with $R_0 := \nabla_x \phi \cdot A x + \nabla_x^* m_s - \partial_t z$. Arguing as in the proof of Lemma 3.5 with this new definition of $R_0$, we obtain that (3.11) holds with

$$R_3 := - \langle R_0 \tilde{\Phi} \rangle, \quad \tilde{\Phi} := \nabla \phi - x - \langle (\nabla \phi - x) \otimes x \rangle \nabla \phi.$$

Observing that

$$R_3 = - \langle \nabla_x \phi \cdot A x \tilde{\Phi} \rangle - \langle T(D \tilde{\Phi}) m_s \rangle + \langle \Omega^{\frac{1}{2}} \tilde{\Phi}, \Omega^{-\frac{1}{2}} \partial_t z \rangle \\ = O(|A| + \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t z\|)$$

and using (4.26b), the proof of (4.35) follows for the same reasons as in the proof of Lemma 3.5. ■

With $\mathcal{D}_1[h]$ defined in (4.22), we obtain the counterpart of Lemma 3.6.

**Lemma 4.11.** There exists a constant $C > 0$ so that

$$\begin{aligned} \frac{d}{dt} \langle -b, b' \rangle &\le -|b'|^2 + C|A|^2 + C \mathcal{D}_1[h], \\ \frac{d}{dt} \langle -c', c'' \rangle &\le -|c''|^2 + C|A|^2 + C \mathcal{D}_1[h]. \end{aligned}$$

**Lemma 4.12.** The following estimates hold

$$|c| \lesssim |b'| + |c''| + \|w_s\|^2 + \|e_s\|^2 + \|h^{\perp}\|^2, \quad (4.36)$$

$$\|r\| \lesssim |b'| + |c''| + \|w_s\|^2 + \|e_s\|^2 + \|h^{\perp}\|^2. \quad (4.37)$$

*Proof.* According to (4.25a), we can write $r$ as

$$r = \sqrt{\frac{2}{d}} \xi_{\phi} c + R_5 \quad (4.38)$$

where $R_5 := z - x \cdot b' + \frac{\xi_2}{2\sqrt{2d}} c'' = O(|b'| + |c''| + \mathcal{D}_1[h])$ because of (4.26a). Using this expression in (2.26) and recalling that $\langle e \rangle = c$ yields

$$\sqrt{\frac{d}{2}} c \left( 1 + \frac{2}{d} \langle \xi_{\phi}^2 \rangle \right) = - \langle \xi_{\phi} R_5 \rangle,$$

from which (4.36) follows. Coming back to (4.38), we establish (4.37). ■

**4.4.4. Second Lyapunov functional.** Let us introduce the Lyapunov function

$$\mathcal{F}_2[h] := \mathcal{F}_1[h] - \varepsilon_5 \langle (X - Y \cdot \nabla_x \phi), \nabla_x \phi \cdot A x \rangle - \varepsilon_6 \langle b, b' \rangle - \varepsilon_6 \langle c', c'' \rangle \quad (4.39)$$

for some additional small parameters $\varepsilon_5$ and $\varepsilon_6$, and the associated dissipation functional

$$\mathcal{D}_2[h] := \mathcal{D}_1[h] + |A|^2 + |b'|^2 + |c''|^2. \quad (4.40)$$

**Lemma 4.13.** For any $0 < \varepsilon_6 < \varepsilon_5 < \varepsilon_4 < \varepsilon_3 < \varepsilon_2 < \varepsilon_1$ with $\varepsilon_1$ small enough, there holds

$$\|h\|^2 \lesssim \mathcal{F}_2[h] \lesssim \mathcal{D}_2[h] \lesssim \|h\|^2. \quad (4.41)$$
Proof of Lemma 4.13. We can control all quantities involved in the definitions of $\mathcal{F}_2$ and $\mathcal{D}_2$ by $\|h\|^2$. Indeed, from (4.1) and (4.23a), we have

$$ \|r\| + \|m\| + \|e\| + \|h^{\perp}\| + |b| + |c| \le \|h\| \quad (4.42) $$

and thus also $\|e_s\| \lesssim \|e\| + |c| \lesssim \|h\|$ from (4.25c). Next, we observe from (4.27) that

$$ |c'| = \sqrt{\frac{2}{d}} |\langle m \cdot \nabla \phi \rangle| \lesssim \|m\| \le \|h\|, $$

from (4.23a) that

$$ |A| = |\langle m \nabla_x^{\text{skew}} \phi \rangle| \lesssim \|m\| \le \|h\|, $$

and thus also $\|m_s\| \lesssim \|m\| + |A| + |b| + |c'| \lesssim \|h\|$ from (4.25b). Similarly, we observe from (4.28) that

$$ |b'| = |\langle r \nabla_x \phi \rangle| \lesssim \|r\| \le \|h\|. $$

Coming back to the definition of $w_s$ and using (4.28), we get

$$ w_s = r - \sqrt{\frac{2}{d}} c \left( \xi_{\phi} - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2 \right) + b' \cdot x - \frac{1}{2d} \left\langle r \left( |\nabla \phi|^2 - \Delta \phi \right) \right\rangle \xi_2, $$

and deduce $\|w_s\| \lesssim \|r\| + |c| + |b'| \lesssim \|h\|$. Similarly, from (4.30), we also have $|c'| \lesssim \|r\| + \|e\| + \|h^{\perp}\| \le \|h\|$. Summing up, we have proved

$$ \|e_s\| + \|m_s\| + \|w_s\| + |A| + |c'| + |b'| + |c'| \le \|h\|. \quad (4.43) $$

We finally have to control the terms $\|\Omega^{-1/2} \partial_t w_s\|$. From (4.2e), (4.2a) and (2.30), we have

$$ \partial_t w_s = \nabla_x^* \cdot m - \langle \nabla_x \nabla_x^* \cdot m \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* \cdot m \rangle \xi_2 - \sqrt{\frac{2}{d}} c' \phi_s $$

and, after performing several integration by parts,

$$ \| \Omega^{-\frac{1}{2}} \partial_t w_s \| \lesssim \|m\| + |c'| \le \|h\|. \quad (4.44) $$

As a consequence of the estimates (4.42), (4.43), (4.44) and of the definition (4.39) of $\mathcal{F}_2$ (also see (4.20)), we have

$$ \|h\|^2 - \mathcal{F}_2[h] \le C \varepsilon_1 \|h\|^2. $$

This completes the proof of the first equivalence in (4.41). For the same reason, we have $\mathcal{D}_2[h] \lesssim \|h\|^2$. On the other way round, from (4.25a) and (4.36), we have

$$ \|r\| \lesssim |b'| + |c'| + |c''| + |c''| + \|w_s\| + \|e_s\| + \|h^{\perp}\| \lesssim |b'| + |c'| + \|w_s\| + \|e_s\| + \|h^{\perp}\| $$

and similarly, from (4.25b) and (4.35), we have

$$ \|m\| \lesssim |A| + |b| + |c'| + \|m_s\| \lesssim |A| + |c'| + \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t w_s\| + \|h^{\perp}\|. $$

Combining the last two estimates, (4.1), (4.25c) and the definition (4.22) of $\mathcal{D}_2$, we deduce the reverse inequality $\|h\|^2 \lesssim \mathcal{D}_2[h]$, which completes the proof of the second equivalence in (4.41). ■
Proof of Lemma 4.13. We can control all quantities involved in the definitions of $\mathcal{F}_2$ and $\mathcal{D}_2$ by $\|h\|^2$. Indeed, from (4.1) and (4.23a), we have

$$ \|r\| + \|m\| + \|e\| + \|h^{\perp}\| + |b| + |c| \le \|h\| \quad (4.42) $$

and thus also $\|e_s\| \lesssim \|e\| + |c| \lesssim \|h\|$ from (4.25c). Next, we observe from (4.27) that

$$ |c'| = \sqrt{\frac{2}{d}} |\langle m \cdot \nabla \phi \rangle| \lesssim \|m\| \le \|h\|, $$

from (4.23a) that

$$ |A| = |\langle m \nabla_x^{\text{skew}} \phi \rangle| \lesssim \|m\| \le \|h\|, $$

and thus also $\|m_s\| \lesssim \|m\| + |A| + |b| + |c'| \lesssim \|h\|$ from (4.25b). Similarly, we observe from (4.28) that

$$ |b'| = |\langle r \nabla_x \phi \rangle| \lesssim \|r\| \le \|h\|. $$

Coming back to the definition of $w_s$ and using (4.28), we get

$$ w_s = r - \sqrt{\frac{2}{d}} c \left( \xi_{\phi} - \frac{1}{2d} \langle \Delta_x \phi \rangle \xi_2 \right) + b' \cdot x - \frac{1}{2d} \left\langle r \left( |\nabla \phi|^2 - \Delta \phi \right) \right\rangle \xi_2, $$

and deduce $\|w_s\| \lesssim \|r\| + |c| + |b'| \lesssim \|h\|$. Similarly, from (4.30), we also have $|c'| \lesssim \|r\| + \|e\| + \|h^{\perp}\| \le \|h\|$. Summing up, we have proved

$$ \|e_s\| + \|m_s\| + \|w_s\| + |A| + |c'| + |b'| + |c'| \le \|h\|. \quad (4.43) $$

We finally have to control the terms $\|\Omega^{-1/2} \partial_t w_s\|$. From (4.2e), (4.2a) and (2.30), we have

$$ \partial_t w_s = \nabla_x^* \cdot m - \langle \nabla_x \nabla_x^* \cdot m \rangle \cdot x - \frac{1}{2d} \langle \Delta_x \nabla_x^* \cdot m \rangle \xi_2 - \sqrt{\frac{2}{d}} c' \phi_s $$

and, after performing several integration by parts,

$$ \| \Omega^{-\frac{1}{2}} \partial_t w_s \| \lesssim \|m\| + |c'| \le \|h\|. \quad (4.44) $$

As a consequence of the estimates (4.42), (4.43), (4.44) and of the definition (4.39) of $\mathcal{F}_2$ (also see (4.20)), we have

$$ \|h\|^2 - \mathcal{F}_2[h] \le C \varepsilon_1 \|h\|^2. $$

This completes the proof of the first equivalence in (4.41). For the same reason, we have $\mathcal{D}_2[h] \lesssim \|h\|^2$. On the other way round, from (4.25a) and (4.36), we have

$$ \|r\| \lesssim |b'| + |c'| + |c''| + |c''| + \|w_s\| + \|e_s\| + \|h^{\perp}\| \le |b'| + |c'| + \|w_s\| + \|e_s\| + \|h^{\perp}\| $$

and similarly, from (4.25b) and (4.35), we have

$$ \|m\| \lesssim |A| + |b| + |c'| + \|m_s\| \lesssim |A| + |c'| + \|m_s\| + \|\Omega^{-\frac{1}{2}} \partial_t w_s\| + \|h^{\perp}\|. $$

Combining the last two estimates, (4.1), (4.25c) and the definition (4.22) of $\mathcal{D}_2$, we deduce the reverse inequality $\|h\|^2 \lesssim \mathcal{D}_2[h]$, which completes the proof of the second equivalence in (4.41). ■
### 4.5. *Proof of Proposition 4.1*

*Proof of Proposition 4.1.* We differentiate with respect to $t$ the Lyapunov function $\mathcal{F}_2[h]$ and use Lemmata 4.6, 4.9 and 4.11 to get

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \kappa_0 \|h^\perp\|^2 - \varepsilon^{1/8} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2 \\
& - \kappa_4 \varepsilon_5 |A|^2 + C \varepsilon_5 (\mathcal{D}_1[h] + C \|h^\perp\| \|h\|) \\
& + \varepsilon_6 (-|b'|^2 - |c''|^2 + 2C|A|^2 + 2C\mathcal{D}_1[h]).
\end{aligned}
$$

Using Young's inequality we have

$$ \varepsilon_5 C^2 \|h^\perp\| \|h\| \le \kappa_0 \|h^\perp\|^2 + \varepsilon_5^2 \kappa_0^{-1} C^4 \|h\|^2 $$

and we deduce for some new constants $C_1, C_2, C_3 > 0$ that

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \varepsilon^{1/8} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2 \\
& - \kappa_4 \varepsilon_5 |A|^2 + C_1 (\varepsilon_5 + \varepsilon_6) \mathcal{D}_1[h] + C_2 \varepsilon_5^2 \|h\|^2 \\
& - \varepsilon_6 (|b'|^2 + |c''|^2) + C_3 \varepsilon_6 |A|^2.
\end{aligned}
$$

As in the proof of Lemma 4.6, we choose appropriately the small parameters $\varepsilon_i$ such that the quantities $\varepsilon_5, \varepsilon_6, \varepsilon_5/\varepsilon_4$ and $\varepsilon_6/\varepsilon_5$ are small enough in terms of $\varepsilon$. With $\varepsilon_5 := \varepsilon^{61/32}$ and $\varepsilon_6 := \varepsilon^{62/32}$, we obtain

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \varepsilon^{15/8} (\kappa - 2\varepsilon^{1/32} C_1) \mathcal{D}_1[h] - \varepsilon^{61/32} (\kappa_4 - \varepsilon^{1/32} C_3) |A|^2 \\
& - \varepsilon^{62/32} (|b'|^2 + |c''|^2) + \varepsilon^2 (C_0 + C_2 \varepsilon^{29/16}) \|h\|^2.
\end{aligned}
$$

Choosing $\varepsilon > 0$ small enough, the differential inequality simplifies into

$$ \frac{d}{dt} \mathcal{F}_2[h] \le - \varepsilon^{62/32} \mathcal{D}_2[h] + 5C \varepsilon^2 \|h\|^2. $$

Because of the equivalences established in Lemma 4.13, there are two constants $K_i > 0$, such that

$$ \frac{d}{dt} \mathcal{F}_2[h] \le - \varepsilon^{62/32} (K_1 - K_2 \varepsilon^{2/32}) \mathcal{F}_2[h]. $$

Choosing $\varepsilon > 0$ smaller if necessary, we obtain

$$ \frac{d}{dt} \mathcal{F}_2[h] \le - \kappa \mathcal{F}_2[h], $$

for some $\kappa > 0$, which implies $\mathcal{F}_2[h(t)] \le e^{-\kappa t} \mathcal{F}_2[h_0]$. This completes the proof of Proposition 4.1, that is, of Part (2) of Theorem 1.1, by using once again the equivalences of Lemma 4.13. ■
### 4.5. *Proof of Proposition 4.1*

*Proof of Proposition 4.1.* We differentiate with respect to $t$ the Lyapunov function $\mathcal{F}_2[h]$ and use Lemmata 4.6, 4.9 and 4.11 to get

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \kappa_0 \|h^\perp\|^2 - \varepsilon^{1/8} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2 \\
& - \kappa_4 \varepsilon_5 |A|^2 + C \varepsilon_5 (\mathcal{D}_1[h] + C \|h^\perp\| \|h\|) \\
& + \varepsilon_6 (-|b'|^2 - |c''|^2 + 2C|A|^2 + 2C \mathcal{D}_1[h]).
\end{aligned}
$$

Using Young's inequality we have

$$ \varepsilon_5 C^2 \|h^\perp\| \|h\| \le \kappa_0 \|h^\perp\|^2 + \varepsilon_5^2 \kappa_0^{-1} C^4 \|h\|^2 $$

and we deduce for some new constants $C_1, C_2, C_3 > 0$ that

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \varepsilon^{1/8} \kappa \mathcal{D}_1[h] + \varepsilon^2 C_0 \|h\|^2 \\
& - \kappa_4 \varepsilon_5 |A|^2 + C_1 (\varepsilon_5 + \varepsilon_6) \mathcal{D}_1[h] + C_2 \varepsilon_5^2 \|h\|^2 \\
& - \varepsilon_6 (|b'|^2 + |c''|^2) + C_3 \varepsilon_6 |A|^2.
\end{aligned}
$$

As in the proof of Lemma 4.6, we choose appropriately the small parameters $\varepsilon_i$ such that the quantities $\varepsilon_5, \varepsilon_6, \varepsilon_5/\varepsilon_4$ and $\varepsilon_6/\varepsilon_5$ are small enough in terms of $\varepsilon$. With $\varepsilon_5 := \varepsilon^{61/32}$ and $\varepsilon_6 := \varepsilon^{62/32}$, we obtain

$$ 
\begin{aligned}
\frac{d}{dt} \mathcal{F}_2[h] \le & - \varepsilon^{15/8} (\kappa - 2\varepsilon^{1/32} C_1) \mathcal{D}_1[h] - \varepsilon^{61/32} (\kappa_4 - \varepsilon^{1/32} C_3) |A|^2 \\
& - \varepsilon^{62/32} (|b'|^2 + |c''|^2) + \varepsilon^2 (C_0 + C_2 \varepsilon^{29/16}) \|h\|^2.
\end{aligned}
$$

Choosing $\varepsilon > 0$ small enough, the differential inequality simplifies into

$$ \frac{d}{dt} \mathcal{F}_2[h] \le -\varepsilon^{62/32} \mathcal{D}_2[h] + 5C \varepsilon^2 \|h\|^2. $$

Because of the equivalences established in Lemma 4.13, there are two constants $K_i > 0$, such that

$$ \frac{d}{dt} \mathcal{F}_2[h] \le -\varepsilon^{62/32} (K_1 - K_2 \varepsilon^{2/32}) \mathcal{F}_2[h]. $$

Choosing $\varepsilon > 0$ smaller if necessary, we obtain

$$ \frac{d}{dt} \mathcal{F}_2[h] \le -\kappa \mathcal{F}_2[h], $$

for some $\kappa > 0$, which implies $\mathcal{F}_2[h(t)] \le e^{-\kappa t} \mathcal{F}_2[h_0]$. This completes the proof of Proposition 4.1, that is, of Part (2) of Theorem 1.1, by using once again the equivalences of Lemma 4.13. ■
5. Proof of hypocoercivity by the commutator method

In this section we give an alternative proof of our main result in Theorem 1.1 using a
commutator method, under the additional hypotheses that

$$
\textit{The linear collision operator } \mathcal{C} \textit{ is bounded in } L^2(\mu^{-1}), \qquad (H9)
$$

$$
\| \nabla_x^2 \phi \|_{L^\infty(\mathbb{R}^d)} < \infty \quad \text{and} \quad \int_{\mathbb{R}^d} x \phi(x) e^{-\phi(x)} dx = 0. \qquad (\text{H10})
$$

Assumption (H9) means that the operator C as defined in (2.24) is bounded on L²(μ)
while (H10) means that the potential φ has bounded second derivatives and is superlinear
at infinity. These assumptions are added merely in order to simplify the computations
but the bound (H9) on the collision operator can be relaxed into just (H2) by using
Ãᵢ := Π*AᵢΠ instead of the Aᵢ's defined in (5.2) in Proposition 5.2 (where Πh = h|| is the
orthogonal projection on the macroscopic part), and the bound in (H10) can be relaxed
into simply |∇²ₓφ| ≤ 1 + |∇ₓφ| thanks to the additional weight [∇ₓφ] in the Poincaré
inequality (H5). We include the commutator method because of its interesting algebraic
properties and potential applications to a larger class of equations. Then part (2) of The-
orem 1.1 writes:

**Proposition 5.1.** Assume that (H0)–(H10) hold and consider a solution h to (2.22)–(2.23) in L²(M). Then there are explicit constants C > 0 and κ > 0 such that

$$
\|h(t)\| \le C e^{-\kappa t} \|h_0\|
$$

where C and κ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to φ such as the rigidity constant defined in (1.14).

While ∇ₓ and ∇_v map scalar functions to vector-valued functions, their adjoints in L²(M) are ∇_x* = −∇_x · + ∇_x φ and ∇_v* = −∇_v · + v · and map vector-valued functions back to scalar functions. The operators ∇_x and ∇_v commute but each does not commute with its adjoint. We have

$$
\begin{align}
[\nabla_v, \mathcal{T}] &= -\nabla_x, & [\nabla_x, \mathcal{T}] &= H_\phi \nabla_v, \tag{5.1} \\
[\nabla_v^*, \mathcal{T}] &= -\nabla_x^*, & [\nabla_x^*, \mathcal{T}] &= (\nabla_x^2 \nabla_v)^*, \nonumber
\end{align}
$$

where $[A, B] = AB - BA$ is the commutator and $H_\phi := (\partial_x^2 x_j \phi)_{i,j}$.

In addition to $\Omega = \nabla_x^* \cdot \nabla_x + 1$ defined in (1.15), we also introduce

$$
\Gamma = \nabla_v^* \cdot \nabla_v + 1, \quad \Lambda = \nabla_v^* \cdot \nabla_v + \nabla_x^* \cdot \nabla_x + 1.
$$

These scalar operators also act, coordinate by coordinate, on tensors.

From, e.g., [9] (see Sections 4.4-4.5) or [25, 27], these operators are self-adjoint in L²(M). As in Section 4, we construct a cascade of estimates.
**5. Proof of hypocoercivity by the commutator method**

In this section we give an alternative proof of our main result in Theorem 1.1 using a commutator method, under the additional hypotheses that

$$ \textit{The linear collision operator } \mathcal{C} \textit{ is bounded in } L^2(\mu^{-1}), \qquad (H9) $$

$$ \| \nabla_x^2 \phi \|_{L^\infty(\mathbb{R}^d)} < \infty \quad \text{and} \quad \int_{\mathbb{R}^d} x \phi(x) e^{-\phi(x)} dx = 0. \qquad (H10) $$

Assumption (H9) means that the operator C as defined in (2.24) is bounded on $L^2(\mu)$ while (H10) means that the potential $\phi$ has bounded second derivatives and is superlinear at infinity. These assumptions are added merely in order to simplify the computations but the bound (H9) on the collision operator can be relaxed into just (H2) by using $\tilde{A}_i := \Pi^* A_i \Pi$ instead of the $A_i$'s defined in (5.2) in Proposition 5.2 (where $\Pi h = h^\|$ is the orthogonal projection on the macroscopic part), and the bound in (H10) can be relaxed into simply $|\nabla_x^2 \phi| \le 1 + |\nabla_x \phi|$ thanks to the additional weight $[\nabla_x \phi]$ in the Poincaré inequality (H5). We include the commutator method because of its interesting algebraic properties and potential applications to a larger class of equations. Then part (2) of Theorem 1.1 writes:

**Proposition 5.1.** Assume that (H0)–(H10) hold and consider a solution $h$ to (2.22)–(2.23) in $L^2(M)$. Then there are explicit constants $C > 0$ and $\kappa > 0$ such that

$$ \|h(t)\| \le C e^{-\kappa t} \|h_0\| $$

where $C$ and $\kappa$ depend only on bounded moments constants, spectral gap constants or explicitly computable quantities associated to $\phi$ such as the rigidity constant defined in (1.14).

While $\nabla_x$ and $\nabla_v$ map scalar functions to vector-valued functions, their adjoints in $L^2(M)$ are $\nabla_x^* = -\nabla_x \cdot + \nabla_x \phi$ and $\nabla_v^* = -\nabla_v \cdot + v \cdot$ and map vector-valued functions back to scalar functions. The operators $\nabla_x$ and $\nabla_v$ commute but each does not commute with its adjoint. We have

$$ \begin{aligned} [\nabla_v, \mathcal{T}] &= -\nabla_x, & [\nabla_x, \mathcal{T}] &= H_\phi \nabla_v, \\ [\nabla_v^*, \mathcal{T}] &= -\nabla_x^*, & [\nabla_x^*, \mathcal{T}] &= (H_\phi \nabla_v)^*, \end{aligned} \qquad (5.1) $$

where $[A, B] = AB - BA$ is the commutator and $H_\phi := (\partial_{x_i x_j}^2)_{i,j}$.

In addition to $\Omega = \nabla_x^* \cdot \nabla_x + 1$ defined in (1.15), we also introduce

$$ \Gamma = \nabla_v^* \cdot \nabla_v + 1, \quad \Lambda = \nabla_v^* \cdot \nabla_v + \nabla_x^* \cdot \nabla_x + 1. $$

These scalar operators also act, coordinate by coordinate, on tensors.

