval $[0, T_0]$. Repeating this argument in every time interval $[nT_0, (n+1)T_0]$ for every $n \in \mathbb{N}$ we deduce the existence of a global solution $\psi$. Furthermore, from the representation formula associated with Equation (6.23), and the previous strong convergence result we have that

$$ (6.27) \quad \psi(t, x, v) = e^{-\nu(v)t} \psi_0(x, v) \mathbf{1}_{t_1 \le 0} + \int_{\max(0, t_1)}^{t} e^{-\nu(v)(t-s)} K \psi(s, x, v) \mathrm{d}s \\ \qquad + \int_{\max(0, t_1)}^{t} e^{-\nu(v)(t-s)} G(s, x, v) \mathrm{d}s + e^{-\nu(v)(t-t_1)} \gamma \psi(t_1, x_1, v) \mathbf{1}_{t_1 > 0}, $$

for every $t \ge 0$ and for almost every $(x, v) \in \mathcal{O}^\varepsilon$.

We can then use the well-posedness from the Step 3 of the proof of Theorem 6.8 and Remark 6.9 to justify the repetition of the hypocoercivity result from Theorem 2.1 applied to Equation (6.26). We may thus repeat the computations leading to the conclusion of Proposition 6.7. Therefore, there are $\varepsilon_{12}, \theta > 0$, given by Proposition 6.1, such that for every $\varepsilon \in (0, \varepsilon_{12})$, there holds the following energy estimate

$$ (6.28) \quad \|\psi_t\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + \|\gamma\psi_t\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} \le C e^{-\theta\varepsilon^2 t} \left( \|\psi_0\|_{L_{\omega_1^A}^{\infty}(\mathcal{O}^\varepsilon)} + \sup_{s \in [0, t]} \|G\|_{L_{\omega_1^{\nu-1}}^{\infty}(\mathcal{U}^\varepsilon)} \right), $$

for every $t \ge 0$, and where $C > 0$ is an universal constant independent of $\varepsilon$. Finally, it is worth remarking that (6.28) is uniform in $\alpha$ from the fact that this estimate comes from Proposition 6.7.
Step 4. We now take a sequence $\alpha_k \nearrow 1$, with $k \in \mathbb{N}$, and we consider the sequence $(f_k)_{k \in \mathbb{N}}$ obtained in Step 3 as the solution to the modified Maxwell reflection boundary condition problem

$$ (6.29) \qquad \left\{ \begin{array}{lll} \partial_t f_k = \mathcal{T} f_k + K f_k + G & \text{in } \mathcal{U}^\varepsilon \\ \gamma_- f_k = \alpha_k \mathcal{R} \gamma_+ f_k & \text{on } \Gamma_-^\varepsilon \\ f_{k,t=0} = f_0 & \text{in } \mathcal{O}^\varepsilon. \end{array} \right. $$

Moreover, it is worth remarking that there holds the representation formula (6.27) for $f_k$.

Using then the energy estimate (6.28) we obtain that $f^k$ and $\gamma f^k$ are bounded sequences in $L_{\omega_1}^\infty(\mathcal{U}^\varepsilon)$ and $L_{\omega_1}^\infty(\Gamma^\varepsilon)$, therefore there is $f \in L_{\omega_1}^\infty(\mathcal{U}^\varepsilon)$ and $\gamma f \in L_{\omega_1}^\infty(\Gamma^\varepsilon)$ such that, as $k \to \infty$, there holds

$$ f^k \to f \text{ weakly-}^* \text{ in } L_{\omega_1}^\infty(\mathcal{U}^\varepsilon) \quad \text{and} \quad \gamma f^k \to \gamma f \text{ weakly-}^* \text{ in } L_{\omega_1}^\infty(\Gamma^\varepsilon). $$

We can thus pass to the limit in the weak formulation associated to Equation (6.29) and we deduce that $f$, with its associated trace $\gamma f$, solves Equation (1.20) in the sense of (6.1).

