Armand Bernou
email: armand.bernou@uniroma1.it

ASYMPTOTIC BEHAVIOR OF DEGENERATE LINEAR KINETIC EQUATIONS WITH NON-ISOTHERMAL BOUNDARY CONDITIONS

Keywords: 2020 Mathematics Subject Classication: 35B40, 35Q20 (82C40, 82D05) Degenerate linear Boltzmann equation, non-isothermal boundaries, steady state, Maxwell boundary condition, Cercignani-Lampis boundary condition, long-time behavior of kinetic PDEs, spacedependent relaxation model

     ∂ t f + v • ∇ x f = Cf, in R + × G, γ -f = Kγ + f, on R + × Σ -, f |t=0 = f 0 , in G, (1)
with the notations G := Ω×R d , and, denoting by n x the unit outward normal vector at x ∈ ∂Ω,

Σ := ∂Ω × R d , Σ ± := (x, v) ∈ Σ, ±(v • n x ) > 0 .
In [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF], the unknown function f = f (t, x, v) is the so-called distribution function. The quantity f (t, x, v)dvdx can be understood as the (non-negative) density at time t of particles whose positions are close to x and velocities close to v. We will study [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] in a L 1 framework, and we denote by γ ± f the trace of f on Σ ± .

1.2. The collision operator. We consider the linear degenerate Boltzmann equation. The corresponding collision operator C is dened, for all f :

G → R, for (x, v) ∈ Ω × R d , by Cf (x, v) = R d k(x, v ′ , v)f (x, v ′ ) -k(x, v, v ′ )f (x, v) dv ′ ,
see below the precise assumptions made on the non-negative function k and on f to make sense of this integral. The so-called collision kernel, k, describes the interactions between the particles and the background. We emphasize that k is modulated in space. Concrete examples of k, including the BGK model and the (non-degenerate) linear Boltzmann model, are presented in Section 2. We may split this collision operator, as

Cf (x, v) = C + f (x, v) + C -f (x, v),
where the gain and loss terms are given respectively by

C + f (x, v) = R d k(x, v ′ , v)f (x, v ′ ) dv ′ , C -f (x, v) = - R d k(x, v, v ′ ) dv ′ f (x, v).
By symmetry, note that the following equality formally holds ∀x ∈ Ω,

R d R d k(x, v ′ , v)f (x, v ′ ) -k(x, v, v ′ )f (x, v) dv ′ dv = 0. (2)
1.3. Boundary conditions. In this paper we model the interaction between the particles and the wall boundary by either the Cercignani-Lampis boundary condition or the Maxwell boundary condition, both with varying temperature. Set, for all x ∈ ∂Ω,

Σ x ± := v ∈ R d , (x, v) ∈ Σ ± .
The boundary operator K is dened, for ϕ supported on (0, ∞) × Σ + , for (t, x, v) belonging to (0, ∞) × Σ -and assuming that ϕ(t, x,

•) ∈ L 1 (Σ x + , R(v ′ → v; x)|v ′ • n x |dv ′ ), by
Kϕ(t, x, v) = Σ x + ϕ(t, x, u) R(u → v; x) |u • n x | du, (3)
with two possible choices for the kernel R(u → v; x) :

• The Cercignani-Lampis boundary condition (CLBC). In this case, R is given, for

x ∈ ∂Ω, u ∈ Σ x + , v ∈ Σ x -, by R(u → v; x) := 1 θ(x)r ⊥ 1 (2πθ(x)r ∥ (2 -r ∥ )) d-1 2 exp - |v ⊥ | 2 2θ(x)r ⊥ - (1 -r ⊥ )|u ⊥ | 2 2θ(x)r ⊥ (4) × I 0 (1 -r ⊥ ) 1 2 u ⊥ • v ⊥ θ(x)r ⊥ exp - |v ∥ -(1 -r ∥ )u ∥ | 2 2θ(x)r ∥ (2 -r ∥ ) ,
with the following notations:

v ⊥ := (v • n x )n x , v ∥ := v -v ⊥ , u ⊥ := (u • n x )n x , u ∥ = u -u ⊥ ,
where I 0 is the modied Bessel function given, for all y ∈ R, by I 0 (y) := 1 π π 0 exp y cos ϕ dϕ, [START_REF]Transport through Diusive and Nondiusive Regions, Embedded Objects, and Clear Layers[END_REF] and where θ(x) > 0 is the wall temperature at x ∈ ∂Ω. The coecients r ⊥ ∈ (0, 1] and r ∥ ∈ (0, 2) are the two accommodation coecients (normal and tangential) at the wall. The value v ⊥ is the normal component of the velocity v at the boundary, while v ∥ is the tangential component. The same interpretation is of course valid for u.

We will heavily use the normalization property, see [START_REF] Chen | Cercignani-Lampis Boundary in the Boltzmann Theory[END_REF]Lemma 10], which, with our notation for R, writes, for all (x, u) ∈ Σ + ,

Σ x - R(u → v; x) |v • n x | dv = 1. (6)
Combined with the symmetry property [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF], this condition will ensure the conservation of mass, as well as the L 1 contraction property of the associated semigroup.

• The Maxwell boundary condition (MBC). For (x, v) ∈ ∂Ω × R d , we set

η x (v) := v -2(v • n x )n x . (7)
In this case, R is given, for x ∈ ∂Ω, u ∈ Σ x + , v ∈ Σ x -, by the following formula

R(u → v; x) = β(x)M (x, v) + (1 -β(x))δ ηx(v) (u) 1 |v • n x | , (8)
where δ y is the Dirac Delta measure at y ∈ R d , where β : ∂Ω → [0, 1] is the accommodation coecient in this setting, and where M is dened on ∂Ω × R d by

M (x, v) := 1 θ(x)(2πθ(x)) d-1 2 e - |v| 2
2θ(x) . (9) 1.4. Assumptions and main results. We denote by L p (E; F ), 1 ≤ p ≤ ∞ the usual L p spaces of applications from E with values in the Banach space F , endowed with the usual norms. We simply write L p (E) when F = R. We present rst our hypotheses regarding the boundary condition:

Hypothesis 1. The boundary operator is dened by [START_REF] Arridge | Optical Tomography in Medical Imaging[END_REF], with the reection operator R given either by [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] case (CLBC): equation ( 4) with (r ⊥ , r ∥ ) ∈ (0, 1] × (0, 2);

(2) case (MBC): equation [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF] with β ≥ β 0 for some 0 < β 0 ≤ 1 on ∂Ω. In both cases, θ : ∂Ω → R * + is continuous.

Regarding the collision operator, we make the following assumptions:

Hypothesis 2.

(

) k ∈ L ∞ (Ω × R d × R d ; R + ) with k ∞ := sup (x,v,v ′ )∈G×R d |k(x, v, v ′ )|; 1
(2) there exist δ k ∈ (0, 1 2 ), M δ k > 0 such that for all

x ∈ Ω, v ∈ R d , R d k(x, v, v ′ )|v ′ | 2δ k dv ′ ≤ M δ k ;
(3) there exists σ ∈ L ∞ (Ω; R + ) such that for all

x ∈ Ω, v ∈ R d , R d k(x, v, v ′ ) dv ′ = σ(x).
We set σ ∞ := ∥σ∥ ∞ in the whole paper. To study the long-time behavior of (1), we will distinguish between two regimes. We prove the existence of a steady state in both cases, however the rates of convergence dier. In the rst setting, only Hypotheses 1 and 2 are assumed. We prove that the rate of convergence is then bounded from above by the (optimal) polynomial rate of (1 + t) -d derived for the free-transport equation in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF][START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF][START_REF] Bernou | A Coupling Approach for the Convergence to Equilibrium for a Collisionless Gas[END_REF]. In the second framework, σ is almost everywhere bounded from below by a positive constant, and exponential convergence towards the steady state is derived. Hypothesis 3. There exists σ 0 > 0 such that for almost all x ∈ Ω, σ(x) ≥ σ 0 .

Ultimately, our upper bounds rely on applications of Harris' theorems, in both the exponential case and the sub-exponential one. Accordingly, we obtain convergence results in the L 1 norm depending on some weighted L 1 norm of the initial data.

To dene our weighted norms, we introduce the function

τ (x, v) := inf{t > 0 : x + tv ∈ ∂Ω} in G ∪ Σ -, 0 in Σ + ∪ Σ 0 , (10)
where Σ 0 = {(x, v) ∈ Σ, v • n x = 0}. The weights considered in this paper will take the following guise: for all (x, v) ∈ Ḡ, the closure of G, for d(Ω) the diameter of Ω (see Subsection 1.8 below)

m α (x, v) = e 2 + d(Ω) |v|c 4 -τ (x, -v) + |v| 2δ α (11)
for 0 < δ < δ k d that will be xed from now on (see Hypothesis 2 for the denition of δ k ), for various α ∈ (0, d) and for c 4 ∈ (0, 1) a constant such that (1 -c 4 ) 4 = 1 -β 0 (see Hypothesis 1). It is to be understood that any value c 4 ∈ (0, 1) can be considered for the case (CLBC) and for the case (MBC) when β ≡ 1. The reader may consider c 4 = 1 2 in the whole paper in those cases.

Remark 1. The form of the weights m α may appear cumbersome at rst sight. They are slight modications of the natural weights of the form (1 + τ (x, v) + |v| 2δ ) α used in [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]: this change from τ (x, v) to d(Ω) |v|c 4 -τ (x, -v) allows to also treat the Maxwell boundary condition in a unied framework.

For all f ∈ L 1 (G), we use the notation

⟨f ⟩ := G f (x, v) dv dx.
We write ∥ • ∥ L 1 for the norm of L 1 (G), and for all w : G → [1, ∞), we set

L 1 w (G) := f ∈ L 1 (G), G |f (x, v)|w(x, v) dv dx < ∞ and ∀f ∈ L 1 w (G), ∥f ∥ w := ∥f w∥ L 1 .
After proving that the problem (1) is well-posed under Hypotheses 1 and 2, see Theorem 13, we introduce the semigroup (S t ) t≥0 such that, for all f ∈ L 1 (G), for all t > 0, S t f is the unique solution of (1) at time t > 0 belonging to L 1 (G). Our main results are written at this semigroup level. Throughout the paper, the constants C, κ > 0 are independent of time and initial data and are allowed to change from line to line. We sometimes write subscripts to emphasize dependencies, for instance C p if C depends on some parameter p.

Theorem 2. Assume that Hypotheses 1 and 2 hold. Then for all p ∈ (0, d), there exists a constant C > 0 such that for all t ≥ 0, for all f, g in L 1 mp (G) with ⟨f ⟩ = ⟨g⟩, there holds:

S t f -S t g L 1 ≤ C (t + 1) p ∥f -g∥ mp . ( 12
)
Under Hypotheses 1-3, for all q ∈ (0, d), there exist two constants C, κ > 0 such that for all t ≥ 0, for all f, g in L 1 mq (G) with ⟨f ⟩ = ⟨g⟩, there holds:

S t f -S t g L 1 ≤ Ce -κt ∥f -g∥ mq . (13)
Three consequences can be drawn from this theorem, which form our main results. Theorem 3. Assume that Hypotheses 1 and 2 hold.

i. There exists a unique f ∞ such that for all ϵ ∈ (0, 1/2),

f ∞ ∈ L 1 m d-ϵ (G), f ∞ ≥ 0, ⟨f ∞ ⟩ = 1 and v • ∇ x f ∞ = Cf ∞ , (x, v) ∈ G, γ -f ∞ = Kγ + f ∞ , (x, v) ∈ Σ -.
ii. For all p ∈ (0, d), there exists a constant C > 0 such that for all t ≥ 0, for all f ∈ L 1 mp (G)

with f ≥ 0 and ⟨f ⟩ = 1, S t (f -f ∞ )∥ L 1 ≤ C (1 + t) p ∥f -f ∞ ∥ mp . (14)
iii. If, additionally, Hypothesis 3 holds, for all q ∈ (0, d), there exists two constants C, κ > 0 such that for all t ≥ 0, for all f ∈ L 1 mq (G) with f ≥ 0 and ⟨f ⟩ = 1,

S t (f -f ∞ ) L 1 ≤ Ce -κt ∥f -f ∞ ∥ mq . (15)
In Section 6, we also assume that f ∞ is uniformly bounded. In this setting, we present a counter-example showing that the exponential convergence can fail when Hypothesis 3 does not hold. We also provide general exponential lower bound for the rate of convergence of (1), as well as a polynomial lower bound for the case where σ cancels on an open ball inside Ω.

Theorem 4. Assume Hypotheses 1 and 2. Let f ∞ given by Theorem 3, and suppose furthermore that f ∞ ∈ L ∞ (G). Then,

(1) for all α ∈ (0, d), the uniform decay rate E(t) such that for all f ∈ L 1 mα (G), with f ≥ 0 and ⟨f ⟩ = 1, t > 0, 1+α)t for all t large enough;

S t f -f ∞ ∥ L 1 ≤ E(t)∥f -f ∞ ∥ mα satises E(t) ≥ C α,∥f∞∥m α e -σ∞(
(2) if σ vanishes on an open ball of Ω, for all α ∈ (0, d), there does not exists any constant C, κ > 0 such that for all f ∈ L 1 mα (G) with ⟨f ⟩ = 1, for all t > 0,

S t f -f ∞ L 1 ≤ Ce -κt ∥f -f ∞ ∥ mα ; (16)
(3) if σ vanishes on an open ball of Ω, for all α ∈ (0, d), the uniform decay rate E(t) such that for all f ∈ L 1 mα (G), with f ≥ 0 and ⟨f ⟩ = 1, t > 0,

S t f -f ∞ ∥ L 1 ≤ E(t)∥f -f ∞ ∥ mα satises E(t) ≥ C α,∥f∞∥m α (t + 1)
-α for t large enough.

Remark 5. A straightforward adaptation of our proof (in the spirit of [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF]) provides those negative results for the time-averaged quantity, that is, in the framework of the theorem, bounds are given for the uniform decay rate E such that

∥S t+• f -f ∞ ∥ L 1 ([0,T ]×G) ≤ E(t)∥f -f ∞ ∥ mα with T > 0 xed. Of course, convergence for ∥S t f -f ∞ ∥ L 1 at rate r(t)
implies convergence for the averaged quantity at rate T r(t). Conversely, those negative results on the time-averaged quantity are stronger than the ones presented in Theorem 4: we chose this exposition to facilitate the comparison with our upper bounds.

(G) to L 1 (G), for α ∈ (0, d) and f ∞ in L ∞ (G).
Bounds are given up to a constant independent of time and initial data.

Before turning to the motivations and to our review of the existing literature, we make a few remarks regarding those results.

Remark 6 (Use of L 1 weighted spaces). Aoki and Golse [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF]Proposition 3.1] showed the nonexistence of a uniform rate of convergence in L 1 (G) for general L 1 (G) initial data in the freetransport case, which is compatible with Hypotheses 1 and 2. The uniform decay is indeed obtained here for initial data in some weighted L 1 spaces instead.

Remark 7 (Constructive constants). The use of deterministic Harris' theorems to study the rate of convergence towards the steady state yields explicit constants [START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF]. Those however depend on the constants appearing in the two conditions from which the proof is derived: the Lyapunov inequality and the Doeblin-Harris condition. While, in the former, constants are transparent, the ones from the latter depend here in a complicated fashion of the domain considered, see Remark 19. Note also that we crucially use the stochastic nature of our boundary conditions: in case (CLBC), as (r ⊥ , r ∥ ) → (0, 0) or as (r ⊥ , r ∥ ) → (0, 2), i.e. as the reection mechanism tends to the specular or the bounce-back boundary condition (see below Subsection 1.5.1), the constants from the Lyapunov conditions explode, see Proposition 14 and its proof. Similarly, in case (MBC), as β 0 → 0, i.e. as the reection mechanism tends to the specular one, our weights construction fails, since we require c 4 ∈ (0, 1) with (1 -c 4 ) 4 = (1 -β 0 ). Regarding the lower bounds from Theorem 4, the constants appearing in front of the convergence rates are also constructive, but depend on the generally unknown values ∥f ∞ ∥ mα , ∥f ∞ ∥ ∞ .