From, e.g., [9] (see Sections 4.4-4.5) or [25, 27], these operators are self-adjoint in $L^2(M)$. As in Section 4, we construct a cascade of estimates.
### 5.1. Cascade of infinite-dimensional correctors

The three following operators play the role of correctors:

$$ egin{cases} A_0 := \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, \\ A_1 := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x, \\ A_2 := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v. \end{cases} \quad (5.2) $$

From [25, 27] or by standard pseudo-differential calculus arguments (see Lemma A.4), $A_0, A_1, A_2$ are bounded operators in $L^2(M)$. Let $\mathbb{R}_n[V]$ be the space of real polynomials of $v$ with degree less or equal than $n \in \mathbb{N}$, then

$$ egin{cases} \nabla_v^{\otimes 3} (\mathbb{R}_2[V]) = \{0\}, & \nabla_v^{\otimes 2} (\mathbb{R}_1[V]) = \{0\}, & \nabla_v (\mathbb{R}_0[V]) = \{0\}, \\ A_2 (\mathbb{R}_2[V]) = \{0\}, & A_1 (\mathbb{R}_1[V]) = \{0\}, & A_0 (\mathbb{R}_0[V]) = \{0\}. \end{cases} $$

This means for instance that $A_2$ can access the local energy without seeing the local density and local momentum, in a descending cascade. Note that the simplest “order 1” corrector $A_0$ is inspired by the corrector $\nabla_x^* \Lambda^{-1/2} : \Lambda^{-1/2} \nabla_v$ introduced in [27] for Fokker-Planck type equations. We define macroscopic deviations from averages quantities that are slightly different from (4.2a)–(4.2c):

$$ egin{aligned} \tilde{e} &:= e - \langle e \rangle, \\ \tilde{m} &:= m - \langle \nabla_x m \rangle x - \langle m \rangle, \\ \tilde{r} &:= r - \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle. \end{aligned} $$

The core commutator estimates are:

**Proposition 5.2.** For all $i \in \{0, 1, 2\}$, we have

$$ \frac{d}{dt} \langle A_i h, h \rangle = - \langle \Lambda_i h, h \rangle + \langle B_i h, h \rangle \quad (5.3) $$

where the $B_i$'s are bounded operators that satisfy

$$ egin{aligned} \langle B_0 h, h \rangle &\lesssim \|h\| (\|h^\perp\| + \|\tilde{e}\| + \|\tilde{m}\|), \\ \langle B_1 h, h \rangle &\lesssim \|h\| (\|h^\perp\| + \|\tilde{e}\|), \\ \langle B_2 h, h \rangle &\lesssim \|h\| \|h^\perp\|, \end{aligned} $$

and where the $\Lambda_i$'s are nonnegative self-adjoint operators that satisfy, for some $C_0, C_1, C_2 > 0$ and $\bar{\lambda}_0, \bar{\lambda}_1, \bar{\lambda}_2 > 0$

$$ egin{aligned} - \langle \Lambda_0 h, h \rangle &\le -\bar{\lambda}_0 \|\tilde{r}\|^2 + C_0 (\|\tilde{m}\| + \|\tilde{e}\| + \|h^\perp\|) \|h\|, \\ - \langle \Lambda_1 h, h \rangle &\le -\bar{\lambda}_1 \|\tilde{m}\|^2 + C_1 (\|\tilde{e}\| + \|h^\perp\|) \|h\|, \\ - \langle \Lambda_2 h, h \rangle &\le -\bar{\lambda}_2 \|\tilde{e}\|^2 + C_2 \|h^\perp\| \|h\|. \end{aligned} $$
### 5.1. Cascade of infinite-dimensional correctors

The three following operators play the role of correctors:

$$ egin{cases} A_0 := \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, \\ A_1 := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x, \\ A_2 := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v. \end{cases} \quad (5.2) $$

From [25, 27] or by standard pseudo-differential calculus arguments (see Lemma A.4), $A_0, A_1, A_2$ are bounded operators in $L^2(M)$. Let $\mathbb{R}_n[V]$ be the space of real polynomials of $v$ with degree less or equal than $n \in \mathbb{N}$, then

$$ egin{cases} \nabla_v^{\otimes 3} (\mathbb{R}_2[V]) = \{0\}, & \nabla_v^{\otimes 2} (\mathbb{R}_1[V]) = \{0\}, & \nabla_v (\mathbb{R}_0[V]) = \{0\}, \\ A_2 (\mathbb{R}_2[V]) = \{0\}, & A_1 (\mathbb{R}_1[V]) = \{0\}, & A_0 (\mathbb{R}_0[V]) = \{0\}. \end{cases} $$

This means for instance that $A_2$ acts at the level of the local energy without seeing the local density and local momentum, in a descending cascade. Note that the simplest “order 1” corrector $A_0$ is inspired by the corrector $\nabla_x^* \Lambda^{-1/2} : \Lambda^{-1/2} \nabla_v$ introduced in [27] for Fokker-Planck type equations. We define macroscopic deviations from averages quantities that are slightly different from (4.2a)–(4.2c):

$$ egin{aligned} \tilde{e} &:= e - \langle e \rangle, \\ \tilde{m} &:= m - \langle \nabla_x m \rangle x - \langle m \rangle, \\ \tilde{r} &:= r - \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle. \end{aligned} $$

The core commutator estimates are:

**Proposition 5.2.** For all $i \in \{0, 1, 2\}$, we have

$$ \frac{d}{dt} \langle A_i h, h \rangle = - \langle \Lambda_i h, h \rangle + \langle B_i h, h \rangle \quad (5.3) $$

where the $B_i$'s are bounded operators that satisfy

$$ egin{aligned} \langle B_0 h, h \rangle &\lesssim \|h\| (\|h^\perp\| + \|\tilde{e}\| + \|\tilde{m}\|), \\ \langle B_1 h, h \rangle &\lesssim \|h\| (\|h^\perp\| + \|\tilde{e}\|), \\ \langle B_2 h, h \rangle &\lesssim \|h\| \|h^\perp\|, \end{aligned} $$

and where the $\Lambda_i$'s are nonnegative self-adjoint operators that satisfy, for some $C_0, C_1, C_2 > 0$ and $\bar{\lambda}_0, \bar{\lambda}_1, \bar{\lambda}_2 > 0$

$$ egin{aligned} - \langle \Lambda_0 h, h \rangle &\le -\bar{\lambda}_0 \|\tilde{r}\|^2 + C_0 (\|\tilde{m}\| + \|\tilde{e}\| + \|h^\perp\|) \|h\|, \\ - \langle \Lambda_1 h, h \rangle &\le -\bar{\lambda}_1 \|\tilde{m}\|^2 + C_1 (\|\tilde{e}\| + \|h^\perp\|) \|h\|, \\ - \langle \Lambda_2 h, h \rangle &\le -\bar{\lambda}_2 \|\tilde{e}\|^2 + C_2 \|h^\perp\| \|h\|. \end{aligned} $$
*Proof of Proposition 5.2.* Since $C$ is self-adjoint and $\mathcal{T}$ is skew-adjoint,

$$ \frac{d}{dt} \langle A_i h, h \rangle = \left\langle \left( [A_i, \mathcal{T}] + A_i C + C A_i \right) h, h \right\rangle \quad \text{for } i \in \{0, 1, 2\}. $$

We can therefore write $[A_i, \mathcal{T}] + A_i C + C A_i =: B_i - \Lambda_i$ where, by using (5.1), we have the following explicit formulas

$$ B_0 := A_0 C + C A_0 $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* [\Lambda^{-\frac{3}{2}}, \mathcal{T}] : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : [\Lambda^{-\frac{3}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes (H_\phi \nabla_v) $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes (H_\phi \nabla_v) \otimes \nabla_x $$

$$ + (H_\phi \nabla_v)^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes (H_\phi \nabla_v)^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes (H_\phi \nabla_v)^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, $$

$$ B_1 := A_1 C + C A_1 $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-1} \Gamma^{-\frac{1}{2}}, \mathcal{T}] : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : [\Lambda^{-1} \Gamma^{-\frac{1}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes (H_\phi \nabla_v) $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes (H_\phi \nabla_v)^* \Lambda^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ - \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + (H_\phi \nabla_v)^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x, $$

$$ B_2 := A_2 C + C A_2 $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{1}{2}} \Gamma^{-1}, \mathcal{T}] : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Lambda^{-\frac{1}{2}} \Gamma^{-1} : [\Gamma^{-1} \Lambda^{-\frac{1}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes (H_\phi \nabla_v)^* \Lambda^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ - \nabla_v^* \otimes \nabla_x^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v, $$

$$ - \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} : \nabla_v \otimes \nabla_v \otimes \nabla_v, $$

and

$$ \Lambda_0 := \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_x \otimes \nabla_x \otimes \nabla_x, $$

$$ \Lambda_1 := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_x \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, $$

$$ \Lambda_2 := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_x \otimes \nabla_v \otimes \nabla_v $$
*Proof of Proposition 5.2.* Since $C$ is self-adjoint and $\mathcal{T}$ is skew-adjoint,

$$ \frac{d}{dt} \langle A_i h, h \rangle = \left\langle \left( [A_i, \mathcal{T}] + A_i C + C A_i \right) h, h \right\rangle \quad \text{for } i \in \{0, 1, 2\}. $$

We can therefore write $[A_i, \mathcal{T}] + A_i C + C A_i =: B_i - \Lambda_i$ where, by using (5.1), we have the following explicit formulas

$$ B_0 := A_0 C + C A_0 $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* [\Lambda^{-\frac{3}{2}}, \mathcal{T}] : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : [\Lambda^{-\frac{3}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes (H_\phi \nabla_v) $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes (H_\phi \nabla_v) \otimes \nabla_x $$

$$ + (H_\phi \nabla_v)^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_x^* \otimes (H_\phi \nabla_v)^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x $$

$$ + \nabla_x^* \otimes (H_\phi \nabla_v)^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, $$

$$ B_1 := A_1 C + C A_1 $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* [\Gamma^{-\frac{1}{2}} \Lambda^{-1}, \mathcal{T}] : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : [\Lambda^{-1} \Gamma^{-\frac{1}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes (H_\phi \nabla_v) $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes (H_\phi \nabla_v)^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ - \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x $$

$$ + (H_\phi \nabla_v)^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_x, $$

$$ B_2 := A_2 C + C A_2 $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{1}{2}} \Gamma^{-1}, \mathcal{T}] : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Lambda^{-\frac{1}{2}} \Gamma^{-1} : [\Gamma^{-1} \Lambda^{-\frac{1}{2}}, \mathcal{T}] \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ + \nabla_v^* \otimes \nabla_v^* \otimes (H_\phi \nabla_v)^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v $$

$$ - \nabla_v^* \otimes \nabla_x^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_v, $$

$$ - \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} : \nabla_v \otimes \nabla_v \otimes \nabla_v, $$

and

$$ \Lambda_0 := \nabla_x^* \otimes \nabla_x^* \otimes \nabla_x^* \Lambda^{-\frac{3}{2}} : \Lambda^{-\frac{3}{2}} \nabla_x \otimes \nabla_x \otimes \nabla_x, $$

$$ \Lambda_1 := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_x \otimes \nabla_v \otimes \nabla_x $$

$$ + \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x, $$

$$ \Lambda_2 := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_x \otimes \nabla_v \otimes \nabla_v $$
$$ 
\begin{aligned}
&+ \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_x \otimes \nabla_v \\
&+ \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x.
\end{aligned}
$$

The $\Lambda_i$'s operators are nonnegative, self-adjoint (see Appendix A.3) and bounded (see Lemma A.4 again).

Note that for all $i \in \{0, 1, 2\}$:

$$ 
\begin{aligned}
|\langle A_i C h, h \rangle| &= |\langle A_i C h^{\perp}, h \rangle| \lesssim \|h^{\perp}\| \|h\|, \\
|\langle C A_i h, h \rangle| &= |\langle A_i h, C h \rangle| = |\langle A_i h, C h^{\perp} \rangle| \lesssim \|h\| \|h^{\perp}\|,
\end{aligned}
$$

because $C$ is self-adjoint and bounded by (H9).

Now we deal with $B_2$. Let us denote by $B_{2,\ell}$ the $\ell$-th line. Since $\Lambda^{1/2} \Gamma [\Lambda^{-1/2} \Gamma^{-1}, \mathcal{T}]$ is bounded by standard pseudo-differential calculus (see [25]),

$$ b_{2,2} := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{1}{2}} \Gamma^{-1}, \mathcal{T}] $$

is bounded and

$$ 
\begin{aligned}
|\langle B_{2,2} h, h \rangle| &= |\langle b_{2,2} : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v h, h \rangle| \\
&= |\langle b_{2,2} : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v h^{\perp}, h \rangle| \lesssim \|h\| \|h^{\perp}\|
\end{aligned}
$$

since the three derivatives in velocity on the right hand side cancel all macroscopic quantities. The other lines are dealt with similarly, using the boundedness of $H_\phi$, and we deduce $|\langle B_2 h, h \rangle| \lesssim \|h\| \|h^{\perp}\|$.

Now we deal with $B_1$. We consider the second line $B_{1,2}$. Then $\Lambda \Gamma^{1/2} [\Lambda^{-1} \Gamma^{-1/2}, \mathcal{T}]$ is bounded by standard pseudo-differential calculus and so

$$ b_{1,2} := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-1} \Gamma^{-\frac{1}{2}}, \mathcal{T}] $$

is bounded. Thus

$$ 
\begin{aligned}
|\langle B_{1,2} h, h \rangle| &= |\langle b_{1,2} : \Gamma^{-\frac{1}{2}} \Lambda^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x h, h \rangle| \\
&\lesssim |\langle B_{1,2} : \Gamma^{-\frac{1}{2}} \Lambda^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x (e \mathcal{E} + h^{\perp}), h \rangle|.
\end{aligned}
$$

The macroscopic quantities $r$ and $m$ are canceled since two derivatives in velocity are involved. We then use $\nabla_v \otimes \nabla_v \otimes \nabla_x e \mathcal{E} = \nabla_x e \otimes \mathrm{Id}_{d \times d}$ and Appendix A.4 shows (using $\Omega \ge 1$ for the first inequality)

$$ \| \Omega^{-1} \nabla_x e \| \lesssim \| \Omega^{-\frac{1}{2}} \nabla_x e \| \lesssim \| \tilde{e} \| \quad \text{so} \quad |\langle B_{1,2} h, h \rangle| \lesssim \|h\| (\| \tilde{e} \| + \|h^{\perp}\|). $$

The other lines are similar and yield the same estimate.

We then deal with $B_0$, and focus on the second line $B_{0,2}$ again. The operator

$$ b_{0,2} := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{3}{2}}, \mathcal{T}] $$
$$ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_x \otimes \nabla_v \\ + \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-1} \Lambda^{-\frac{1}{2}} : \Lambda^{-\frac{1}{2}} \Gamma^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x. $$

The $\Lambda_i$'s operators are nonnegative, self-adjoint (see Appendix A.3) and bounded (see Lemma A.4 again).

Note that for all $i \in \{0, 1, 2\}$:

$$ \begin{aligned} |\langle A_i C h, h \rangle| &= |\langle A_i C h^{\perp}, h \rangle| \lesssim \|h^{\perp}\| \|h\|, \\ |\langle C A_i h, h \rangle| &= |\langle A_i h, C h \rangle| = |\langle A_i h, C h^{\perp} \rangle| \lesssim \|h\| \|h^{\perp}\|. \end{aligned} $$

because $C$ is self-adjoint and bounded by (H9).

Now we deal with $B_2$. Let us denote by $B_{2,\ell}$ the $\ell$-th line. Since $\Lambda^{1/2} \Gamma [\Lambda^{-1/2} \Gamma^{-1}, \mathcal{T}]$ is bounded by standard pseudo-differential calculus (see [25]),

$$ b_{2,2} := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{1}{2}} \Gamma^{-1}, \mathcal{T}] $$

is bounded and

$$ \begin{aligned} |\langle B_{2,2} h, h \rangle| &= |\langle b_{2,2} : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v h, h \rangle| \\ &= |\langle b_{2,2} : \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_v \otimes \nabla_v h^{\perp}, h \rangle| \lesssim \|h\| \|h^{\perp}\| \end{aligned} $$

since the three derivatives in velocity on the right hand side cancel all macroscopic quantities. The other lines are dealt with similarly, using the boundedness of $H_\phi$, and we deduce $|\langle B_2 h, h \rangle| \lesssim \|h\| \|h^{\perp}\|$.

Now we deal with $B_1$. We consider the second line $B_{1,2}$. Then $\Lambda \Gamma^{1/2} [\Lambda^{-1} \Gamma^{-1/2}, \mathcal{T}]$ is bounded by standard pseudo-differential calculus and so

$$ b_{1,2} := \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-1} \Gamma^{-\frac{1}{2}}, \mathcal{T}] $$

is bounded. Thus

$$ \begin{aligned} |\langle B_{1,2} h, h \rangle| &= |\langle b_{1,2} : \Gamma^{-\frac{1}{2}} \Lambda^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x h, h \rangle| \\ &\lesssim |\langle b_{1,2} : \Gamma^{-\frac{1}{2}} \Lambda^{-1} \nabla_v \otimes \nabla_v \otimes \nabla_x (e \mathcal{E} + h^{\perp}), h \rangle|. \end{aligned} $$

The macroscopic quantities $r$ and $m$ are canceled since two derivatives in velocity are involved. We then use $\nabla_v \otimes \nabla_v \otimes \nabla_x e \mathcal{E} = \nabla_x e \otimes \mathrm{Id}_{d \times d}$ and Appendix A.4 shows (using $\Omega \ge 1$ for the first inequality)

$$ \| \Omega^{-1} \nabla_x e \| \lesssim \| \Omega^{-\frac{1}{2}} \nabla_x e \| \lesssim \| \tilde{e} \| \quad \text{so} \quad |\langle B_{1,2} h, h \rangle| \lesssim \|h\| (\| \tilde{e} \| + \|h^{\perp}\|). $$

The other lines are similar and yield the same estimate.

We then deal with $B_0$, and focus on the second line $B_{0,2}$ again. The operator

$$ b_{0,2} := \nabla_v^* \otimes \nabla_v^* \otimes \nabla_x^* [\Lambda^{-\frac{3}{2}}, \mathcal{T}] $$
is bounded arguing as before, and

$$ \left|\langle B_{0,2}h, h \rangle\right| = \left|\left\langle b_{0,2} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x h, h \right\rangle\right| \lesssim \|h\| (\|\tilde{e}\| + \|\tilde{m}\| + \|h^{\perp}\|). \quad (5.4) $$

Indeed a direct computation gives

$$ \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x h = \Omega^{-\frac{3}{2}} \nabla_x \otimes \nabla_x m + \tilde{\Omega}^{-\frac{3}{2}} \nabla_x \otimes \nabla_x (v e) $$

with $\tilde{\Omega} := (\nabla_x^* \nabla_v + 2)$. The factor 2 comes from the fact that for all $\alpha \in \mathbb{R}$,

$$ \Lambda^{\alpha}(ve) = (\nabla_x^* \nabla_x + \nabla_v^* \nabla_v + 1)^{\alpha}(ve) = (\nabla_x^* \nabla_x + 1 + 1)^{\alpha}(ve) $$

since $v$ is an eigenfunction of $\nabla_v^* \nabla_v$ with eigenvalue 1. To complete the proof of (5.4), it is sufficient to notice that $\|\Omega^{-\frac{3}{2}} \nabla_x^2 e\| \le \|\Omega^{-1} \nabla_x^2 e\| \le \|\tilde{e}\|$ and also that, by Appendix A.4, $\|\Omega^{-\frac{3}{2}} \nabla_x^2 m\| \le \|\Omega^{-1} \nabla_x^2 m\| \le \|\tilde{m}\|$. The other lines in $B_0$ are treated similarly again.

Now we deal with the main nonnegative terms $\langle \Lambda_i h, h \rangle$ in (5.3). We first compute the contribution of $\Lambda_0$, $\Lambda_1$ and $\Lambda_2$ acting respectively on $r$, $m$ and $e$. Due to the number of derivatives in velocity in the right hand side of the expressions giving the $\Lambda_i$'s, we have $\Lambda_i(\mathbb{R}_{i-1}[V]) = \{0\}$ for $i \in \{1, 2\}$ and $\Lambda_i(\mathbb{R}_i[V]) \subset \mathbb{R}_i[V]$ for $i \in \{0, 1, 2\}$ and

$$ \langle \Lambda_0 r, r \rangle = \iint_{\mathbb{R}^d \times \mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \mathcal{M} \, dx \, dv = \int_{\mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \rho \, dx, \quad (5.5) $$

$$ \begin{aligned} \langle \Lambda_1 (m(x) \cdot v), (m(x) \cdot v) \rangle &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} \left( (\Omega^{-1} \partial_{x_i x_k}^2 m_j) (\Omega^{-1} \partial_{x_i x_k}^2 m_j) \right. \\ & \qquad \left. + (\Omega^{-1} \partial_{x_i x_k}^2 m_j) (\Omega^{-1} \partial_{x_j x_k}^2 m_i) \right) \mathcal{M} \, dx \, dv \quad (5.6) \\ &= 2 \int_{\mathbb{R}^d} |\Omega^{-1} \nabla_x \nabla_x^{\text{sym}} m|^2 \rho \, dx, \end{aligned} $$

$$ \begin{aligned} \langle \Lambda_2 (e(x) \mathfrak{E}(v)), (e(x) \mathfrak{E}(v)) \rangle &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} \left( (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \right. \\ & \qquad \left. + (\Omega^{-\frac{1}{2}} \partial_{x_j} e) (\partial_{v_i}^2 v_k) \mathfrak{E} (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \right) \\ & \qquad + (\Omega^{-\frac{1}{2}} \partial_{x_i} e) (\partial_{v_j}^2 v_k) \mathfrak{E} (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \right) \mathcal{M} \, dx \, dv \\ &= \int_{\mathbb{R}^d} \left( 2 (\Omega^{-\frac{1}{2}} \partial_{x_k} e)^2 + \frac{2}{d} (\Omega^{-\frac{1}{2}} \partial_{x_i} e)^2 + \frac{2}{d} (\Omega^{-\frac{1}{2}} \partial_{x_j} e)^2 \right) \rho \, dx \\ &= \left( \frac{4}{d} + 2 \right) \int_{\mathbb{R}^d} |\Omega^{-\frac{1}{2}} \nabla_x e|^2 \rho \, dx. \end{aligned} \quad (5.7) $$

Next we use the cascade of Poincaré inequalities of Lemma A.6. For the density $r$, this implies that there is a constant $\bar{\lambda}_0 > 0$ such that

$$ \begin{aligned} \langle \Lambda_0 r, r \rangle &= \int_{\mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \rho \, dx \\ &\ge 2 \bar{\lambda}_0 \|r - \langle r \rangle - \langle \nabla_x r \rangle \cdot x - \frac{1}{2} \langle \nabla_x^2 r \rangle : (x \otimes x - \langle x \otimes x \rangle)\|^2 = 2 \bar{\lambda}_0 \|\tilde{r}\|^2. \end{aligned} \quad (5.8) $$
is bounded arguing as before, and

$$ \left| \langle B_{0,2}h, h \rangle \right| = \left| \left\langle b_{0,2} : \Lambda^{-\frac{3}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x h, h \right\rangle \right| \lesssim \|h\| (\|\tilde{e}\| + \|\tilde{m}\| + \|h^{\perp}\|). \quad (5.4) $$

Indeed a direct computation gives

$$ \Gamma^{-1} \Lambda^{-\frac{1}{2}} \nabla_v \otimes \nabla_x \otimes \nabla_x h = \Omega^{-\frac{3}{2}} \nabla_x \otimes \nabla_x m + \tilde{\Omega}^{-\frac{3}{2}} \nabla_x \otimes \nabla_x (v e) $$

with $\tilde{\Omega} := (\nabla_x^* \nabla_v + 2)$. The factor 2 comes from the fact that for all $\alpha \in \mathbb{R}$,

$$ \Lambda^{\alpha}(ve) = (\nabla_x^* \nabla_x + \nabla_v^* \nabla_v + 1)^{\alpha}(ve) = (\nabla_x^* \nabla_x + 1 + 1)^{\alpha}(ve) $$

since $v$ is an eigenfunction of $\nabla_v^* \nabla_v$ with eigenvalue 1. To complete the proof of (5.4), it is sufficient to notice that $\|\Omega^{-\frac{3}{2}} \nabla_x^2 e\| \le \|\Omega^{-1} \nabla_x^2 e\| \le \|\tilde{e}\|$ and also that, by Appendix A.4, $\|\Omega^{-\frac{3}{2}} \nabla_x^2 m\| \le \|\Omega^{-1} \nabla_x^2 m\| \le \|\tilde{m}\|$. The other lines in $B_0$ are treated similarly again.