Furthermore, we remark that by taking the representation formula (6.27) for $f^k$, multiplying it by any arbitrary function $g \in L_{m_1}^1(\mathcal{U}^\varepsilon)$, where $m_1 := \omega_1^{-1}$, we may pass to the limit as $k \to \infty$ and by density we deduce that $f$ also satisfies the representation formula (6.27).

Finally, we conclude by remarking that the conclusion of Proposition 6.1 holds from the fact that we have access to a Duhamel-type formulation and using Remark 6.9. $\square$

6.4. **Proof of Theorem 6.3.** The proof of Theorem 6.3 follows exactly the same arguments used for the proof of Theorem 6.2, using Proposition 5.1 instead of Proposition 6.7. $\square$

## 7. PROOF OF THEOREM 1.3 FOR STRONGLY CONFINING WEIGHTS

We dedicate this section to prove a first part of Theorem 1.3.

**Theorem 7.1.** Consider either Assumption (RH1) or (RH2) to hold, and let $\omega_1 \in \mathfrak{W}_1$ a strongly confining admissible weight function. For every $\varepsilon \in (0, \varepsilon_{12})$, where we recall that $\varepsilon_{12} > 0$ is given by Proposition 6.1, there is $\eta_0^1(\varepsilon) \in (0, 1)$ satisfying $\eta_0^1(\varepsilon) \to 0$ as $\varepsilon \to 0$, such that if $f_0 \in L_{\omega_1}^\infty(\mathcal{O}^\varepsilon)$ satisfies

$$ \|f_0\|_{L_{\omega_1}^{\infty}(\mathcal{O}^{\varepsilon})} \leq (\eta_0^1(\varepsilon))^2, $$

there is a function $f \in L_{\omega_1}^\infty(\mathcal{U}^\varepsilon)$ with an associated trace $\gamma f \in L_{\omega_1}^\infty(\Gamma^\varepsilon)$, unique solution to the linearized rescaled Boltzmann Equation (1.5)-(1.9)-(1.10) in the distributional sense, i.e for every test function $\varphi \in \mathcal{D}(\mathcal{U}^\varepsilon)$ there holds

$$ (7.1) \quad \int_{\mathcal{O}^\varepsilon} f(t, \cdot) \varphi(t, \cdot) dv dx - \int_0^t \int_{\mathcal{O}^\varepsilon} f (\partial_t \varphi + v \cdot \nabla_x \varphi + K^* \varphi - \nu \varphi) + \varphi \mathcal{Q}(f, f) dv dx ds \\ \qquad + \int_0^t \int_{\Sigma^\varepsilon} \gamma f \varphi (n_x \cdot v) dv d\sigma_x = \int_{\mathcal{O}^\varepsilon} f_0 \varphi(0, \cdot) dv dx, $$

for every $t \ge 0$, and we recall that $K^*$ is defined in (6.2). Furthermore, there holds

$$ (7.2) \qquad \|f_t\|_{L_{\omega_1}^{\infty}(\mathcal{O})} \le e^{-\theta\varepsilon^2t} \eta_0^1(\varepsilon) \quad \forall t \ge 0. $$

7.1. **Estimate on the bilinear Boltzmann collision operator.** Before studying the non-linear Boltzmann equation we state the following control on the bilinear Boltzmann collision operator.

**Lemma 7.2.** Let $\omega$ be an admissible weight function, and let $g, h \in L_\omega^\infty(\mathcal{O}^\varepsilon)$. There holds

$$ \|\mathcal{Q}(g, h)\|_{L_{\omega}^{\infty}(\omega_{\nu-1}(\mathcal{O}^{\varepsilon}))} \le C_{\mathcal{Q}} \|g\|_{L_{\omega}^{\infty}(\mathcal{O}^{\varepsilon})} \|h\|_{L_{\omega}^{\infty}(\mathcal{O}^{\varepsilon})}. $$

for some constant $C_{\mathcal{Q}} = C_{\mathcal{Q}}(\omega) > 0$.
Remark 7.3. This type of estimate is classical in the study of the Boltzmann equation, see for instance [38, 42, 20, 31]. Thus we do not prove it, the interested reader may find a prove in either of the above references.