Remark 8 (About the boundedness hypothesis in Theorem 4). It is known for both boundary conditions considered (see [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF] for case (MBC), [START_REF] Chen | Cercignani-Lampis Boundary in the Boltzmann Theory[END_REF] for case (CLBC)) that, in the case of small temperature variations at the wall, the steady state of the full Boltzmann equation exists, is unique in the class of suciently regular functions, and is bounded. The boundedness hypothesis from Theorem 4 thus appears natural.

Remark 9 (About the connectedness assumption). We assume that Ω is connected for simplicity. The case where Ω has nitely many connected components can also be dealt with, by splitting the densities and the corresponding steady states on each of those components. Further extensions seem really involved.

Remark 10 (About a Doeblin condition). It might be possible to derive the exponential convergence from L 1 (G) to L 1 (G) under the additional Hypothesis 3, for instance by showing that, in this setting, the semigroup satises a Doeblin condition (rather than what we call a Doeblin-Harris one): for some T > 0 and a non-negative measure ν ̸ ≡ 0, for all

(x, v) ∈ G and f ∈ L 1 (G; R + ), S T f (x, v) ≥ ν(x, v) G f (y, w) dy dw,
which is to be compared with the statement of Theorem 18 which only gives an upper bound to a restricted integral. Such a strategy was successful in [START_REF] Evans | Quantitative Rates of Convergence to Equilibrium for the Degenerate Linear Boltzmann equation on the Torus[END_REF] for the study of the degenerate linear Boltzmann equation in the torus. This could upgrade very slightly our results, since we only obtain exponential convergence from L 1 mϵ (G) to L 1 (G) for any ϵ > 0. We found however dicult to adapt the argument of [START_REF] Evans | Quantitative Rates of Convergence to Equilibrium for the Degenerate Linear Boltzmann equation on the Torus[END_REF] to a framework including our boundary conditions.

Remark 11 (Absence of perturbative arguments). We emphasize that our proofs do not rely on any perturbative arguments. We can thus treat the whole spectrum of accommodation coecients for case (CLBC), that is (r ⊥ , r ∥ ) in (0, 1] × (0, 2), and, for case (MBC), β ∈ (β 0 , 1] for any β 0 > 0 xed. Similarly, we only assume continuity and positivity of the temperature, without requiring small variations around a constant.

1.5. Context, previous results and motivations. The linear Boltzmann equation is a fundamental one in kinetic theory and statistical physics. It describes the behavior of a dilute gas of particles encountering collisions with some background [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF][START_REF] Dautray | Mathematical Analysis and Numerical Methods for Science and Technology[END_REF][START_REF] Dautray | Mathematical Analysis and Numerical Methods for Science and Technology[END_REF]. Applications of this model span a wide range of disciplines: in physics, it is used to investigate neutron transport [START_REF] Cercignani | Rareed Gas Dynamics[END_REF], quantum scattering [START_REF] Erd®s | Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation[END_REF] and semiconductor device modeling [START_REF] Markowich | Semiconductor Equations[END_REF]. The linear Boltzmann equation has been derived in several contexts, see [START_REF] Breteaux | A Geometric Derivation of the Linear Boltzmann Equation for a Particle Interacting With a Gaussian Random Field, Using a Fock Space Approach[END_REF] for the case of a particle interacting with a random eld, [START_REF] Bodineau | The Brownian Motion as the Limit of a Deterministic System of Hard-Spheres[END_REF] for a study of hard-spheres, representing gas molecules. The degenerate linear Boltzmann equation is a generalized version, adapted for instance to the study of radiative transfer systems inside which dierent parts of the space may have dierent transparencies. Our model set inside a bounded domain with stochastic boundary conditions is also reminiscent of the one presented in [START_REF]Transport through Diusive and Nondiusive Regions, Embedded Objects, and Clear Layers[END_REF] for the study of photon migration within the skull, with applications in imagining of tumors and cerebral oxygenation [START_REF] Arridge | Optical Tomography in Medical Imaging[END_REF][START_REF] Arridge | Optical Imaging in Medicine: II. Modelling and Reconstruction[END_REF].

In the past few years, the study of the linear Boltzmann equation, and of the BGK model [START_REF] Bhatnagar | A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems[END_REF] (also called the BKW model [START_REF] Welander | On the Temperature Jump in a Rareed Gas[END_REF] in the physics literature) where k

(x, v, v ′ ) = M 1 (v ′ ) with M 1 (v) = e -|v| 2 2 (2π) d 2 , v ∈ R d , (17)
combined with some boundary conditions have drawn a lot of interest within the mathematical community. There are two main reasons for this:

• those models have some physical relevance, with several well-identied applications;

• they present strong mathematical challenges, due to the delicate interaction between the transport operator with boundary conditions and the collision operator.

We develop those two aspects in the next paragraphs.

1.5.1. Physical features and boundary conditions. We present some key facts, and refer to [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF][START_REF] Cercignani | Rareed Gas Dynamics[END_REF] for more details.

When modeling a gas inside a bounded domain Ω, several choices of boundary conditions at ∂Ω are at disposal. The most simple ones are:

(1) the bounce-back boundary condition : for all

(t, x, v) ∈ R + × Σ -, f (t, x, v) = f (t, x, -v); (18)
(2) the specular reection: for all

(t, x, v) ∈ R + × Σ -, f (t, x, v) = f t, x, η x (v) . ( 19
)
Those conditions are unable to render the stress exerted by the gas on the wall, and for this reason, Maxwell [START_REF] Maxwell | On Stresses in Raried Gases Arising from Inequalities of Temperature[END_REF]Appendix] introduced the pure diuse reection: for all (t, x, v) ∈ R + × Σ -, taking the temperature θ ≡ 1 independent of x,

f (t, x, v) = 1 (2π) d-1 2 e -|v| 2 2 Σ x + f (t, x, w)|w • n x | dw. ( 20
)
As opposed to [START_REF] Cañizo | Hypocoercivity of Linear Kinetic Equations via Harris's Theorem[END_REF] and [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF], there is no correlation between the incoming velocities of particles hitting the wall and their outgoing ones in [START_REF] Cercignani | Rareed Gas Dynamics[END_REF]. A rst possible correction is to consider instead the Maxwell boundary condition [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF], a convex combination between the pure diuse reection and the specular one.

Introduced at the beginning of the 1970's by Cercignani and Lampis [START_REF] Cercignani | Kinetic Models for Gas-Surface Interactions[END_REF], condition (CLBC) provides a more delicate way to modify [START_REF] Cercignani | Rareed Gas Dynamics[END_REF] to obtain those correlations. Its superior accuracy over the aforementioned models was exhibited numerous times, both from numerical computations performed in the 1980's, and from physical experiments [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF][START_REF] Yu | Poiseuille's Flow and Thermal Creep for Dierent Scattering Kernels for a Gas Scattered by a Channel Surface[END_REF][START_REF] Pantazis | Gas-Surface Scattering Eect on Vacuum Gas Flows Through Rectangular Channels[END_REF][START_REF] Sharipov | Application of the Cercignani-Lampis Scattering Kernel to Calculations of Rareed Gas Flows. I. Plane Flow Between Two Parallel Plates[END_REF], see in particular the recent work of Yamaguchi et al. [START_REF] Yamaguchi | Mass Flow Rate Measurement of Thermal Creep Flow From Transitional to Slip Flow Regime[END_REF]. This paper was followed by a theoretical derivation of the coecients in the context of hard spheres from Nguyen et al. [START_REF] Nguyen | Variational Derivation of Thermal Slip Coecients on the Basis of the Boltzmann Equation for Hard-Sphere Molecules and Cercignani-Lampis Boundary Conditions: Comparison With Experimental Results[END_REF], who showed that the accommodation coecients are independent of the shape of the domain, depend on the gas species considered, and can, for some of those, be very dierent from the values (1, 1) corresponding to [START_REF] Cercignani | Rareed Gas Dynamics[END_REF]. For instance, in a setting controlling temperature variations and pressure, an estimation for He was given in [56, Table II], with values (r ⊥ , r ∥ ) = (0.15, 0.8). It is thus important to obtain mathematical results for the whole spectrum (0, 1] × (0, 2) of accommodation coecients. 1.5.2. Mathematical motivations and previous results. Equation [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] combines a rst-order transport dynamics with two subtle relaxation eects in the velocity variable:

• the degenerate collision mechanism;

• a stochastic boundary operator. Several results are already known regarding the long-time behavior of this kind of model.

Consider rst the sole transport dynamics with boundary conditions. Aoki and Golse [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF] where the rst to question whether the thermalisation eect at the wall alone was enough to produce a spectral gap. For the diuse reection [START_REF] Cercignani | Rareed Gas Dynamics[END_REF], they identify the lack of uniform convergence for L 1 (G) initial data, and proved a convergence rate of (1 + t) -1 , from some weighted L 1 space to L 1 . This result was improved up to the optimal rate 1 (1+t) d-in several subsequent articles by Kuo, Liu and Tsai [START_REF] Kuo | Free Molecular Flow with Boundary Eect[END_REF][START_REF] Kuo | Equilibrating Eects of Boundary and Collision in Rareed Gases[END_REF] and Kuo [START_REF] Kuo | Equilibrating Eect of Maxwell-Type Boundary Condition in Highly Rareed Gas[END_REF] in a radial domain, and by Bernou-Fournier [START_REF] Bernou | A Coupling Approach for the Convergence to Equilibrium for a Collisionless Gas[END_REF] and Bernou [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF] in a C 2 bounded one, and ultimately culminated in the treatment of the more general Cercignani-Lampis boundary condition [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF], for which the same rate of convergence was obtained. The key outcome of those research is that stochastic boundary conditions ((MBC) and (CLBC)) provide only a polynomial rate of convergence in the L 1 (G) distance: there is no spectral gap for those dynamics.

Next, we turn to hypocoercive equations, that is, dynamics combining a relaxation in the velocity variable with a transport operator. Those have been heavily studied in the past twenty years, and we will restrain to the equations closest to our framework. The BGK model was studied in the torus by Mouhot and Neumann [START_REF] Mouhot | Quantitative Perturbative Study of Convergence to Equilibrium for Collisional Kinetic Models in the Torus[END_REF] who proved the existence of a spectral gap in H 1 norm. This toroidal case was also investigated, along with the case of the whole space with a connement potential, by Dolbeault-Mouhot-Schmeiser [START_REF] Dolbeault | Hypocoercivity for Kinetic Equations With Linear Relaxation Terms[END_REF][START_REF] Dolbeault | Hypocoercivity for Linear Kinetic Equations Conserving Mass[END_REF], who gave a beautiful, simple proof of exponential decay using L 2 hypocoercivity which applies to a whole range of linear operators, including the linear Boltzmann equation. Those articles are part of the growing literature regarding hypocoercivity, which in some sense started from the work of Desvillettes, Hérau, Nier, Mouhot and Villani [START_REF] Desvillettes | On the Trend to Global Equilibrium in Spatially Inhomogeneous Entropy-Dissipating Systems: The Linear Fokker-Planck Equation[END_REF][START_REF] Desvillettes | On the Trend to Global Equilibrium for Spatially Inhomogeneous Kinetic Systems: The Boltzmann Equation[END_REF][START_REF] Hérau | Hypocoercivity and Exponential Time Decay for the Linear Inhomogeneous Relaxation Boltzmann Equation[END_REF][START_REF] Hérau | Isotropic Hypoellipticity and Trend to Equilibrium for the Fokker-Planck Equation with a High-Degree Potential[END_REF][START_REF] Mouhot | The Princeton Companion to Applied Mathematics, chapter Areas of Applied Mathematics: Kinetic Theory[END_REF][START_REF] Villani | Hypocoercivity[END_REF] among others, and beneted from earlier approaches, in particular the high-order Sobolev energy method of Guo [START_REF] Guo | The Vlasov-Poisson-Boltzmann System Near Maxwellians[END_REF]. At last, the degenerate linear Boltzmann equation investigated in this paper was studied in great details in the toroidal setting. In the case where v ∈ V with V bounded from below and above, Bernard and Salvarini [START_REF] Bernard | On the Exponential Decay to Equilibrium of the Degenerate Linear Boltzmann Equation[END_REF] obtained exponential convergence towards the equilibrium under a geometric control condition. They also built in [START_REF] Bernard | On the Convergence to Equilibrium for Degenerate Transport Problems[END_REF] a counter-example showing that exponential convergence is not true in general. Later, Han-Kwan and Léautaud [START_REF] Han-Kwan | Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium[END_REF] used tools from control theory to deal with the case v ∈ R d with a connement potential, and obtained conditions about the spatial behavior of k under which exponential convergence to the steady state occurs. They also characterized the latter under some extra hypotheses on k. In some sense, our paper extends the results of [START_REF] Bernard | On the Exponential Decay to Equilibrium of the Degenerate Linear Boltzmann Equation[END_REF] to the case where x ∈ Ω, v ∈ R d with (stochastic) boundary conditions. This paper uses deterministic strategies inspired from probabilistic methods. Those tools, namely Doeblin and Harris theorems, were already used by Cañizo-Cao-Evans-Yoldas [START_REF] Cañizo | Hypocoercivity of Linear Kinetic Equations via Harris's Theorem[END_REF] to derive convergence rates towards equilibrium for the relaxation operator and the linear Boltzmann equation, in the torus and in the whole space with a connement potential, some forms of the latter leading to polynomial rates of convergence, rather than exponential ones. Evans and Moyano [START_REF] Evans | Quantitative Rates of Convergence to Equilibrium for the Degenerate Linear Boltzmann equation on the Torus[END_REF] also recently used Doeblin's theorem to derive quantitative exponential convergence of the degenerate linear Boltzmann equation in the torus.