Now we deal with the main nonnegative terms $\langle \Lambda_i h, h \rangle$ in (5.3). We first compute the contribution of $\Lambda_0$, $\Lambda_1$ and $\Lambda_2$ acting respectively on $r$, $m$ and $e$. Due to the number of derivatives in velocity in the right hand side of the expressions giving the $\Lambda_i$'s, we have $\Lambda_i(\mathbb{R}_{i-1}[V]) = \{0\}$ for $i \in \{1, 2\}$ and $\Lambda_i(\mathbb{R}_i[V]) \subset \mathbb{R}_i[V]$ for $i \in \{0, 1, 2\}$ and

$$ \langle \Lambda_0 r, r \rangle = \iint_{\mathbb{R}^d \times \mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \mathcal{M} \, dx \, dv = \int_{\mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \rho \, dx, \quad (5.5) $$

$$ \begin{aligned} \langle \Lambda_1 (m(x) \cdot v), (m(x) \cdot v) \rangle &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} \left( (\Omega^{-1} \partial_{x_i x_k}^2 m_j) (\Omega^{-1} \partial_{x_i x_k}^2 m_j) \right. \\ & \qquad \left. + (\Omega^{-1} \partial_{x_i x_k}^2 m_j) (\Omega^{-1} \partial_{x_j x_k}^2 m_i) \right) \mathcal{M} \, dx \, dv \quad (5.6) \\ &= 2 \int_{\mathbb{R}^d} |\Omega^{-1} \nabla_x \nabla_x^{\text{sym}} m|^2 \rho \, dx, \end{aligned} $$

$$ \begin{aligned} \langle \Lambda_2 (e(x) \mathfrak{E}(v)), (e(x) \mathfrak{E}(v)) \rangle &= \iint_{\mathbb{R}^d \times \mathbb{R}^d} \left( (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \right) (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \\ & \quad + (\Omega^{-\frac{1}{2}} \partial_{x_j} e) (\partial_{v_i}^2 v_k) \mathfrak{E} (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \\ & \quad + (\Omega^{-\frac{1}{2}} \partial_{x_i} e) (\partial_{v_j}^2 v_k) \mathfrak{E} (\Omega^{-\frac{1}{2}} \partial_{x_k} e) (\partial_{v_i}^2 v_j) \mathfrak{E} \right) \mathcal{M} \, dx \, dv \\ &= \int_{\mathbb{R}^d} \left( 2 (\Omega^{-\frac{1}{2}} \partial_{x_k} e)^2 + \frac{2}{d} (\Omega^{-\frac{1}{2}} \partial_{x_i} e)^2 + \frac{2}{d} (\Omega^{-\frac{1}{2}} \partial_{x_j} e)^2 \right) \rho \, dx \\ &= \left( \frac{4}{d} + 2 \right) \int_{\mathbb{R}^d} |\Omega^{-\frac{1}{2}} \nabla_x e|^2 \rho \, dx. \end{aligned} \quad (5.7) $$

Next we use the cascade of Poincaré inequalities of Lemma A.6. For the density $r$, this implies that there is a constant $\bar{\lambda}_0 > 0$ such that

$$ \begin{aligned} \langle \Lambda_0 r, r \rangle &= \int_{\mathbb{R}^d} |\Omega^{-\frac{3}{2}} \nabla_x^3 r|^2 \rho \, dx \\ &\ge 2 \bar{\lambda}_0 \|r - \langle r \rangle - \langle \nabla_x r \rangle \cdot x - \frac{1}{2} \langle \nabla_x^2 r \rangle : (x \otimes x - \langle x \otimes x \rangle)\|^2 = 2 \bar{\lambda}_0 \|\tilde{r}\|^2. \end{aligned} \quad (5.8) $$
Regarding the momentum $m$, one first observes that

$$|\Omega^{-1} \nabla_x \nabla_x^{\text{sym}} m|^2 \geq \frac{1}{9} |\Omega^{-1} \nabla_x^2 m|^2$$

thanks to the Schwarz lemma written as

$$\forall i, j, k \in \{1, \dots, d\}, \partial_{ij}^2 m_k = \partial_i (\nabla^{\text{sym}} m)_{jk} + \partial_j (\nabla_x^{\text{sym}} m)_{ik} - \partial_k (\nabla_x^{\text{sym}} m)_{ij}.$$

Applied to (5.5)–(5.7), the cascade of Poincaré inequalities at order 2 stated in Lemma A.6 implies that there exists a constant $\bar{\lambda}_1$ such that

$$\begin{aligned} \langle \Lambda_1 (m \cdot v), (m \cdot v) \rangle &\geq \frac{2}{9} \| \Omega^{-1} \nabla_x^2 m \|^2 \\ &\geq 2 \bar{\lambda}_1 \| m - \langle m \rangle - \langle \nabla_x m \rangle x \|^2 = 2 \bar{\lambda}_1 \| \tilde{m} \|^2. \end{aligned} \quad (5.9)$$

Regarding the energy $e$, we use the standard Poincaré inequality in $L^2(\rho)$ (the order 1 inequality of Lemma A.6) to get $\bar{\lambda}_2 > 0$ such that

$$\langle \Lambda_2 (e \mathfrak{E}(v)), (e \mathfrak{E}(v)) \rangle = \left(\frac{4}{d} + 2\right) \| \Omega^{-\frac{1}{2}} \nabla_x e \|^2 \geq 2 \bar{\lambda}_2 \|e - \langle e \rangle\|^2 = 2 \bar{\lambda}_2 \|\tilde{e}\|^2. \quad (5.10)$$

With the above estimates in hand, we can investigate all terms appearing in $\langle \Lambda_i h, h \rangle$. According to the number of velocity and space gradients in $\Lambda_2$, we get that

$$\langle \Lambda_2 h, h \rangle = \langle \Lambda_2 (e \mathfrak{E}(v)), e \mathfrak{E}(v) \rangle + \langle \Lambda_2 (e \mathfrak{E}(v)), h^{\perp} \rangle + \langle \Lambda_2 h^{\perp}, h \rangle,$$

from which one obtains with (5.10) that

$$-\langle \Lambda_2 h, h \rangle \leq -\bar{\lambda}_2 \|\tilde{e}\|^2 + O \left( \|h^{\perp}\| \|h\| \right).$$

Similarly for $m$, using in addition that $\Lambda_1$ is self-adjoint and $\Lambda_1 (e \mathfrak{E}(v)) = \Lambda_1 (\tilde{e} \mathfrak{E}(v))$, one has

$$\langle \Lambda_1 h, h \rangle = \langle \Lambda_1 (m \cdot v), m \cdot v \rangle + \langle \Lambda_1 (m \cdot v), (e - \langle e \rangle) \mathfrak{E}(v) + h^{\perp} \rangle + \langle \Lambda_1 ((e - \langle e \rangle) \mathfrak{E}(v) + h^{\perp}), h \rangle,$$

which implies using (5.9) that

$$-\langle \Lambda_1 h, h \rangle \leq -\bar{\lambda}_1 \|\tilde{m}\|^2 + O(\|\tilde{e}\| \|h\|) + O(\|h^{\perp}\| \|h\|).$$

Finally, regarding the local density $r$, we get similarly

$$\langle \Lambda_0 h, h \rangle = \langle \Lambda_0 r, r \rangle + \langle \Lambda_0 r, \tilde{m} \cdot v + \tilde{e} \mathfrak{E}(v) + h^{\perp} \rangle + \langle \Lambda_0 (\tilde{m} \cdot v + \tilde{e} \mathfrak{E}(v) + h^{\perp}), h \rangle$$

and it follows from (5.8) that

$$-\langle \Lambda_0 h, h \rangle \leq -\bar{\lambda}_0 \|\tilde{r}\|^2 + O(\|\tilde{m}\| \|h\|) + O(\|\tilde{e}\| \|h\|) + O(\|h^{\perp}\| \|h\|).$$

The proof of the proposition is complete.

<figure><img src="image_7.png" /></figure>
Regarding the momentum $m$, one first observes that

$$|\Omega^{-1} \nabla_x \nabla_x^{\text{sym}} m|^2 \geq \frac{1}{9} |\Omega^{-1} \nabla_x^2 m|^2$$

thanks to the Schwarz lemma written as

$$\forall i, j, k \in \{1, \dots, d\}, \partial_{ij}^2 m_k = \partial_i (\nabla^{\text{sym}} m)_{jk} + \partial_j (\nabla_x^{\text{sym}} m)_{ik} - \partial_k (\nabla_x^{\text{sym}} m)_{ij}.$$

Applied to (5.5)–(5.7), the cascade of Poincaré inequalities at order 2 stated in Lemma A.6 implies that there exists a constant $\bar{\lambda}_1$ such that

$$\begin{aligned} \langle \Lambda_1 (m \cdot v), m \cdot v \rangle &\ge \frac{2}{9} \| \Omega^{-1} \nabla_x^2 m \|^2 \\ &\ge 2 \bar{\lambda}_1 \| m - \langle m \rangle - \langle \nabla_x m \rangle x \|^2 = 2 \bar{\lambda}_1 \| \tilde{m} \|^2. \end{aligned} \quad (5.9)$$

Regarding the energy $e$, we use the standard Poincaré inequality in $L^2(\rho)$ (the order 1 inequality of Lemma A.6) to get $\bar{\lambda}_2 > 0$ such that

$$\langle \Lambda_2 (e \mathfrak{E}(v)), e \mathfrak{E}(v) \rangle = \left(\frac{4}{d} + 2\right) \| \Omega^{-\frac{1}{2}} \nabla_x e \|^2 \ge 2 \bar{\lambda}_2 \|e - \langle e \rangle\|^2 = 2 \bar{\lambda}_2 \|\tilde{e}\|^2. \quad (5.10)$$

With the above estimates in hand, we can investigate all terms appearing in $\langle \Lambda_i h, h \rangle$. According to the number of velocity and space gradients in $\Lambda_2$, we get that

$$\langle \Lambda_2 h, h \rangle = \langle \Lambda_2 (e \mathfrak{E}(v)), e \mathfrak{E}(v) \rangle + \langle \Lambda_2 (e \mathfrak{E}(v)), h^{\perp} \rangle + \langle \Lambda_2 h^{\perp}, h \rangle,$$

from which one obtains with (5.10) that

$$-\langle \Lambda_2 h, h \rangle \le -\bar{\lambda}_2 \|\tilde{e}\|^2 + O \left( \|h^{\perp}\| \|h\| \right).$$

Similarly for $m$, using in addition that $\Lambda_1$ is self-adjoint and $\Lambda_1 (e \mathfrak{E}(v)) = \Lambda_1 (\tilde{e} \mathfrak{E}(v))$, one has

$$\langle \Lambda_1 h, h \rangle = \langle \Lambda_1 (m \cdot v), m \cdot v \rangle + \langle \Lambda_1 (m \cdot v), (e - \langle e \rangle) \mathfrak{E}(v) + h^{\perp} \rangle + \langle \Lambda_1 ((e - \langle e \rangle) \mathfrak{E}(v) + h^{\perp}), h \rangle,$$

which implies using (5.9) that

$$-\langle \Lambda_1 h, h \rangle \le -\bar{\lambda}_1 \|\tilde{m}\|^2 + O(\|\tilde{e}\| \|h\|) + O(\|h^{\perp}\| \|h\|).$$

Finally, regarding the local density $r$, we get similarly

$$\langle \Lambda_0 h, h \rangle = \langle \Lambda_0 r, r \rangle + \langle \Lambda_0 r, \tilde{m} \cdot v + \tilde{e} \mathfrak{E}(v) + h^{\perp} \rangle + \langle \Lambda_0 (\tilde{m} \cdot v + \tilde{e} \mathfrak{E}(v) + h^{\perp}), h \rangle$$

and it follows from (5.8) that

$$-\langle \Lambda_0 h, h \rangle \le -\bar{\lambda}_0 \|\tilde{r}\|^2 + O(\|\tilde{m}\| \|h\|) + O(\|\tilde{e}\| \|h\|) + O(\|h^{\perp}\| \|h\|).$$

The proof of the proposition is complete.

<figure><img src="image_8.png" /></figure>
We collect the previous estimates into a first partial Lyapunov inequality:

**Lemma 5.3.** Define the following norm

$$ \|h\|_{H_1}^2 := \|h\|^2 + \varepsilon_0 \langle A_0 h, h \rangle + \varepsilon_1 \langle A_1 h, h \rangle + \varepsilon_2 \langle A_2 h, h \rangle $$

for $\varepsilon_0, \varepsilon_1, \varepsilon_2 > 0$, then for $c_\mathscr{C} \gg \varepsilon_2 \gg \varepsilon_1 \gg \varepsilon_0$, we have

$$ \frac{1}{2} \frac{d}{dt} \|h\|_{H_1}^2 \le - \frac{c_\mathscr{C}}{2} \|h^\perp\|^2 - \frac{\varepsilon_2}{2} \bar{\lambda}_2 \|\bar{e}\|^2 - \frac{\varepsilon_1}{2} \bar{\lambda}_1 \|\bar{m}\|^2 - \varepsilon_0 \bar{\lambda}_0 \|\bar{r}\|^2 + \eta_1 \|h\|^2, $$

for some $0 < \eta_1 \ll \varepsilon_0$.

*Proof of Lemma 5.3.* Propositions 5.2 combined with Lemma 4.2 imply

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_1}^2 &\le - c_\mathscr{C} \|h^\perp\|^2 - \varepsilon_2 \bar{\lambda}_2 \|\bar{e}\|^2 - \varepsilon_1 \bar{\lambda}_1 \|\bar{m}\|^2 - \varepsilon_0 \bar{\lambda}_0 \|\bar{r}\|^2 \\ &\quad + C \varepsilon_2 \|h\| \|h^\perp\| + C \varepsilon_1 \|h\| (\|h^\perp\| + \|\bar{e}\|) \\ &\quad + C \varepsilon_0 \|h\| (\|h^\perp\| + \|\bar{m}\| + \|\bar{e}\|) \end{aligned} $$

for some constant $C > 0$. The statement then follows from repeated uses of Young's inequality for products. ■

In fact the time derivatives of the local density, momentum and energy can also be controlled as follows:

$$ \left\{ \begin{aligned} \frac{d}{dt} \langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle &\ge \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 - O(\|h\| \|\tilde{r}\|), \\ \frac{d}{dt} \langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle &\ge \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 - O(\|h\| \|\tilde{m}\|), \\ \frac{d}{dt} \langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle &\ge \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 - O(\|h\| \|\tilde{e}\|). \end{aligned} \right. \quad (5.11) $$

This leads to second improved partial Lyapunov inequality:

**Lemma 5.4.** Given $1 \gg \varepsilon_2 \gg \varepsilon'_2 \gg \varepsilon_1 \gg \varepsilon'_1 \gg \varepsilon_0 \gg \varepsilon'_0 \gg \eta_1$, the norm

$$ \|h\|_{H_2}^2 := \|h\|_{H_1}^2 - \varepsilon'_2 \langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle - \varepsilon'_1 \langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle - \varepsilon'_0 \langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle. $$

satisfies

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_2}^2 &\le - \frac{c_\mathscr{C}}{4} \|h^\perp\|^2 - \frac{\varepsilon_2}{4} \bar{\lambda}_2 \|\bar{e}\|^2 - \frac{\varepsilon_1}{4} \bar{\lambda}_1 \|\bar{m}\|^2 - \frac{\varepsilon_0}{2} \bar{\lambda}_0 \|\bar{r}\|^2 \\ &\quad - \varepsilon'_2 \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \bar{e}\|^2 - \varepsilon'_1 \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \bar{m}\|^2 - \varepsilon'_0 \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \bar{r}\|^2 \\ &\quad + \eta_2 \|h\|^2 \end{aligned} $$

for some $0 < \eta_2 \ll \varepsilon'_0$.

*Proof of Lemma 5.4.* This follows from (5.11) and

$$ |\langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle| + |\langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle| + |\langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle| \lesssim \|h\|^2 $$

and the fact that second order time derivatives of the macroscopic quantities can be controlled by $\Omega$ (as in Section 4). ■
We collect the previous estimates into a first partial Lyapunov inequality:

**Lemma 5.3.** *Define the following norm*

$$ \|h\|_{H_1}^2 := \|h\|^2 + \varepsilon_0 \langle A_0 h, h \rangle + \varepsilon_1 \langle A_1 h, h \rangle + \varepsilon_2 \langle A_2 h, h \rangle $$

for $\varepsilon_0, \varepsilon_1, \varepsilon_2 > 0$, then for $c_\mathscr{C} \gg \varepsilon_2 \gg \varepsilon_1 \gg \varepsilon_0$, we have

$$ \frac{1}{2} \frac{d}{dt} \|h\|_{H_1}^2 \le - \frac{c_\mathscr{C}}{2} \|h^\perp\|^2 - \frac{\varepsilon_2}{2} \bar{\lambda}_2 \|\tilde{e}\|^2 - \frac{\varepsilon_1}{2} \bar{\lambda}_1 \|\tilde{m}\|^2 - \varepsilon_0 \bar{\lambda}_0 \|\tilde{r}\|^2 + \eta_1 \|h\|^2, $$

for some $0 < \eta_1 \ll \varepsilon_0$.

*Proof of Lemma 5.3.* Propositions 5.2 combined with Lemma 4.2 imply

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_1}^2 &\le - c_\mathscr{C} \|h^\perp\|^2 - \varepsilon_2 \bar{\lambda}_2 \|\tilde{e}\|^2 - \varepsilon_1 \bar{\lambda}_1 \|\tilde{m}\|^2 - \varepsilon_0 \bar{\lambda}_0 \|\tilde{r}\|^2 \\ &\quad + C \varepsilon_2 \|h\| \|h^\perp\| + C \varepsilon_1 \|h\| (\|h^\perp\| + \|\tilde{e}\|) \\ &\quad + C \varepsilon_0 \|h\| (\|h^\perp\| + \|\tilde{m}\| + \|\tilde{e}\|) \end{aligned} $$

for some constant $C > 0$. The statement then follows from repeated uses of Young's inequality for products. ■

In fact the time derivatives of the local density, momentum and energy can also be controlled as follows:

$$ \left\{ \begin{aligned} \frac{d}{dt} \langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle &\ge \| \Omega^{-\frac{1}{2}} \partial_t \tilde{r} \|^2 - O(\|h\| \|\tilde{r}\|), \\ \frac{d}{dt} \langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle &\ge \| \Omega^{-\frac{1}{2}} \partial_t \tilde{m} \|^2 - O(\|h\| \|\tilde{m}\|), \\ \frac{d}{dt} \langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle &\ge \| \Omega^{-\frac{1}{2}} \partial_t \tilde{e} \|^2 - O(\|h\| \|\tilde{e}\|). \end{aligned} \right. \qquad (5.11) $$

This leads to second improved partial Lyapunov inequality:

**Lemma 5.4.** *Given $1 \gg \varepsilon_2 \gg \varepsilon'_2 \gg \varepsilon_1 \gg \varepsilon'_1 \gg \varepsilon_0 \gg \varepsilon'_0 \gg \eta_1$, the norm*

$$ \|h\|_{H_2}^2 := \|h\|_{H_1}^2 - \varepsilon'_2 \langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle - \varepsilon'_1 \langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle - \varepsilon'_0 \langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle. $$

satisfies

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_2}^2 &\le - \frac{c_\mathscr{C}}{4} \|h^\perp\|^2 - \frac{\varepsilon_2}{4} \bar{\lambda}_2 \|\tilde{e}\|^2 - \frac{\varepsilon_1}{4} \bar{\lambda}_1 \|\tilde{m}\|^2 - \frac{\varepsilon_0}{2} \bar{\lambda}_0 \|\tilde{r}\|^2 \\ &\quad - \varepsilon'_2 \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 - \varepsilon'_1 \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 - \varepsilon'_0 \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 \\ &\quad + \eta_2 \|h\|^2 \end{aligned} $$

for some $0 < \eta_2 \ll \varepsilon'_0$.

*Proof of Lemma 5.4.* This follows from (5.11) and

$$ |\langle \partial_t \tilde{e}, \Omega^{-1} \tilde{e} \rangle| + |\langle \partial_t \tilde{m}, \Omega^{-1} \tilde{m} \rangle| + |\langle \partial_t \tilde{r}, \Omega^{-1} \tilde{r} \rangle| \lesssim \|h\|^2 $$

and the fact that second order time derivatives of the macroscopic quantities can be controlled by $\Omega$ (as in Section 4). ■
## 5.2. Cascade of finite-dimensional correctors

In view of Lemma 5.3, what remains to be controlled are the finite dimensional terms

$$ \langle r \rangle, \quad \langle \nabla_x m \rangle x + \langle m \rangle \quad \text{and} \quad \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle. $$

### 5.2.1. Control of moments of the local momentum. We compute

$$ \frac{d}{dt} \langle v_i v_j h \rangle = 2 \left\langle \left( \nabla_x^{\text{sym}} m \right)_{i,j} \right\rangle + \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle, $$

which yields

$$ \begin{aligned} & \frac{d}{dt} \left\langle v_i v_j h \right\rangle \frac{d}{dt} \left\langle v_i v_j h \right\rangle \\ & \ge 2 \left\langle \left( \nabla_x^{\text{sym}} m \right)_{i,j} \right\rangle^2 - \left| \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle^2 \right| + \langle v_i v_j h \rangle \frac{d^2}{d^2 t} \langle v_i v_j h \rangle \end{aligned} $$

and

$$ \left\{ \begin{aligned} & \langle v_i v_j h \rangle \frac{d}{dt} \langle v_i v_j h \rangle \lesssim \|h\| \|h^{\perp}\|, \\ & \left| \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle \right|^2 + \langle v_i v_j h \rangle \frac{d^2}{d^2 t} \langle v_i v_j h \rangle \lesssim \|h\| \|h^{\perp}\|. \end{aligned} \right. $$

Define for all $i \in \{1, \dots, d\}$

$$ \psi_i(v) := 1 + \sqrt{\frac{d}{2}} \left(1 + \frac{4}{d}\right) \mathfrak{E}(v) - \sqrt{\frac{d}{2}} |v_i|^2 \mathfrak{E}(v) $$

which is orthogonal to 1, $v$, $|v|^2$. We then compute

$$ \frac{d}{dt} \langle \psi_i h \rangle = 4 \left\langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right\rangle + \left\langle \left[ \mathcal{T}(\psi_i) + C(\psi_i) \right] h^{\perp} \right\rangle, $$

which yields

$$ \begin{aligned} & \frac{d}{dt} \left\langle \psi_i h \right\rangle \frac{d}{dt} \left\langle \psi_i h \right\rangle \\ & \ge 8 \left\langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right\rangle^2 - \left\langle \left[ \mathcal{T}(\psi_i(v)) + C(\psi_i(v)) \right] h^{\perp} \right\rangle^2 + \langle \psi_i h \rangle \frac{d^2}{d^2 t} \langle \psi_i h \rangle \end{aligned} $$

and

$$ \left\{ \begin{aligned} & \langle \psi_i h \rangle \frac{d}{dt} \langle \psi_i h \rangle \lesssim \|h\|^2, \\ & \left| \left\langle \left[ \mathcal{T}(\psi_i(v)) + C(\psi_i(v)) \right] h^{\perp} \right\rangle \right|^2 + \langle \psi_i h \rangle \frac{d^2}{d^2 t} \langle \psi_i h \rangle \lesssim \|h\| \|h^{\perp}\|. \end{aligned} \right. $$

We finally introduce the third norm

$$ \|h\|^2_{H_3} := \|h\|^2_{H_2} - \varepsilon_3 \sum_{i \neq j} \langle v_i v_j h \rangle \frac{d}{dt} \langle v_i v_j h \rangle - \varepsilon_3 \langle \psi_i h \rangle \frac{d}{dt} \langle \psi_i h \rangle $$
## 5.2. Cascade of finite-dimensional correctors

In view of Lemma 5.3, what remains to be controlled are the finite dimensional terms

$$ \langle r \rangle, \quad \langle \nabla_x m \rangle x + \langle m \rangle \quad \text{and} \quad \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle. $$

### 5.2.1. Control of moments of the local momentum. We compute

$$ \frac{d}{dt} \langle v_i v_j h \rangle = 2 \left\langle \left( \nabla_x^{\text{sym}} m \right)_{i,j} \right\rangle + \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle, $$

which yields

$$ \begin{aligned} & \frac{d}{dt} \left\langle v_i v_j h \right\rangle \frac{d}{dt} \left\langle v_i v_j h \right\rangle \\ & \ge 2 \left\langle \left( \nabla_x^{\text{sym}} m \right)_{i,j} \right\rangle^2 - \left| \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle^2 \right| + \left\langle v_i v_j h \right\rangle \frac{d^2}{dt^2} \left\langle v_i v_j h \right\rangle \end{aligned} $$

and

$$ \left\{ \begin{aligned} & \left\langle v_i v_j h \right\rangle \frac{d}{dt} \left\langle v_i v_j h \right\rangle \lesssim \|h\| \|h^{\perp}\|, \\ & \left| \left\langle \left( \mathcal{T}(v_i v_j) + C(v_i v_j) \right) h^{\perp} \right\rangle \right| + \left\langle v_i v_j h \right\rangle \frac{d^2}{dt^2} \left\langle v_i v_j h \right\rangle \lesssim \|h\| \|h^{\perp}\|. \end{aligned} \right. $$