7.2. **Proof of Theorem 7.1.** We define the norm

$$ [[f]]_1 := \sup_{s \in [0, \infty)} \left[ e^{\theta \epsilon^2 s} \|f_s\|_{L_{\omega_1}^{\infty}(\mathcal{O}^{\epsilon})} \right], $$

where we recall that $\theta > 0$ is given by Proposition 6.1. We consider a $\lambda > 0$ to be fixed later, and we define the Banach space

$$ \mathcal{Z}_1 := \{g \in L_{\omega_1}^{\infty}(\mathcal{U}^{\epsilon}), [[g]]_1 \le \lambda\}, $$

equipped with the strong topology in $L_{\omega_1}^{\infty}(\mathcal{U}^{\epsilon})$.

We denote $\Psi_1$ as the map that to $g \in \mathcal{Z}_1$ assigns $f$ the solution of

$$ (7.3) \qquad \left\{ \begin{array}{lll} \partial_t f = & \mathcal{L} f + \mathcal{Q}(g, g) & \text{in } \mathcal{U}^{\epsilon} \\ \gamma_- f = & \mathcal{R}\gamma_+ f, & \text{on } \Gamma_-^{\epsilon} \\ f_{t=0} = & f_0, & \text{in } \mathcal{O}^{\epsilon}, \end{array} \right. $$

which is given by Theorem 6.2. Indeed, we observe that Theorem 6.2 holds by using Lemma 7.2 and the fact that $g \in \mathcal{Z}_1$. We remark then that Proposition 6.1 implies that for every $\epsilon \in (0, \epsilon_{12})$ there is $C_0 = C_0(\epsilon) > 0$ satisfying $C_0(\epsilon) \to \infty$ as $\epsilon \to 0$, such that

$$ (7.4) \qquad \begin{aligned} [[f]]_1 &\le C_0 \|f_0\|_{L_{\omega_1}^{\infty}(\mathcal{O}^{\epsilon})} + C_0 \sup_{s \in [0,t]} \left[ e^{\theta \epsilon^2 s} \|\mathcal{Q}(g_s, g_s)\|_{L_{\omega_1 \nu^{-1}}^{\infty}(\mathcal{O}^{\epsilon})} \right] \\ &\le C_0 \|f_0\|_{L_{\omega_1}^{\infty}(\mathcal{O}^{\epsilon})} + C_0 C_{\mathcal{Q}} [[g]]_1^2 \end{aligned} $$

where we have used Lemma 7.2 to obtain the second line and the constant $C_{\mathcal{Q}} = C_{\mathcal{Q}}(\omega_1) > 0$ is given by Lemma 7.2.

We then choose $\eta_0^1(\epsilon) = \lambda(\epsilon)$ and

$$ \lambda = \lambda(\epsilon) := \min \left( \frac{1}{C_0(\epsilon)(1 + C_{\mathcal{Q}})}, \frac{1}{4C_0(\epsilon)C_{\mathcal{Q}}}, 1 \right) \xrightarrow{\epsilon \to 0} 0. $$

In particular, this choice of $\lambda$ together with (7.4) implies that $f \in \mathcal{Z}_1$, thus $\Psi_1 : \mathcal{Z}_1 \to \mathcal{Z}_1$.