To conclude this literature review, we focus on models involving both a hypocoercive structure for the equation and non-deterministic boundary conditions. For deterministic boundary conditions (specular or bounce-back), we simply quote, among others [START_REF] Duan | Global Mild Solutions of the Landau and Non-Cuto Boltzmann Equations[END_REF][START_REF] Guo | Decay and Continuity of the Boltzmann Equation in Bounded Domains[END_REF][START_REF] Kim | The Boltzmann Equation with Specular Boundary Condition in Convex Domains[END_REF][START_REF] Kim | Decay of the Boltzmann Equation with the Specular Boundary Condition in Non-convex Cylindrical Domains[END_REF]. For the diuse reection with constant temperature, Guo [START_REF] Guo | Decay and Continuity of the Boltzmann Equation in Bounded Domains[END_REF] obtained exponential convergence in some weighted L ∞

x,v space for the linearized Boltzmann equation when Ω is smooth and convex, using his famous L 2 -L ∞ approach. Briant [START_REF] Briant | Perturbative Theory for the Boltzmann Equation in Bounded Domains With Dierent Boundary Conditions[END_REF] extended this result to more general weights. Guo-Briant [START_REF] Briant | Asymptotic Stability of the Boltzmann Equation With Maxwell Boundary Conditions[END_REF] upgraded those ndings to get explicit constants and handle the Maxwell boundary condition. Regarding those topics, we also mention the recent [START_REF] Bernou | Hypocoercivity for Kinetic Linear Equations in Bounded Domains With General Maxwell Boundary Condition[END_REF]. When the reection at the wall is diuse with small temperature variations, Esposito-Guo-Kim-Marra [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF] showed the existence of a steady state and gave an exponential result of convergence in L ∞ norm, see also [START_REF] Duan | Eects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rareed Gas[END_REF], but virtually nothing is none outside this case. The study of condition (CLBC) in those collisional contexts has been very sparse, with the notable exception of the work of Chen [START_REF] Chen | Cercignani-Lampis Boundary in the Boltzmann Theory[END_REF], who extended the results from [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF], under again an assumption of small temperature variations and strong hypotheses on the accommodation coecients, that must be close to [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF][START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF]. The recent article of Dietert et al. [START_REF] Dietert | Quantitative Geometric Control in Linear Kinetic Theory[END_REF] is the closest to our framework, as it considers degenerate linear equations (namely, the linear Boltzmann equation and the linear Fokker-Planck equation) with connement mechanisms that include the case of the diuse reection (20) at constant temperature. Using trajectorial methods and tools from control theory, the authors give conditions under which exponential convergence towards the equilibrium is achieved, in some L 2 norm, with constructive rates. The approach allows to treat several models and connement mechanisms in a unied way. 1.6. Contributions. The L 2 -hypocoercivity tools mentioned above require the knowledge of the equilibrium and some form of separation of variables for it, as the velocity distribution is used as a weight to tailor appropriate functional spaces. So far those methods have given limited insights about the asymptotic behavior of the solutions when the temperature varies at the boundary. This framework is however meaningful from a physical point of view in our context: considering degenerate models implies that the thermalization eects are dierent in various regions of space. Extending these features up to the boundary, which amounts to considering wall temperatures that also change with the position, is thus a very natural assumption. The linear Boltzmann equation with variable temperature at the boundary is also an interesting framework for the study of the Fourier law in the kinetic regime [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF].

In this paper, our main contributions apply to all three stochastic boundary conditions considered (diuse, Maxwell and Cercignani-Lampis) with no assumptions on the temperature variations, and in a general C 2 bounded domain. We obtain ve main results:

(1) the existence and uniqueness of the steady state for the (degenerate) linear Boltzmann equation;

(2) an exponential rate of convergence towards this steady state, in L 1 (G), for the linear Boltzmann equation (i.e. σ ≡ 1); (3) an exponential rate of convergence towards the steady state, in L 1 (G), for the degenerate linear Boltzmann equation under an additional control condition; (4) a polynomial rate of convergence towards the steady state, in L 1 (G), for the degenerate linear Boltzmann equation without the additional control condition; (5) a precise picture of the convergence, including lower bounds on the rates, under an additional boundedness assumption of the steady state that is known to hold for the full Boltzmann equation in the case of small temperature variations.

At last, we present in Section 2 a linear relaxation model that can be seen as the counterpart to the degenerate linear BGK one in the case of varying temperature at the boundary. This model is very natural in the study of multi-species interaction combined with boundary eects, and we provide quantitative estimates of convergence in Corollary 12.

In our opinion, our results regarding the convergence rates should be read as follows:

• Under the sole Hypotheses 1 and 2, the decay is due to the interplay between the freetransport dynamics and the boundary condition, hence the rate of convergence can not beat the one obtained for the free-transport problem in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF][START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]. The stochastic nature of the boundary condition is key to the mixing in both space and velocity: in terms of trajectories this is best understood at the level of the Doeblin-Harris condition, Theorem 18, where it is shown, roughly, that particles with controlled velocity and next boundary collision time span the whole phase space. • Under the additional Hypothesis 3, the collision operator is suciently involved into the dynamics to provide further mixing, and therefore additional decay, eventually leading to some exponential convergence.

1.7. Strategy and plan of the paper.

Section 2 presents three applications with given choices of k. We start with the linear BGK equation and a general linear Boltzmann model. Of particular interest is our third setting, namely the interaction between two gas species, which corresponds to k

(x, v, v ′ ) = f∞ (x, v ′ ), (x, v, v ′ ) ∈ G × R d ,
for f∞ the steady state of the full Boltzmann equation set inside the domain with variable boundary temperature. This provides a relaxation model which is more relevant than the usual linear BGK one for the case where the temperature varies both inside the domain and at the boundary.

Ultimately, this paper relies on a deterministic Doeblin-Harris type argument, in the spirit of [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF][START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF], see also [START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF] and the recent review [START_REF] Yolda³ | On Quantitative Hypocoercivity Estimates Based on Harris-Type Theorems[END_REF]. The core of our strategy builds upon the structure (and known results) of the underlying free-transport operator. Under Hypotheses 1 and 2, Problem (1) is a bounded perturbation of the two models studied in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF] and [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]. This is the key argument providing our well-posedness result and important features of the trace, in Section 3. Section 4 is devoted to our derivation of the Lyapunov conditions. The main point is as follows:

when dierentiating ∥S t f ∥ mα for some f ∈ L 1 mα (G), t ≥ 0 and α ∈ (1, d), one obtains d dt ∥S t f ∥ mα ≤ Q α (S t f ) - G v • ∇ x m α |S t f | dv dx - G σ(x)|S t f | m α dv dx, (21)
where Q α (S t f ) represents the sum of the boundary and gain terms. One shows the equality -v • ∇ x m α = -α m α-1 on G by construction of the weights. Under Hypotheses 1 and 2 we just ignore the last term on the right-hand-side (r.h.s.) of ( 21) and the decay of the norm is given by the term -α∥S t f ∥ m α-1 originating from the free-transport dynamics rather than from the collision operator. Under the additional Hypothesis 3, we ignore this term in m α-1 and get the decay from the loss term of the collision operator, in the form of -σ 0 ∥S t f ∥ mα . The treatment of the boundary terms in Q α within ( 21) is a delicate point, for which we adapt the previous strategies introduced in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF][START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]. We integrate ( 21) on [0, T ], T > 0, as one can show that integrated boundary uxes of f are controlled by C(1 + T )∥f ∥ L 1 for both boundary conditions. We conclude that for all α ∈ (1, d), there exist K 1 , K 2 > 0 two constants such that:

• under Hypotheses 1 and 2, for all T > 0, for all f ∈ L 1 mα (G),

∥S T f ∥ mα + α T 0 ∥S s f ∥ m α-1 ds ≤ ∥f ∥ mα + K(1 + T )∥f ∥ L 1 . (22)
• If Hypothesis 3 also holds, for all T > 0, all f ∈ L 1 mα (G),

∥S T f ∥ mα + σ 0 T 0 ∥S s f ∥ mα ds ≤ ∥f ∥ mα + K 2 (1 + T )∥f ∥ L 1 . ( 23
)
Section 5 focuses rst on the Doeblin-Harris condition in Subsection 5.1. There, the Duhamel formula (55) renders very concretely our use of the free-transport dynamics. Indeed, we rst show that, for all

(t, x, v) ∈ R + × G, S t f (x, v) ≥ 1 {τ (x,-v)≤t} e -τ (x,-v) 0 σ(x-sv) ds S t-τ (x,-v) f x -τ (x, -v)v, v ,
which allows us to ignore the gain collision mechanism (we only need some boundedness of σ) to derive the minoration condition: for all Λ large enough, there exist T (Λ) > 0 and a non-negative measure ν ̸ ≡ 0 on G such that for all (x, v) ∈ G, for all

f 0 ∈ L 1 (G), f 0 ≥ 0, S T (Λ) f 0 (x, v) ≥ ν(x, v)
{(y,w)∈G: m 1 (y,w)≤Λ} f 0 (y, w) dy dw. [START_REF] Dautray | Mathematical Analysis and Numerical Methods for Science and Technology[END_REF] Once conditions ( 22)-( 23) and ( 24) are established, we follow in Subsection 5.2 a strategy reminiscent of the one in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF][START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF][START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF]. Roughly, the core mechanism is as follows: inside the sublevel sets of the weight functions, (24) provides some contraction. Outside of those sublevel sets, the Lyapunov conditions ( 22)-( 23) tell us how fast the dynamics return to them. The speed of convergence can thus be read at this level. This strategy is in some sense analogous to a probabilistic coupling one such for the free-transport dynamics is performed in [START_REF] Bernou | A Coupling Approach for the Convergence to Equilibrium for a Collisionless Gas[END_REF] but relying on the framework introduced by Hairer-Mattingly [START_REF] Hairer | Convergence of Markov Processes[END_REF][START_REF] Hairer | Yet Another Look at Harris' Ergodic Theorem for Markov Chains[END_REF] and rened by Cañizo-Mischler [START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF] allows to escape the corresponding cumbersome construction especially in models like the ones investigated here, whose probabilistic writing would involve several sources of randomness by playing with weighted norms instead. We obtain the existence and uniqueness of the steady state, and some rate of convergence towards it. This is one of the main strengths of Doeblin-Harris type arguments, particularly with respect to hypocoercivity methods, which makes them well-tailored for the study of models whose parameterization is more involved: the knowledge of the steady state is not required a priori. Section 6 is devoted to the proof of Theorem 4. By building an appropriate initial data f ϵ depending on some parameter ϵ ∈ (0, 1), and by using a comparison principle with the solution of the problem

     ∂ t Φ + v • ∇Φ = -σ(x)Φ in R + × G, γ -Φ = 0 on R + × Σ -, Φ |t=0 = f ϵ , in G,
whose solution is explicitly given by the method of characteristics, we derive a general inequality on the uniform convergence rate E(t). We then draw conclusions by choosing appropriately ϵ. This strategy is in part inspired from [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF].

1.8. Notations. We write B for the closure of any set B. We denote by C 1 c (E) and C ∞ c (E) the space of test functions, C 1 and C ∞ c with compact support, respectively, on E. We write dζ(x) for the surface measure at x ∈ ∂Ω. For a function ϕ on (0, ∞) × Ḡ, we denote γ ± ϕ its trace on (0, ∞) × Σ ± , under the assumption that this object is well-dened. We write W 1,∞ (R d ; R) for the space of functions g admitting a weak derivative, ∇g, such that both g and ∇g belong to L ∞ (R d ). We write |||H||| A→B for the operator norm of H acting between the two Banach spaces A and B.

Throughout the paper, 0 < δ < δ k d is xed, with δ k given by Hypothesis 2. We denote d(Ω) = sup x,y∈ Ω |x -y| the diameter of Ω, which is nite by assumption. For h ∈ R, we dene ⌊h⌋ := inf{z ≤ h, z ∈ Z}. In the whole paper, the positive constants C and κ depend only on the parameters (and not on the time nor on the initial data), and are allowed to change from line to line. We write subscripts when we wish to emphasize some dependency, e.g. C α is a constant depending on α which can vary from line to line. We write σ ∞ for the upper bound of σ, which is well-dened under Hypothesis 2. For two random variables X, Y dened on a probability space with the same distribution, we write

X L = Y . For all (x, v) ∈ ∂Ω × R d , η x (v) := v -2(v • n x )n x .
In particular η x maps Σ x ± to Σ x ∓ and v → η x (v) has Jacobian 1. We sometimes have to distinguish between both boundary conditions, in which case we write (MBC) and (CLBC) to refer to the two settings of Hypothesis 1.

Applications

We detail in this section several collision kernels tting into our framework, in growing order of complexity.

2.1. Linear relaxation kernel. We set, for all

(x, v, v ′ ) ∈ G × R d , k(x, v, v ′ ) = σ(x)M 1 (v ′ ) with σ ∈ L ∞ (Ω; R + )
and M 1 given by [START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF]. This corresponds to the so-called degenerate linear BGK model, whose collision operator is given,

for f ∈ L 1 (G), (x, v) ∈ G, by Cf (x, v) = σ(x) M 1 (v) R d f (x, v ′ ) dv ′ -f (x, v) .

Linear Boltzmann equation. We set, for all

(x, v, v ′ ) ∈ G × R d , k(x, v, v ′ ) = σ(x)p(v, v ′ ), with R d p(v, v ′ ) dv ′ = P , P > 0 constant, sup v∈R d R d p(v, v ′ )|v ′ | 2δ k dv ′ < 0 for some δ k ∈ (0, 1
2 ), and σ ∈ L ∞ (Ω; R + ). This generalizes the previous model and includes the case σ ≡ 1, which corresponds to the (non-degenerate) linear Boltzmann equation, and the case where p(v, •) ⊂ V with V ⊂ R d bounded from above and below, as considered by Bernard-Salvarani [START_REF] Bernard | On the Exponential Decay to Equilibrium of the Degenerate Linear Boltzmann Equation[END_REF] on the torus.

2.3.

Relaxation against the steady state of a Boltzmann equation. We present here a model which captures specic features of the case when the temperature varies at the boundary. The linear Boltzmann equation is a classical model for the interaction between two species of gas, say (A) and (B), when one of the species is more dense than the other [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF]. Consider the following setting: species (A) is more dense, and has already reached a steady state inside the domain Ω. Species (B) is less dense, to the point where inner collisions between particles of species (B) can be neglected in its evolution. In the case (MBC) with β ≡ 1 and small temperature variations, it is known [33, Theorem 1.1], see also [START_REF] Guiraud | Problème aux limites intérieur pour l'équation de Boltzmann en régime stationnaire, faiblement non linéaire[END_REF][START_REF] Guiraud | An H-Theorem for a Gas of Rigid Spheres in a Bounded Domain. Théories cinétiques classiques et relativistes[END_REF], that species (A), whose dynamics can be described by a full Boltzmann equation, admits a steady state, which depends on both x and v and that we denote f A,∞ . Furthermore,

(1) f A,∞ ∈ L ∞ (G), (2) sup x∈Ω R d f A,∞ (x, v)|v| 2δ k dv ≲ 1 for all δ k ∈ (0, 1/2).
Upon imposing more precise moment conditions on f A,∞ , its uniqueness is again known [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF]. In case (CLBC), analogous results are available, provided that the temperature variations are small and that the accommodation coecients are close to the values (1, 1) [22, Corollary 2].

In the study of space-dependent thermal exchanges, it is thus natural to study the model (1) for the density f B of the species (B) with the choice

k(x, v, v ′ ) = f A,∞ (x, v ′ ), (x, v, v ′ ) ∈ G × R d (note that of course, R d k(x, v, v ′ ) dv ′ = σ(x) is independent of v in this framework)
that is, to study the following evolution problem for f B :

         ∂ t f B (t, x, v) + v • ∇ x f B (t, x, v) = f A,∞ (x, v) R d f B (t, x, v ′ ) dv ′ -f B (t, x, v) R d f A,∞ (x, v ′ ) dv ′ , (t, x, v) ∈ R + × G, γ -f B = Kγ + f B , (t, x, v) ∈ R + × Σ, f B (0, x, v) = f 0 (x, v), (x, v) ∈ G, (25)
for some initial data

f 0 ∈ L 1 mα (G), α ∈ (0, d).
It is clear from the conditions detailed above and satised by f A,∞ that this choice of k satises Hypothesis 2.

Our results directly lead to the following corollary.

Corollary 12. Under Hypothesis 1, the problem [START_REF] Desvillettes | On the Trend to Global Equilibrium in Spatially Inhomogeneous Entropy-Dissipating Systems: The Linear Fokker-Planck Equation[END_REF] is well-posed. We write (S B,t ) t≥0 for the associated C 0 -stochastic semigroup given by Theorem 13.