Define for all $i \in \{1, \dots, d\}$

$$ \psi_i(v) := 1 + \sqrt{\frac{d}{2}} \left(1 + \frac{4}{d}\right) \mathfrak{E}(v) - \sqrt{\frac{d}{2}} |v_i|^2 \mathfrak{E}(v) $$

which is orthogonal to 1, $v$, $|v|^2$. We then compute

$$ \frac{d}{dt} \langle \psi_i h \rangle = 4 \left\langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right\rangle + \left\langle \left[ \mathcal{T}(\psi_i) + C(\psi_i) \right] h^{\perp} \right\rangle, $$

which yields

$$ \begin{aligned} & \frac{d}{dt} \left\langle \psi_i h \right\rangle \frac{d}{dt} \left\langle \psi_i h \right\rangle \\ & \ge 8 \left\langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right\rangle^2 - \left\langle \left[ \mathcal{T}(\psi_i(v)) + C(\psi_i(v)) \right] h^{\perp} \right\rangle^2 + \left\langle \psi_i h \right\rangle \frac{d^2}{dt^2} \left\langle \psi_i h \right\rangle \end{aligned} $$

and

$$ \left\{ \begin{aligned} & \left\langle \psi_i h \right\rangle \frac{d}{dt} \left\langle \psi_i h \right\rangle \lesssim \|h\|^2, \\ & \left| \left\langle \left[ \mathcal{T}(\psi_i(v)) + C(\psi_i(v)) \right] h^{\perp} \right\rangle \right|^2 + \left\langle \psi_i h \right\rangle \frac{d^2}{dt^2} \left\langle \psi_i h \right\rangle \lesssim \|h\| \|h^{\perp}\|. \end{aligned} \right. $$

We finally introduce the third norm

$$ \|h\|_{\mathcal{H}_3}^2 := \|h\|_{\mathcal{H}_2}^2 - \varepsilon_3 \sum_{i \neq j} \langle v_i v_j h \rangle \frac{d}{dt} \langle v_i v_j h \rangle - \varepsilon_3 \langle \psi_i h \rangle \frac{d}{dt} \langle \psi_i h \rangle $$
$$ - \epsilon'_3 \sum_{i \neq j} \langle \frac{d}{dt} (\nabla_x^{\text{sym}} m)_{i,j} \rangle \langle (\nabla_x^{\text{sym}} m)_{i,j} \rangle \\ - \epsilon'_3 \langle \frac{d}{dt} \left( \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right) \rangle \langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \rangle $$

for $1 \gg \epsilon_3 \gg \epsilon'_3 \gg \epsilon_2 \gg \epsilon'_2 \gg \epsilon_1 \gg \epsilon'_1 \gg \epsilon_0 \gg \epsilon'_0$. Therefore, defining as in (4.2b) and (4.2c) the space inhomogeneous terms

$$ m_s := m - \langle \nabla_x^{\text{skew}} m \rangle x - \frac{1}{d} \langle \nabla_x \cdot m \rangle x - \langle m \rangle \quad \text{and} \quad e_s := \tilde{e} = e - \langle e \rangle, $$

we obtain

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_3}^2 & \le - \frac{c \mathcal{C}}{8} \|h^+\|^2 - \frac{\epsilon_2}{4} \bar{\lambda}_2 \|e_s\|^2 \\ & \quad - \frac{\epsilon_1}{8} \bar{\lambda}_1 \|\tilde{m}\|^2 - \frac{\epsilon_1}{8} \bar{\lambda}_1 \|m_s\|^2 - \frac{\epsilon_0}{2} \bar{\lambda}_0 \|\tilde{r}\|^2 \\ & \quad - \frac{\epsilon'_2}{2} \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 \\ & \quad - \frac{\epsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 - \frac{\epsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 \\ & \quad - \epsilon'_0 \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 + \eta_3 \|h\|^2 \end{aligned} $$

for another $0 < \eta_3 \ll \epsilon'_0$.

### 5.2.2. Control of moments of the local density and energy.
We now control the difference between the finite dimensional quantities

$$ r - \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle $$

and

$$ w_s := r - \frac{1}{2d} \langle \Delta_x r \rangle (|x|^2 - \langle |x|^2 \rangle) - \langle \nabla_x r \rangle \cdot x \\ - \sqrt{\frac{2}{d}} \langle e \rangle \left[ \phi - \langle \phi \rangle - \frac{1}{2d} \langle \Delta_x \phi \rangle (|x|^2 - \langle |x|^2 \rangle) \right] $$

defined in (4.2e) in Section 4, which is made of the two terms (using that $\langle r \rangle = 0$)

$$ \left\{ \begin{aligned} I_1 &:= -\frac{1}{2} \sum_{1 \le i,j \le d} \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) (x_i x_j - \langle x_i x_j \rangle), \\ I_2 &:= \sqrt{\frac{2}{d}} \langle e \rangle \left[ \phi - \langle \phi \rangle - \frac{1}{2d} \langle \Delta_x \phi \rangle (|x|^2 - \langle |x|^2 \rangle) \right]. \end{aligned} \right. $$

The first term is controlled by using

$$ \frac{d}{dt} \left( \langle (\nabla_x^{\text{sym}} m)_{ij} \rangle - \frac{1}{d} \langle \nabla_x \cdot m \rangle \delta_{ij} \right) $$
$$ - \epsilon_3' \sum_{i \neq j} \langle \frac{d}{dt} (\nabla_x^{\text{sym}} m)_{i,j} \rangle \langle (\nabla_x^{\text{sym}} m)_{i,j} \rangle \\ - \epsilon_3' \langle \frac{d}{dt} \left( \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \right) \rangle \langle \frac{1}{d} \nabla_x \cdot m - \partial_{x_i} m_i \rangle $$

for $1 \gg \epsilon_3 \gg \epsilon_3' \gg \epsilon_2 \gg \epsilon_2' \gg \epsilon_1 \gg \epsilon_1' \gg \epsilon_0 \gg \epsilon_0'$. Therefore, defining as in (4.2b) and (4.2c) the space inhomogeneous terms

$$ m_s := m - \langle \nabla_x^{\text{skew}} m \rangle x - \frac{1}{d} \langle \nabla_x \cdot m \rangle x - \langle m \rangle \quad \text{and} \quad e_s := \tilde{e} = e - \langle e \rangle, $$

we obtain

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{H_3}^2 & \le - \frac{c \mathscr{C}}{8} \|h^+\|^2 - \frac{\epsilon_2}{4} \bar{\lambda}_2 \|e_s\|^2 \\ & \quad - \frac{\epsilon_1}{8} \bar{\lambda}_1 \|\tilde{m}\|^2 - \frac{\epsilon_1}{8} \bar{\lambda}_1 \|m_s\|^2 - \frac{\epsilon_0}{2} \bar{\lambda}_0 \|\tilde{r}\|^2 \\ & \quad - \frac{\epsilon_2'}{2} \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 \\ & \quad - \frac{\epsilon_1'}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 - \frac{\epsilon_1'}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 \\ & \quad - \epsilon_0' \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 + \eta_3 \|h\|^2 \end{aligned} $$

for another $0 < \eta_3 \ll \epsilon_0'$.

### 5.2.2. Control of moments of the local density and energy.
We now control the difference between the finite dimensional quantities

$$ r - \frac{1}{2} \langle \nabla_x^{\otimes 2} r \rangle : (x \otimes x - \langle x \otimes x \rangle) - \langle \nabla_x r \rangle \cdot x - \langle r \rangle $$

and

$$ w_s := r - \frac{1}{2d} \langle \Delta_x r \rangle (|x|^2 - \langle |x|^2 \rangle) - \langle \nabla_x r \rangle \cdot x \\ - \sqrt{\frac{2}{d}} \langle e \rangle \left[ \phi - \langle \phi \rangle - \frac{1}{2d} \langle \Delta_x \phi \rangle (|x|^2 - \langle |x|^2 \rangle) \right] $$

defined in (4.2e) in Section 4, which is made of the two terms (using that $\langle r \rangle = 0$)

$$ \left\{ \begin{aligned} I_1 &:= -\frac{1}{2} \sum_{1 \le i,j \le d} \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) (x_i x_j - \langle x_i x_j \rangle), \\ I_2 &:= \sqrt{\frac{2}{d}} \langle e \rangle \left[ \phi - \langle \phi \rangle - \frac{1}{2d} \langle \Delta_x \phi \rangle (|x|^2 - \langle |x|^2 \rangle) \right]. \end{aligned} \right. $$

The first term is controlled by using

$$ \frac{d}{dt} \left( \langle (\nabla_x^{\text{sym}} m)_{ij} \rangle - \frac{1}{d} \langle \nabla_x \cdot m \rangle \delta_{ij} \right) $$
$$ = - \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) + \text{controlled terms} $$

since the left hand side is already under control, and the second term is controlled by observing that $\langle e \rangle = \langle r \phi \rangle$ due to the energy conservation, and

$$ \langle r \phi \rangle = \langle \tilde{r} \phi \rangle + \frac{1}{2} \langle \nabla_x^2 r \rangle : \langle (x \otimes x - (x \otimes x)) \rangle \phi $$

provided that $\int_{\mathbb{R}^d} x \phi e^{-\phi} dx = 0$, and $\nabla_x^2 r = \nabla_x^2 \tilde{r}$, so finally

$$ |\langle e \rangle| = |\langle r \phi \rangle| \lesssim \|\tilde{r}\|. $$

This allows to define the final and fourth norm

$$ \|h\|_{\mathcal{H}_4}^2 := \|h\|_{\mathcal{H}_3}^2 \\ + \varepsilon_w \frac{d}{dt} \left( \langle (\nabla_x^{\text{sym}} m)_{ij} \rangle - \frac{1}{d} \langle \nabla_x \cdot m \rangle \delta_{ij} \right) \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) \\ - \varepsilon_w' \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) \frac{d}{dt} \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) $$

for $1 \gg \varepsilon_3 \gg \varepsilon'_3 \gg \varepsilon_2 \gg \varepsilon'_2 \gg \varepsilon_1 \gg \varepsilon'_1 \gg \varepsilon_0 \gg \varepsilon'_0 \gg \varepsilon_w \gg \varepsilon'_w$, with

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{\mathcal{H}_4}^2 &\le - \frac{c_c}{16} \|h^\perp\|^2 - \frac{\varepsilon_2}{8} \bar{\lambda}_2 \|e_s\|^2 - \frac{\varepsilon_w}{2} \|w_s\|^2 \\ &\quad - \frac{\varepsilon_1}{8} \bar{\lambda}_1 \|\tilde{m}\|^2 - \frac{\varepsilon_1}{16} \bar{\lambda}_1 \|m_s\|^2 - \frac{\varepsilon_0}{2} \bar{\lambda}_0 \|\tilde{r}\|^2 \\ &\quad - \frac{\varepsilon'_2}{2} \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 - \frac{\varepsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 \\ &\quad - \frac{\varepsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 - \varepsilon'_0 \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 \\ &\quad - \varepsilon'_w \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \eta_4 \|h\|^2 \end{aligned} $$

for $0 < \eta_4 \ll \varepsilon'_w$. Denote by $C_4[h]$ the semi-norm of the controlled quantities

$$ C_4[h] := \left( \|h^\perp\|^2 + \|\tilde{e}\|^2 + \|\tilde{m}\|^2 + \|m_s\|^2 + \|\tilde{r}\|^2 + \|w_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t e_s\|^2 \right. \\ \left. + \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 \right)^{1/2} $$

and adjust the constants to get, for some $0 < \eta \ll \varepsilon \ll 1$,

$$ \frac{1}{2} \frac{d}{dt} \|h\|_{\mathcal{H}_4}^2 \le - \varepsilon C_4[h]^2 + \eta \|h\|^2. \qquad (5.12) $$

### 5.3. Control of the remaining finite-dimensional quantities related to the special macroscopic modes

Estimate (5.12) controls the same microscopic and macroscopic parts of the solution as in Lemma 4.6 in the micro-macro method. The remaining finite-dimensional quantities related to the special macroscopic modes can then be treated exactly as in Sections 4.4 and 4.5. This completes the proof of Proposition 5.1.
$$ = - \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) + \text{controlled terms} $$

since the left hand side is already under control, and the second term is controlled by observing that $\langle e \rangle = \langle r \phi \rangle$ due to the energy conservation, and

$$ \langle r \phi \rangle = \langle \tilde{r} \phi \rangle + \frac{1}{2} \langle \nabla_x^2 r \rangle : \langle (x \otimes x - (x \otimes x)) \rangle \phi $$

provided that $\int_{\mathbb{R}^d} x \phi e^{-\phi} dx = 0$, and $\nabla_x^2 r = \nabla_x^2 \tilde{r}$, so finally

$$ |\langle e \rangle| = |\langle r \phi \rangle| \lesssim \|\tilde{r}\|. $$

This allows to define the final and fourth norm

$$ \|h\|_{\mathcal{H}_4}^2 := \|h\|_{\mathcal{H}_3}^2 \\ + \varepsilon_w \frac{d}{dt} \left( \langle (\nabla_x^{\text{sym}} m)_{ij} \rangle - \frac{1}{d} \langle \nabla_x \cdot m \rangle \delta_{ij} \right) \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) \\ - \varepsilon_w' \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) \frac{d}{dt} \left( \langle \partial_{x_i x_j}^2 r \rangle - \frac{1}{d} \langle \Delta_x r \rangle \delta_{ij} \right) $$

for $1 \gg \varepsilon_3 \gg \varepsilon'_3 \gg \varepsilon_2 \gg \varepsilon'_2 \gg \varepsilon_1 \gg \varepsilon'_1 \gg \varepsilon_0 \gg \varepsilon'_0 \gg \varepsilon_w \gg \varepsilon'_w$, with

$$ \begin{aligned} \frac{1}{2} \frac{d}{dt} \|h\|_{\mathcal{H}_4}^2 &\le - \frac{c_c}{16} \|h^\perp\|^2 - \frac{\varepsilon_2}{8} \bar{\lambda}_2 \|e_s\|^2 - \frac{\varepsilon_w}{2} \|w_s\|^2 \\ &\quad - \frac{\varepsilon_1}{8} \bar{\lambda}_1 \|\tilde{m}\|^2 - \frac{\varepsilon_1}{16} \bar{\lambda}_1 \|m_s\|^2 - \frac{\varepsilon_0}{2} \bar{\lambda}_0 \|\tilde{r}\|^2 \\ &\quad - \frac{\varepsilon'_2}{2} \bar{\lambda}_2 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{e}\|^2 - \frac{\varepsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 \\ &\quad - \frac{\varepsilon'_1}{2} \bar{\lambda}_1 \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 - \varepsilon'_0 \bar{\lambda}_0 \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 \\ &\quad - \varepsilon'_w \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 + \eta_4 \|h\|^2 \end{aligned} $$

for $0 < \eta_4 \ll \varepsilon'_w$. Denote by $C_4[h]$ the semi-norm of the controlled quantities

$$ C_4[h] := \left( \|h^\perp\|^2 + \|\tilde{e}\|^2 + \|\tilde{m}\|^2 + \|m_s\|^2 + \|\tilde{r}\|^2 + \|w_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t e_s\|^2 \right. \\ \left. + \|\Omega^{-\frac{1}{2}} \partial_t \tilde{m}\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t m_s\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t \tilde{r}\|^2 + \|\Omega^{-\frac{1}{2}} \partial_t w_s\|^2 \right)^{1/2} $$

and adjust the constants to get, for some $0 < \eta \ll \varepsilon \ll 1$,

$$ \frac{1}{2} \frac{d}{dt} \|h\|_{\mathcal{H}_4}^2 \le - \varepsilon C_4[h]^2 + \eta \|h\|^2. \qquad (5.12) $$

### 5.3. Control of the remaining finite-dimensional quantities related to the special macroscopic modes

Estimate (5.12) controls the same microscopic and macroscopic parts of the solution as in Lemma 4.6 in the micro-macro method. The remaining finite-dimensional quantities related to the special macroscopic modes can then be treated exactly as in Sections 4.4 and 4.5. This completes the proof of Proposition 5.1.
# Appendix A. Some technical computations

## A.1. Momentum conservation versus infinitesimal rotations

Here we prove (2.19) for a solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With $x \mapsto Ax := \mathbb{P}_\phi m_0(x)$,

$$ 
\begin{align*}
m_0(x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} v f_0(x, v) \, dv, & m_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} v f(t, x, v) \, dv, \\
r_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} f(t, x, v) \, dv, & e_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} \mathfrak{E}(v) f(t, x, v) \, dv,
\end{align*}
$$

let us define $h^\perp$ such that $f = r_f \mathcal{M} + m_f \cdot v \mathcal{M} + e_f \mathfrak{E} \mathcal{M} + h^\perp \mathcal{M}$.

**Lemma A.1.** With the above notations, we have $\mathbb{P}(m_f - m_0) \in \mathcal{R}_\phi^\perp$.

*Proof of Lemma A.1.* At $t=0$, we have $\mathbb{P}m_f(0) - \mathbb{P}m_0 = 0$. Let $B \in \mathcal{R}_\phi$. To prove that $\mathbb{P}(m_f - m_0)$ is orthogonal to $x \mapsto Bx$, it is sufficient to prove that

$$ \forall t \ge 0, \quad \langle m_f(t) - m_0, Bx \rangle = 0. $$

By direct computation, we have

$$ \partial_t m_f = - \nabla_x r_f + \sqrt{\frac{2}{d}} \nabla_x^* e_f + \nabla_x^* \cdot E[h^\perp] $$

where $E[h^\perp] = \int_{\mathbb{R}^d} (v \otimes v - \mathrm{Id}_{d \times d}) h^\perp \mu \, dv$, and use it to compute

$$ 
\begin{align*}
\frac{d}{dt} \langle m_f - m_0, Bx \rangle \\
&= \langle \partial_t m_f, Bx \rangle \\
&= \left\langle -\nabla_x r_f + \sqrt{\frac{2}{d}} \nabla_x^* e_f + \nabla_x^* \cdot E(h^\perp), Bx \right\rangle \\
&= -\langle r_f, \nabla_x^* \cdot Bx \rangle + \sqrt{\frac{2}{d}} \langle e_f, \nabla_x \cdot Bx \rangle + \langle E[h^\perp] : \nabla \otimes Bx \rangle
\end{align*}
$$

where the last line follows from an integration by parts. The first term in the right hand side vanishes because $\nabla_x^* \cdot Bx = -\nabla_x \cdot Bx + \nabla_\phi \cdot Bx = 0$ since $B$ is skew-symmetric and $(x \mapsto Bx) \in \mathcal{R}_\phi$. The second term vanishes as well because $\nabla_x \cdot Bx = 0$. Since $E[h^\perp] : \nabla \otimes Bx = -E[h^\perp] : B = 0$ because $E[h^\perp]$ is symmetric and $B$ is skew-symmetric, the third term also vanishes. This proves that $\frac{d}{dt} \langle m(t) - m_0, Bx \rangle = 0$ and completes the proof. ■

## A.2. Special macroscopic modes: the invertibility and rank

We state and prove two results used in Section 3.3 and implicitly in Section 4.5. The first result deals with the invertibility of the matrices $M_\phi$ and $\tilde{M}_\phi$ defined respectively in (3.10) and (3.16).
**Appendix A. Some technical computations**

A.1. Momentum conservation versus infinitesimal rotations

Here we prove (2.19) for a solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$. With $x \mapsto Ax := \mathbb{P}_\phi m_0(x)$,

$$
\begin{align*}
m_0(x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} v f_0(x, v) \, dv, & m_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} v f(t, x, v) \, dv, \\
r_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} f(t, x, v) \, dv, & e_f(t, x) &:= e^{\phi(x)} \int_{\mathbb{R}^d} \mathfrak{E}(v) f(t, x, v) \, dv,
\end{align*}
$$

let us define $h^{\perp}$ such that $f = r_f \mathcal{M} + m_f \cdot v \mathcal{M} + e_f \mathfrak{E} \mathcal{M} + h^{\perp} \mathcal{M}$.

**Lemma A.1.** With the above notations, we have $\mathbb{P}(m_f - m_0) \in \mathcal{R}_\phi^\perp$.

*Proof of Lemma A.1.* At $t=0$, we have $\mathbb{P}m_f(0) - \mathbb{P}m_0 = 0$. Let $B \in \mathcal{R}_\phi$. To prove that $\mathbb{P}(m_f - m_0)$ is orthogonal to $x \mapsto Bx$, it is sufficient to prove that

$$
\forall t \ge 0, \quad \langle m_f(t) - m_0, Bx \rangle = 0.
$$

By direct computation, we have

$$
\partial_t m_f = - \nabla_x r_f + \sqrt{\frac{2}{d}} \nabla_x^* e_f + \nabla_x^* \cdot E[h^\perp]
$$

where $E[h^\perp] = \int_{\mathbb{R}^d} (v \otimes v - \mathrm{Id}_{d \times d}) h^\perp \mu \, dv$, and use it to compute

$$
\begin{align*}
\frac{d}{dt} \langle m_f - m_0, Bx \rangle \\
&= \langle \partial_t m_f, Bx \rangle \\
&= \left\langle -\nabla_x r_f + \sqrt{\frac{2}{d}} \nabla_x^* e_f + \nabla_x^* \cdot E(h^\perp), Bx \right\rangle \\
&= -\langle r_f, \nabla_x^* \cdot Bx \rangle + \sqrt{\frac{2}{d}} \langle e_f, \nabla_x \cdot Bx \rangle + \langle E[h^\perp] : \nabla \otimes Bx \rangle
\end{align*}
$$

where the last line follows from an integration by parts. The first term in the right hand side vanishes because $\nabla_x^* \cdot Bx = -\nabla_x \cdot Bx + \nabla_\phi \cdot Bx = 0$ since B is skew-symmetric and $(x \mapsto Bx) \in \mathcal{R}_\phi$. The second term vanishes as well because $\nabla_x \cdot Bx = 0$. Since $E[h^\perp] : \nabla \otimes Bx = -E[h^\perp] : B = 0$ because $E[h^\perp]$ is symmetric and B is skew-symmetric, the third term also vanishes. This proves that $\frac{d}{dt}\langle m(t) - m_0, Bx \rangle = 0$ and completes the proof. ■

A.2. Special macroscopic modes: the invertibility and rank

We state and prove two results used in Section 3.3 and implicitly in Section 4.5. The first result deals with the invertibility of the matrices $M_\phi$ and $\tilde{M}_\phi$ defined respectively in (3.10) and (3.16).
**Lemma A.2.** If $d_{\phi} = d$, the matrix $M_{\phi}$ is invertible. If $1 \le d_{\phi} \le d - 1$, the matrix $\tilde{M}_{\phi}$ is invertible.

*Proof of Lemma A.2.* Assume that $d_{\phi} = d$ in (1.9). Let $u \in \mathbb{R}^d$ be such that $M_{\phi}u = 0$. Then $M_{\phi}u \cdot u = \langle |\Phi \cdot u|^2 \rangle = 0$, which implies that $\Phi(x) \cdot u = 0$ for any $x \in \mathbb{R}^d$, hence $u = 0$. This means that $\text{Ker } M_{\phi} = \{0\}$. The proof in the case $d_{\phi} \le d - 1$ follows exactly the same scheme. ■

The second result deals with the linear independence of the two functions $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$ defined in (3.14), and similarly for $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$ defined in (3.19).

**Lemma A.3.** If $d_{\phi} = d$, we have $\text{Rank}(\tilde{\Psi}_1, \tilde{\Psi}_2) = 2$. If $1 \le d_{\phi} \le d - 1$, we have $\text{Rank}(\tilde{\Psi}_1, \tilde{\Psi}_2) = 2$.