Furthermore, if we take $g_1, g_2 \in \mathcal{Z}_1$ and denoting $f_i = \Psi_1(g_i)$ for $i = 1, 2$, we have that $\psi = f_1 - f_2$ is the weak solution of the evolution equation

$$ \left\{ \begin{array}{lll} \partial_t \psi = & \mathcal{L} \psi + \mathcal{Q}(g_1 + g_2, g_1 - g_2) & \text{in } \mathcal{U}^{\epsilon} \\ \gamma_{-} \psi = & \mathcal{R} \gamma_{+} \psi & \text{on } \Gamma_{-}^{\epsilon} \\ \psi_{t=0} = & 0 & \text{in } \mathcal{O}^{\epsilon}, \end{array} \right. $$

in the sense of Theorem 6.2. Using then Proposition 6.1 and Lemma 7.2 as before we have that

$$ [[\psi]]_1 \le C_0 C_{\mathcal{Q}} [[g_1 + g_2]]_1 [[g_1 - g_2]]_1 \le 2 C_0 C_{\mathcal{Q}} \lambda [[g_1 - g_2]]_1 \le \frac{1}{2} [[g_1 - g_2]]_1. $$

This implies that $\Psi_1$ is a contraction in $\mathcal{Z}_1$, thus there is a unique fixed point $f \in \mathcal{Z}_1$ for this map. We deduce that $f$ is a weak solution of the nonlinear Boltzmann Equation (1.5)-(1.9)-(1.10) in the sense of (7.1). Furthermore, (7.2) comes from the very fact that $f \in \mathcal{Z}_1$. $\square$

8. PROOF OF THEOREM 1.3 FOR WEAKLY CONFINING WEIGHTS

We dedicate this section to prove the remaining part of Theorem 1.3 on the well-posedness of the Boltzmann equation for weakly confining admissible weight functions.
**Theorem 8.1.** Consider either Assumption (RH1) or (RH2) to hold, and let $\omega_0 \in \mathfrak{W}_0$ a weakly confining admissible weight function. Define furthermore $\varepsilon_{13} := \min(\varepsilon_9, \varepsilon_{12})$, where we recall that $\varepsilon_9 > 0$ is given by Proposition 5.1 and $\varepsilon_{12} > 0$ is given by Proposition 6.1.

For every $\varepsilon \in (0, \varepsilon_{13})$ there is $\eta_0^0(\varepsilon) \in (0, 1)$ such that $\eta_0^0(\varepsilon) \to 0$ as $\varepsilon \to 0$, such that for every $f_0 \in L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})$ satisfying

$$ \|f_0\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \leq (\eta_0^0(\varepsilon))^2, $$

there is $f \in L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon})$ with an associated trace $\gamma f \in L_{\omega_0}^{\infty}(\Gamma^{\varepsilon})$, unique solution to the linearized rescaled Boltzmann Equation (1.5)-(1.9)-(1.10) in the distribution sense, i.e for every test function $\varphi \in \mathcal{D}(\bar{\mathcal{U}}^{\varepsilon})$ there holds

$$ (8.1) \quad \int_{\mathcal{O}^{\varepsilon}} f(t, \cdot) \varphi(t, \cdot) dv dx - \int_{0}^{t} \int_{\mathcal{O}^{\varepsilon}} f (\partial_t \varphi + v \cdot \nabla_x \varphi + K^* \varphi - \nu \varphi) + \varphi \mathcal{Q}(f, f) dv dx ds \\ \qquad + \int_{0}^{t} \int_{\Sigma^{\varepsilon}} \gamma f \varphi (n_x \cdot v) dv d\sigma_x = \int_{\mathcal{O}^{\varepsilon}} f_0 \varphi(0, \cdot) dv dx, $$

for every $t \ge 0$, and we recall that $K^*$ is defined in (6.2). Furthermore, there is $\theta > 0$ such that there holds

$$ (8.2) \qquad \|f_t\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \le e^{-\theta \varepsilon^2 t} \eta_0^0(\varepsilon) \quad \forall t \ge 0, $$