(1) There exists a unique steady state for the problem [START_REF] Desvillettes | On the Trend to Global Equilibrium in Spatially Inhomogeneous Entropy-Dissipating Systems: The Linear Fokker-Planck Equation[END_REF],

f ∞,B such that for all α ∈ (0, d), f ∞,B in L 1 mα (G), f ∞,B ≥ 0 and ⟨f ∞,B ⟩ = 1.
(2) For all p ∈ (0, d), there exists a constant C > 0 such that for all t ≥ 0, for all f ∈ L 1 mp (G)

with f ≥ 0 and ⟨f ⟩ = 1, S B,t (f -f ∞,B ) L 1 ≤ C (1 + t) p f -f ∞,B mp . (3) Assume that f A,∞ is continuous in G, with, for all x ∈ Ω, R d f A,∞ (x, v) dv > 0.
Then for all q ∈ (0, d), there exist two constants C, κ > 0 such that for all t ≥ 0, all f ∈ L 1 mq (G)

with f ≥ 0 and ⟨f ⟩ = 1, S B,t (f -f ∞,B ) L 1 ≤ Ce -κt f -f ∞,B mq .
Proof. The well-posedness and the existence of the associated C 0 -stochastic semigroup are given by Theorem 13. Points [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] and [START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF] are given by Theorem 3. Point [START_REF] Arridge | Optical Tomography in Medical Imaging[END_REF] follows from the fact that, by compactness, those hypotheses on f A,∞ imply inf x∈Ω σ(x) > 0, so that Hypothesis 3 is satised, and Point iii. of Theorem 3 applies.

□

It is worth noting that in case (MBC) with β ≡ 1, when Ω is convex, it is known that f A,∞ is continuous in G. A further renement is provided in [33, Theorem 1.2] and shows that, at least in the case where the temperature variations are very small (that is δ ≪ 1 in the notations of [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF]), one should expect

R d f A,∞ (x, v) dv > 0 for all x ∈ Ω. Point (3) of Corollary 12 is thus relevant in this situation.
3. Setting, well-posedness and trace theory 3.1. Associated semigroup. We gather our well-posedness result and some key elementary properties in the next theorem. Note that the boundary operator K given by ( 3) is non-negative and has norm 1. In case (CLBC), it follows from the normalization property [START_REF] Bernard | On the Convergence to Equilibrium for Degenerate Transport Problems[END_REF]. In case (MBC), it is easily obtained: for all (x, u) ∈ Σ + ,

Σ x - R(u → v; x)|v • n x | dv = β(x) Σ x - M (x, v)|v • n x | dv + (1 -β(x)) |η x (u) • n x | |u • n x | = β(x) + (1 -β(x)) = 1, (26)
where we used |u

• n x | = |η x (u) • n x | and that Σ x - M (x, v)|v • n x | dv = 1
by denition of M (whose normalisation is exactly tailored for this property).

Theorem 13 (Well-posedness, mass conservation, contraction property and trace equality).

Assume Hypotheses 1 and 2 hold. There exists a C 0 -stochastic semigroup (S t ) t≥0 associated to the problem [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] in [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF] with initial condition f . Moreover, for all f ∈ L 1 (G),

L 1 (G). That is, for all t ≥ 0, f ∈ L 1 (G), (S t f ) t≥0 is the unique solution in L ∞ ([0, ∞), L 1 (G)) of
i. for all t ≥ 0, the trace of S t f , denoted γ t f , is well-dened, with (γ t f ) t≥0 an element of

L 1 loc ([0, ∞) × Σ, (v • n x ) 2 dv dζ(x) dt
) such that the Green's formula is satised: for all t 0 , t

1 in R + and φ ∈ C 1 c (R + × Ḡ) with φ ≡ 0 on R + × Σ 0 : t 1 t 0 G S t f ∂ t + v • ∇ x φ + φ Cf dv dx dt = G S t f φ dv dx t 1 t 0 + t 1 t 0 Σ (γ t f )(v • n x ) φ dv dζ(x) dt.
We also have the renormalization property: for all

β ∈ W 1,∞ (R), t ≥ 0 γ t β(f ) = β(γ t f ).
ii. The mass is conserved: for all t ≥ 0,

G S t f (x, v) dx dv = G f (x, v) dx dv. (27)
iii. For all t ≥ 0,

∥S t f ∥ L 1 ≤ ∥f ∥ L 1 . (28)
iv. The semigroup (S t ) t≥0 is non-negative.

Proof.

Step 1: well-posedness. As the boundary operator is conservative and stochastic, one can show that the associated free-transport problem, corresponding to (1) with C ≡ 0, is governed by a C 0 -stochastic semigroup (T t ) t≥0 , i.e. a non-negative, mass-conservative semigroup such that, for f 0 ∈ L 1 (G), for all t ≥ 0,

T t f 0 = f (t, •) is the unique solution in L ∞ ([0, ∞); L 1 (G))
to the free-transport problem taken at time t. In case (CLBC), this was obtained by Cercignani and Lampis [START_REF] Cercignani | Kinetic Models for Gas-Surface Interactions[END_REF], along with the fact that (S t ) t≥0 is a contraction semigroup, see also [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]. For case (MBC) a proof can be found in [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF].

Turning to [START_REF] Alexandrychev | Mass and Heat Transfer in the Free-Molecular Regime Flow of Gas Through a Channel of Finite Length[END_REF], note that the corresponding operator is nothing but a perturbation of the free-transport equation, with either boundary conditions, by the operator C. According to Pazy [58, Chapter 3, Theorem 1.1], since C is linear and bounded in L 1 (G), which follows easily by Hypothesis 2, one can associate a C 0 -stochastic semigroup (S t ) t≥0 such that, for all

f 0 ∈ L 1 (G), t ≥ 0, S t f 0 = f (t, •) is the unique solution in L ∞ ([0, ∞); L 1 (G)) to (1) at time t.
Step 2: proof of i. and ii. Point i. follows from a mutis mutandis adaptation of the detailed proof of Mischler [53, Theorem 1 and Corollary 1]. The latter deals with a sole source term on the r.h.s. of the equation, but, as also pointed out by Dietert-Hérau-Hutridurga-Mouhot [27, Appendix B], the result can be easily extended to bounded linear operator in L 1 , as is the case of our operator C under Hypothesis 2. We refer the interested reader to [START_REF] Mischler | On The Trace Problem For Solutions Of The Vlasov Equation[END_REF].

Once the trace is well-dened, point ii. follows from a direct computation:

d dt G S t f (x, v) dv dx = - G v • ∇ x S t f (x, v) dv dx + G R d k(x, v ′ , v)S t f (x, v ′ ) dv ′ dv dx - G S t f (x, v) R d k(x, v, v ′ ) dv ′ dv dx = - Σ (v • n x )γ t f (x, v) dv dζ(x) = 0,
where the second equality follows from Green's formula for the boundary term, and using that the two collision terms cancel out, thanks to Fubini's theorem, using Hypothesis 2 and that S t f ∈ L 1 (G). The last equality follows from the normalization property ( 6) and [START_REF] Desvillettes | On the Trend to Global Equilibrium for Spatially Inhomogeneous Kinetic Systems: The Boltzmann Equation[END_REF].

Step 3: contraction property. Let t ≥ 0. By Kato's inequality, one has

d dt G |S t f | dv dx ≤ - G v • ∇ x |S t f | dv dx + G R d k(x, v, v ′ )|S t f |(v ′ ) dv ′ dv dx - G σ(x)|S t f | dv dx = - ∂Ω×R d γ|S t f | (v • n x ) dv dζ(x)
where we used Tonelli's theorem to prove that the last two terms on the right-hand-side of the rst inequality cancel out. By i.

∂Ω×R d γ|S t f ||v • n x | dζ(x) dv = ∂Ω×R d |γ t f ||v • n x | dζ(x) dv,
and it follows from the boundary condition and the triangle inequality that

- Σ |γ t f |(v • n x ) dv dζ(x) = - Σ + |γ t f ||v • n x | dv dζ(x) + Σ - Σ x + R(u → v; x)|u • n x |γ t f (x, u) du |v • n x | dv dζ(x) ≤ - Σ + |γ t f |(x, v) |v • n x | dv dζ(x) + Σ + |γ t f |(x, u) |u • n x | R d R(u → v; x)|v • n x | dv du dζ(x) = 0,
where we also used the normalization property to obtain the last equality.

Step 4. Proof of (iv) The positivity property is a classical consequence of the contraction in L 1 and of the linearity, see for instance [8, Proof of Theorem 3, Step 4]. □

Lyapunov Conditions

Recall the denition of the weights m α from [START_REF] Bernou | A Coupling Approach for the Convergence to Equilibrium for a Collisionless Gas[END_REF], α ∈ (0, d), and that 0 < δ ≪ 1 is xed throughout the paper. The goal of this section is to prove the following Lyapunov conditions: Proposition 14.

(1) For α ∈ (1, d), under Hypotheses 1 and 2, there exists a constant K > 0 such that for all

T > 0, all f ∈ L 1 mα (G), ∥S T f ∥ mα + α T 0 ∥S s f ∥ m α-1 ds ≤ ∥f ∥ mα + K(1 + T )∥f ∥ L 1 . (29)
(2) Under Hypothesis 1-3, for all α ∈ (1, d), there exists a constant K 2 > 0 such that for all T > 0, all f ∈ L 1 mα (G),

∥S T f ∥ mα + σ 0 T 0 ∥S s f ∥ mα ds ≤ ∥f ∥ mα + K 2 (1 + T )∥f ∥ L 1 . (30)
We will make use of both the function τ , see [START_REF] Bernou | Hypocoercivity for Kinetic Linear Equations in Bounded Domains With General Maxwell Boundary Condition[END_REF], and q dened for all (x, v) ∈ Ḡ by q(x, v) := x + τ (x, v)v. [START_REF] Duan | Global Mild Solutions of the Landau and Non-Cuto Boltzmann Equations[END_REF] To derive the Lyapunov conditions, we rst need to obtain some control of the ux. Using the general Cercignani-Lampis boundary condition rather than the diuse one generates additional diculty, see [9, Remark 17]. We start by deriving the following.

Lemma 15 (Control of the ux). Under Hypotheses 1 and 2, we have (CLBC) for all Λ > 0, there exists an explicit constant

C Λ > 0 s.t. for all f ∈ L 1 (G), T > 0, T 0 ∂Ω {v•nx>0,|v|≤Λ} |v ⊥ | γ + |S s f |(x, v) dv dζ(x) ds ≤ C Λ (1 + T )∥f ∥ L 1 ; (32)
(MBC) there exists an explicit constant C > 0 such that for all f ∈ L 1 (G), T > 0,

T 0 Σ + |v ⊥ | γ + |S s f |(x, v) dv dζ(x) ds ≤ C(1 + T )∥f ∥ L 1 . (33)
Proof.

Step 1: an inequality for a boundary term. We have, by denition of (S t ) t≥0 , by linearity of (1) and by positivity of the semigroup, that

∂ t |S t f | + v • ∇ x |S t f | = C |S t f | , a.e. in [0, T ] × G.
Recall that x → n x is a W 1,∞ (Ω) map by hypothesis. Multiplying this equation by

(v • n x ) and integrating on [0, T ] × Ω × {v ∈ R d , |v| ≤ 1}, we nd T 0 Ω {|v|≤1} (v • n x ) ∂ t + v • ∇ x |S t f |(x, v) dv dx dt = T 0 Ω {|v|≤1} (v • n x ) R d k(x, v ′ , v) |S t f |(x, v ′ ) dv ′ dv dx dt - T 0 Ω {|v|≤1} (v • n x ) |S t f |(x, v) σ(x) dv dx dt
where we used the denition of σ. Integrating by parts in both time and space on the left-hand side, we nd

Ω {|v|≤1} (v • n x ) |S t f |(x, v) dv dx T 0 - T 0 Ω {|v|≤1} |S t f |(x, v) v • ∇ x (v • n x ) dv dx dt + T 0 ∂Ω {|v|≤1} |v • n x | 2 γ|S t f |(x, v) dv dζ(x) dt = T 0 Ω {|v|≤1} (v • n x ) R d k(x, v ′ , v) |S t f |(x, v ′ ) dv ′ dv dx dt - T 0 Ω {|v|≤1} (v • n x ) |S t f |(x, v) σ(x) dv dx dt (34)
where we used that, according to Theorem 13, point i.,

|γS t f (x, v)| = γ|S t f |(x, v) a.e. in (0, ∞) × Σ + ∪ (0, ∞) × Σ -. (35)
We note rst that

T 0 Ω {|v|≤1} (v • n x ) R d k(x, v ′ , v) |S t f |(x, v ′ ) dv ′ dv dx dt - T 0 Ω {|v|≤1} (v • n x ) |S t f |(x, v) σ(x) dv dx dt ≤ k ∞ + σ ∞ T 0 ∥S s f ∥ L 1 ds, (36)
where we used Tonelli's theorem and Hypothesis 2. Isolating the integral on the boundary ∂Ω and throwing away the integral included in Σ + in (34), using that x → n x belongs to W 1,∞ (Ω), the triangle inequality and [START_REF] Guiraud | An H-Theorem for a Gas of Rigid Spheres in a Bounded Domain. Théories cinétiques classiques et relativistes[END_REF], this leads to

T 0 {(x,v)∈Σ -,|v|≤1} |v ⊥ | 2 γ -|S t f |(x, v) dv dζ(x) dt ≤ 2∥f ∥ L 1 + C ∥n • ∥ W 1,∞ + k ∞ + σ ∞ T 0 ∥S s f ∥ L 1 ds ≤ C(1 + T )∥f ∥ L 1 ,
where we used the L 1 contraction from Theorem 13.

Step 2: Conclusion in case (MBC). Using the boundary condition, Tonelli's theorem and the positivity

T 0 {(x,v)∈Σ -,|v|≤1} |v ⊥ | 2 γ -|S t f |(x, v) dv dζ(x) dt ≥ T 0 Σ + β(x)|u ⊥ |γ + |S t f |(x, u) {v∈Σ x -,|v|≤1} |v ⊥ | 2 e - |v| 2 2θ(x) (2π) d-1 2 θ(x) d+1 2 dv du dζ(x) dt. ( 37
)
We use the notation M from ( 9) in what follows. Note that

x → {v∈Σ x + ,|v|≤1} M (x, v)(v ⊥ ) 2 dv is continuous and positive, since x → M (x, v) and x → n x are continuous for all v ∈ R d . Since ∂Ω is compact, letting B 0 = min x∈∂Ω {v∈Σ x + ,|v|≤1} M (x, v)(v ⊥ ) 2 dv > 0,
we deduce from (37), Step 1 and since β(x) ≥ β 0 that

T 0 Σ + |v ⊥ |γ + |S t f |(x, v) dv dζ(x) dt ≤ 1 B 0 β 0 C(1 + T )∥f ∥ L 1 ,
which concludes the proof of (33).