*Proof of Lemma A.3.* Let us assume that $d_{\phi} = d$ and argue by contradiction. Assume that $\tilde{\Psi}_1 = \lambda \tilde{\Psi}_2$ for some $\lambda \in \mathbb{R}^*$, that is, there are constants $\alpha, \beta \in \mathbb{R}^d$ and $\gamma \in \mathbb{R}$ such that

$$ \phi + \frac{1}{2} \nabla_x \phi \cdot x + \alpha \cdot \nabla_x \phi = \frac{\lambda}{4} |x|^2 + \beta \cdot x + \gamma. \quad (A.1) $$

We first look for quadratic solutions to (A.1) of the form $\phi_0 = x \cdot M_0 x + b_0 \cdot x + c_0$ with $M_0 \in \mathfrak{M}_{d \times d}(\mathbb{R})$, $b_0 \in \mathbb{R}^d$ and $c_0 \in \mathbb{R}$. Plugging $\phi_0$ into (A.1), one obtains $M_0 = \frac{1}{8} \text{Id}_{d \times d}$, $b_0 = \frac{2}{3}(\beta - \frac{\lambda}{4}\alpha)$ and $c_0 = \gamma - \frac{2}{3}\alpha \cdot (\beta - \frac{\lambda}{4}\alpha)$. Now let $\phi$ be a solution to (A.1). Define $\psi_0(x) = \phi(x) - \phi_0(x)$ and then $\psi(y) = \psi_0(y - 2\alpha)$, which hence verifies

$$ \psi(y) - \frac{1}{2} \nabla_y \psi(y) \cdot y = 0. $$

Let $\zeta(y) = |y|^2 \psi(y)$ so that $\nabla_y \zeta(y) \cdot y = 2|y|^2 (\psi(y) - \frac{1}{2} \nabla_y \psi(y) \cdot y) = 0$ for any $y \in \mathbb{R}^d$. In polar coordinates $(r, \theta)$, this implies that $\zeta(y) = \zeta(\theta)$ and hence

$$ \forall r > 0, \quad \psi(r, \theta) = \frac{\zeta(\theta)}{r^2}. $$

But $\psi$ is by assumption continuous at the origin, therefore $\lim_{r \to 0} \psi(r, \theta)$ is finite, which in turn implies that $\psi(r, \theta) = 0$. Finally one gets $\phi = \phi_0$. Thanks to the normalizations (H3) and (H7), one gets $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ and, by definition (1.9), $E_{\phi} = \{0\}$, which contradicts the hypothesis $d_{\phi} = d$. This completes the proof when $d_{\phi} = d$. When $d_{\phi} \le d - 1$, we argue similarly on $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$. ■

### A.3. Some computations for the commutator method

Here we prove technical claims used in Section 5. The first result is concerned with boundedness of the operators defined in Section 5.1 under Assumptions (H9) and (H10).

**Lemma A.4.** Assume that (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8)– (H9)– (H10) hold. Then the operators $\Lambda_i$, $A_i$ and $B_i$, $i \in \{1, \dots, 3\}$, are bounded.
**Lemma A.2.** If $d_{\phi} = d$, the matrix $M_{\phi}$ is invertible. If $1 \le d_{\phi} \le d - 1$, the matrix $\tilde{M}_{\phi}$ is invertible.

*Proof of Lemma A.2.* Assume that $d_{\phi} = d$ in (1.9). Let $u \in \mathbb{R}^d$ be such that $M_{\phi}u = 0$. Then $M_{\phi}u \cdot u = \langle |\Phi \cdot u|^2 \rangle = 0$, which implies that $\Phi(x) \cdot u = 0$ for any $x \in \mathbb{R}^d$, hence $u = 0$. This means that $\text{Ker } M_{\phi} = \{0\}$. The proof in the case $d_{\phi} \le d - 1$ follows exactly the same scheme. ■

The second result deals with the linear independence of the two functions $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$ defined in (3.14), and similarly for $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$ defined in (3.19).

**Lemma A.3.** If $d_{\phi} = d$, we have Rank($\tilde{\Psi}_1, \tilde{\Psi}_2$) = 2.
If $1 \le d_{\phi} \le d - 1$, we have Rank($\tilde{\Psi}_1, \tilde{\Psi}_2$) = 2.

*Proof of Lemma A.3.* Let us assume that $d_{\phi} = d$ and argue by contradiction. Assume that $\tilde{\Psi}_1 = \lambda \tilde{\Psi}_2$ for some $\lambda \in \mathbb{R}^*$, that is, there are constants $\alpha, \beta \in \mathbb{R}^d$ and $\gamma \in \mathbb{R}$ such that

$$ \phi + \frac{1}{2} \nabla_x \phi \cdot x + \alpha \cdot \nabla_x \phi = \frac{\lambda}{4} |x|^2 + \beta \cdot x + \gamma. \quad (A.1) $$

We first look for quadratic solutions to (A.1) of the form $\phi_0 = x \cdot M_0 x + b_0 \cdot x + c_0$ with $M_0 \in \mathfrak{M}_{d \times d}(\mathbb{R})$, $b_0 \in \mathbb{R}^d$ and $c_0 \in \mathbb{R}$. Plugging $\phi_0$ into (A.1), one obtains $M_0 = \frac{1}{8} \text{Id}_{d \times d}$, $b_0 = \frac{2}{3}(\beta - \frac{\lambda}{4}\alpha)$ and $c_0 = \gamma - \frac{2}{3}\alpha \cdot (\beta - \frac{\lambda}{4}\alpha)$. Now let $\phi$ be a solution to (A.1). Define $\psi_0(x) = \phi(x) - \phi_0(x)$ and then $\psi(y) = \psi_0(y - 2\alpha)$, which hence verifies

$$ \psi(y) - \frac{1}{2} \nabla_y \psi(y) \cdot y = 0. $$

Let $\zeta(y) = |y|^2 \psi(y)$ so that $\nabla_y \zeta(y) \cdot y = 2|y|^2 (\psi(y) - \frac{1}{2} \nabla_y \psi(y) \cdot y) = 0$ for any $y \in \mathbb{R}^d$.
In polar coordinates $(r, \theta)$, this implies that $\zeta(y) = \zeta(\theta)$ and hence

$$ \forall r > 0, \quad \psi(r, \theta) = \frac{\zeta(\theta)}{r^2}. $$

But $\psi$ is by assumption continuous at the origin, therefore $\lim_{r \to 0} \psi(r, \theta)$ is finite, which in turn implies that $\psi(r, \theta) = 0$. Finally one gets $\phi = \phi_0$. Thanks to the normalizations (H3) and (H7), one gets $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ and, by definition (1.9), $E_{\phi} = \{0\}$, which contradicts the hypothesis $d_{\phi} = d$. This completes the proof when $d_{\phi} = d$. When $d_{\phi} \le d - 1$, we argue similarly on $\tilde{\Psi}_1$ and $\tilde{\Psi}_2$. ■

### A.3. Some computations for the commutator method

Here we prove technical claims used in Section 5. The first result is concerned with boundedness of the operators defined in Section 5.1 under Assumptions (H9) and (H10).

**Lemma A.4.** Assume that (H0)–(H1)–(H2)–(H3)–(H4)–(H5)–(H6)–(H7)–(H8)– (H9)– (H10) hold. Then the operators $\Lambda_i$, $A_i$ and $B_i$, $i \in \{1, \dots, 3\}$, are bounded.
*Proof of Lemma A.4.* As a typical example, we focus on $\Lambda_1$ for which it is sufficient to show that $\Lambda^{-1} \Gamma^{-1/2} \partial_{x_i} \partial_{v_j} \partial_{v_k}$ is bounded in $L^2(\mathcal{M})$. Adopting the point of view of [25, Proposition A.7], we first conjugate with $\mathcal{M}^{1/2}$ and only have to check that $\tilde{\Gamma}^{-1/2} (\partial_{v_k} + v_k/2)$ and $\tilde{\Lambda}^{-1} (\partial_{x_i} + x_i/2) (\partial_{v_j} + v_j/2)$ are bounded in $L^2(\mathbb{R}^d, dxdv)$, where

$$ 
\begin{aligned}
\tilde{\Lambda} &= \sum_i \left( -\partial_{x_i} + \frac{x_i}{2} \right) \left( \partial_{x_i} + \frac{x_i}{2} \right) + \left( -\partial_{v_i} + \frac{v_i}{2} \right) \left( \partial_{v_i} + \frac{v_i}{2} \right), \\
\tilde{\Gamma} &= \sum_i \left( -\partial_{v_i} + \frac{v_i}{2} \right) \left( \partial_{v_i} + \frac{v_i}{2} \right).
\end{aligned}
$$

For $\tilde{\Gamma}^{-1/2}(\partial_{v_k} + v_k/2)$ this is due to the fact that $\tilde{\Gamma}^{-1/2}$ is of order $-1$ and $\partial_{v_k} + v_k/2$ of order $1$ in the pseudo-differential calculus associated to the metric $(dv^2 + d\eta^2)/(1 + |v|^2 + |\eta|^2)$, $\eta$ being the dual variable of $v$. The composition is then of order $0$ and the Calderón-Vaillancourt Theorem (see [8]) implies the boundedness. For $\tilde{\Lambda}^{-1}(\partial_{x_i} + x_i/2)(\partial_{v_j} + v_j/2)$, the result is also true because $\Lambda^{-1}$ is of order $-2$ and $(\partial_{x_i} + x_i/2)(\partial_{v_j} + v_j/2)$ is of order $2$ in the pseudo-differential calculus associated to the metric $(dx^2 + dv^2 + d\xi^2 + d\eta^2)/(1 + |\eta|^2 + |v|^2 + |\nabla\phi|^2 + |\xi|^2)$, $\xi$ being the dual variable of $v$. This implies the desired boundedness. Such calculus with two levels (involving $\Lambda$ in all variable and $\Gamma$ only in velocity variables) is also at the core of the boundedness of terms like $\Lambda \Gamma^{1/2} [\Lambda^{-1} \Gamma^{-1/2}, \mathcal{T}]$ where we use that $\Lambda$ and $\Gamma$ commute and that the commutation decrease the order by $1$, so that this operator is of order $0$ and therefore bounded by the Calderón-Vaillancourt Theorem. Note in addition that $H_\phi$ (which appears, e.g., in the $B_i$'s) is of order $0$ which greatly simplifies the proofs. For all other terms $\Lambda_i$, $A_i$ and $B_i$, similar computations give the result. ■

The second result deals with the symmetry and nonnegativity of $\Lambda_1$.

**Lemma A.5.** The operator $\Lambda_1$ is symmetric and nonnegative.

*Proof of Lemma A.5.* First we check that

$$ \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x $$

is symmetric, since for the other part of $\Lambda_1$ this is obvious:

$$ 
\begin{aligned}
\langle \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x f, g \rangle \\
&= \sum_{i,j,k} \langle \partial_{x_i}^* \partial_{v_j}^* \partial_{x_k}^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, g \rangle \\
&= \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} g \rangle \\
&= \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_j} \partial_{v_k} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_j} \partial_{x_k} \partial_{x_i} g \rangle \\
&= \sum_{i,j,k} \langle f, \partial_{x_i}^* \partial_{v_k}^* \partial_{x_j}^* \Gamma^{-\frac{1}{2}} \Lambda^{-2} \Gamma^{-\frac{1}{2}} \partial_{v_j} \partial_{x_k} \partial_{x_i} g \rangle
\end{aligned}
$$
**Proof of Lemma A.4.** As a typical example, we focus on $\Lambda_1$ for which it is sufficient to show that $\Lambda^{-1} \Gamma^{-1/2} \partial_{x_i} \partial_{v_j} \partial_{v_k}$ is bounded in $L^2(\mathcal{M})$. Adopting the point of view of [25, Proposition A.7], we first conjugate with $\mathcal{M}^{1/2}$ and only have to check that $\tilde{\Gamma}^{-1/2} (\partial_{v_k} + v_k/2)$ and $\tilde{\Lambda}^{-1} (\partial_{x_i} + x_i/2) (\partial_{v_j} + v_j/2)$ are bounded in $L^2(\mathbb{R}^d, dxdv)$, where

$$ \tilde{\Lambda} = \sum_i \left( -\partial_{x_i} + \frac{x_i}{2} \right) \left( \partial_{x_i} + \frac{x_i}{2} \right) + \left( -\partial_{v_i} + \frac{v_i}{2} \right) \left( \partial_{v_i} + \frac{v_i}{2} \right), \\ \tilde{\Gamma} = \sum_i \left( -\partial_{v_i} + \frac{v_i}{2} \right) \left( \partial_{v_i} + \frac{v_i}{2} \right). $$

For $\tilde{\Gamma}^{-1/2}(\partial_{v_k} + v_k/2)$ this is due to the fact that $\tilde{\Gamma}^{-1/2}$ is of order $-1$ and $\partial_{v_k} + v_k/2$ of order $1$ in the pseudo-differential calculus associated to the metric $(dv^2 + d\eta^2)/(1+|v|^2+|\eta|^2)$, $\eta$ being the dual variable of $v$. The composition is then of order $0$ and the Calderón-Vaillancourt Theorem (see [8]) implies the boundedness. For $\tilde{\Lambda}^{-1}(\partial_{x_i} + x_i/2)(\partial_{v_j} + v_j/2)$, the result is also true because $\Lambda^{-1}$ is of order $-2$ and $(\partial_{x_i} + x_i/2)(\partial_{v_j} + v_j/2)$ is of order $2$ in the pseudo-differential calculus associated to the metric $(dx^2 + dv^2 + d\xi^2 + d\eta^2)/(1+|\eta|^2+|v|^2+|\nabla\phi|^2+|\xi|^2)$, $\xi$ being the dual variable of $v$. This implies the desired boundedness. Such calculus with two levels (involving $\Lambda$ in all variables and $\Gamma$ only in velocity variables) is also at the core of the boundedness of terms like $\Lambda \Gamma^{1/2} [\Lambda^{-1} \Gamma^{-1/2}, \mathcal{T}]$ where we use that $\Lambda$ and $\Gamma$ commute and that the commutation with $\mathcal{T}$ decreases the order of derivatives by $1$, so that this operator is of order $0$ and therefore bounded by the Calderón-Vaillancourt Theorem. Note in addition that $H_\phi$ (which appears, e.g., in the $B_i$'s) is of order $0$ which greatly simplifies the proofs. For all other terms $\Lambda_i$, $A_i$ and $B_i$, similar computations give the result. ■

The second result deals with the symmetry and nonnegativity of $\Lambda_1$.

**Lemma A.5.** The operator $\Lambda_1$ is symmetric and nonnegative.

*Proof of Lemma A.5.* First we check that

$$ \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x $$

is symmetric, since for the other part of $\Lambda_1$ this is obvious:

$$ \begin{aligned} \langle \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x f, g \rangle & = \sum_{i,j,k} \langle \partial_{x_i}^* \partial_{v_j}^* \partial_{x_k}^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, g \rangle \\ & = \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} g \rangle \\ & = \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_j} \partial_{v_k} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_j} \partial_{x_k} \partial_{x_i} g \rangle \\ & = \sum_{i,j,k} \langle f, \partial_{x_i}^* \partial_{v_k}^* \partial_{x_j}^* \Gamma^{-\frac{1}{2}} \Lambda^{-2} \Gamma^{-\frac{1}{2}} \partial_{v_j} \partial_{x_k} \partial_{x_i} g \rangle \end{aligned} $$
$$ = \langle f, \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x g \rangle. $$

Next check that $\Lambda_1$ is indeed a nonnegative operator:

$$
\begin{align*}
\langle \Lambda_1 f, f \rangle &= \langle \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* f, f \rangle \\
&\quad + \langle \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v^* \otimes \nabla_x^* \otimes \nabla_x^* f, f \rangle \\
&= \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f \rangle \\
&\quad + \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f \rangle \\
&= \sum_{i,j,k} \frac{1}{2} \| \Lambda^{-1} \Gamma^{-\frac{1}{2}} (\partial_{x_k} \partial_{v_j} + \partial_{v_k} \partial_{x_j}) \partial_{x_i} f \|^2.
\end{align*}
$$

This completes the proof.

<figure><img src="image_8.png" /></figure>

### A.4. A cascade of Poincaré-Lions inequalities

Under Assumptions (H9) and (H10), we prove several inequalities used in Section 5.1. Let $\varphi$ a smooth function in $L^2(\rho)$ with compact support and

$$ P_0(\varphi) := \langle \varphi \rangle, $$

$$ P_1(\varphi) := \langle \varphi \rangle + \langle \nabla_x \varphi \rangle \cdot x, $$

$$ P_2(\varphi) := \langle \varphi \rangle + \langle \nabla_x \varphi \rangle \cdot x + \frac{1}{2} \langle (\nabla_x \otimes \nabla_x) \varphi \rangle : (x \otimes x - \langle x \otimes x \rangle). $$

**Lemma A.6.** Let $n \in \{1, 2, 3\}$. Then there exists a constant $c_{P,n} > 0$ such that for all smooth $\varphi$ with compact support we have

$$ c_{P,n} \|\varphi - P_{n-1}(\varphi)\|^2 \le \|\Omega^{-n/2} \nabla_x^n \varphi\|^2. $$

*Proof of Lemma A.6.* For $n=1$ this is exactly the Poincaré-Lions Theorem as stated in [9, Proposition 5] and recalled in (4.6). Let us prove the result for $n=2$. Let $\varphi$ be smooth and with compact support. We have

$$ \langle \varphi - P_1(\varphi) \rangle = \langle \varphi \rangle - \langle \varphi \rangle - \langle \nabla_x \varphi \rangle \cdot \langle x \rangle = 0 $$

because $\langle x \rangle = 0$. We therefore apply the Poincaré-Lions inequality (i.e., the case $n=1$) to $\varphi - P_1(\varphi)$, which gives

$$ \|\varphi - P_1(\varphi)\|^2 \le c_{P,1}^{-1} \|\Omega^{-\frac{1}{2}} \nabla_x (\varphi - P_1(\varphi))\|^2 = c_{P,1}^{-1} \|\Omega^{-\frac{1}{2}} (\nabla_x \varphi - \langle \nabla_x \varphi \rangle)\|^2. $$

We then apply the “$-1$-order” Poincaré-Lions inequality in [9, Lemma 10] recalled in (4.7) to $\nabla_x \varphi$ to get, for some $C_{LPL} > 0$ depending only on $\phi$,

$$ \|\Omega^{-\frac{1}{2}} (\nabla_x \varphi - \langle \nabla_x \varphi \rangle)\|^2 \le C_{LPL} \|\Omega^{-1} \nabla_x^2 \varphi\|^2. $$
$$ = \langle f, \nabla_x^* \nabla_v^* \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v \nabla_x \nabla_x g \rangle. $$

Next check that $\Lambda_1$ is indeed a nonnegative operator:

$$
\begin{aligned}
\langle \Lambda_1 f, f \rangle &= \langle \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* f, f \rangle \\
&\quad + \langle \nabla_x^* \otimes \nabla_v^* \otimes \nabla_x^* \Gamma^{-\frac{1}{2}} \Lambda^{-1} : \Lambda^{-1} \Gamma^{-\frac{1}{2}} \nabla_v^* \otimes \nabla_x^* \otimes \nabla_x^* f, f \rangle \\
&= \sum_{i,j,k} \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f \rangle \\
&\quad + \langle \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{v_k} \partial_{x_j} \partial_{x_i} f, \Lambda^{-1} \Gamma^{-\frac{1}{2}} \partial_{x_k} \partial_{v_j} \partial_{x_i} f \rangle \\
&= \sum_{i,j,k} \frac{1}{2} \| \Lambda^{-1} \Gamma^{-\frac{1}{2}} (\partial_{x_k} \partial_{v_j} + \partial_{v_k} \partial_{x_j}) \partial_{x_i} f \|^2.
\end{aligned}
$$

This completes the proof.

<figure><img src="image_9.png" /></figure>

### A.4. A cascade of Poincaré-Lions inequalities

Under Assumptions (H9) and (H10), we prove several inequalities used in Section 5.1. Let $\varphi$ a smooth function in $L^2(\rho)$ with compact support and

$$ P_0(\varphi) := \langle \varphi \rangle, $$

$$ P_1(\varphi) := \langle \varphi \rangle + \langle \nabla_x \varphi \rangle \cdot x, $$

$$ P_2(\varphi) := \langle \varphi \rangle + \langle \nabla_x \varphi \rangle \cdot x + \frac{1}{2} \langle (\nabla_x \otimes \nabla_x) \varphi \rangle : (x \otimes x - \langle x \otimes x \rangle). $$

**Lemma A.6.** Let $n \in \{1, 2, 3\}$. Then there exists a constant $c_{P,n} > 0$ such that for all smooth $\varphi$ with compact support we have

$$ c_{P,n} \|\varphi - P_{n-1}(\varphi)\|^2 \le \|\Omega^{-n/2} \nabla_x^n \varphi\|^2. $$

*Proof of Lemma A.6.* For $n=1$ this is exactly the Poincaré-Lions Theorem as stated in [9, Proposition 5] and recalled in (4.6). Let us prove the result for $n=2$. Let $\varphi$ be smooth and with compact support. We have

$$ \langle \varphi - P_1(\varphi) \rangle = \langle \varphi \rangle - \langle \varphi \rangle - \langle \nabla_x \varphi \rangle \cdot \langle x \rangle = 0 $$

because $\langle x \rangle = 0$. We therefore apply the Poincaré-Lions inequality (i.e., the case $n=1$) to $\varphi - P_1(\varphi)$, which gives

$$ \|\varphi - P_1(\varphi)\|^2 \le c_{P,1}^{-1} \|\Omega^{-\frac{1}{2}} \nabla_x (\varphi - P_1(\varphi))\|^2 = c_{P,1}^{-1} \|\Omega^{-\frac{1}{2}} (\nabla_x \varphi - \langle \nabla_x \varphi \rangle)\|^2. $$

We then apply the “$-1$-order” Poincaré-Lions inequality in [9, Lemma 10] recalled in (4.7) to $\nabla_x \varphi$ to get, for some $C_{LPL} > 0$ depending only on $\phi$,

$$ \|\Omega^{-\frac{1}{2}} (\nabla_x \varphi - \langle \nabla_x \varphi \rangle)\|^2 \le C_{LPL} \|\Omega^{-1} \nabla_x^2 \varphi\|^2. $$
This proves the case $n = 2$ with $c_{P,2} = c_{P,1}/C_{LPL}$.

In the case $n = 3$, we define $\psi := \varphi - \frac{1}{2} \langle \nabla^2 \varphi \rangle : x \otimes x$ and we compute

$$ 
\begin{aligned}
\psi - P_1(\psi) &= \psi - \langle \psi \rangle - \langle \nabla_x \psi \rangle \cdot x \\
&= \varphi - \frac{1}{2} \langle \nabla^2 \varphi \rangle : x \otimes x - \langle \varphi \rangle \\
&\quad + \frac{1}{2} \langle \nabla^2 \varphi \rangle : \langle x \otimes x \rangle - \langle \nabla_x \varphi \rangle \cdot x + \langle \nabla_x^2 \varphi \rangle : \langle x \rangle \otimes x \\
&= \varphi - P_2(\varphi)
\end{aligned}
$$

since $\langle x \rangle = 0$. We apply the inequality for $n = 2$ and obtain

$$ 
\begin{aligned}
\|\varphi - P_2(\varphi)\|^2 &= \|\psi - P_1(\psi)\|^2 \\
&\le c_{P,2}^{-1} \|\Omega^{-1} \nabla_x^2 \psi\|^2 = c_{P,2}^{-1} \|\Omega^{-1}(\nabla_x^2 \varphi - \langle \nabla_x^2 \varphi \rangle)\|^2. \quad (A.2)
\end{aligned}
$$

Arguing as for the proof of the “$-1$ order” Poincaré-Lions inequality in [9, Lemma 10], we prove the “$-2$ order” Poincaré-Lions inequality with constant $C_{LPL} > 0$:

$$ \|\Omega^{-1}(f - \langle f \rangle)\|^2 \le C_{LPL} \|\Omega^{-\frac{3}{2}} \nabla_x f\|^2 $$

for any $f \in C_c^\infty$. Applying this estimate in (A.2) to $\nabla_x^2 \varphi$ gives

$$ \|\varphi - P_2(\varphi)\|^2 \le c_{P,2}^{-1} C_{LPL} \|\Omega^{-\frac{3}{2}} \nabla_x^3 \varphi\|^2 $$

and concludes for $n = 3$ with $c_{P,3} = c_{P,2}/C_{LPL}$, and completes the proof. ■

## Appendix B. Extension to weakly coercive collision operators

Our method covers the case of collision operators $\mathcal{C}$ that do not possess a spectral gap (assumption (H1) in Section 1) but only satisfy a weaker coercivity property (see (H1') below). In this appendix, we state some results and changes to be done in the proofs.

### B.1. Results on decay rates

We assume that $\mathcal{C}$ satisfies, for some $\alpha > 0$, the weak coercivity property

$$ - \int_{\mathbb{R}^d} (\mathcal{C} f(v)) f(v) \mu(v)^{-1} dv \ge c_{\mathcal{C}} \|f - \Pi f\|_{L^2(\lfloor v \rfloor^{-\alpha} \mu^{-1})}^2 \quad (H1') $$

for some constant $c_{\mathcal{C}} > 0$, and for all $f$ in the domain of $\mathcal{C}$, where $\Pi$ is the $L^2(\mu^{-1})$-orthogonal projection onto Ker $\mathcal{C}$. Here $\lfloor v \rfloor$ denotes the weight $\sqrt{1 + |v|^2}$. Moreover, we suppose that for any polynomial function $p(v) : \mathbb{R}^d \to \mathbb{R}$ of degree at most 4, the function $p \mu$ is in the domain of $\mathcal{C}$ and

$$ C(p) := \|\mathcal{C}(p \mu)\|_{L^2(\lfloor v \rfloor^{\alpha} \mu^{-1})} < \infty. \quad (H2') $$

The analog of our main result in Theorem 1.1 then becomes:
This proves the case $n = 2$ with $c_{p,2} = c_{p,1}/C_{LPL}$.