**Remark 8.2.** The proof follows the main ideas of [10, Section 6].

**8.1. Dissipative equation.** Let $h : \mathcal{U}^{\varepsilon} \to \mathbb{R}$, we study the following equation

$$ (8.3) \quad \left\{ \begin{array}{lll} \partial_t f_1 & = & \mathcal{T} f_1 + \mathcal{A}_\delta f_1 + \mathcal{Q}(f_1 + h, f_1 + h) & \text{in } \mathcal{U}^{\varepsilon} \\ \gamma_- f_1 & = & \mathcal{R} \gamma_+ f_1 & \text{on } \Gamma_-^{\varepsilon} \\ f_{1,t=0} & = & f_0 & \text{in } \mathcal{O}^{\varepsilon}. \end{array} \right. $$

We dedicate this subsection to prove the following well-posedness result.

**Proposition 8.3.** Consider either Assumption (RH1) or (RH2) to hold, let $\omega_0 \in \mathfrak{W}_0$ a weakly confining admissible weight function, and let $f_0 \in L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})$ and such that

$$ \|f_0\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \le \eta_1, $$

some $\eta_1 > 0$ small enough. Assume furthermore that for every $\varepsilon \in (0, \varepsilon_8)$ and $\delta \in (0, \delta_0)$ there holds

$$ (8.4) \qquad \sup_{s \in [0, \infty)} \left[ e^{\theta \varepsilon^2 s} \|h_s\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \right] \le \eta_1, $$

where $\varepsilon_9, \delta_0 > 0$ are given by Proposition 5.1, and $\theta \in (0, \nu_0/2)$ is given by Proposition 6.1. For every $\varepsilon \in (0, \varepsilon_8)$ and $\delta \in (0, \delta_0)$ there is $f_1 \in L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon})$ with an associated trace $\gamma f_1 \in L_{\omega_0}^{\infty}(\Gamma^{\varepsilon})$, unique solution to Equation (8.3) in the weak sense, i.e for any $\varphi \in \mathcal{D}(\bar{\mathcal{U}}^{\varepsilon})$ there holds

$$ (8.5) \quad \int_{\mathcal{O}^{\varepsilon}} f_1(t, \cdot) \varphi(t, \cdot) \, dx dv - \int_{0}^{t} \int_{\mathcal{O}^{\varepsilon}} f_1 A_{\delta}^{*} \varphi + f (\partial_t \varphi + v \cdot \nabla_x \varphi - \nu \varphi) \, dx dv ds \\ - \int_{0}^{t} \int_{\mathcal{O}^{\varepsilon}} \mathcal{Q}(f_1 + h, f_1 + h) \varphi \, dx dv ds + \int_{0}^{t} \int_{\Sigma^{\varepsilon}} \gamma f_1 \varphi (n_x \cdot v) \, d\sigma_x dv = \int_{\mathcal{O}^{\varepsilon}} f_0(\cdot) \varphi(0, \cdot) \, dx dv, $$

for every $t \ge 0$, and where we recall that $A_{\delta}^{*}$ has been defined in (6.4). Furthermore, there holds

$$ (8.6) \qquad \|f_{1,t}\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \le C e^{-\theta \varepsilon^2 t} \left( \|f_0\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} + \sup_{s \in [0, \infty)} \left[ e^{\theta \varepsilon^2 s} \|h_s\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \right]^2 \right) \quad \forall t \ge 0, $$

for some constant $C > 0$, independent of $\varepsilon$.

**Proof.** The proof is mainly a repetition of the arguments from the proof of Theorem 7.1, thus we skip it. $\square$
8.2. **Control on the operator $K_\delta$.** We recall the definition of $K_\delta$ given by (1.25) and from [38, Section 4] we have the following control.

**Lemma 8.4.** Let $\omega$ be an admissible weight function. For every $\delta > 0$ there holds

$$ (8.7) \qquad \|K_\delta f\|_{L^\infty(\mathcal{O}^\varepsilon)} \le C_\delta \|f\|_{L^\infty(\mathcal{O}^\varepsilon)}, $$

for some constructive constant $C_\delta > 0$.