Step 3: Proof of [START_REF] Erd®s | Linear Boltzmann Equation as the Weak Coupling Limit of a Random Schrödinger Equation[END_REF]. Using the boundary condition and Tonelli's theorem,

T 0 {(x,v)∈Σ -,|v|≤1} |v ⊥ | 2 γ -|S t f |(x, v) dv dζ(x) dt = T 0 Σ + |u ⊥ | γ + |S t f |(x, u) {v∈Σ x -,|v|≤1} |v ⊥ | 2 R(u → v; x) dv du dζ(x) dt
and nally we obtain from Step 1

T 0 {(x,u)∈Σ + ,|u|≤Λ} |u ⊥ | γ + |S t f |(x, u) {v∈Σ x -,|v|≤1} |v ⊥ | 2 R(u → v; x) dv du dζ(x) dt ≤ T 0 Σ + |u ⊥ | γ + |S t f |(x, u) {v∈Σ x -,|v|≤1} |v ⊥ | 2 R(u → v; x) dv du dζ(x) dt ≤ C(1 + T )∥f ∥ L 1 , (38)
where we used that {(x, u) ∈ Σ + , |u| ≤ Λ} ⊂ Σ + and the positivity of the integrand. We claim that there exists c Λ > 0 such that for all (x, u) ∈ Σ + with |u| ≤ Λ,

J u,x := {v∈Σ x -,|v|≤1} |v ⊥ | 2 R(u → v; x) dv ≥ c Λ .
Indeed,

J u,x = {v∈Σ x -,|v|≤1} |v ⊥ | 2 θ(x)r ⊥ (2πθ(x)r ∥ (2 -r ∥ )) d-1 2 exp - |v ⊥ | 2 2θ(x)r ⊥ - (1 -r ⊥ )|u ⊥ | 2 2θ(x)r ⊥ × I 0 (1 -r ⊥ ) 1 2 u ⊥ • v ⊥ θ(x)r ⊥ exp - |v ∥ -(1 -r ∥ )u ∥ | 2 2θ(x)r ∥ (2 -r ∥ ) dv,
and, since x → n x and x → θ(x) are continuous, (x, u) → J u,x is continuous with J u,x > 0 on the compact set {(x, u) ∈ Σ + , |u| ≤ Λ}. Therefore, there exists c Λ > 0 such that for all (x, u) ∈ Σ + with |u| ≤ Λ,

J u,x ≥ c Λ .
Note that, for any given Λ, the value of c Λ can be computed explicitly. Inserting this into (38), we nd

c Λ T 0 {(x,v)∈Σ + ,|v|≤Λ} |v ⊥ | γ + |S t f |(x, v) dv dζ(x) dt ≤ C(1 + T )∥f ∥ L 1
and the conclusion follows by setting C Λ = C c λ > 0. □

In the case (CLBC), we will also need the following result, whose proof in the case Lemma 18] and can be adapted directly to treat any small δ and any L 1 , L 2 > 0. We also emphasize that the proof carries on as long as

δ = 1 4 , α ∈ (1, d + 1) and L 1 = 1, L 2 = d(Ω) is given in [9,
max((1 -r ⊥ ), (1 -r ∥ ) 2 ) < 1, which encompasses the case r ⊥ = 1, r ∥ ̸ = 1 and r ∥ = 1, r ⊥ ̸ = 1. Lemma 16. Let δ ∈ (0, 1 2 ), α ∈ (1, d + 1). Set, for L 1 , L 2 > 0, (x, u) ∈ Σ + , I u,x,L 1 ,L 2 := Σ x - |v ⊥ | L 1 + L 2 + |v| 2 δ α -L 1 + |u| 2 δ α R(u → v; x) dv.
In case (CLBC), for any L 1 , L 2 > 0, for all P > 0, there exists Λ > 0 such that for all x ∈ ∂Ω,

u ∈ Σ x + with |u| ≥ Λ, I u,x,L 1 ,L 2 ≤ -P. ( 39
)
Proof of Proposition 14. It is known, see [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF]Equation (25)] and [START_REF] Esposito | Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law[END_REF] for a detailed derivation, that for all

(x, v) ∈ G, v • ∇ x τ (x, v) = -1. Hence, for all α ∈ (1, d), v • ∇ x m α (x, v) = -α(v • ∇ x τ (x, -v))m α-1 = -α m α-1 . (40)
In the whole proof, we write Case (1) and ( 2) when we wish to distinguish between the proof of ( 29) and the one of (30), respectively.

Step

1. Let α ∈ (1, d), f ∈ L 1 mα (G).
We dierentiate the m α -norm of f . First, since n x is the unit outward normal at x ∈ ∂Ω, for T > 0, we apply Green's formula to nd

d dT G |S T f | m α dv dx ≤ G |S T f | (v • ∇ x m α ) dv dx - Σ (v • n x ) m α (γ|S T f |) dv dζ(x) (41) + G m α (x, v) R d k(x, v ′ , v)|S T f |(x, v ′ ) dv ′ dv dx - G m α (x, v)|S T f |(x, v) R d k(x, v, v ′ ) dv ′ dv dx.
By Theorem 13,

|γS t |f (x, v) = γ|S t f |(x, v), a.e. in (R * + × Σ + ) ∪ (R * + × Σ -)
, hence, we will not distinguish between both values in what follows.

Step 2: First term on the r.h.s. of [START_REF] Han-Kwan | Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium[END_REF]. Using [START_REF] Hairer | Yet Another Look at Harris' Ergodic Theorem for Markov Chains[END_REF], we immediately obtain, for α ∈ (1, d),

G |S T f |(v • ∇ x m α ) dv dx = -α∥S T f ∥ m α-1 . ( 42
)
Step 3: Ante-penultimate term on the r.h.s. of [START_REF] Han-Kwan | Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium[END_REF]. By Tonelli's theorem, for any

α ∈ (1, d) G m α (x, v) R d k(x, v ′ , v)|S T f |(x, v ′ ) dv ′ dv dx = G |S T f |(x, v) R d k(x, v, v ′ )m α (x, v ′ ) dv ′ dv dx. ( 43
)
We claim that sup

(x,v)∈G R d k(x, v, v ′ )m α (x, v ′ ) dv ′ < ∞. ( 44
)
Indeed, we clearly have, for all (x, v) ∈ G,

m α (x, v) ≤ e 2 + d(Ω) |v|c 4 + |v| 2δ α , hence, for all K ≫ 1, R d k(x, v, v ′ )m α (x, v ′ ) dv ′ ≤ R d k(x, v, v ′ ) e 2 + d(Ω) |v ′ |c 4 + |v ′ | 2δ α dv ′ ≤ {v ′ ∈R d :|v ′ |≤K} k(x, v, v ′ ) e 2 + K 2δ + d(Ω) |v ′ |c 4 α dv ′ + {v ′ ∈R d :|v ′ |≥K} k(x, v, v ′ ) e 2 + d(Ω) c 4 K + |v ′ | 2δ α dv ′ . Since α ∈ (1, d), by convexity, (x+y) α ≤ C α (x α +y α ) for some constant C α > 0 for all x, y ∈ R + . Thus, R d k(x, v, v ′ )m α (x, v ′ ) dv ′ ≤ C K,α,d(Ω),c 4 σ(x) + {|v ′ |≤K} k(x, v, v ′ ) 1 |v ′ | α dv ′ + {|v ′ |≥K} |v ′ | 2δα k(x, v, v ′ ) dv ′ ≤ C σ ∞ + k ∞ + zM δ k
where we use Hypothesis 2, the fact that 2αδ ≤ 2δ k by choice of δ, and that, by a change to hyperspherical coordinates,

{v∈R d ,|v|≤K} 1 |v| α dv = C d K 0 1 r α+1-d dr ≤ C d,α,K since α + 1 -d < 1
by assumption on α. This concludes the proof of [START_REF] Kim | The Boltzmann Equation with Specular Boundary Condition in Convex Domains[END_REF]. Injecting this into (43), we nd, for some constant C independent of T and f , using the

L 1 contraction G m α (x, v) R d k(x, v ′ , v)|S T f |(x, v ′ ) dv ′ dv dx ≤ C∥S T f ∥ L 1 ≤ C∥f ∥ L 1 . ( 45
)
Step 4: Control of the last term on the r.h.s. of [START_REF] Han-Kwan | Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium[END_REF]. Case (1): Since k is non-negative, for any α ∈ (1, d), one easily obtains

- G m α (x, v)|S T f |(x, v) R d k(x, v, v ′ ) dv ′ dv dx ≤ 0. ( 46
)
Case (2): We use Hypothesis 3. We simply obtain

- G σ(x)m α (x, v)|S T f |(x, v) dv dx ≤ -σ 0 ∥S T f ∥ mα . ( 47
)
Step 5: Control of the boundary term in [START_REF] Han-Kwan | Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium[END_REF]. Let

B := - Σ (v • n x )m α (x, v) γ|S T f | dv dζ(x).
We show that, in case (CLBC), for some Λ > 0, [START_REF] Kuo | Free Molecular Flow with Boundary Eect[END_REF] while in the case (MBC),

B ≤ C Λ {(x,v)∈Σ + ,|v|≤Λ} γ + |S T f |(x, v) |v ⊥ | dv dζ(x),
B ≤ C Σ + γ + |S T f |(x, v) |v ⊥ | dv dζ(x). ( 49
)
This step is divided into two further substeps, the rst one treating [START_REF] Kuo | Free Molecular Flow with Boundary Eect[END_REF], the second one focusing on (49).

Step 5.1: case (CLBC). By denition of B, B = -

Σ + γ + |S T f | |v ⊥ | m α (x, v) dv dζ(x) + Σ - γ -|S T f | |v ⊥ | m α (x, v) dv dζ(x) =: -B 1 + B 2 ,
the last equality standing for a denition of B 1 and B 2 . Using the boundary condition and Tonelli's theorem, it is straightforward to see that

B 2 = Σ + γ + |S T f |(x, u) |u ⊥ | Σ x - m α (x, v) |v ⊥ | R(u → v; x) dv du dζ(x).
Set, for all x ∈ ∂Ω, u ∈ Σ x + ,

P u,x := Σ x - m α (x, v) |v ⊥ | R(u → v; x) dv.
We will split the integral in P u,x between an integral on {v ∈ Σ x -, |v| ≤ 1} and one on the set {v ∈ Σ x -, |v| ≥ 1}. We start with the treatment of the former. Note rst that, for all v ∈ Σ x -, u ⊥ • v ⊥ ≤ 0 so that, using the denition of I 0 (5),

I 0 (1 -r ⊥ ) 1 2 u ⊥ • v ⊥ θ(x)r ⊥ ≤ exp - 2(1 -r ⊥ ) 1 2 u ⊥ • v ⊥ 2θ(x)r ⊥ ,
hence, using θ(x) ≥ θ 0 for some θ 0 > 0 for all x ∈ ∂Ω (by positivity and continuity assumptions)

R(u → v; x) = exp - |v ∥ -(1-r ∥ )u ∥ | 2 2θ(x)r ∥ (2-r ∥ ) (2πθ(x)r ∥ (2 -r ∥ )) d-1 2 exp -|v ⊥ | 2 2θ(x)r ⊥ -(1-r ⊥ )|u ⊥ | 2 2θ(x)r ⊥ θ(x)r ⊥ I 0 (1 -r ⊥ ) 1 2 u ⊥ • v ⊥ θ(x)r ⊥ ≤ 1 (2πθ(x)r ∥ (2 -r ∥ )) d-1 2 exp -|v ⊥ +(1-r ⊥ ) 1 2 u ⊥ | 2 2θ(x)r ⊥ - |v ∥ -(1-r ∥ )u ∥ | 2 2θ(x)r ∥ (2-r ∥ ) θ(x)r ⊥ ≤ 1 θ 0 r ⊥ (2πθ 0 r ∥ (2 -r ∥ )) d-1 2 ≤ C,
where we used the upper bound 1 for both exponentials. We clearly have, for all (x, v) ∈ Ḡ,

m α (x, v) ≤ e 2 + d(Ω) c 4 |v| + |v| 2δ α
by denition of m α , and using that |v ⊥ | ≤ |v|, we get

{v∈Σ x -,|v|≤1} m α (x, v) |v ⊥ | R(u → v; x) dv ≤ {v∈Σ x -,|v|≤1} e 2 + 1 + d(Ω) |v|c 4 α |v ⊥ | R(u → v; x) dv ≤ C {v∈Σ x -,|v|≤1} e 2 + 1 + d(Ω) c 4 |v| α |v| dv ≤ C α
for some constant C α > 0 independent of u and x. We used that α < d to obtain the existence of such nite C α (as can be checked by using an hyperspherical change of variable, see Step 3). On the other hand,

{v∈Σ x -,|v|≥1} m α (x, v) |v ⊥ | R(u → v; x) dv ≤ {v∈Σ x -,|v|≥1} e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv ≤ Σ x - e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv.
Overall, we proved that

P u,x ≤ C α + Σ x - e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv. ( 50
)
On the other hand, for all (x, u) 6) and ( 26). Since we

∈ Σ + , Σ x - |v ⊥ | R(u → v; x) dv = 1 by (
also have τ (x, -v) ≤ d(Ω)/|v| and c 4 < 1, d(Ω) c 4 |v| -τ (x, -v) ≥ 0, so that m α (x, u) ≥ (e 2 + |u| 2δ ) α , we get -B 1 ≤ - Σ + |u ⊥ | γ + |S T f |(x, u) (e 2 + |u| 2δ ) α Σ x - |v ⊥ | R(u → v; x) dv du dζ(x). (51)
Gathering ( 50), ( 51) and the denition of B, we nd

B ≤ Σ + |u ⊥ | γ + |S T f |(x, u) × C α + Σ x - e 2 + d(Ω) c 4 + |v| 2δ α -(e 2 + |u| 2δ ) α |v ⊥ | R(u → v; x) dv du dζ(x) ≤ Σ + |u ⊥ | γ + |S T f |(x, u) C α + I u,x,e 2 , d(Ω) c 4 du dζ(x),
where I u,x,e 2 ,d(Ω)/c 4 is dened as in Lemma 16 with δ = δ, α as before and Lemma 16 applied with P = C α , we nd that

L 1 = e 2 , L 2 = d(Ω) c 4 . Splitting Σ x + as Σ x + = u ∈ Σ x + : |u| < Λ ∪ u ∈ Σ x + : |u| ≥ Λ with Λ > 0 given by
{(x,u)∈Σ + ,|u|≥Λ} |u ⊥ | |γ + S T f |(x, u) C α + I u,x,e 2 , d(Ω) c 4 du dζ(x) ≤ 0, (52) leading to B ≤ ∂Ω {u∈Σ x + ,|u|≤Λ} |u ⊥ | |γ + S T f |(x, u) C α + I u,x,e 2 , d(Ω) c 4 du dζ(x) ≤ ∂Ω {u∈Σ x + ,|u|≤Λ} |u ⊥ | |γ + S T f |(x, u) × C α + Σ x - e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv du dζ(x). (53)
We claim that

sup x∈∂Ω,u∈Σ x + ,|u|≤Λ Σ x - e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv ∈ (0, ∞). (54)
Note that the proof of (48) directly follows from this claim and [START_REF] Maxwell | On Stresses in Raried Gases Arising from Inequalities of Temperature[END_REF]. It thus only remains to prove [START_REF] Mouhot | Quantitative Perturbative Study of Convergence to Equilibrium for Collisional Kinetic Models in the Torus[END_REF]. Let x ∈ ∂Ω, u ∈ Σ x + with |u| ≤ Λ. For simplicity we will rely on the probabilistic tools introduced in [9, Section 2.2]. We may write the integral inside the supremum as

E e 2 + d(Ω) c 4 + |X| 2 + |Y | 2 δ α , for Y ∼ Ri((1 -r ⊥ ) 1 2 |u ⊥ |, θ(x)r ⊥ ) a Rice distribution (see [9, Denition 12]) of parameters (1 -r ⊥ ) 1 2 |u ⊥ | and θ(x)r ⊥ , and X ∼ N ((1 -r ∥ )u ∥ , θ(x)r ∥ (2 -r ∥ )I d-1
) a Gaussian random variable, where I k denotes the identity matrix of size k × k. Using [9, Proposition 13] (taken from [START_REF] Kobayashi | Probability, Random Processes, and Statistical Analysis[END_REF]), we have, for any ϑ ∈ [0, 2π),

Y L = Y 2 1 + Y 2 2 with Y 1 ∼ N (1 -r ⊥ ) 1 2 |u ⊥ | cos(ϑ), θ(x)r ⊥ , Y 2 ∼ N (1 -r ⊥ ) 1 2 |u ⊥ | sin(ϑ), θ(x)r ⊥
two random variables independent of everything else. The integral thus rewrites

E e 2 + d(Ω) c 4 + |X| 2 + |Y 1 | 2 + |Y 2 | 2 δ α ,
and is nite for any (x, u) in Σ + with |u| ≤ Λ by standard property of the moments of Gaussian random variables. Since x → n x and x → θ(x) are continuous, for

(x n , u n ) → (x, u) in {(x, u) ∈ Σ + , |u| ≤ Λ}, v ∈ R d , lim n→∞ 1 {v•nx n <0} |v • n xn |R(u n → v; x n ) = 1 {v•nx<0} |v • n x |R(u → v; x)
almost everywhere, hence

(x, u) → Σ x - e 2 + d(Ω) c 4 + |v| 2δ α |v ⊥ | R(u → v; x) dv
is continuous by dominated convergence theorem. Since {(x, u) ∈ Σ + , |u| ≤ Λ} is compact, the proof of (54) is complete.