In the case $n = 3$, we define $\psi := \varphi - \frac{1}{2} \langle \nabla^2 \varphi \rangle : x \otimes x$ and we compute

$$ 
\begin{aligned}
\psi - P_1(\psi) &= \psi - \langle \psi \rangle - \langle \nabla_x \psi \rangle \cdot x \\
&= \varphi - \frac{1}{2} \langle \nabla^2 \varphi \rangle : x \otimes x - \langle \varphi \rangle \\
&\quad + \frac{1}{2} \langle \nabla^2 \varphi \rangle : \langle x \otimes x \rangle - \langle \nabla_x \varphi \rangle \cdot x + \langle \nabla_x^2 \varphi \rangle : \langle x \rangle \otimes x \\
&= \varphi - P_2(\varphi)
\end{aligned}
$$

since $\langle x \rangle = 0$. We apply the inequality for $n = 2$ and obtain

$$ 
\begin{aligned}
\|\varphi - P_2(\varphi)\|^2 &= \|\psi - P_1(\psi)\|^2 \\
&\le c_{p,2}^{-1} \|\Omega^{-1} \nabla_x^2 \psi\|^2 = c_{p,2}^{-1} \|\Omega^{-1}(\nabla_x^2 \varphi - \langle \nabla_x^2 \varphi \rangle)\|^2. \quad (A.2)
\end{aligned}
$$

Arguing as for the proof of the “$-1$ order” Poincaré-Lions inequality in [9, Lemma 10], we prove the “$-2$ order” Poincaré-Lions inequality with constant $C_{LPL} > 0$:

$$ \|\Omega^{-1}(f - \langle f \rangle)\|^2 \le C_{LPL} \|\Omega^{-\frac{3}{2}} \nabla_x f\|^2 $$

for any $f \in C_c^\infty$. Applying this estimate in (A.2) to $\nabla_x^2 \varphi$ gives

$$ \|\varphi - P_2(\varphi)\|^2 \le c_{p,2}^{-1} C_{LPL} \|\Omega^{-\frac{3}{2}} \nabla_x^3 \varphi\|^2 $$

and concludes for $n = 3$ with $c_{p,3} = c_{p,2}/C_{LPL}$, and completes the proof. ■

## Appendix B. Extension to weakly coercive collision operators

Our method covers the case of collision operators $\mathcal{C}$ that do not possess a spectral gap (assumption (H1) in Section 1) but only satisfy a weaker coercivity property (see (H1') below). In this appendix, we state some results and changes to be done in the proofs.

### B.1. Results on decay rates

We assume that $\mathcal{C}$ satisfies, for some $\alpha > 0$, the weak coercivity property

$$ - \int_{\mathbb{R}^d} (\mathcal{C} f(v)) f(v) \mu(v)^{-1} \, dv \ge c_{\mathcal{C}} \|f - \Pi f\|_{L^2(\lfloor v \rfloor^{-\alpha} \mu^{-1})}^2 \quad (H1') $$

for some constant $c_{\mathcal{C}} > 0$, and for all $f$ in the domain of $\mathcal{C}$, where $\Pi$ is the $L^2(\mu^{-1})$-orthogonal projection onto Ker $\mathcal{C}$. Here $\lfloor v \rfloor$ denotes the weight $\sqrt{1 + |v|^2}$. Moreover, we suppose that for any polynomial function $p(v) : \mathbb{R}^d \to \mathbb{R}$ of degree at most 4, the function $p \mu$ is in the domain of $\mathcal{C}$ and

$$ C(p) := \|\mathcal{C}(p \mu)\|_{L^2(\lfloor v \rfloor^{\alpha} \mu^{-1})} < \infty. \quad (H2') $$

The analog of our main result in Theorem 1.1 then becomes:
**Theorem B.1.** Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy assumptions (H0)-(H1')-(H2')-(H3)-(H4)-(H5)-(H6)-(H7)-(H8). Then

(1) All special macroscopic modes of (1.4) are given by (1.12), i.e., are linear combinations of the Maxwellian, the energy mode, rotation modes compatible with $\phi$, and harmonic directional or pulsating modes if allowed by $\phi$.

(2) There exists a norm $\|\cdot\|_{L^2(M^{-1})}$ on $L^2(M^{-1})$, which is equivalent to $\|\cdot\|_{L^2(M^{-1})}$ (with quantitative comparison constants), and some explicit $\lambda > 0$ such that, for any solution $f \in C(\mathbb{R}^+; L^2(M^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(M^{-1})$, there exists a unique special macroscopic mode $F$ (determined by $f_0$) such that

$$ \forall t \ge 0, \quad \frac{1}{2} \frac{d}{dt} \|f(t) - F(t)\|_{L^2(M^{-1})}^2 \le -\lambda \|f(t) - F(t)\|_{L^2([v]^{-\alpha} M^{-1})}^2 . \quad (B.1) $$

The differential inequality (B.1) alone is not sufficient to prove a decay estimate when $f_0$ is merely in $L^2(M^{-1})$. In order to get such an estimate, one needs to assume more decay at infinity for $f_0$ and the differential inequality (B.1) has to be replaced by an inequality in spaces with stronger weights. For instance, assume that, for some $\beta > 0$,

$t \mapsto e^{t\mathcal{L}}$ is a strongly continuous uniformly bounded semigroup on $L^2(M^{-1-\beta})$, (H9') with $\mathcal{L}$ as in (1.1). In the spirit of [43], we obtain the following decay rate.

**Corollary B.2.** Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy assumptions (H0)-(H1')-(H2')-(H3)-(H4)-(H5)-(H6)-(H7)-(H8)-(H9'). Then there are explicit constants $C_0 > 0$ and $\Lambda > 0$ such that, for any solution $f \in C(\mathbb{R}^+; L^2(M^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(M^{-1-\beta})$, there exists a unique special macroscopic mode $F$ (determined by $f_0$) such that

$$ \forall t \ge 0, \quad \|f(t) - F(t)\|_{L^2(M^{-1})} \le C_0 \exp\left(-\Lambda t^{\frac{2}{2\alpha}}\right) \|f_0 - F(0)\|_{L^2(M^{-1-\beta})} $$

As in the proof of Theorem 1.1, it is convenient to work with the function $h$ defined by (2.22). We use various norms: $\|\cdot\|$ defined by (1.13) and also

$$ \|h\|_{\star}^{2} := \iint_{\mathbb{R}^{d} \times \mathbb{R}^{d}} |h|^2 |v|^{-\alpha} \mathcal{M} \, dx \, dv \quad \text{and} \quad \|h\|_{1-\beta}^{2} := \iint_{\mathbb{R}^{d} \times \mathbb{R}^{d}} |h|^2 \mathcal{M}^{1-\beta} \, dx \, dv. $$

Observe that $\|\cdot\| = \|\cdot\|_1 \le \|\cdot\|_{1-\beta}$, $\|\cdot\|_{\star} = \|\cdot\|_{L^2([v]^{-\alpha} M^{-1})}^2 \le \|\cdot\|$ and

$$ \|h\|_{\star}^{2} \approx \|h^{\perp}\|_{\star}^{2} + \|r\|^{2} + \|m\|^{2} + \|e\|^{2}. $$

## B.2. Proof of Theorem B.1

Proposition 3.1 applies: the proof of Part (1) is the same as Part (1) of Theorem 1.1. To prove Part (2), we argue as in the proof of Proposition 4.1, using the new assumptions. Thanks to (H1') and using that $\langle \mathcal{T}h, h \rangle = 0$, the function $h$ defined by (2.22) satisfies

$$ \frac{1}{2} \frac{d}{dt} \|h\|^2 = \langle Ch, h \rangle + \langle \mathcal{T}h, h \rangle \le -c_{\mathcal{C}} \|h^{\perp}\|_{\star}^2. $$
**Theorem B.1.** Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy assumptions (H0)-(H1')-(H2')-(H3)-(H4)-(H5)-(H6)-(H7)-(H8). Then

(1) All special macroscopic modes of (1.4) are given by (1.12), i.e., are linear combinations of the Maxwellian, the energy mode, rotation modes compatible with $\phi$, and harmonic directional or pulsating modes if allowed by $\phi$.

(2) There exists a norm $\|\cdot\|_{L^2(\mathcal{M}^{-1})}$ on $L^2(\mathcal{M}^{-1})$, which is equivalent to $\|\cdot\|_{L^2(\mathcal{M}^{-1})}$ (with quantitative comparison constants), and some explicit $\lambda > 0$ such that, for any solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1})$, there exists a unique special macroscopic mode $F$ (determined by $f_0$) such that

$$ \forall t \ge 0, \quad \frac{1}{2} \frac{d}{dt} \|f(t) - F(t)\|_{L^2(\mathcal{M}^{-1})}^2 \le -\lambda \|f(t) - F(t)\|_{L^2([v]^{-\alpha} \mathcal{M}^{-1})}^2. \quad (B.1) $$

The differential inequality (B.1) alone is not sufficient to prove a decay estimate when $f_0$ is merely in $L^2(\mathcal{M}^{-1})$. In order to get such an estimate, one needs to assume more decay at infinity for $f_0$ and the differential inequality (B.1) has to be replaced by an inequality in spaces with stronger weights. For instance, assume that, for some $\beta > 0$,

$t \mapsto e^{t\mathcal{L}}$ is a strongly continuous uniformly bounded semigroup on $L^2(\mathcal{M}^{-1-\beta})$, (H9') with $\mathcal{L}$ as in (1.1). In the spirit of [43], we obtain the following decay rate.

**Corollary B.2.** Assume that the potential $\phi$ and the collision operator $\mathcal{C}$ satisfy assumptions (H0)-(H1')-(H2')-(H3)-(H4)-(H5)-(H6)-(H7)-(H8)-(H9'). Then there are explicit constants $C_0 > 0$ and $\Lambda > 0$ such that, for any solution $f \in C(\mathbb{R}^+; L^2(\mathcal{M}^{-1}))$ to (1.1) with initial datum $f_0 \in L^2(\mathcal{M}^{-1-\beta})$, there exists a unique special macroscopic mode $F$ (determined by $f_0$) such that

$$ \forall t \ge 0, \quad \|f(t) - F(t)\|_{L^2(\mathcal{M}^{-1})} \le C_0 \exp\left(-\Lambda t^{\frac{2}{2\alpha}}\right) \|f_0 - F(0)\|_{L^2(\mathcal{M}^{-1-\beta})}. $$

As in the proof of Theorem 1.1, it is convenient to work with the function $h$ defined by (2.22). We use various norms: $\|\cdot\|$ defined by (1.13) and also

$$ \|h\|_{\star}^{2} := \iint_{\mathbb{R}^{d} \times \mathbb{R}^{d}} |h|^2 |v|^{-\alpha} \mathcal{M} \, dx \, dv \quad \text{and} \quad \|h\|_{1-\beta}^{2} := \iint_{\mathbb{R}^{d} \times \mathbb{R}^{d}} |h|^2 \mathcal{M}^{1-\beta} \, dx \, dv. $$

Observe that $\|\cdot\| = \|\cdot\|_1 \le \|\cdot\|_{1-\beta}$, $\|\cdot\|_{\star} = \|\cdot\|_{L^2([v]^{-\alpha} \mathcal{M}^{-1})} \le \|\cdot\|$ and

$$ \|h\|_{\star}^{2} \approx \|h^{\perp}\|_{\star}^{2} + \|r\|^{2} + \|m\|^{2} + \|e\|^{2}. $$

## B.2. Proof of Theorem B.1

Proposition 3.1 applies: the proof of Part (1) is the same as Part (1) of Theorem 1.1. To prove Part (2), we argue as in the proof of Proposition 4.1, using the new assumptions. Thanks to (H1') and using that $\langle \mathcal{T}h, h \rangle = 0$, the function $h$ defined by (2.22) satisfies

$$ \frac{1}{2} \frac{d}{dt} \|h\|^2 = \langle Ch, h \rangle + \langle \mathcal{T}h, h \rangle \le -c_{\mathcal{C}} \|h^{\perp}\|^2_{\star}. $$
This replaces the estimate of Lemma 4.2. We can then use (H2') and the above estimate to prove counterparts of estimates between Lemma 4.3 and Lemma 4.12 with $||h^\perp||$ and $||h||$ respectively replaced by $||h^\perp||_\star$ and $||h||_\star$. Then $\mathcal{F}_2$ defined in (4.39) using (4.20), with an appropriate choice of parameters $0 \ll \varepsilon_6 \ll \varepsilon_5 \ll \varepsilon_4 \ll \varepsilon_3 \ll \varepsilon_2 \ll \varepsilon_1 \ll 1$, is equivalent to $||\cdot||^2$ and satisfies

$$ \frac{d}{dt} \mathcal{F}_2[h] \leq -\kappa \overline{\mathcal{D}}_2[h] $$

for some constant $\kappa > 0$, where $\overline{\mathcal{D}}_2[h]$ is defined as $\mathcal{D}_2[h]$ in (4.40) with $||h^\perp||^2$ replaced by $||h^\perp||_\star^2$ in (4.22). This concludes the proof with

$$ \|f - F\|_{L^2(M^{-1})}^2 := \mathcal{F}_2[h] $$

since $\overline{\mathcal{D}}_2$ is equivalent to $||\cdot||_\star^2$.

<figure><img src="image_9.png" /></figure>

### B.3. Proof of Corollary B.2

Let $h_0 \in L^2(M^{1-\beta})$ and consider the solution $t \mapsto h(t)$ to the equation $\partial_t h = \mathcal{L}h$ with initial datum $h(0) = h_0$. Thanks to (H9'), there is some $C > 0$ such that

$$ \forall t \geq 0, \quad \|h(t)\|_{1-\beta} \leq C \|h_0\|_{1-\beta}. $$

We now observe that, for any $R > 0$, the following interpolation inequality holds

$$ \|g\|^2 \leq (1 + R^2)^{\alpha/2} \|g\|_{\star}^2 + \bar{\mu}^{\beta}(R) \|g\|_{1-\beta}^2 $$

with $\bar{\mu}(R) := (2\pi)^{-d/2} \|e^{-\phi}\|_{L^\infty(\mathbb{R}^d)} e^{-R^2/2}$. Therefore, thanks to (B.1) and using fact that $h \mapsto \|f - F\|_{L^2(M^{-1})}^2 = \mathcal{F}_2[h]$ is equivalent to $h \mapsto \|h\|^2$, one deduces that

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_2[h(t)] &\leq -\Lambda (1 + R^2)^{-\alpha/2} \mathcal{F}_2[h(t)] + \lambda (1 + R^2)^{-\alpha/2} \bar{\mu}^{\beta}(R) \|h(t)\|_{1-\beta}^2 \\ &\leq -\Lambda (1 + R^2)^{-\alpha/2} \mathcal{F}_2[h(t)] + \lambda C (1 + R^2)^{-\alpha/2} \bar{\mu}^{\beta}(R) \|h_0\|_{1-\beta}^2 \end{aligned} $$

for any $R > 0$ and all $t \geq 0$, and for some positive constant $\Lambda$. This yields, for all $t \geq 0$,

$$ \mathcal{F}_2[h(t)] \leq \exp\left(-\Lambda (1 + R^2)^{-\alpha/2} t\right) \mathcal{F}_2[h_0] + \frac{\lambda}{\Lambda} C \bar{\mu}^{\beta}(R) \|h_0\|_{1-\beta}^2. $$

Taking $R > 0$ such that $1 + R^2 = t^{2/(2+\alpha)}$, we obtain

$$ \forall t \geq 0, \quad \mathcal{F}_2[h(t)] \leq \exp\left(-\Lambda t^{\frac{2}{2+\alpha}}\right) \|h_0\|_{1-\beta}^2, $$

which completes the proof.

<figure><img src="image_10.png" /></figure>
This replaces the estimate of Lemma 4.2. We can then use (H2') and the above estimate to prove counterparts of estimates between Lemma 4.3 and Lemma 4.12 with $||h^\perp||$ and $||h||$ respectively replaced by $||h^\perp||_\star$ and $||h||_\star$. Then $\mathcal{F}_2$ defined in (4.39) using (4.20), with an appropriate choice of parameters $0 \ll \varepsilon_6 \ll \varepsilon_5 \ll \varepsilon_4 \ll \varepsilon_3 \ll \varepsilon_2 \ll \varepsilon_1 \ll 1$, is equivalent to $|| \cdot ||^2$ and satisfies

$$ \frac{d}{dt} \mathcal{F}_2[h] \leq -\kappa \overline{\mathcal{D}}_2[h] $$

for some constant $\kappa > 0$, where $\overline{\mathcal{D}}_2[h]$ is defined as $\mathcal{D}_2[h]$ in (4.40) with $||h^\perp||^2$ replaced by $||h^\perp||_\star^2$ in (4.22). This concludes the proof with

$$ \|f - F\|_{L^2(M^{-1})}^2 := \mathcal{F}_2[h] $$

since $\overline{\mathcal{D}}_2$ is equivalent to $|| \cdot ||_\star^2$.

<figure><img src="image_10.png" /></figure>

### B.3. Proof of Corollary B.2

Let $h_0 \in L^2(M^{1-\beta})$ and consider the solution $t \mapsto h(t)$ to the equation $\partial_t h = \mathcal{L}h$ with initial datum $h(0) = h_0$. Thanks to (H9'), there is some $C > 0$ such that

$$ \forall t \geq 0, \quad \|h(t)\|_{1-\beta} \leq C \|h_0\|_{1-\beta}. $$

We now observe that, for any $R > 0$, the following interpolation inequality holds

$$ \|g\|^2 \leq (1 + R^2)^{\alpha/2} \|g\|_{\star}^2 + \bar{\mu}^{\beta}(R) \|g\|_{1-\beta}^2 $$

with $\bar{\mu}(R) := (2\pi)^{-d/2} \|e^{-\phi}\|_{L^\infty(\mathbb{R}^d)} e^{-R^2/2}$. Therefore, thanks to (B.1) and using fact that $h \mapsto \|f - F\|_{L^2(M^{-1})}^2 = \mathcal{F}_2[h]$ is equivalent to $h \mapsto \|h\|^2$, one deduces that

$$ \begin{aligned} \frac{d}{dt} \mathcal{F}_2[h(t)] &\leq -\Lambda (1 + R^2)^{-\alpha/2} \mathcal{F}_2[h(t)] + \lambda (1 + R^2)^{-\alpha/2} \bar{\mu}^{\beta}(R) \|h(t)\|_{1-\beta}^2 \\ &\leq -\Lambda (1 + R^2)^{-\alpha/2} \mathcal{F}_2[h(t)] + \lambda C (1 + R^2)^{-\alpha/2} \bar{\mu}^{\beta}(R) \|h_0\|_{1-\beta}^2 \end{aligned} $$

for any $R > 0$ and all $t \geq 0$, and for some positive constant $\Lambda$. This yields, for all $t \geq 0$,

$$ \mathcal{F}_2[h(t)] \leq \exp\left(-\Lambda (1 + R^2)^{-\alpha/2} t\right) \mathcal{F}_2[h_0] + \frac{\lambda}{\Lambda} C \bar{\mu}^{\beta}(R) \|h_0\|_{1-\beta}^2. $$

Taking $R > 0$ such that $1 + R^2 = t^{2/(2+\alpha)}$, we obtain

$$ \forall t \geq 0, \quad \mathcal{F}_2[h(t)] \leq \exp\left(-\Lambda t^{\frac{2}{2+\alpha}}\right) \|h_0\|_{1-\beta}^2, $$

which completes the proof.

<figure><img src="image_11.png" /></figure>
### B.4. Comments and open questions

In order to apply Theorem B.1 to the linearized Boltzmann and Landau operators with very soft potentials, one has to establish (H9'), which is so far an open question. Instead of proving stretched exponential decay rates as in Corollary B.2, polynomial decay rates could also be achieved with (H9') replaced, for some $k > 0$ large enough, by

$$t \mapsto e^{t\mathcal{L}} \text{ is a strongly continuous uniformly bounded semigroup on } L^2(\mathcal{H}^k M^{-1}) \qquad (\text{H9''})$$

where $\mathcal{H}(x, v) = \phi(x) + \frac{1}{2}|v|^2 - \min_{\mathbb{R}^d} \phi$. Such a condition is also open in the case of the linearized Boltzmann and Landau operators with very soft potentials, but might be easier to prove in the spirit of [34, Appendix A].

## Appendix C. Examples and remarks

### C.1. Examples of collision operators

We list some examples of linear collision operators $\mathcal{C}$ satisfying the hypotheses of Theorem 1.1, in particular the spectral gap property (H1) and the bounded moment property (H2).

**Example C.1 (The full linear Boltzmann operator).** Consider

$$\mathcal{C}f := -(f - r_f M - m_f \cdot v M - e_f \mathfrak{E} M)$$

where $r_f, m_f$ and $e_f$ are defined by

$$r_f(t, x) := \left( \int_{\mathbb{R}^d} f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad (\text{local density}),$$

$$m_f(t, x) := \left( \int_{\mathbb{R}^d} v f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad (\text{local momentum}),$$

$$e_f(t, x) := \left( \int_{\mathbb{R}^d} \mathfrak{E}(v) f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad (\text{local thermal energy}).$$

By construction, $\mathcal{C}$ satisfies the spectral gap condition (H1) and since it is bounded, it satisfies also the bounded moment property (H2).

**Example C.2 (The linearized Boltzmann collision operator).** Consider

$$\mathcal{C}f := \int_{\mathbb{R}^d} \int_{\mathbb{Q}^{d-1}} (f' \mu_*' + f_*' \mu' - f \mu_* - f_* \mu) |v - v_*|^\gamma b(\theta) \, d\sigma \, dv_*$$

with the notation $f' = f(v')$, $f_* = f(v_*)$ and $f_*' = f(v_*')$ and

$$v' := \frac{v + v_*}{2} + \sigma \frac{|v - v_*|}{2}, \quad v_*' := \frac{v + v_*}{2} - \sigma \frac{|v - v_*|}{2}, \qquad (\text{C.1})$$
### B.4. Comments and open questions

In order to apply Theorem B.1 to the linearized Boltzmann and Landau operators with very soft potentials, one has to establish (H9'), which is so far an open question. Instead of proving stretched exponential decay rates as in Corollary B.2, polynomial decay rates could also be achieved with (H9') replaced, for some $k > 0$ large enough, by

$$t \mapsto e^{t\mathcal{L}} \text{ is a strongly continuous uniformly bounded semigroup on } L^2(\mathcal{H}^k M^{-1}) \qquad (\text{H9''})$$

where $\mathcal{H}(x, v) = \phi(x) + \frac{1}{2}|v|^2 - \min_{\mathbb{R}^d} \phi$. Such a condition is also open in the case of the linearized Boltzmann and Landau operators with very soft potentials, but might be easier to prove in the spirit of [34, Appendix A].

## Appendix C. Examples and remarks

### C.1. Examples of collision operators

We list some examples of linear collision operators $\mathcal{C}$ satisfying the hypotheses of Theorem 1.1, in particular the spectral gap property (H1) and the bounded moment property (H2).

**Example C.1 (The full linear Boltzmann operator).** Consider

$$\mathcal{C}f := -(f - r_f M - m_f \cdot v M - e_f \mathfrak{E} M)$$

where $r_f, m_f$ and $e_f$ are defined by

$$r_f(t, x) := \left( \int_{\mathbb{R}^d} f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad \text{(local) density,}$$

$$m_f(t, x) := \left( \int_{\mathbb{R}^d} v f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad \text{(local) momentum,}$$

$$e_f(t, x) := \left( \int_{\mathbb{R}^d} \mathfrak{E}(v) f(t, x, v) \, dv \right) e^{\phi(x)}, \qquad \text{(local) thermal energy.}$$

By construction, $\mathcal{C}$ satisfies the spectral gap condition (H1) and since it is bounded, it satisfies also the bounded moment property (H2).

**Example C.2 (The linearized Boltzmann collision operator).** Consider

$$\mathcal{C}f := \int_{\mathbb{R}^d} \int_{\mathbb{Q}^{d-1}} (f' \mu_*' + f_*' \mu' - f \mu_* - f_* \mu) |v - v_*|^\gamma b(\theta) \, d\sigma \, dv_*$$

with the notation $f' = f(v')$, $f_* = f(v_*)$ and $f_*' = f(v_*')$ and

$$v' := \frac{v + v_*}{2} + \sigma \frac{|v - v_*|}{2}, \quad v_*' := \frac{v + v_*}{2} - \sigma \frac{|v - v_*|}{2}, \qquad (\text{C.1})$$
and $\theta$ is the deviation angle defined by $\cos \theta := \frac{(v-v_*)}{|v-v_*|} \cdot \sigma$, and where $\gamma \in (-d, +\infty)$. We assume that $b$ is positive, smooth away from $\theta = 0$ and bounded by $b(\theta) \lesssim \theta^{-(d-1)-s}$ with $s \in [0, 2)$. This framework includes the short-range so-called hard spheres interactions, as well as the long-range so-called hard potentials and moderately soft potentials interactions. This operator satisfies the spectral gap property (H1) when $\gamma + s \ge 0$ but only satisfies the weaker coercivity property (H1') with $\alpha = \gamma + s$ when $\gamma + s < 0$ (see [2,35,37] for quantitative estimates). It is in general not bounded on $L^2(\mu^{-1})$. Polynomials multiplied by $\mu$ are however in the domain of $\mathcal{C}$ and it satisfies the boundedness property (H2).