8.3. **Residual linear equation.** We consider $g : \mathcal{U}^\varepsilon \to \mathbb{R}$, and we study the following linear evolution equation

$$ (8.8) \qquad \left\{ \begin{array}{lll} \partial_t f_2 & = & \mathcal{L} f_2 + K_\delta g \quad \text{in } \mathcal{U}^\varepsilon \\ \gamma_- f_2 & = & \mathcal{R} \gamma_+ f_2 \quad \text{on } \Gamma_-^\varepsilon \\ f_{2,t=0} & = & 0 \quad \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$

where we recall that $K_\delta$ is defined in (1.25). We define the projection operator

$$ \mathcal{P} f(t, x, v) := \mathcal{M}(v) \int_{\mathcal{O}^\varepsilon} f(t, x, u) du, $$

and we dedicate this subsection to prove the following well-posedness and long-time behavior result

**Proposition 8.5.** Consider either Assumption (RH1) or (RH2) to be satisfied, let $g \in L_{\omega_0}^\infty(\mathcal{U}^\varepsilon)$, for some weakly confining admissible weight function $\omega_0 \in \mathfrak{W}_0$.

For every strongly confining admissible weight function $\omega_1 \in \mathfrak{W}_1$, every $\delta > 0$, and every $\varepsilon \in (0, \varepsilon_{13})$, where we recall that $\varepsilon_{13} > 0$ is given in Theorem 8.1, there is $f_2 \in L_{\omega_1}^\infty(\mathcal{U}^\varepsilon)$ unique weak solution of Equation (8.8) in the sense of the Theorem 6.2.

Moreover, if $\mathcal{P}(f_2 + g) = 0$, and for every $\varepsilon \in (0, \varepsilon_{13})$ there holds

$$ (8.9) \qquad \|g_t\|_{L_{\omega_0}^\infty(\mathcal{O}^\varepsilon)} \le C_g e^{-\theta\varepsilon^2 t} \quad \forall t \ge 0, $$

for some constant $C_g > 0$, and where $\theta > 0$ is given by Proposition 6.1, then there is a constant $C = C(\varepsilon) > 0$ satisfying $C(\varepsilon) \to \infty$ as $\varepsilon \to 0$, such that

$$ (8.10) \qquad \|f_{2,t}\|_{L_{\omega_1}^\infty(\mathcal{O}^\varepsilon)} \le C C_g e^{-\theta\varepsilon^2 t}, $$

for every $t \ge 0$.

*Proof.* The proof follows the main ideas from [10, Proposition 6.8] by using Lemma 8.4, the existence results from Theorem 6.2 and the decay estimate from Proposition 6.1, thus we skip it. □

8.4. **Proof of Theorem 8.1.** We consider $h \in L_{\omega_0}^\infty(\mathcal{O}^\varepsilon)$ such that for every $\varepsilon \in (0, \varepsilon_{13})$ and $\delta \in (0, \delta_0)$ there holds

$$ (8.11) \qquad \sup_{s \in [0, \infty)} \left[ e^{\theta \varepsilon^2 s} \|h_s\|_{L_{\omega_0}^\infty(\mathcal{O}^\varepsilon)} \right] \le \eta_1, $$

where $\delta_0 > 0$ is given by Proposition 5.1, $\theta \in (0, \nu_0/2)$ is given by Proposition 6.1, and $\eta_1 > 0$ is given by Proposition 8.3. We define $f_1 \in L_{\omega_0}^\infty(\mathcal{U}^\varepsilon)$ as the solution of the evolution equation