Step 5.2: case (MBC). We prove here [START_REF] Kuo | Equilibrating Eects of Boundary and Collision in Rareed Gases[END_REF]. Using the boundary condition, we have

B = Σ - |v ⊥ |m α (x, v)β(x)M (x, v) Σ x + |v ′ • n x ||S T f |(x, v ′ ) dv ′ dv dζ(x) + Σ - |v ⊥ |(1 -β(x))m α (x, v)|S T f |(x, η x (v)) dv dζ(x) - Σ + |v ⊥ |m α (x, v)|S T f | dv dζ(x) ≤ Σ - |v ⊥ |m α (x, v)β(x)M (x, v) Σ x + |v ′ • n x ||S T f |(x, v ′ ) dv ′ dv dζ(x) + Σ + |v ⊥ ||S T f |(x, v) (1 -β 0 )m α (x, η x (v)) -m α (x, v) dv dζ(x),
where we used the change of variable v → η x (v) with Jacobian 1 and that |v ⊥ | = |η x (v) • n x | to get the inequality. Let us focus on the last term on the right-hand side of this inequality. For all

(x, v) ∈ Σ + , since c 4 ∈ (0, 1), (1 -c 4 ) 4 = (1 -β 0 ), α < d < 4, τ (x, -η x (v)) = 0 and |η x (v)| = |v|, (1 -β 0 )m α (x, η x (v)) = (1 -c 4 ) 4 e 2 + d(Ω) |v|c 4 + |v| 2δ α ≤ (1 -c 4 ) e 2 + d(Ω) |v|c 4 + |v| 2δ α ≤ e 2 + d(Ω) c 4 |v| - d(Ω) |v| + |v| 2δ α ≤ e 2 + d(Ω) c 4 |v| -τ (x, -v) + |v| 2δ α = m α (x, v),
where we used τ (x, -v) ≤ d(Ω)/|v| by denition of d(Ω). We get, as a rst conclusion,

B ≤ Σ - |v ⊥ |m α (x, v)β(x)M (x, v) Σ x + |v ′ • n x ||S T f |(x, v ′ ) dv ′ dv dζ(x) ≤ Σ + |v ⊥ ||S T f |(x, v) Σ x - |u ⊥ |m α (x, u)M (x, u) du dv dζ(x).
Observing that sup x,v∈∂Ω×R d M (x, v) < ∞ and that sup x∈∂Ω R d |v| r M (x, v) dv < ∞ for all r > 0, one can split the integral over v ∈ R d exactly as in the proof of [START_REF] Kim | The Boltzmann Equation with Specular Boundary Condition in Convex Domains[END_REF]. It follows that

sup x∈∂Ω Σ x - |v ′ • n x |M (x, v ′ )m α (x, v ′ ) dv ′ ≤ C,
which concludes the proof of (49).

Step 6: conclusion under Hypotheses 1 and 2. Using ( 42), ( 45), [START_REF] Kobayashi | Probability, Random Processes, and Statistical Analysis[END_REF] and Step 5 inside (41), we obtain, for all α ∈ (1, d), for Λ > 0 given by Step 5, for some constant C > 0 allowed to depend on α, d, Λ, d dT

∥S T f ∥ mα ≤ -α∥S T f ∥ m α-1 + C∥f ∥ L 1 + CB T ,
where,

B T = Σ + |v ⊥ |γ + |S T f |(x, v) dv dζ(x) case (MBC), {(x,v)∈Σ + ,|v|≤Λ} γ + |S T f |(x, v)|v ⊥ | dv dζ(x) case (CLBC).
Integrating this inequality on [0, T ], we nd

∥S T f ∥ mα + α T 0 ∥S s f ∥ m α-1 ds ≤ ∥f ∥ mα + CT ∥f ∥ L 1 + C T 0 B s ds.
The conclusion follows by noticing that Lemma 15 implies, in both cases (MBC) and (CLBC)

T 0 B s ds ≤ C(1 + T )∥f ∥ L 1 .
Step 7: conclusion under Hypotheses 1-3. Using ( 42), ( 45), [START_REF] Kuo | Equilibrating Eect of Maxwell-Type Boundary Condition in Highly Rareed Gas[END_REF] and Step 5 inside (41), throwing away the negative term from [START_REF] Hérau | Hypocoercivity and Exponential Time Decay for the Linear Inhomogeneous Relaxation Boltzmann Equation[END_REF], for all α ∈ (1, d), for Λ > 0 given by Step 5, for some constant C > 0 allowed to depend on α, d, Λ,

d dT ∥S T f ∥ mα ≤ -σ 0 ∥S T f ∥ mα + C∥f ∥ L 1 + CB T ,
where again,

B T = Σ + |v ⊥ |γ + |S T f |(x, v) dv dζ(x) case (MBC), {(x,v)∈Σ + ,|v|≤Λ} γ + |S T f |(x, v)|v ⊥ | dv dζ(x) case (CLBC).
Integrating this inequality on [0, T ], we nd

∥S T f ∥ mα + σ 0 T 0 ∥S s f ∥ mα ds ≤ ∥f ∥ mα + CT ∥f ∥ L 1 + C T 0 B s ds.

The conclusion follows again by noticing that Lemma 15 implies

T 0 B s ds ≤ C(1 + T )∥f ∥ L 1 . □ 5.
Proof of Theorems 2 and 3

In this section, we rst prove a Doeblin-Harris condition, that, along with the Lyapunov conditions obtained in Section 4, provide the proof of Theorem 2. Theorem 3 then follows by a usual Cauchy sequence argument, and by applying Theorem 2. Recall the denitions of τ and q from (10) and ( 31) respectively. 5.1. Doeblin-Harris condition. We start with the proof of the Doeblin-Harris condition. The key point is that, since σ ∈ L ∞ (Ω), one can nd a natural lower bound of the dynamics by the free-transport ones. For this, we establish rst a Duhamel formula. Lemma 17. For all f ∈ L 1 (G), for all (t, x, v) ∈ R + × Ḡ, the following formula holds:

S t f (x, v) = 1 {τ (x,-v)≤t} e -t t-τ (x,-v) σ(x-(t-s)v) ds S t-τ (x,-v) f q(x, -v), v (55)
+ 1 {t<τ (x,-v)} e -t 0 σ(x-(t-s)v) ds f (x -tv, v) + t max(0,t-τ (x,-v)) e -t s σ(x-(t-u)v) du × R d k x -(t -s)v, v ′ , v S s f x -(t -s)v, v ′ dv ′ ds.
As a consequence, for all f ∈ L 1 (G) with f ≥ 0, for all (x, v) ∈ Ḡ,

S t f (x, v) ≥ 1 {τ (x,-v)≤t} e -τ (x,-v) 0 σ(x-sv) ds S t-τ (x,-v) f q(x, -v), v . (56)
Proof. For (x, v) ∈ Σ -∪ Σ 0 , we have τ (x, -v) = 0 and q(x, -v) = x so that the formula is obviously true.

Step 1. Consider the problem

     ∂ t g + v • ∇ x g + σ(x)g = 0, in R + × G, γ -g = Kγ + g, on R + × Σ -, g(0, x, v) = g 0 (x, v), in G, (57)
with K given by (3). Problem ( 57) is a bounded perturbation of the corresponding free-transport problem, and therefore (see the proof of Theorem 13, Step 1) admits a unique solution g such that for all

g 0 ∈ L 1 (G), t ≥ 0, g(t, •) is the unique solution in L ∞ ([0, ∞); L 1 (G)
) to the equation taken at time t. We write (e tT ) t≥0 for the associated C 0 -stochastic semigroup. Assume rst that (e tT ) t≥0 satises, for

(t, x, v) ∈ R + × Ḡ, e tT g (x, v) = e -t 0 σ(x-(t-s)v) ds g(x -tv, v)1 {t<τ (x,-v)} (58) + e -t t-τ (x,-v) σ(x-(t-s)v) ds e (t-τ (x,-v))T g x -τ (x, -v)v, v 1 {t≥τ (x,-v)} .
Adding the source operator C + and setting s = max(0, t -τ (x, -v)), we obtain a solution of the form

f (t, x, v) = e (t-s)T f (s, •, •) (x, v) + t s e (t-u)T C + f (u, •, •) (x, v) du which rewrites as f (t, x, v) = e -t s σ(x-(t-u)v) du f (s, x -(t -s)v, v) + t s e -t u σ(x-(t-r)v) dr R d k x -(t -u)v, v ′ , v f (u, x -(t -u)v, v ′ ) dv ′ du.
Expanding on the two possible values of max(0, t -τ (x, -v)) and recalling that, by denition, q(x, -v) = x -τ (x, -v)v concludes the proof. Moreover, [START_REF] Nguyen | Variational Derivation of Thermal Slip Coecients on the Basis of the Boltzmann Equation for Hard-Sphere Molecules and Cercignani-Lampis Boundary Conditions: Comparison With Experimental Results[END_REF] follows by using that (S t ) t≥0 is a non-negative semigroup and a change of variable in the integral inside the exponential.

Step 2. We prove [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Dierential Equations[END_REF]. We keep, for all t ≥ 0, the notation g(t, •, •) for the unique solution at time t of (57) in the remaining part of the proof. Note that, as a solution in L 1 (G), g solves [START_REF] Pantazis | Gas-Surface Scattering Eect on Vacuum Gas Flows Through Rectangular Channels[END_REF] in the sense of distributions. To prove [START_REF] Pazy | Semigroups of Linear Operators and Applications to Partial Dierential Equations[END_REF], we consider a test function

ϕ ∈ C ∞ c ([0, ∞) × Ḡ). Then ∞ 0 G ϕ(t, x, v)g(t, x, v) dv dx dt = ∞ 0 G ϕ(t, x, v) t max(0,t-τ (x,-v)) d ds g s, x -(t -s)v, v e -t s σ(x-(t-u)v)du ds dv dx dt + ∞ 0 G ϕ(t, x, v) g max 0, t -τ (x, -v) , x -t -max(0, t -τ (x, -v) )v, v × e -t max(0,t-τ (x,-v)) σ(x-(t-u)v) du dv dx dt.
Expanding the bracket in the rst term on the right-hand side gives

d ds g s, x -(t -s)v, v e -t s σ(x-(t-u)v)du = ∂ s + v • ∇ x + σ g (s, x -(t -s)v, v)e -t s σ(x-(t-u)v)du = 0,
since g is a solution in the sense of distributions of [START_REF] Pantazis | Gas-Surface Scattering Eect on Vacuum Gas Flows Through Rectangular Channels[END_REF]. This concludes the proof of (58) in the sense of distributions and the conclusion in L 1 follows by density. □

Recall that δ ∈ (0, 1 2 ) is xed and set, for all (x, v) ∈ Ḡ, ⟨x, v⟩ := (1 + τ (x, v) + |v| 2δ ). Our rst Doeblin-Harris condition is the following. Theorem 18. Under Hypotheses 1 and 2, for any Λ ≥ 2, there exist T (Λ) > 0 and a nonnegative measure ν on G, depending on Λ, with ν ̸ ≡ 0, such that for all (x, v) ∈ G, for all

f 0 ∈ L 1 (G), f 0 ≥ 0, S T (Λ) f 0 (x, v) ≥ ν(x, v)
{(y,w)∈G,⟨y,w⟩≤Λ} f 0 (y, w) dy dw. [START_REF] Sharipov | Application of the Cercignani-Lampis Scattering Kernel to Calculations of Rareed Gas Flows. I. Plane Flow Between Two Parallel Plates[END_REF] Moreover, ν satises ⟨ν⟩ ≤ 1 and there exists ξ > 0 such that for all Λ ≥ 2, T (Λ) = ξΛ.

The proof follows from a direct adaptation of the one of [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]Theorem 21]. We only give the rst step to emphasize how the latter should be modied. Sketch of proof. For all t > 0, (x, v) ∈ Ḡ, we write f (t, x, v) = S t f 0 (x, v). For the sake of simplicity we simply write f (t, x, v) for γf (t, x, v) for (t, x, v) ∈ R + × Σ.

We let (t, x, v) ∈ (0, ∞) × G and compute a rst lower-bound for f (t, x, v). Recall the denitions of τ from [START_REF] Bernou | Hypocoercivity for Kinetic Linear Equations in Bounded Domains With General Maxwell Boundary Condition[END_REF] and q from [START_REF] Duan | Global Mild Solutions of the Landau and Non-Cuto Boltzmann Equations[END_REF]. By [START_REF] Nguyen | Variational Derivation of Thermal Slip Coecients on the Basis of the Boltzmann Equation for Hard-Sphere Molecules and Cercignani-Lampis Boundary Conditions: Comparison With Experimental Results[END_REF], we have

f (t, x, v) ≥ e -τ (x,-v) 0 σ(x-sv) ds f (t -τ (x, -v), q(x, -v), v)1 {t≥τ (x,-v)} .
Set y 0 = q(x, -v), τ 0 = τ (x, -v). We have, using the boundary conditions and (56) again,

f (t, x, v) ≥ 1 {τ 0 ≤t} e -σ∞τ 0 f (t -τ 0 , y 0 , v) ≥ 1 {τ 0 ≤t} e -σ∞τ 0 Σ y 0 + f (t -τ 0 , y 0 , v 0 ) |v 0 • n y 0 | R(v 0 → v; y 0 ) dv 0 ≥ 1 {τ 0 ≤t} e -σ∞τ 0 Σ y 0 + e -σ∞τ (y 0 ,-v 0 ) f (t -τ 0 -τ (y 0 , -v 0 ), q(y 0 , -v 0 ), v 0 ) × 1 {τ 0 +τ (y 0 ,-v 0 )≤t} |v 0 • n y 0 | R(v 0 → v; y 0 ) dv 0 ≥ 1 {τ 0 ≤t} e -σ∞τ 0 Σ y 0 + 1 {τ 0 +τ (y 0 ,-v 0 )≤t} |v 0 • n y 0 | R(v 0 → v; y 0 ) × e -σ∞τ (y 0 ,-v 0 ) Σ q(y 0 ,-v 0 ) + |v 1 • n q(y 0 ,-v 0 ) | R v 1 → v 0 ; q(y 0 , -v 0 ) × f (t -τ 0 -τ (y 0 , -v 0 ), q(y 0 , -v 0 ), v 1 ) dv 1 dv 0 .
From there, the proof in case (CLBC) (including in the case (r ⊥ , r ∥ ) = (1, 1)) is a straightforward adaptation of [9, Proof of Theorem 21], the only dierence being the presence of extra constants e -σ∞τ i for various time intervals τ i appearing from the repeated use of the Duhamel formula [START_REF] Mouhot | The Princeton Companion to Applied Mathematics, chapter Areas of Applied Mathematics: Kinetic Theory[END_REF]. Those are easy to treat since the proof ultimately uses a truncation of the space of integration of those times on a nite interval.