**Example C.3** (The linearized Landau collision operator). With same convention as in Example C.2, consider $\mathcal{C}f = \mu Ch$ with $h = f/\mu$ and

$$ Ch := \nabla_v \cdot \left( \int_{\mathbb{R}^d} B_\gamma(v, v_*) (\nabla h - \nabla h_*) \mu \mu_* dv_* \right) $$

where the cross-section is defined by

$$ B_{\gamma}(v, v_{*}) := |v - v_{*}|^{\gamma+2} \left( \mathrm{Id} - \frac{v - v_{*}}{|v - v_{*}|} \otimes \frac{v - v_{*}}{|v - v_{*}|} \right) $$

with parameter $\gamma \in [-d, 1]$. This operator is non-local, of order 2 in velocity (of diffusive type) and therefore not bounded. It satisfies the spectral gap condition (H1) when $\gamma \in [-2, 1]$ but only the weaker coercivity property (H1') with $\alpha = \gamma + 2$ when $\gamma \in [-d, -2)$ (see, e.g., [2, 37] for constructive estimates). Again all polynomials in velocity multiplied by $\mu$ are in its domain and it satisfies the boundedness property (H2). Note that the main physical case, the linearisation of the so-called Landau-Coulomb collision operator (describing statistically collisions for a gas of electrons with Coulomb interactions) corresponds to $\gamma = -3$ in dimension $d = 3$ and is covered by our (extended) assumption (H1').

**Remark C.4.** Examples C.2 and C.3 are obtained after a linearization of the bilinear form associated with the original nonlinear collision kernel around the Gaussian $\mu$ and not around the Maxwellian $M$: when linearizing the full nonlinear inhomogeneous kinetic models around a Maxwellian $M$, one gets an additional term $\rho(x)$ in front of the collision operator that goes to zero at infinity. We have not considered this degeneracy in the present paper: it is likely to create significant difficulties since there is then no uniform-in-$x$ spectral gap for $\rho \in \mathcal{C}$.

## C.2. Examples of potentials

Let us discuss and illustrate the hypotheses (H5) and (H6) on the potential $\phi$. The bounded moment hypothesis (H6) is not restrictive. Functions like $\phi(x) = \frac{d+5}{2} \ln(1 + |x|^2) - Z_\phi$ which are very slowly increasing at infinity satisfy this hypothesis, as well as fast increasing ones like $\phi(x) = e^{|x|^4} - Z_\phi$ (here $Z_\phi$ is the constant of normalization of $e^{-\phi}$ in $L^1$). Regarding the Poincaré inequality (H5), many works have been devoted to the study of sufficient conditions in order to guarantee the existence of a spectral gap. Here are some examples.
and $\theta$ is the deviation angle defined by $\cos \theta := \frac{(v-v_*)}{|v-v_*|} \cdot \sigma$, and where $\gamma \in (-d, +\infty)$. We assume that $b$ is positive, smooth away from $\theta = 0$ and bounded by $b(\theta) \lesssim \theta^{-(d-1)-s}$ with $s \in [0, 2)$. This framework includes the short-range so-called hard spheres interactions, as well as the long-range so-called hard potentials and moderately soft potentials interactions. This operator satisfies the spectral gap property (H1) when $\gamma + s \ge 0$ but only satisfies the weaker coercivity property (H1') with $\alpha = \gamma + s$ when $\gamma + s < 0$ (see [2,35,37] for quantitative estimates). It is in general not bounded on $L^2(\mu^{-1})$. Polynomials multiplied by $\mu$ are however in the domain of $\mathcal{C}$ and it satisfies the boundedness property (H2).

**Example C.3** (The linearized Landau collision operator). With same convention as in Example C.2, consider $\mathcal{C}f = \mu Ch$ with $h = f/\mu$ and

$$ Ch := \nabla_v \cdot \left( \int_{\mathbb{R}^d} B_\gamma(v, v_*) (\nabla h - \nabla h_*) \mu \mu_* dv_* \right) $$

where the cross-section is defined by

$$ B_{\gamma}(v, v_{*}) := |v - v_{*}|^{\gamma+2} \left( \mathrm{Id} - \frac{v - v_{*}}{|v - v_{*}|} \otimes \frac{v - v_{*}}{|v - v_{*}|} \right) $$

with parameter $\gamma \in [-d, 1]$. This operator is non-local, of order 2 in velocity (of diffusive type) and therefore not bounded. It satisfies the spectral gap condition (H1) when $\gamma \in [-2, 1]$ but only the weaker coercivity property (H1') with $\alpha = \gamma + 2$ when $\gamma \in [-d, -2)$ (see, e.g., [2, 37] for constructive estimates). Again all polynomials in velocity multiplied by $\mu$ are in its domain and it satisfies the boundedness property (H2). Note that the main physical case, the linearisation of the so-called Landau-Coulomb collision operator (describing statistically collisions for a gas of electrons with Coulomb interactions) corresponds to $\gamma = -3$ in dimension $d = 3$ and is covered by our (extended) assumption (H1').

**Remark C.4.** Examples C.2 and C.3 are obtained after a linearization of the bilinear form associated with the original nonlinear collision kernel around the Gaussian $\mu$ and not around the Maxwellian $M$: when linearizing the full nonlinear inhomogeneous kinetic models around a Maxwellian $M$, one gets an additional term $\rho(x)$ in front of the collision operator that goes to zero at infinity. We have not considered this degeneracy in the present paper: it is likely to create significant difficulties since there is then no uniform-in-$x$ spectral gap for $\rho \in \mathcal{C}$.

## C.2. Examples of potentials

Let us discuss and illustrate the hypotheses (H5) and (H6) on the potential $\phi$. The bounded moment hypothesis (H6) is not restrictive. Functions like $\phi(x) = \frac{d+5}{2} \ln(1 + |x|^2) - Z_\phi$ which are very slowly increasing at infinity satisfy this hypothesis, as well as fast increasing ones like $\phi(x) = e^{|x|^4} - Z_\phi$ (here $Z_\phi$ is the constant of normalization of $e^{-\phi}$ in $L^1$). Regarding the Poincaré inequality (H5), many works have been devoted to the study of sufficient conditions in order to guarantee the existence of a spectral gap. Here are some examples.
**Example C.5.** The harmonic potential $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ satisfies the Poincaré inequality with constant $c_P = 1$. The inequality is equivalent to the spectral gap inequality for the operator $\Omega$ defined in (1.15). In the flat $L^2$ space, the change of unknown $u = v e^{-\phi/2}$ shows that the Poincaré inequality is also equivalent to the spectral gap inequality for the quantum harmonic oscillator operator $-\Delta_x + \frac{1}{4}|x|^2 - \frac{d}{2}$.

**Example C.6.** For a general $\phi$, the change of unknown $u = v e^{-\phi/2}$ yields the following Schrödinger-type operator

$$P_{\phi} = -\Delta_{x} + \frac{1}{4} |\nabla_{x} \phi|^{2} - \frac{1}{2} \Delta_{x} \phi.$$

According to the so-called Bakry-Emery theory (see for instance to [1]), there is a spectral gap as soon as the Hessian $\nabla_x^2\phi$ is uniformly strictly positive at infinity. When it is uniformly strictly positive everywhere the following estimate is available on the spectral gap $c_P$:

$$c_P \geq \frac{1}{2} \inf_{x \in \mathbb{R}^d} \lambda_1(\nabla_x^2 \phi)$$

where $\lambda_1(\nabla_x^2 \phi) > 0$ is the lowest eigenvalue of $\nabla_x^2 \phi$.

**Example C.7.** All potentials $\phi$ such that $P_{\phi}$ has compact resolvent satisfy the Poincaré inequality (H5). This happens in particular when

$$\lim_{|x| \to \infty} \left( \frac{1}{4} |\nabla_x \phi|^2 - \frac{1}{2} \Delta_x \phi \right) = +\infty,$$

which is implied for instance by the stronger assumption

$$\lim_{|x| \to \infty} |\nabla_x \phi| = +\infty, \quad \text{and} \quad \lim_{|x| \to \infty} \frac{\Delta_x \phi(x)}{|\nabla_x \phi(x)|^2} = 0. \qquad (C.2)$$

This is a standard result on Schrödinger operators, see for instance [38, Theorem XIII.67 p. 249], and 0 is then a simple discrete eigenvalue. The argument in the latter reference is not constructive, and for a simpler constructive argument we refer for instance to [46, Theorem A.1] or the IMS truncation method in [41].

**Example C.8.** Here is an exotic example of potential that does not satisfy (C.2) nor the Bakry-Émery criterion (uniform convexity of $\phi$) and for which the Poincaré inequality holds. Consider on $\mathbb{R}^2$

$$\phi(x, y) = x^2 (1 + y^2)^2 - Z_{\phi}$$

where $Z_{\phi}$ is the normalization constant so that $\rho = e^{-\phi}$ is a probability density. One can check that $P_{\phi}$ has a spectral gap, although $\phi$ is constant on the unbounded set $\{x = 0\}$.
**Example C.5.** The harmonic potential $\phi(x) = \frac{1}{2}|x|^2 + \frac{d}{2}\log(2\pi)$ satisfies the Poincaré inequality with constant $c_p = 1$. The inequality is equivalent to the spectral gap inequality for the operator $\Omega$ defined in (1.15). In the flat $L^2$ space, the change of unknown $u = v e^{-\phi/2}$ shows that the Poincaré inequality is also equivalent to the spectral gap inequality for the quantum harmonic oscillator operator $-\Delta_x + \frac{1}{4}|x|^2 - \frac{d}{2}$.

**Example C.6.** For a general $\phi$, the change of unknown $u = v e^{-\phi/2}$ yields the following Schrödinger-type operator

$$P_{\phi} = -\Delta_{x} + \frac{1}{4} |\nabla_{x} \phi|^{2} - \frac{1}{2} \Delta_{x} \phi.$$

According to the so-called Bakry-Emery theory (see for instance to [1]), there is a spectral gap as soon as the Hessian $\nabla_x^2\phi$ is uniformly strictly positive at infinity. When it is uniformly strictly positive everywhere the following estimate is available on the spectral gap $c_p$:

$$c_p \geq \frac{1}{2} \inf_{x \in \mathbb{R}^d} \lambda_1(\nabla_x^2 \phi)$$

where $\lambda_1(\nabla_x^2 \phi) > 0$ is the lowest eigenvalue of $\nabla_x^2 \phi$.

**Example C.7.** All potentials $\phi$ such that $P_{\phi}$ has compact resolvent satisfy the Poincaré inequality (H5). This happens in particular when

$$\lim_{|x| \to \infty} \left( \frac{1}{4} |\nabla_x \phi|^2 - \frac{1}{2} \Delta_x \phi \right) = +\infty,$$

which is implied for instance by the stronger assumption

$$\lim_{|x| \to \infty} |\nabla_x \phi| = +\infty, \quad \text{and} \quad \lim_{|x| \to \infty} \frac{\Delta_x \phi(x)}{|\nabla_x \phi(x)|^2} = 0. \qquad (C.2)$$

This is a standard result on Schrödinger operators, see for instance [38, Theorem XIII.67 p. 249], and 0 is then a simple discrete eigenvalue. The argument in the latter reference is not constructive, and for a simpler constructive argument we refer for instance to [46, Theorem A.1] or the IMS truncation method in [41].

**Example C.8.** Here is an exotic example of potential that does not satisfy (C.2) nor the Bakry-Émery criterion (uniform convexity of $\phi$) and for which the Poincaré inequality holds. Consider on $\mathbb{R}^2$

$$\phi(x, y) = x^2 (1 + y^2)^2 - Z_{\phi}$$

where $Z_{\phi}$ is the normalization constant so that $\rho = e^{-\phi}$ is a probability density. One can check that $P_{\phi}$ has a spectral gap, although $\phi$ is constant on the unbounded set $\{x = 0\}$.
### C.3. Change of coordinates

Let us discuss the reduction to the normalization (H7). Note that the formulas for $Ker \mathcal{C}$ are invariant by orthonormal change of coordinates in the velocity variable. By orthonormal change of coordinates in both the velocity and space variables, we can then reduce to the case when $\phi$ satisfies

$$ \langle \nabla_x^2 \phi \rangle = \begin{pmatrix} p_1^2 & 0 & 0 & \cdots & 0 \\ 0 & p_2^2 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & p_d^2 \end{pmatrix}. \qquad (C.3) $$

where we suppose without loss of generality that all $p_j$'s are positive. The analysis of the present paper can be adapted to this case, including the main Theorem 1.1, with the following changes. We define the set of adapted centred rotational modes compatible with $\phi$ as in (1.8):

$$ \mathcal{R}_{\phi} = \{(x, v) \mapsto (Ax \cdot v) \mathcal{M} : A \in \mathcal{R}_{\phi}\}. \qquad (C.4) $$

We then choose orthonormal coordinates $x = (x_1, x_2, \dots, x_d)$ such that $\partial_{x_j} \phi = p_j^2 x_j$ for some $p_j > 0$ if and only if $j \in I_{\phi} := \{d_{\phi} + 1, \dots, d\}$, and $x_j = 0$ for any $j \in I_{\phi}$ if $x \in E_{\phi}$ (the linear subspace defined in (1.9)). We define the set of harmonic directional modes by

$$ \mathcal{D}_{\phi} = \operatorname{Span} \{f_j^{-}(t, x, v), f_j^{+}(t, x, v)\}_{j \in I_{\phi}}, \qquad (C.5) $$

where

$$ \begin{aligned} f_j^{-}(t, x, v) &:= (p_j x_j \cos(p_j t) - v_j \sin(p_j t)) \mathcal{M}(x, v), \\ f_j^{+}(t, x, v) &:= (p_j x_j \sin(p_j t) + v_j \cos(p_j t)) \mathcal{M}(x, v). \end{aligned} $$

If $d_{\phi} = 0$ and for some $p > 0$, $p_j = p$ for all $j \in \{1, \dots, d\}$, we define the set of harmonic pulsating modes by

$$ \mathcal{P}_{\phi} = \operatorname{Span} \{f^{-}(t, x, v), f^{+}(t, x, v)\} $$

where

$$ \begin{aligned} f^{-}(t, x, v) &:= \left( \frac{1}{2} \left( |p x|^2 - |v|^2 \right) \cos(2 p t) - p x \cdot v \sin(2 p t) \right) \mathcal{M}(x, v), \\ f^{+}(t, x, v) &:= \left( \frac{1}{2} \left( |p x|^2 - |v|^2 \right) \sin(2 p t) + p x \cdot v \cos(2 p t) \right) \mathcal{M}(x, v). \end{aligned} $$

The functions in $\mathcal{R}_{\phi}$, $\mathcal{D}_{\phi}$ and $\mathcal{P}_{\phi}$ are special macroscopic modes of (1.1). With these definitions, the proof of Theorem 1.1 can be adapted to prove a hypocoercivity result taking into account all special macroscopic modes.
### C.3. Change of coordinates

Let us discuss the reduction to the normalization (H7). Note that the formulas for $Ker \mathcal{C}$ are invariant by orthonormal change of coordinates in the velocity variable. By orthonormal change of coordinates in both the velocity and space variables, we can then reduce to the case when $\phi$ satisfies

$$ \langle \nabla_x^2 \phi \rangle = \begin{pmatrix} p_1^2 & 0 & 0 & \cdots & 0 \\ 0 & p_2^2 & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & p_d^2 \end{pmatrix}. \qquad (C.3) $$

where we suppose without loss of generality that all $p_j$'s are positive. The analysis of the present paper can be adapted to this case, including the main Theorem 1.1, with the following changes. We define the set of adapted centred rotational modes compatible with $\phi$ as in (1.8):

$$ \mathcal{R}_{\phi} = \{(x, v) \mapsto (Ax \cdot v) \mathcal{M} : A \in \mathcal{R}_{\phi}\}. \qquad (C.4) $$

We then choose orthonormal coordinates $x = (x_1, x_2, \dots, x_d)$ such that $\partial_{x_j} \phi = p_j^2 x_j$ for some $p_j > 0$ if and only if $j \in I_{\phi} := \{d_{\phi} + 1, \dots, d\}$, and $x_j = 0$ for any $j \in I_{\phi}$ if $x \in E_{\phi}$ (the linear subspace defined in (1.9)). We define the set of harmonic directional modes by

$$ \mathcal{D}_{\phi} = \operatorname{Span} \{f_j^{-}(t, x, v), f_j^{+}(t, x, v)\}_{j \in I_{\phi}}, \qquad (C.5) $$

where

$$ \begin{aligned} f_j^{-}(t, x, v) &:= (p_j x_j \cos(p_j t) - v_j \sin(p_j t)) \mathcal{M}(x, v), \\ f_j^{+}(t, x, v) &:= (p_j x_j \sin(p_j t) + v_j \cos(p_j t)) \mathcal{M}(x, v). \end{aligned} $$

If $d_{\phi} = 0$ and for some $p > 0$, $p_j = p$ for all $j \in \{1, \dots, d\}$, we define the set of harmonic pulsating modes by

$$ \mathcal{P}_{\phi} = \operatorname{Span} \{f^{-}(t, x, v), f^{+}(t, x, v)\} $$

where

$$ \begin{aligned} f^{-}(t, x, v) &:= \left( \frac{1}{2} \left( |p x|^2 - |v|^2 \right) \cos(2 p t) - p x \cdot v \sin(2 p t) \right) \mathcal{M}(x, v), \\ f^{+}(t, x, v) &:= \left( \frac{1}{2} \left( |p x|^2 - |v|^2 \right) \sin(2 p t) + p x \cdot v \cos(2 p t) \right) \mathcal{M}(x, v). \end{aligned} $$

The functions in $\mathcal{R}_{\phi}$, $\mathcal{D}_{\phi}$ and $\mathcal{P}_{\phi}$ are special macroscopic modes of (1.1). With these definitions, the proof of Theorem 1.1 can be adapted to prove a hypocoercivity result taking into account all special macroscopic modes.
### C.4. Spectral interpretation

We have focused so far on real solutions to (1.1), which is natural since physical solutions (densities of probability) are real valued. By considering complex solutions, we can interpret the results in terms of the complex spectrum of the nonnegative operator

$$-\mathcal{L} = v \cdot \nabla_x - \nabla_x \phi \cdot \nabla_v - \mathcal{C}$$

in $L^2_{\mathcal{C}}(\mathcal{M}^{-1})$, the complexification of $L^2(\mathcal{M}^{-1})$. We consider $\phi$ as in (C.3). We can then describe precisely the spectrum of $-\mathcal{L}$ and obtain resolvent estimates in a half-plane that includes the imaginary axis. Notice first that 0 is in the spectrum of $-\mathcal{L}$ with associated eigenspace

$$\operatorname{Span}(\mathcal{M}) \oplus \operatorname{Span}(\mathcal{H} \mathcal{M}) \oplus \mathfrak{R}_{\phi, \mathcal{C}}$$

where $\mathfrak{R}_{\phi, \mathcal{C}}$ is the set of rotation modes as defined in (C.4) but extended to the corresponding $\mathcal{C}$-vector space. This set is then of (complex) dimension $2 + \dim(\mathfrak{R}_{\phi})$. Depending on the harmonicity of $\phi$ we have three cases which are summarized in Figure 1.

(a) Case with no harmonic modes ($d_{\phi} = d$). In this case $\phi$ has no harmonic directions and there no non-zero eigenvalue on the imaginary axis.

(b) Case with harmonic directional modes but no pulsating modes ($1 \le d_{\phi} \le d - 1$). In this case, the real vector space of functions $\mathcal{D}_{\phi}$ in (C.5) yields the complex set

$$\mathcal{D}_{\phi, \mathcal{C}} = \operatorname{Span}_{\mathcal{C}} \left\{ (p_j x_j - i v_j) e^{-i p_j t} \mathcal{M}(x, v), (p_j x_j + i v_j) e^{i p_j t} \mathcal{M}(x, v) \right\}_{j \in I_{\phi}}$$

where $I_{\phi} := \{d_{\phi} + 1, \dots, d\}$, to which we can associate the eigenfunctions of $(-\mathcal{L})$ corresponding to the eigenvalues $\mp i p_j$ and given by

$$(x, v) \mapsto f_j^{\pm}(x, v) = (p_j x_j \pm i v_j) \mathcal{M}(x, v).$$

(c) Case with harmonic directional and pulsating modes ($d_{\phi} = 0$). In this last case necessarily all $p_j$'s are equal to a common value $p > 0$ and $\phi(x) = \frac{1}{2}|p x|^2 + \frac{d}{2} \log(2\pi) - d \log(p)$. All possible harmonic directional modes exist, as well as all possible infinitesimal rotational modes $\mathfrak{R}_{\phi, \mathcal{C}}$ with $\mathfrak{R}_{\phi} = \mathfrak{R}_{\phi, d \times d}(\mathcal{C})$. The complexification of the set $\mathfrak{P}_{\phi}$ defined in (C.5) is

$$\mathfrak{P}_{\phi, \mathcal{C}} = \operatorname{Span} \{e^{2ip_t} f^+(x, v), e^{-2ip_t} f^-(x, v)\}$$

where $f^{\pm}(x, v) := [p x \cdot v \pm \frac{i}{2}(|p x|^2 - |v|^2)] \mathcal{M}(x, v)$ are eigenfunctions of $(-\mathcal{L})$ with eigenvalues $\pm 2 i p$.

The analysis of the paper can be extended to the complex Hilbertian space $L^2_{\mathcal{C}}(\mathcal{M}^{-1})$ with a set of special macroscopic modes defined by

$$S := \operatorname{Span}(\mathcal{M}) \oplus \operatorname{Span}(\mathcal{H} \mathcal{M}) \oplus \mathfrak{R}_{\phi, \mathcal{C}} \oplus \operatorname{Span} \{f_j^{\pm}\}_{j \in I_{\phi}} \oplus \operatorname{Span} \{f^{\pm}\}$$

where the $f_j$'s and the $f^{\pm}$'s are defined above (when $\phi$ has the relevant harmonicity). Let $S^{\perp}$ be the orthogonal of $S$ in $L^2_{\mathcal{C}}(\mathcal{M}^{-1})$. We note that since $\mathcal{L}$ is a real operator, both $S$
### C.4. Spectral interpretation

We have focused so far on real solutions to (1.1), which is natural since physical solutions (densities of probability) are real valued. By considering complex solutions, we can interpret the results in terms of the complex spectrum of the nonnegative operator

$$-\mathcal{L} = v \cdot \nabla_x - \nabla_x \phi \cdot \nabla_v - \mathcal{C}$$

in $L^2_{\mathbb{C}}(\mathcal{M}^{-1})$, the complexification of $L^2(\mathcal{M}^{-1})$. We consider $\phi$ as in (C.3). We can then describe precisely the spectrum of $-\mathcal{L}$ and obtain resolvent estimates in a half-plane that includes the imaginary axis. Notice first that 0 is in the spectrum of $-\mathcal{L}$ with associated eigenspace

$$\mathrm{Span}(\mathcal{M}) \oplus \mathrm{Span}(\mathcal{H} \mathcal{M}) \oplus \mathfrak{R}_{\phi, \mathbb{C}}$$

where $\mathfrak{R}_{\phi, \mathbb{C}}$ is the set of rotation modes as defined in (C.4) but extended to the corresponding $\mathbb{C}$-vector space. This set is then of (complex) dimension $2 + \dim(\mathfrak{R}_{\phi})$. Depending on the harmonicity of $\phi$ we have three cases which are summarized in Figure 1.

(a) Case with no harmonic modes ($d_{\phi} = d$). In this case $\phi$ has no harmonic directions and there no non-zero eigenvalue on the imaginary axis.