$$ (8.12) \qquad \left\{ \begin{array}{lll} \partial_t f_1 & = & \mathcal{P} f_1 + \mathcal{A}_\delta f_1 + \mathcal{Q}(f_1 + h, f_1 + h) \quad \text{in } \mathcal{U}^\varepsilon \\ \gamma_- f_1 & = & \mathcal{R} \gamma_+ f_1, \quad \text{on } \Gamma_-^\varepsilon \\ f_{1,t=0} & = & f_0, \quad \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$
which is given by Proposition 8.3. Moreover, we fix $\omega_1 \in \mathfrak{M}_1$ a strongly confining admissible weight function, and we define $f_2 \in L_{\omega_1}^{\infty}(\mathcal{U}^{\varepsilon})$ as the solution of the evolution equation

$$ (8.13) \qquad \left\{ \begin{array}{lll} \partial_t f_2 & = & \mathcal{L} f_2 + \mathcal{K}_\delta f_1 \quad \text{in } \mathcal{U}^\varepsilon \\ \gamma_- f_2 & = & \mathcal{R} \gamma_+ f_2 \quad \text{on } \Gamma_-^\varepsilon \\ f_{2,t=0} & = & 0 \quad \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$

which is given by Proposition 8.5. We emphasize that, defined this way, $f_2$ depends of $h$.

We define the norm

$$ [[g]]_0 := \sup_{s \in [0, \infty)} \left[ e^{\theta \varepsilon^2 s} \|g_s\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} \right], $$

and, for $\lambda \in (0, \eta_1)$ to be fixed later, we define the Banach space $\mathcal{Z}_0 := \{g \in L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon}), [[g]]_0 \le \lambda\}$, equipped with the strong topology in $L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon})$, which makes $\mathcal{Z}_0$ a bounded, convex, closed subset of $L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon})$. Furthermore, we denote $\Psi$ as the map that to $h \in \mathcal{Z}_0$ assigns $f_2$ as defined above.

We remark that $f_1 + f_2$ solves Equation (1.20) with $G = \mathcal{Q}(f_1 + h, f_1 + h)$, which together with (1.15) implies that $\mathcal{P}(f_1 + f_2) = 0$. We set then the constant $C_{\omega_1} > 0$ such that

$$ \|g\|_{L_{\omega_0}^{\infty}(\mathcal{U}^{\varepsilon})} \le C_{\omega_1} \|g\|_{L_{\omega_1}^{\infty}(\mathcal{U}^{\varepsilon})}, $$

and using (8.11), (8.10) and (8.6) we obtain that

$$ [[f_1]]_0 \le C_1 (\|f_0\|_{L_{\omega_0}^{\infty}(\mathcal{O}^{\varepsilon})} + [[h]]_0^2) \le C_1 ((\eta_0^0)^2 + \lambda^2), \quad \text{and} \quad [[f_2]]_0 \le C_{\omega_1} C_2 C_1 ((\eta_0^0)^2 + \lambda^2), $$

where $C_1 > 0$ is given by (8.6) in Proposition 8.3, and $C_2 = C_2(\varepsilon) > 0$, satisfying $C_2(\varepsilon) \to \infty$ as $\varepsilon \to 0$, is given by (8.10) in Proposition 8.5.

We define $\eta_0^0 = \lambda$ and

$$ \lambda = \lambda(\varepsilon) := \min \left( \frac{1}{8C_{\omega_1}C_1C_2(\varepsilon)}, \frac{1}{6C_1}, \frac{1}{8C_QC_3}, \frac{1}{16C_3C_Q}, \frac{\eta_1}{2}, 1 \right) \underset{\varepsilon \to 0}{\longrightarrow} 0 $$

where $C_Q > 0$ is given by Lemma 7.2 and $C_3 > 0$ is the constant given by Proposition 5.1. This choices of parameters implies in particular that $[[f_1]] \le \lambda$ and $[[f_2]]_0 \le \lambda$, thus $f_1, f_2 \in \mathcal{Z}_0$, therefore $\Psi: \mathcal{Z}_0 \to \mathcal{Z}_0$.