Since, in case (CLBC) with (r ⊥ , r ∥ ) = 1, for all x ∈ ∂Ω, u ∈ Σ x + , v ∈ Σ x -, one has the equality R(u → v; x) = M (x, v), the proof also allows to handle the case (MBC), by bounding from below all terms related to the specular reection by 0 and using that β(x) ≥ β 0 > 0 by Hypothesis 1. □

Remark 19 (Constructive property of ν). As in [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]Remark 22] and [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF]Remark 8], even though some compactness arguments are used in the previous proof, constructive lower bounds can be derived at least in the simple case where Ω is the unit disk, see [START_REF] Bernou | A Semigroup Approach to the Convergence Rate of a Collisionless Gas[END_REF]Remark 8]. Note that the control of the additional factors due to the jumps (the ones such as e -σ∞τ 0 ) are explicit and do not rely on a compactness argument. In general, we thus expect to be able to nd a constructive lower bound for any given Ω.

The following corollary allows us to relate the Doeblin-Harris condition with the weights used in Section 4.

Corollary 20. Under Hypotheses 1 and 2, there exists Λ 0 > 0 such that for all Λ ≥ Λ 0 , there exist T (Λ) > 0 and a non-negative measure ν on G, depending on Λ, with ν ̸ ≡ 0, such that for all (x, v) ∈ G, for all

f 0 ∈ L 1 (G), f 0 ≥ 0, S T (Λ) f 0 (x, v) ≥ ν(x, v) D Λ f 0 (y, w) dy dw, (60)
where D Λ = {(y, w) ∈ G : m 1 (y, w) ≤ Λ}. Moreover, ν satises ⟨ν⟩ ≤ 1 and there exists ξ > 0 such that for all Λ ≥ 2, T (Λ) = ξΛ.

Proof. For all x ∈ G, m 1 (x, v) → ∞ as |v| → ∞. Hence there exists Λ 0 > 0 such that, denoting by λ the Lebesgue measure on

R d × R d , λ{(y, w) ∈ G, m 1 (y, w) ≤ Λ 0 } > 0. Since c 4 < 1 and τ (x, v) + τ (x, -v) ≤ d(Ω) |v| ≤ d(Ω) |v|c 4
by denition of τ , we have

m 1 (x, v) = e 2 + d(Ω) |v|c 4 -τ (x, -v) + |v| 2δ ≥ 1 + τ (x, v) + |v| 2δ = ⟨x, v⟩.
Hence, for Λ ≥ Λ 0 , we have D Λ ⊂ {(x, v) ∈ G : ⟨x, v⟩ ≤ Λ} and D Λ ̸ = ∅. The conclusion then follows from Theorem 18. □ 5.2. Proof of Theorem 2. From the Lyapunov conditions, Proposition 14 and the Doeblin-Harris condition, Corollary 20, the proof of Theorem 2 follows from Harris-type theorems. We assume for simplicity that g ≡ 0, so that f ∈ L 1 mα (G) with ⟨f ⟩ = 0 in what follows. More precisely, the demonstration of the polynomial result ( 12) is obtained exactly as for the free-transport case [9, Section 5]: note that this proof only uses that the semigroup is stochastic, that the weights considered are superlinear, and that the subgeometric Lyapunov inequality [START_REF] Dolbeault | Hypocoercivity for Linear Kinetic Equations Conserving Mass[END_REF] and the Doeblin-Harris condition (60) hold. In fact, the only dierence is that we use, for (x, v) ∈ G, α ∈ (1, d), weights of the form For completeness, and because the argument is less redundant with the one in [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF] in this case, we provide a proof of [START_REF] Bodineau | The Brownian Motion as the Limit of a Deterministic System of Hard-Spheres[END_REF]. We use the approach of [17, Proof of Theorem 3.2]. We start with the following lemma: Lemma 21. Under Hypothesis 1-3, for all α ∈ (1, d), there exist T > 0, µ > 0, γ 0 ∈ (0, 1) such that, setting

m α (x, v) = e 2 + d(Ω) |v|c 4 -τ (x, -v) + |v| 2δ α rather than w α (x, v) = (1 + τ (x, v) + |v| 2δ ) α ,
|||•||| µ := ∥ • ∥ L 1 + µ∥ • ∥ mα , one has, for all f ∈ L 1 mα (G) with ⟨f ⟩ = 0, |||S T f ||| µ ≤ γ 0 |||f ||| µ . (61)
Proof of Lemma 21. Step 1: Reformulation of the Lyapunov inequality. Let t > 0. Recall that α ∈ (1, d) is given. By Proposition 14, and more precisely equation [START_REF] Duan | Eects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rareed Gas[END_REF], we have, for any f ∈ L 1 mα (G),

∥S t f ∥ mα + σ 0 t 0 ∥S s f ∥ mα ds ≤ ∥f ∥ mα + K 2 (1 + t)∥f ∥ L 1 . (62)
Note that in particular, for all s ∈ (0, t)

∥S t-s S s f ∥ mα ≤ ∥S s f ∥ mα + K 2 (1 + t -s)∥S s f ∥ L 1 which rewrites ∥S t f ∥ mα -K 2 (1 + t -s)∥S s f ∥ L 1 ≤ ∥S s f ∥ mα , (63)
and injecting ( 63) inside [START_REF] Yamaguchi | Mass Flow Rate Measurement of Thermal Creep Flow From Transitional to Slip Flow Regime[END_REF] gives

∥S t f ∥ mα + σ 0 t 0 ∥S t f ∥ mα -K 2 (1 + t -s)∥S s f ∥ L 1 ds ≤ ∥f ∥ mα + K 2 (1 + t)∥f ∥ L 1 .
Using also the L 1 contraction from Theorem 13, we obtain

∥S t f ∥ mα ≤ 1 1 + σ 0 t ∥f ∥ mα + K 2 1 + σ 0 t σ 0 t 2 2 + (1 + σ 0 )t + 1 ∥f ∥ L 1 ,
and ultimately, for some constant K 3 > 0,

∥S t f ∥ mα ≤ 1 1 + σ 0 t ∥f ∥ mα + K 3 (1 + t)∥f ∥ L 1 . (64)
We note that the combination of (64) and Theorem 18 already ts [17, Section 3], so that one can readily apply their results. To facilitate the task of the reader, we nevertheless present a proof starting from those two results, in particular because our Doeblin-Harris condition, Corollary 20 is slightly non-standard.

Step 2: describing two alternatives. According to Corollary 20, for all ρ > 2, there exists T (ρ) = ξρ for some constant ξ > 0 and a non-negative measure ν on G with ν ̸ ≡ 0, ⟨ν⟩ ≤ 1 such that

S T (ρ) h ≥ ν {(x,v)∈G,m 1 (x,v)≤ρ} h dv dx, for all h ∈ L 1 (G) with h ≥ 0.
By assumption, f ∈ L 1 mα (G) and ⟨f ⟩ = 0. We set, for any ρ > 0, κ(ρ

) = K 3 (1 + T (ρ)). Since T (ρ) = ξρ for some constant ξ > 0, κ(ρ) ∼ ρ→∞ Cρ for some C > 0. Since α ∈ (1, d), one can nd ρ 0 such that, for all ρ > ρ 0 , T (ρ) > 1, κ(ρ) > 1 and ρ α ≥ 4κ(ρ) 1- 1 1+σ 0
. We x ρ > ρ 0 , T = T (ρ) > T (ρ 0 ) =: T 0 for the remaining part of the proof. Note that, since T (ρ) = ξρ for some constant ξ, any choice of T > T (ρ 0 ) is possible. We set A := ρ α 4 and dene, for µ > 0 to be chosen, the µ-norm by

|||f ||| µ := ∥f ∥ L 1 + µ∥f ∥ mα .
We distinguish two cases. Indeed, we have the alternative:

∥f ∥ mα ≤ A∥f ∥ L 1 , (65a) or ∥f ∥ mα > A∥f ∥ L 1 . (65b)
Step 3: alternative (65a). We prove a convergence result in the µ-norm in the case of the rst alternative, (65a). Set, for all Λ > 0,

D Λ = {(x, v) ∈ G, m 1 (x, v) ≤ Λ}. Using ⟨f ⟩ = 0, that m α ≡ m α
1 and Corollary 20, we have, for all (x, v) ∈ G,

S T f ± (x, v) ≥ ν(x, v) G f ± (x ′ , v ′ ) dv ′ dx ′ -ν(x, v) D c ρ f ± (x ′ , v ′ ) dv ′ dx ′ ≥ ν(x, v) 2 G |f (x ′ , v ′ )| dv ′ dx ′ -ν(x, v) D c ρ |f (x ′ , v ′ )| dv ′ dx ′ ≥ ν(x, v) 2 G |f (x ′ , v ′ )| dv ′ dx ′ - ν(x, v) ρ α G |f (x ′ , v ′ )|m α (x ′ , v ′ ) dv ′ dx ′ ≥ ν(x, v) 2 G |f (x ′ , v ′ )| dv ′ dx ′ - ν(x, v) 4 G |f (x ′ , v ′ )| dv ′ dx ′ = ν(x, v) 4 G |f (x ′ , v ′ )| dv ′ dx ′ =: ν(x, v),
where the third inequality is given by the fact that

D c ρ = {(x, v) ∈ G, m α (x, v)/ρ α ≥ 1}.
The last inequality is obtained by condition (65a). The nal equality stands for a denition of ν(x, v) for all (x, v) ∈ G. Note that ν ≥ 0 on G. We deduce,

|S T f | = |S T f + -ν -(S T f --ν)| ≤ |S T f + -ν| + |S T f --ν| = S T f + + S T f --2ν = S T |f | -2ν,
and, integrating over G, we have, using the contraction property, that ν = ν 4 ∥f ∥ L 1 , and that ν is non-negative with ⟨ν⟩ ≤ 1,

(66) ∥S T f ∥ L 1 ≤ ∥f ∥ L 1 -2∥ν∥ L 1 = 1 - ⟨ν⟩ 2 ∥f ∥ L 1 = ν∥f ∥ L 1 ,
with ν ∈ (0, 1). Hence, S T is a strict contraction in L 1 in the case where f satises (65a). Writing γ := 1/(1 + σ 0 T ) < 1 in (64) and using the denition of κ(ρ), we derive an inequality on the

µ-norm of S T f , |||S T f ||| µ = ∥S T f ∥ L 1 + µ∥S T f ∥ mα ≤ ν∥f ∥ L 1 + µ γ∥f ∥ mα + κ(ρ)∥f ∥ L 1 ≤ ν + µκ(ρ) ∥f ∥ L 1 + µγ∥f ∥ mα .
Finally, we choose 0 < µ ≤ 1-ν 2κ < 1 and deduce (67)

|||S T f ||| µ ≤ γ 1 |||f ||| µ with γ 1 := min(γµ, 1+ν
2 ) < 1.

Step 4: alternative (65b). By choice of T > 1 and ρ in Step 2, we have, with γ as before,

ρ α 4κ(ρ) > 1 1 -γ . ( 68
)
By choice of A, a direct use of (64) leads to

∥S T f ∥ mα ≤ γ∥f ∥ mα + κ(ρ)∥f ∥ L 1 ≤ γ∥f ∥ mα + κ(ρ) A ∥f ∥ mα ≤ γ + 4 κ(ρ) ρ α ∥f ∥ mα ≤ γ∥f ∥ mα , with 0 < γ := 4κ(ρ)/ρ α + γ < 1 by (68). Hence |||S T f ||| µ = ∥S T f ∥ L 1 + µ∥S T f ∥ mα ≤ ∥f ∥ L 1 + µγ∥f ∥ mα ≤ (1 -µϵ 0 )∥f ∥ L 1 + µ(γ + ϵ 0 )∥f ∥ mα ,
where we used m α ≥ 1 to obtain the last inequality. Using that γ < 1, we can choose ϵ 0 > 0 small enough (µ is xed by the previous step) so that

|||S T f ||| µ ≤ γ 2 |||f ||| µ ,
with γ 2 := min(1 -µϵ 0 , γ + ϵ 0 ) < 1.

Step 5: conclusion. We set γ 0 = max(γ 1 , γ 2 ) < 1 (which depends on our choice of α) to complete the proof. □

From there, the proof follows by a semigroup argument. The extension to α in (0, d) is obtained by an interpolation argument. The key tool for this is the following corollary applicable to spaces of the form {f ∈ L 1 w (G), ⟨f ⟩ = 0} with w ≥ 1 some weight on G. We denote |||H||| A→B the operator norm of H acting between the two Banach spaces A and B.

Corollary 22. [8, Corollary 3] Let ϕ 1 , ϕ 2 , φ1 , φ2 be four measurable functions on G positive almost everywhere. Let also

A 1 = L 1 ϕ 1 (G), A 2 = L 1 ϕ 2 (G), Ã1 = L 1 φ1 (G), Ã2 = L 1 φ2 (G).
Let, for all γ ∈ (0, 1), ϕ γ and φγ be dened by

ϕ γ := ϕ γ 1 ϕ 1-γ 2 , φγ := φγ 1 φ1-γ 2 ,
respectively, and

A γ = L 1 ϕγ (G), Ãγ = L 1 φγ (G).
Assume that there exists a bounded projection

Π : (A i , Ãi ) → (A ′ i , Ã′ i ) for i ∈ {1, 2} with A ′ i ⊂ A i , Ã′ i ⊂ Ãi . Let also A ′ γ = (A ′ 1 + A ′ 2 ) ∩ A γ , Ã′ γ = ( Ã′ 1 + Ã′ 2 ) ∩ Ãγ . Assume that S is a linear operator from A ′ 1 to Ã′ 1 and from A ′ 2 to Ã′ 2 with |||S||| A ′ 1 → Ã′ 1 ≤ N 1 , |||S||| A ′ 2 → Ã′ 2 ≤ N 2 ,
for N 1 , N 2 > 0. Then S is a linear operator from A ′ γ to Ã′ γ and there exists C > 0 depending only on Π such that

|||S||| A ′ γ → Ã′ γ ≤ CN γ 1 N 1-γ 2 .
the existence of T, µ > 0 and γ 0 ∈ (0, 1) such that (61) holds for all f ∈ L 1 mα,0 (G). Since both f ∞ and g ∞ belong to L 1 mα (G), with ⟨f ∞ -g ∞ ⟩ = 0 by linearity, we obtain

|||S T (f ∞ -g ∞ )||| µ ≤ γ 0 |||f ∞ -g ∞ ||| µ . (70) Since S T (f ∞ -g ∞ ) = f ∞ -g ∞
by linearity and since both are steady states, (70) rewrites

|||f ∞ -g ∞ ||| µ ≤ γ 0 |||f ∞ -g ∞ ||| µ .
It follows that |||f ∞ -g ∞ ||| µ = 0, and thus ∥f ∞ -g ∞ ∥ L 1 = 0, which proves the uniqueness.