(b) Case with harmonic directional modes but no pulsating modes ($1 \le d_{\phi} \le d - 1$). In this case, the real vector space of functions $\mathcal{D}_{\phi}$ in (C.5) yields the complex set

$$\mathcal{D}_{\phi, \mathbb{C}} = \mathrm{Span}_{\mathbb{C}} \left\{ (p_j x_j - i v_j) e^{-i p_j t} \mathcal{M}(x, v), (p_j x_j + i v_j) e^{i p_j t} \mathcal{M}(x, v) \right\}_{j \in I_{\phi}}$$

where $I_{\phi} := \{d_{\phi} + 1, \dots, d\}$, to which we can associate the eigenfunctions of $(-\mathcal{L})$ corresponding to the eigenvalues $\mp i p_j$ and given by

$$(x, v) \mapsto f_j^{\pm}(x, v) = (p_j x_j \pm i v_j) \mathcal{M}(x, v).$$

(c) Case with harmonic directional and pulsating modes ($d_{\phi} = 0$). In this last case necessarily all $p_j$'s are equal to a common value $p > 0$ and $\phi(x) = \frac{1}{2}|p x|^2 + \frac{d}{2} \log(2\pi) - d \log(p)$. All possible harmonic directional modes exist, as well as all possible infinitesimal rotational modes $\mathfrak{R}_{\phi, \mathbb{C}}$ with $\mathfrak{R}_{\phi} = \mathfrak{R}^{\text{ skew}}_{d \times d}(\mathbb{C})$. The complexification of the set $\mathfrak{P}_{\phi}$ defined in (C.5) is

$$\mathfrak{P}_{\phi, \mathbb{C}} = \mathrm{Span} \left\{ e^{2i p t} f^{+}(x, v), e^{-2i p t} f^{-}(x, v) \right\}$$

where $f^{\pm}(x, v) := [p x \cdot v \pm \frac{i}{2}(|p x|^2 - |v|^2)] \mathcal{M}(x, v)$ are eigenfunctions of $(-\mathcal{L})$ with eigenvalues $\pm 2 i p$.

The analysis of the paper can be extended to the complex Hilbert space $L^2_{\mathbb{C}}(\mathcal{M}^{-1})$ with a set of special macroscopic modes defined by

$$S := \mathrm{Span}(\mathcal{M}) \oplus \mathrm{Span}(\mathcal{H} \mathcal{M}) \oplus \mathfrak{R}_{\phi, \mathbb{C}} \oplus \mathrm{Span} \{f_j^{\pm}\}_{j \in I_{\phi}} \oplus \mathrm{Span} \{f^{\pm}\}$$

where the $f_j$'s and the $f^{\pm}$'s are defined above (when $\phi$ has the relevant harmonicity). Let $S^{\perp}$ be the orthogonal of $S$ in $L^2_{\mathbb{C}}(\mathcal{M}^{-1})$. We note that since $\mathcal{L}$ is a real operator, both $S$
<figure><img src="image_11.png" /><figcaption>Fig. 1. Complex spectrum of $-\mathcal{L}$.</figcaption></figure>

and $S^{\perp}$ are stable by conjugation and therefore stable by $\mathcal{L}$ and $\mathcal{L}^*$. Using the Laplace transform, we obtain from Theorem 1.1 the following resolvent estimate for $-\mathcal{L}|_{S^{\perp}}$:

$$ \forall z \in \mathbb{C} \text{ with } \mathcal{R}(z) < \kappa, \quad \|(z\mathrm{Id} + \mathcal{L}|_{S^{\perp}})^{-1}\|_{\mathcal{B}(S^{\perp})} \le \frac{\tilde{C}}{\kappa - \mathcal{R}(z)} $$

where $\tilde{C}$ is an explicit constant depending on $\kappa$ and $C$ in Theorem 1.1 and $\|\cdot\|_{\mathcal{B}(S^{\perp})}$ stands for the operator norm on $S^{\perp}$. The provides the resolvent estimates in the left half-planes in Figure 1.
<figure><img src="image_12.png" /><figcaption>Fig. 1. Complex spectrum of $-\mathcal{L}$.</figcaption></figure>

and $S^{\perp}$ are stable by conjugation and therefore stable by $\mathcal{L}$ and $\mathcal{L}^*$. Using the Laplace transform, we obtain from Theorem 1.1 the following resolvent estimate for $-\mathcal{L}|_{S^{\perp}}$:

$$ \forall z \in \mathbb{C} \text{ with } \mathcal{R}(z) < \kappa, \quad \|(z\mathrm{Id} + \mathcal{L}|_{S^{\perp}})^{-1}\|_{\mathcal{B}(S^{\perp})} \le \frac{\tilde{C}}{\kappa - \mathcal{R}(z)} $$

where $\tilde{C}$ is an explicit constant depending on $\kappa$ and $C$ in Theorem 1.1 and $\|\cdot\|_{\mathcal{B}(S^{\perp})}$ stands for the operator norm on $S^{\perp}$. The provides the resolvent estimates in the left half-planes in Figure 1.
### C.5. Special macroscopic modes for the full nonlinear Boltzmann equation

The special macroscopic modes which minimize the entropy for the full nonlinear Boltzmann equation are the nonlinear counterparts to the linearized special macroscopic modes studied in the present paper. They appear for the first time in the literature in Boltzmann's paper [5] as mentioned in the introduction. The full nonlinear inhomogeneous Boltzmann equation is

$$ \partial_t F + v \cdot \nabla_x F - \nabla_x \phi \cdot \nabla_v F = \partial_t F + \mathcal{T} F = \mathcal{Q}(F, F) \quad (C.6) $$

where, with the classical notations $F' = F(v')$, $F_* = F(v_*)$ and $F'_* = F(v'_*)$ associated to elastic collisions $(v, v_*) \mapsto (v', v'_*)$, such that the microscopic conservation of momentum $v' + v'_* = v + v_*$ and energy $|v'|^2 + |v'_*|^2 = |v|^2 + |v_*|^2$ hold, the Boltzmann collision operator writes

$$ \mathcal{Q}(F, F) := \int_{\mathbb{R}^d} \int_{\mathbb{S}^{d-1}} \mathbf{B}(v - v_*, \sigma) (F'F_* - FF_*) \, d\sigma \, dv_* . $$

Here $\mathbf{B} \ge 0$ is the cross-section. We refer to [10] for more details. Let us assume the normalization (H7) on $\phi$. We consider the functions in the space $\mathcal{S}$ of special macroscopic modes generated by

▷ the set $\mathfrak{R}_\phi$ of rotation modes compatible with $\phi$ if $\phi$ admits any,

▷ the set $\mathfrak{D}_\phi$ of harmonic directional modes if $\phi$ has harmonic directions,

▷ the set $\mathfrak{P}_\phi$ of harmonic pulsating modes if $\phi$ is fully harmonic.

For any $f \in \mathcal{S}$, the function $F(t, x, v) := e^{h(t,x,v)} \mathcal{M}(x, v)$ with $h = f/\mathcal{M}$ is a time-periodic solution to (C.6). Indeed $h(t, x, \cdot)$ is a linear combination of $1, v_i, i \in \{1, \dots, d\}$ and $|v|^2$ for each $t, x$, and therefore the microscopic conservation of momentum and energy imply

$$ \forall t, x, v, \sigma, \quad h(t, x, v') + h(t, x, v_*) = h(t, x, v) + h(t, x, v_*) $$

where the four velocities $v, v_*, v', v'_*$ satisfy (C.1). This proves the identity $e^{h'} \mathcal{M}' e^{h_*} \mathcal{M}_* = e^h \mathcal{M} e^{h_*} \mathcal{M}_*$ and thus $\mathcal{Q}(e^h \mathcal{M}, e^h \mathcal{M}) = 0$. Finally we obtain $\mathcal{T}(e^h \mathcal{M}) = \mathcal{T}(e^h) \mathcal{M} + e^h \mathcal{T}(\mathcal{M}) = e^h [\mathcal{T}(h) \mathcal{M} + \mathcal{T}(\mathcal{M})] = 0$, where we have used that $\mathcal{T}$ is a first order operator and $\mathcal{T}(\mathcal{M}) = \mathcal{T}(h) = 0$ as calculated before.

**Acknowledgements.** The authors warmly thank L. Desvillettes, S. Vũ Ngoc, C. Cheverry and M. Rodrigues for fruitful discussions and remarks which led to the observations of Appendices C.4 and C.5.

**Funding.** KC and JD have been partially supported by the Project EFI (ANR-17-CE40-0030). FH benefits from the support of the France 2030 framework programme, through the Centre Henri Lebesgue Mathematical Center. CM is partially supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme MAFRAN (grant agreement No. 726386).
### C.5. Special macroscopic modes for the full nonlinear Boltzmann equation

The special macroscopic modes which minimize the entropy for the full nonlinear Boltzmann equation are the nonlinear counterparts to the linearized special macroscopic modes studied in the present paper. They appear for the first time in the literature in Boltzmann’s paper [5] as mentioned in the introduction. The full nonlinear inhomogeneous Boltzmann equation is

$$ \partial_t F + v \cdot \nabla_x F - \nabla_x \phi \cdot \nabla_v F = \partial_t F + \mathcal{T} F = \mathcal{Q}(F, F) \quad (C.6) $$

where, with the classical notations $F' = F(v')$, $F_* = F(v_*)$ and $F'_* = F(v'_*)$ associated to elastic collisions $(v, v_*) \mapsto (v', v'_*)$, such that the microscopic conservation of momentum $v' + v'_* = v + v_*$ and energy $|v'|^2 + |v'_*|^2 = |v|^2 + |v_*|^2$ hold, the Boltzmann collision operator writes

$$ \mathcal{Q}(F, F) := \int_{\mathbb{R}^d} \int_{\mathbb{S}^{d-1}} \mathbf{B}(v - v_*, \sigma) (F'F_* - FF_*) \, d\sigma \, dv_* . $$

Here $\mathbf{B} \ge 0$ is the cross-section. We refer to [10] for more details. Let us assume the normalization (H7) on $\phi$. We consider the functions in the space $\mathcal{S}$ of special macroscopic modes generated by

▷ the set $\mathfrak{R}_\phi$ of rotation modes compatible with $\phi$ if $\phi$ admits any,

▷ the set $\mathfrak{D}_\phi$ of harmonic directional modes if $\phi$ has harmonic directions,

▷ the set $\mathfrak{P}_\phi$ of harmonic pulsating modes if $\phi$ is fully harmonic.

For any $f \in \mathcal{S}$, the function $F(t, x, v) := e^{h(t,x,v)} \mathcal{M}(x, v)$ with $h = f/\mathcal{M}$ is a time-periodic solution to (C.6). Indeed $h(t, x, \cdot)$ is a linear combination of $1, v_i, i \in \{1, \dots, d\}$ and $|v|^2$ for each $t, x$, and therefore the microscopic conservation of momentum and energy imply

$$ \forall t, x, v, \sigma, \quad h(t, x, v') + h(t, x, v_*) = h(t, x, v) + h(t, x, v_*) $$

where the four velocities $v, v_*, v', v'_*$ satisfy (C.1). This proves the identity $e^{h'} \mathcal{M}' e^{h_*} \mathcal{M}'_* = e^h \mathcal{M} e^{h_*} \mathcal{M}_*$ and thus $\mathcal{Q}(e^h \mathcal{M}, e^h \mathcal{M}) = 0$. Finally we obtain $\mathcal{T}(e^h \mathcal{M}) = \mathcal{T}(e^h) \mathcal{M} + e^h \mathcal{T}(\mathcal{M}) = e^h [\mathcal{T}(h) \mathcal{M} + \mathcal{T}(\mathcal{M})] = 0$, where we have used that $\mathcal{T}$ is a first order operator and $\mathcal{T}(\mathcal{M}) = \mathcal{T}(h) = 0$ as calculated before.

**Acknowledgements.** The authors warmly thank L. Desvillettes, S. Vũ Ngoc, C. Cheverry and M. Rodrigues for fruitful discussions and remarks which led to the observations of Appendices C.4 and C.5.

**Funding.** KC and JD have been partially supported by the Project EFI (ANR-17-CE40-0030). FH benefits from the support of the France 2030 framework programme, through the Centre Henri Lebesgue Mathematical Center. CM is partially supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme MAFRAN (grant agreement No. 726386).
References

[1] Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348, Springer, Cham (2014) MR 3155209

[2] Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana 21, 819–841 (2005) MR 2231011

[3] Bardos, C., Golse, F., Levermore, C. D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46, 667–753 (1993) MR 1213991

[4] Binney, J., Tremaine, S.: Galactic dynamics. Princeton university press (2011)

[5] Boltzmann, L.: Über die Aufstellung und Integration der Gleichungen, welche die Molekularbewegung in Gasen bestimmen. In: Wissenschaftliche Abhandlungen von L. Boltzmann, vol. 2, Barth, Leipzig, 55–102 (1876) (1909)

[6] Bosi, R., Cáceres, M. J.: The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria. J. Stat. Phys. 136, 297–330 (2009) MR 2525248

[7] Briant, M., Merino-Aceituno, S., Mouhot, C.: From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight. Anal. Appl. (Singap.) 17, 85–116 (2019) MR 3894734

[8] Calderón, A.-P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. U.S.A. 69, 1185–1187 (1972) MR 298480

[9] Carrapatoso, K., Dolbeault, J., Hérau, F., Mischler, S., Mouhot, C.: Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators. Archive for Rational Mechanics and Analysis 243, 1565–1596 (2022) MR 4381147

[10] Cercignani, C.: The Boltzmann equation and its applications. Applied Mathematical Sciences 67, Springer-Verlag, New York (1988) MR 1313028

[11] Desvillettes, L., Villani, C.: On a variant of Korn's inequality arising in statistical mechanics. ESAIM Control Optim. Calc. Var. 8, 603–619 (electronic) (2002) MR 1932965

[12] Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159, 245–316 (2005) MR 2116276

[13] Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Acad. Sci. Paris, Ser. I 347, 511–516 (2009)

[14] Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Amer. Math. Soc. 367, 3807–3828 (2015) MR 3324910

[15] Duan, R.: Hypocoercivity of linear degenerately dissipative kinetic equations. Nonlinearity 24, 2165–2189 (2011) MR 2813582

[16] Duan, R., Li, W.-X.: Hypocoercivity for the linear Boltzmann equation with confining forces. J. Stat. Phys. 148, 306–324 (2012) MR 2966364

[17] Grad, H.: On Boltzmann's *H*-theorem. J. Soc. Indust. Appl. Math. 13, 259–277 (1965) MR 0180278

[18] Gualdani, M. P., Mischler, S., Mouhot, C.: Factorization of non-symmetric operators and exponential *H*-theorem. Mém. Soc. Math. Fr. (N.S.) 153, 137 (2017) MR 3779780

[19] Guéry-Odelin, D., Muga, J. G., Ruiz-Montero, M. J., Trizac, E.: Nonequilibrium solutions of the Boltzmann equation under the action of an external force. Phys. Rev. Lett. 112, 180602 (2014)

[20] Guo, Y.: The Landau equation in a periodic box. Comm. Math. Phys. 231, 391–434 (2002) MR MR1946444

[21] Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55, 1104–1135 (2002) MR MR1908664
References

[1] Bakry, D., Gentil, I., Ledoux, M.: Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348, Springer, Cham (2014) MR 3155209

[2] Baranger, C., Mouhot, C.: Explicit spectral gap estimates for the linearized Boltzmann and Landau operators with hard potentials. Rev. Mat. Iberoamericana 21, 819–841 (2005) MR 2231011

[3] Bardos, C., Golse, F., Levermore, C. D.: Fluid dynamic limits of kinetic equations. II. Convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math. 46, 667–753 (1993) MR 1213991

[4] Binney, J., Tremaine, S.: Galactic dynamics. Princeton university press (2011)

[5] Boltzmann, L.: Über die Aufstellung und Integration der Gleichungen, welche die Molekularbewegung in Gasen bestimmen. In: Wissenschaftliche Abhandlungen von L. Boltzmann, vol. 2, Barth, Leipzig, 55–102 (1876) (1909)

[6] Bosi, R., Cáceres, M. J.: The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria. J. Stat. Phys. 136, 297–330 (2009) MR 2525248

[7] Briant, M., Merino-Aceituno, S., Mouhot, C.: From Boltzmann to incompressible Navier-Stokes in Sobolev spaces with polynomial weight. Anal. Appl. (Singap.) 17, 85–116 (2019) MR 3894734

[8] Calderón, A.-P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. U.S.A. 69, 1185–1187 (1972) MR 298480

[9] Carrapatoso, K., Dolbeault, J., Hérau, F., Mischler, S., Mouhot, C.: Weighted Korn and Poincaré-Korn inequalities in the Euclidean space and associated operators. Archive for Rational Mechanics and Analysis 243, 1565–1596 (2022) MR 4381147

[10] Cercignani, C.: The Boltzmann equation and its applications. Applied Mathematical Sciences 67, Springer-Verlag, New York (1988) MR 1313028

[11] Desvillettes, L., Villani, C.: On a variant of Korn's inequality arising in statistical mechanics. ESAIM Control Optim. Calc. Var. 8, 603–619 (electronic) (2002) MR 1932965

[12] Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Invent. Math. 159, 245–316 (2005) MR 2116276

[13] Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for kinetic equations with linear relaxation terms. C. R. Acad. Sci. Paris, Ser. I 347, 511–516 (2009)

[14] Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. Trans. Amer. Math. Soc. 367, 3807–3828 (2015) MR 3324910

[15] Duan, R.: Hypocoercivity of linear degenerately dissipative kinetic equations. Nonlinearity 24, 2165–2189 (2011) MR 2813582

[16] Duan, R., Li, W.-X.: Hypocoercivity for the linear Boltzmann equation with confining forces. J. Stat. Phys. 148, 306–324 (2012) MR 2966364

[17] Grad, H.: On Boltzmann's *H*-theorem. J. Soc. Indust. Appl. Math. 13, 259–277 (1965) MR 0180278

[18] Gualdani, M. P., Mischler, S., Mouhot, C.: Factorization of non-symmetric operators and exponential *H*-theorem. Mém. Soc. Math. Fr. (N.S.) 153, 137 (2017) MR 3779780

[19] Guéry-Odelin, D., Muga, J. G., Ruiz-Montero, M. J., Trizac, E.: Nonequilibrium solutions of the Boltzmann equation under the action of an external force. Phys. Rev. Lett. 112, 180602 (2014)

[20] Guo, Y.: The Landau equation in a periodic box. Comm. Math. Phys. 231, 391–434 (2002) MR MR1946444

[21] Guo, Y.: The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55, 1104–1135 (2002) MR MR1908664
[22] Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. **153**, 593–630 (2003) MR MR2000470

[23] Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. **59**, 626–687 (2006) MR MR2172804

[24] Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. **169**, 89–117 (2003) MR MR2005637

[25] Helffer, B., Nier, F.: Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians. Lecture Notes in Mathematics 1862, Springer-Verlag, Berlin (2005) MR MR2130405

[26] Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. **46**, 349–359 (2006) MR MR2215889

[27] Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. **171**, 151–218 (2004) MR MR2034753

[28] Herda, M., Rodrigues, L. M.: Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit. J. Stat. Phys. **170**, 895–931 (2018) MR MR3767000

[29] Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. **119**, 147–171 (1967) MR MR222474

[30] Kawashima, S., Shizuta, Y.: On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J. (2) **40**, 449–464 (1988) MR MR957056

[31] Knauf, A.: Mathematical physics: classical mechanics. Unitext 109, Springer-Verlag, Berlin (2018) MR MR3752660

[32] Korn, A.: Die Eigenschwingungen eines elastischen Körpers mit ruhender Oberfläche. Akad. der Wissenschaften, Munich, Math. phys. KI. **36**, 351 (1906)

[33] Korn, A.: Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Krak. Anz., 705-724 (1909) (1909)

[34] Mischler, S., Mouhot, C.: Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. Arch. Ration. Mech. Anal. **221**, 677–723 (2016) MR MR3488535

[35] Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Comm. Partial Differential Equations **31**, 1321–1348 (2006) MR MR2254617

[36] Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity **19**, 969–998 (2006) MR MR2214953

[37] Mouhot, C., Strain, R. M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) **87**, 515–535 (2007) MR MR2322149

[38] Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978) MR MR0493421

[39] Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quart. Appl. Math. **62**, 163–179 (2004) MR MR2032577

[40] Sideris, T. C., Thomases, B., Wang, D.: Long time behavior of solutions to the 3D compressible Euler equations with damping. Comm. Partial Differential Equations **28**, 795–816 (2003) MR MR1978315

[41] Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) **38**, 295–308 (1983) MR MR708966

[42] Spivak, M.: Physics for mathematicians—mechanics I. Publish or Perish, Inc., Houston, TX (2010) MR MR2761185
[22] Guo, Y.: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. **153**, 593–630 (2003) MR MR2000470

[23] Guo, Y.: Boltzmann diffusive limit beyond the Navier-Stokes approximation. Comm. Pure Appl. Math. **59**, 626–687 (2006) MR MR2172804

[24] Hanouzet, B., Natalini, R.: Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. Arch. Ration. Mech. Anal. **169**, 89–117 (2003) MR MR2005637

[25] Helffer, B., Nier, F.: Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians. Lecture Notes in Mathematics 1862, Springer-Verlag, Berlin (2005) MR MR2130405

[26] Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. **46**, 349–359 (2006) MR MR2215889

[27] Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. **171**, 151–218 (2004) MR MR2034753

[28] Herda, M., Rodrigues, L. M.: Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit. J. Stat. Phys. **170**, 895–931 (2018) MR MR3767000

[29] Hörmander, L.: Hypoelliptic second order differential equations. Acta Math. **119**, 147–171 (1967) MR MR222474

[30] Kawashima, S., Shizuta, Y.: On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J. (2) **40**, 449–464 (1988) MR MR957056

[31] Knauf, A.: Mathematical physics: classical mechanics. Unitext 109, Springer-Verlag, Berlin (2018) MR MR3752660

[32] Korn, A.: Die Eigenschwingungen eines elastischen Körpers mit ruhender Oberfläche. Akad. der Wissenschaften, Munich, Math. phys. KI. **36**, 351 (1906)

[33] Korn, A.: Über einige Ungleichungen, welche in der Theorie der elastischen und elektrischen Schwingungen eine Rolle spielen. Krak. Anz., 705-724 (1909) (1909)

[34] Mischler, S., Mouhot, C.: Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation. Arch. Ration. Mech. Anal. **221**, 677–723 (2016) MR MR3488535

[35] Mouhot, C.: Explicit coercivity estimates for the linearized Boltzmann and Landau operators. Comm. Partial Differential Equations **31**, 1321–1348 (2006) MR MR2254617

[36] Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity **19**, 969–998 (2006) MR MR2214953

[37] Mouhot, C., Strain, R. M.: Spectral gap and coercivity estimates for linearized Boltzmann collision operators without angular cutoff. J. Math. Pures Appl. (9) **87**, 515–535 (2007) MR MR2322149

[38] Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978) MR MR0493421

[39] Ruggeri, T., Serre, D.: Stability of constant equilibrium state for dissipative balance laws system with a convex entropy. Quart. Appl. Math. **62**, 163–179 (2004) MR MR2032577

[40] Sideris, T. C., Thomases, B., Wang, D.: Long time behavior of solutions to the 3D compressible Euler equations with damping. Comm. Partial Differential Equations **28**, 795–816 (2003) MR MR1978315

[41] Simon, B.: Semiclassical analysis of low lying eigenvalues. I. Nondegenerate minima: asymptotic expansions. Ann. Inst. H. Poincaré Sect. A (N.S.) **38**, 295–308 (1983) MR MR708966

[42] Spivak, M.: Physics for mathematicians—mechanics I. Publish or Perish, Inc., Houston, TX (2010) MR MR2761185
[43] Strain, R. M., Guo, Y.: Almost exponential decay near Maxwellian. Comm. Partial Differential Equations **31**, 417–429 (2006) MR [2209761](https://mathSciDirect.com/doi/abs/10.1016/j.cdv.2006.06.006)

[44] Uhlenbeck, G. E., Ford, G. W.: Lectures in statistical mechanics. With an appendix on quantum statistics of interacting particles by E.M.Montroll. Lectures in Applied Mathematics 1, American Mathematical Society, Providence (1963)

[45] Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 71–305 (2002) MR [1942465](https://mathSciDirect.com/abs/10.1016/bs.hfd.2001.08.003)

[46] Villani, C.: Hypocoercivity. Mem. Amer. Math. Soc. **202**, iv+141 (2009) MR [2562709](https://mathSciDirect.com/abs/10.1090/ memory/2009/202/iv/141)
[43] Strain, R. M., Guo, Y.: Almost exponential decay near Maxwellian. Comm. Partial Differential Equations **31**, 417–429 (2006) MR MR2209761

[44] Uhlenbeck, G. E., Ford, G. W.: Lectures in statistical mechanics. With an appendix on quantum statistics of interacting particles by E.M.Montroll. Lectures in Applied Mathematics 1, American Mathematical Society, Providence (1963)

[45] Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 71–305 (2002) MR 1942465

[46] Villani, C.: Hypocoercivity. Mem. Amer. Math. Soc. **202**, iv+141 (2009) MR 2562709