Furthermore, we consider $h_1, h_2 \in \mathcal{Z}_0$ and we define $f_1, g_1 \in \mathcal{Z}_0$ as the solutions of Equation (8.12) associated with $h_1$ and $h_2$ respectively. We also define $f_2 = \Psi(h_1)$ and $g_2 = \Psi(h_2)$, i.e the solutions of Equation (8.13) associated with $f_1$ and $g_1$ respectively. We denote $\phi = f_1 - g_1$, $\psi := f_2 - g_2$, and we observe that $\phi$ and $\psi$ are the weak solutions of

$$ \left\{ \begin{array}{lll} \partial_t \phi = & \mathcal{T} \phi + \mathcal{A}_\delta \phi + \mathcal{Q}(f_1 + g_1 + h_1 + h_2, \phi + h_1 - h_2) & \text{in } \mathcal{U}^\varepsilon \\ \gamma_- \phi = & \mathcal{R} \gamma_+ \phi, & \text{on } \Gamma_-^\varepsilon \\ \phi_{t=0} = & 0, & \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$

and

$$ \left\{ \begin{array}{lll} \partial_t \psi = & \mathcal{L} \psi + \mathcal{K}_\delta \phi & \text{in } \mathcal{U}^\varepsilon \\ \gamma_- \psi = & \mathcal{R} \gamma_+ \psi & \text{on } \Gamma_-^\varepsilon \\ \psi_{t=0} = & 0 & \text{in } \mathcal{O}^\varepsilon, \end{array} \right. $$

respectively. On the one hand using Proposition 5.1 and Lemma 7.2 we have that

$$ (8.14) \qquad [[\phi]]_0 \le C_3 C_Q [[f_1 + g_1 + h_1 + h_2]]_0 [[\phi + h_1 - h_2]]_0 \le \frac{1}{2} [[\phi]]_0 + 4\lambda C_3 C_Q [[h_1 - h_2]]_0 $$

where we have used the triangular inequality and the definition of $\lambda$ to obtain the last inequality. In particular, (8.14) and our very definition of $\lambda$ implies that

$$ [[\phi]]_0 \le 8\lambda C_3 C_Q [[h_1 - h_2]]_0 \le \frac{1}{2} [[h_1 - h_2]]_0. $$

This implies that $\Psi$ is a contraction in $\mathcal{Z}_0$, thus using the Banach fixed point theorem we deduce that there is a unique fixed point $f_2 \in \mathcal{Z}_0$ for this map. Moreover, from the very definition
of the map $\Psi$ we deduce that $f_2$ is a weak solution of Equation (8.13), and Proposition 8.3 further gives the uniqueness of $f_1$, weak solution of Equation (8.12) with $h = f_2$.

Putting together the aforementioned weak formulations we deduce that $f = f_1 + f_2$ is a weak solution of the Boltzmann equation (1.5)-(1.9)-(1.10) in the sense of (8.1). Finally, (8.2) comes from the very fact that $f = f_1 + f_2$ with $f_1, f_2 \in \mathcal{Z}_0$. This concludes the proof. $\square$

**Acknowledgements.** The author deeply thanks Kleber Carrapatoso and Stéphane Mischler for presenting the problem, for pointing important bibliographical references, for the discussions and their comments during the process of the paper.

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 945332.

The author has no conflicts of interest to declare that are relevant to the content of this article. No datasets were generated or analyzed during the current study.

For the purpose of open access, the author has applied a Creative Commons Attribution (CC-BY) license to any Author Accepted Manuscript version arising from this submission.

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(R. Medina) CENTRE DE RECHERCHE EN Mathématiques de la Décision (CEREMADE, CNRS UMR 7534), UNIVERSITÉS PSL & PARIS-DAUPHINE, PLACE DE LATTRE DE TASSIGNY, 75775 PARIS 16, FRANCE
Email address: richard.medina-rodriguez@dauphine.psl.eu