Step 2: Existence. Set α = d -ϵ, let g ∈ L 1 mα (G) with ⟨g⟩ = 1 and let again T, µ > 0 and γ 0 ∈ (0, 1) given by Lemma 21. Dene, for all h ≥ 1,

g h = S T h g, f h = g h+1 -g h .
Note that for all h ≥ 1, ⟨f h ⟩ = 0 by mass conservation. The inequality (61) applied to f h reads

|||f h+1 ||| µ ≤ γ 0 |||f h ||| µ . (71)
It follows that (|||f h ||| µ ) h∈N * is a non-negative, decreasing sequence converging towards 0. Hence, for 0 < ω ≪ 1 xed, one can set N > 0 such that for all r > N ,

|||f r ||| µ ≤ µ γ 0 (1 -γ 0 )ω. Next, recalling ∥ • ∥ mα ≤ 1 µ |||•||| µ , we have, for q > r > N , µ∥g q+1 -g r+1 ∥ mα = µ q h=r+1 f h mα ≤ µ q-1 h=r ∥S T f h ∥ mα ≤ q-1 h=r |||S T f h ||| µ ≤ |||f r ||| µ q-r h=1 γ h 0 ≤ |||f r ||| µ γ 0 1 -γ 0 ≤ µω,
by denition of N , where we used repeatedly (71). We deduce that (g h ) h≥0 is a Cauchy sequence in the Banach space L 1 mα (G), and thus converges towards a limit f ∞ with ⟨f ∞ ⟩ = ⟨g⟩ by mass conservation. A similar argument to the one used in the proof of uniqueness shows that f ∞ is independent of the starting function g ∈ L 1 mα (G).

Counter-example for Hypothesis 3 and lower bounds on the convergence rate

In this section we present the proof of Theorem 4. We draw inspiration from Aoki and Golse [2, Section 3].

We let B := B(0, 1) be the unit ball in R d centered at 0 and use |B| to denote its volume. Upon translating and rescaling, we may assume 0 ∈ Ω, and R = 1 2 d(∂Ω, 0) > 1. In the whole section, we pick the following initial data: for 0 < ϵ ≪ 1 to be chosen, (x, v) ∈ G, f (x, v) = 1 ϵ 2d |B| 2 1 ϵB (x)1 ϵB (v). (72) Note that f ∈ L 1 mα (G) for all α ∈ (0, d) and that ⟨f ⟩ = 1. Throughout the proof, f ∞ is given by Theorem 3, and we set H 0 = ∥f ∞ ∥ L ∞ (G) , which is nite by assumption. We start by establishing a preliminary lemma, deduced from bounds for the convergence of (S t f ) t≥0 towards f ∞ . This leads to an inequality parameterized by ϵ. We then deduce all three results of Theorem 4 by making dierent choices of ϵ in the various settings.

In the whole section, we write |B| for the volume of B(0, 1), x + denotes max(0, x) for x ∈ R and for all A ⊂ R d , u ∈ R d , A + u = {z + u : z ∈ A}.

6.1. A preliminary lemma. We prove the following: Lemma 23. Let α ∈ (0, d). Assume there exists a uniform decay rate E : R + → R + such that E(t) → 0 as t → ∞ and for all g ∈ L 1 mα (G) with ⟨g⟩ = 1, for all t ≥ 0,

S t g -f ∞ ∥ L 1 (G) ≤ E(t)∥g -f ∞ ∥ mα .
Then, there exist C α > 0 allowed to depend on ∥f ∞ ∥ mα and ϵ 0 ∈ (0, 1) such that for all ϵ ∈ (0, ϵ 0 ),

G 1 ϵ 2d |B| 2 1 ϵB (v)1 ϵB+tv (x)1 {0≤t|v|≤R-ϵ} × e -t 0 σ(x-(t-u)v) du -ϵ 2d |B| 2 H 0 + dv dx ≤ C α E(t)ϵ -α . ( 73
)
Proof. We choose g = f with f given by (72) in the whole proof.

Step 1: comparison principle. We introduce the problem

     ∂ t Φ + v • ∇Φ = -σ(x)Φ in R + × G, γ -Φ = 0 on R + × Σ -, Φ |t=0 = f, in G, (74)
which corresponds to the evolution problem for the density of particles killed when a clock parameterized by σ rings, and when they reach the boundary. By the methods of characteristics (see also Step 2 of the proof of ( 55)), we have that, for (t, x, v) ∈ R + × G, Φ(t, x, v) = 1 {τ (x,-v)≥t} e -t 0 σ(x-(t-s)v) ds f (x -tv, v)

is the unique solution with Φ in L ∞ ([0, ∞); L 1 (G)). Using the Duhamel principle [START_REF] Mouhot | The Princeton Companion to Applied Mathematics, chapter Areas of Applied Mathematics: Kinetic Theory[END_REF] and the fact that (S t ) t≥0 is non-negative, S t f (x, v) ≥ Φ(t, x, v).

Step 2: a lower bound for the left-hand-side of [START_REF] Briant | Asymptotic Stability of the Boltzmann Equation With Maxwell Boundary Conditions[END_REF]. We have, by monotonicity of x → x + , for t > 0 Step 3: An upper bound for the right-hand-side. For C α > 0 a constant depending on α, ∥f ∞ ∥ mα allowed to change from line to line, using that for all (x, v) ∈ G, by convexity, m α (x, v) ≤ C α 1 + 1 |v| α + |v| 2δα , we get Choosing ϵ 0 = 2 -1-1 α < 1 concludes the proof. □

S t f -f ∞ L 1 (G) ≥ G S t f (x, v) -f ∞ (x, v) + dv dx ≥ Φ(t, x, v) -f ∞ (x, v) + dv dx ≥ G 1 ϵ 2d |B| 2 e -
E(t)∥f -f ∞ ∥ mα ≤ E(t)
The next subsections are devoted to the proof of Theorem 4.

6.2. Proof of point (1) of Theorem 4.

Step 1: improved lower bound. Starting from (73), we note rst that, since σ ≤ σ ∞ and by monotonicity of x → x + , one obtains e -σ∞t -ϵ 2d Step 2: conclusion. With the choice ϵ = e -σ∞t , one gets from (75), for t large enough so that ϵ < ϵ 0 , e -σ∞t -e -2dσ∞t |B| 2 H 0 ≥ e -σ∞t /2 and e tσ∞ R t

-1 t > 2,
e -σ∞t -e -2dσ∞t |B| 2 H 0 + ≤ C α e σ∞αt E(t).

We conclude that, for t large enough, E(t) ≥ C α e -σ∞(1+α)t .

6.3. Proof of point ( 2) and (3) of Theorem 4. We rewrite slightly dierently the previous setting. Once again, this is done without loss of generality by rescaling and translating, in view of the hypotheses. We assume that 0 ∈ Ω, that R = 1 2 d(∂Ω, 0) > 1, and that σ ≡ 0 on B(0, 1). Note that σ may cancel on a larger region of Ω, but we can always reduce the considered ball to t this framework. We consider again the initial data f given by (72).

Step 1: a lower bound for the killing term. Let v ∈ ϵB, t > 0, x ∈ Ω with x ∈ ϵB + tv. Then, for all u ∈ [0, t], by assumptions on σ We thus get e -t 0 σ(x-(t-u)v) du ≥ e -σ∞ t-1-ϵ ϵ + .

Step 2: improved version of (73). Injecting Step 1 into (73), we nd by monotonicity of x → x + e -σ∞ t-

1 ϵ +1 + -ϵ 2d |B| 2 H 0 + G 1 ϵB (v) ϵ 2d |B| 2 1 ϵB+tv (x)1 {0≤t|v|≤R-ϵ} dv dx ≤ C α E(t)ϵ -α .
Computing the integration in space and using that the integral in v lies again in the domain {v ∈ R d : |v| ≤ ϵ and |v| ≤ R-ϵ t }, we get e -σ∞ t- and R-1 t ≥ 1, we get, using also x + ≥ x ∈ R, e -σ∞ 2 ≤ C α e -κt t α .

Since the right-hand-side tends to 0 as t → ∞, (16) can not hold.

Step 4: Conclusion for point [START_REF] Arridge | Optical Tomography in Medical Imaging[END_REF]. Starting again from (76) and choosing ϵ = 1 t+1 for t large enough so that ϵ < ϵ 0 , |B| 2 H 0 (t+1) 2d < 1 2 and t+1 t (R -1 t+1 ) > 1, one nds 1 2 ≤ C α E(t)(t + 1) α .

It easily follows that E(t) ≥ C α (t + 1) -α for all t large enough.

1 .

 1 Model. In this article, we study the degenerate linear Boltzmann equation set inside a C 2 bounded domain (open, connected) Ω ⊂ R d , d ∈ {2, 3}, with some boundary conditions that we detail below. The initial boundary value problem writes

1 ϵ

 1 t 0 σ(x-(t-u)v) du 1 ϵB (v)1 ϵB+tv (x)1 {τ (x,-v)≥t} -2d |B| 2 e -t 0 σ(x-(t-u)v) du 1 ϵB (v)1 ϵB+tv (x)1 {0≤t|v|≤R-ϵ} -H 0 + dv dxwhere we have used that, on {t ≤ τ (x, -v)},τ (x, -v) = t + τ (x -tv, -v) ≥ (2R -ϵ)/|v| if x -tv ∈ ϵB by denition of R.Using the properties of x → x + , we then haveS t f -f ∞ L 1 (G) ≥ G 1 ϵ 2d |B| 2 1 ϵB (v)1 ϵB+tv (x)1 {0≤t|v|≤R-ϵ} × e -t 0 σ(x-(t-u)v) du -ϵ 2d |B| 2 H 0 + dv dx.

ϵB ϵB 1 ϵ 1 ϵ

 11 2d |B| 2 e 2 + d(Ω) c 4 |v| + |v| 2δ α + ∥f ∞ ∥ mα ≤ C α E(t) 1 + ϵ 2δα + 1 ϵ α . Indeed, note that by use of hyperspherical coordinates, since α < d ϵB 1 ϵ d |B| d(Ω) α c α 4 |v| α dv ≤ C α ϵ 0 d r d-1-α dr ≤ C α ϵ -α .

|B| 2 H 0 + G 1 ϵ

 01 2d |B| 2 1 ϵB (v)1 ϵB+tv (x)1 {0≤t|v|≤R-ϵ} dv dx ≤ C α E(t)ϵ -α .Using Tonelli's theorem to perform rst the integration in space, we nde -σ∞t -ϵ 2d |B| 2 H 0 + R d 1 ϵ d |B| 1 ϵB (v)1 {0≤t|v|≤R-ϵ} dv dx ≤ C α E(t)ϵ -α .The integral in v lies in the domain {v ∈ R d : |v| ≤ ϵ and |v| ≤ R-ϵ t } so one getse -σ∞t -ϵ 2d |B| 2 H 0 + 1 ϵ d min ϵ d , (R -ϵ) d t d ≤ C α E(t)ϵ -α ,and we conclude thate -σ∞t -ϵ 2d |B| 2 H 0 + min 1, R -ϵ ϵt d ≤ C α E(t)ϵ -α . (75)

  σ(x -(t -u)v) = σ(x -(t -u)v)1 {x-(t-u)v̸ ∈B(0,1)} ≤ σ ∞ 1 {x-(t-u)v̸ ∈B(0,1)} . Moreover, on {u : x -(t -u)v ̸ ∈ B(0, 1)}, |uv| ≥ |x -(t -u)v| -|x -tv| ≥ 1 -ϵ and since |v| ≤ ϵ, we nd u ≥ 1-ϵ ϵ . Hence u ∈ [0, t] : x -(t -u)v ̸ ∈ B(0, 1) ⊂ u ∈ [0, t] : u ≥ 1 -ϵ ϵ .

2

 2

Table 1

 1

	Hypothesis	Lower bound	Upper bound
	σ ≡ 0 on a ball B ⊂ Ω	(t + 1) -α	(t + 1) -α
	without Assumption 3	e -σ∞(1+α)t	(t + 1) -α
	under Assumption 3	e -σ∞(1+α)t	e -κt

summarizes our ndings for this specic framework:

Table 1 .

 1

Convergence rate E(t) from L 1 mα

  but this dierence is not seen at the level of the proof, which only uses the asymptotic behavior of the weights as |v| → +∞ once[START_REF] Dolbeault | Hypocoercivity for Linear Kinetic Equations Conserving Mass[END_REF] and Corollary 20 are established.Alternatively, the polynomial result can be obtained by applying[START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF] Theorem 5.6], since Proposition 14 provides, in the words of those authors, a weak generator Lyapunov condition, and Corollary 20 gives a Harris irreducibility condition.

  Step 3: Conclusion for point[START_REF] Aoki | On the Speed of Approach to Equilibrium for a Collisionless Gas[END_REF]. Assume that E(t) = Ce -κt . From Step 2, the following inequality should check e -σ∞ t-1 With the choice ϵ = 1 t and t large enough so that ϵ < ϵ 0 , e -σ∞ -1 t 2d |B| 2 H 0 ≥ e -σ∞

		1 ϵ +1	+	-ϵ 2d |B| 2 H 0	+ 1 ϵ d min ϵ d ,	(R -ϵ) d t d	≤ C α E(t)ϵ -α ,
	which leads to
	(76)	e -σ∞ t-1 ϵ +1	+	-ϵ 2d |B| 2 H 0	+	min 1,	(R -ϵ) ϵt
	(77)	ϵ +1	+	-ϵ 2d |B| 2 H 0	+	min 1,	(R -ϵ) ϵt

d ≤ C α E(t)ϵ -α . d ≤ C α e -κt ϵ -α .

Acknowledgments: The author thanks Silvia Lorenzani for useful discussions and for pointing out the references [56,62]. This project received funding from the European Union's Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie Grant Agreement No. 10103324.

Proof of Theorem 2. Step 1: Proof in the case α ∈ (1, d). Let T, µ and γ 0 ∈ (0, 1) be given by Lemma 21. Let t > 0 and write j = ⌊t/T ⌋. Then j ∈ (t/T -1, t/T ] and using the L 1 contraction and ( 61)

where we used

Thus, for some constant C > 0 (possibly larger than e -ln(γ 0 ) (1 + µ) to handle the case t < T ), for κ = -ln(γ 0 )/T > 0, we obtain

Step 2: Interpolation. We derive a convergence result for ∥ • ∥ mq for q ∈ (0, 1]. Set

for any weight w on Ḡ. We recall the notation M 1 from [START_REF] Cañizo | Harris-Type Results on Geometric and Subgeometric Convergence to Equilibrium for Stochastic Semigroups[END_REF]. Note that

where |Ω| denotes the volume of Ω. By use of hyperspherical coordinates, it is straightforward to check that Πh ∈ L 1

Since ⟨h⟩ = 0 implies Πh = h, and ⟨Πh⟩ = 0 for all h ∈ L 1 (G), Π is a bounded projection as claimed. Let t > 0. From Theorem 13, we have andfrom Step 1., since 3/2 ∈ (1, d), there exist C, κ > 0 such that

We apply Corollary 22 with the projection Π and the values:

, and, using the denition of Π,

, and, using the denition of Π, we have

We conclude from the corollary that

The same argument can be applied for all t ≥ 0. The conclusion follows. □ 5.3. Proof of Theorem 3. We rst note that the proofs of ( 14) and ( 15) are straightforward applications of Theorem 2 once i. is established. Thus, i. is the sole point of the statement whose proof is lacking.

As before, we only detail the exponential case: the polynomial one can be established exactly as in [START_REF] Bernou | Convergence Toward the Steady State of a Collisionless Gas With Cercignani-Lampis Boundary Condition[END_REF]Section 5.3].

Step 1: Uniqueness. Let ϵ ∈ (0, 1/2), α = d -ϵ. We will recycle Lemma 21. Assume there exists two steady states f ∞ , g ∞ with the desired properties. Applying Lemma 21 with α gives