Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the \textit{a priori} bounds on the solution.


Introduction
The Boltzmann equation rules the dynamics of rarefied gas particles moving in a domain Ω of R d with velocities in R d (d 2) when the only interactions taken into account are elastic binary collisions. More precisely, the Boltzmann equation describes the time evolution of f (t, x, v), the distribution of particles in position and velocity, starting from an initial distribution f 0 (x, v) .
In the present article we are interested in the case where the gas stays inside a domain of which walls are heated at a constant temperature T ∂ . Contrary to the classical specular (billiard balls) or bounce-back reflections boudary conditions, the temperature of the boundary generates a diffusion towards the inside of the domain which prevents the usual preservation of energy of the gas.
We investigate the case where Ω is a C 2 convex bounded domain and that the boundary conditions are Maxwellian diffusion. The Boltzmann equation reads with f satisfying the Maxwellian diffusion boundary condition: with n(x) denoting the outwards normal to Ω at x on ∂Ω. This boundary condition expresses the physical process where particles are absorbed by the wall and then emitted back into Ω according to the thermodynamical equilibrium distribution between the wall and the gas.
The operator Q(f, f ) encodes the physical properties of the interactions between two particles. This operator is quadratic and local in time and space. It is given by where f ′ , f * , f ′ * and f are the values taken by f at v ′ , v * , v ′ * and v respectively. Define: We recognise here the conservation of kinetic energy and momentum when two particles of velocities v and v * collide to give two particles of velocities v ′ and v ′ * . The collision kernel B 0 contains all the information about the interaction between two particles and is determined by physics (see [3] or [4] for a formal derivation for the hard sphere model of particles). In this paper we shall only be interested in the case of B satisfying the following product form which is a common assumption as it is more convenient and also covers a wide range of physical applications. Moreover, we shall assume that Φ satisfies either (1.4) ∀z ∈ R, c Φ |z| γ Φ(z) C Φ |z| γ or a mollified assumption c Φ and C Φ being strictly positive constants and γ in (−d, 1]. The collision kernel is said to be "hard potential" in the case of γ > 0, "soft potential" if γ < 0 and "Maxwellian" if γ = 0. Finally, we shall consider b to be a continuous function on θ in (0, π], strictly positive near θ ∼ π/2, which satisfies (1.6) b (cos θ) sin d−2 θ ∼ θ→0 + b 0 θ −(1+ν) for b 0 > 0 and ν in (−∞, 2). The case when b is locally integrable, ν < 0, is referred to by the Grad's cutoff assumption (first introduce in [7]) and therefore B will be said to be a cutoff collision kernel. The case ν 0 will be designated by non-cutoff collision kernel.
1.1. Motivations and comparison with previous results. The aim of this article is to show and to quantify the strict positivity of the solutions to the Boltzmann equation when the gas particles move in a domain with boundary conditions. In that sense, it continues the study started in [1] about exponential lower bounds for solutions to the Boltzmann equation when specular refletions boundary conditions were taken into account.
We would like to emphasize that, in the spirit of [1], our results show that the gas will instantaneously fill up the whole domain even if the initial configuration contains vacuum. Indeed, they only require some regularity on the solution and no further assumption on its local density. Previous studies assumed the latter to be uniformly bounded from below, which is equivalent of assuming a priori either that there is no vacuum or that the solution is strictly positive.
Moreover, the present results only require solutions to the Boltzmann equation to be continuous away from the grazing set which is a property that is known to hold in the case of Maxwellian diffusion boundary conditions [9].
The issue of quantifying the positivity of solutions has been investigated for a long time since it not only presents a great physical interest but also appears to be of significant importance for the mathematical study of the Boltzmann equation. Indeed, exponential lower bounds are essential for entropy-entropy production methods used to describe long-time behaviour for kinetic equations [5] [6]. More recently, such lower bounds were needed to prove uniqueness of solutions to the Boltzmann equation in L 1 v L ∞ x 1 + |v| 2+0 [8]. Several works quantified the study of an explicit lower bound for solutions to the Boltzmann equation. We give here a brief overview and we refer the interested reader to the more detailed description in [1].
The first result about the strict positivity of solutions to the Boltzmann equation was derived by Carleman [2]. Noticing that a part Q + of the Boltzmann operator Q satisfies a spreading property, roughly speaking with C + < 1 (see Lemma 3.2 for an exact statement), he proved the immediate creation of an exponential lower bound for a certain class of solutions (radially symmetric in velocity) to the spatially homogeneous equation with hard potential kernel with angular cutoff. The latter result was improved to a Maxwellian lower bound and extended to the case of non-radially symmetric solutions to the spatially homogeneous equation with hard potential and cutoff by Pulvirenti and Wennberg [11].
Finally, the study in the case of the full equation has been tackled by Mouhot [10] in the case of the torus Ω = T d , and more recently by the author [1] in C 2 convex bounded domains with specular boundary conditions in all dimension d. In both articles, a Maxwellian lower bound is derived for solutions to the full Boltzmann equation with angular cutoff (both with hard of soft potentials) and they showed the appearance of an exponential lower bound in the non-cutoff case.
Our present results show that the previous properties proven for the full Boltzmann equation [10][1] still hold in the physically relevant case of the Maxwellian diffusion generated by the boundary of the domain Ω. This is done for all the physically relevant collision kernels such as with and without angular cutoff and hard and soft potentials. Moreover, in the case of a solution that has uniformly bounded local mass and entropy, the proofs are entirely constructive and the constants are explicit and only depend on the a priori bounds on the solution and the geometry of the domain, which is of great physical interest for the study of the spreading of gas into a domain with heated walls.
There are two key contributions in the present article. The main one is a quantification of the strict positivity of the Maxwellian diffusion process thanks to a combination of a localised positivity of the solution and a geometrical study of the rebounds against a convex boundary. Roughly speaking, we show that the wall instantaneously starts to diffuse in all directions and that its diffusion is uniformly bounded from below. The second one is a spreading method combining the effects used in previous studies and the exponential diffusive process (see next section for details).
1.2. Our strategy. The main strategy to tackle this result relies on the breakthrough of Carleman [2], namely finding an "upheaval point" (a first minoration uniform in time but localised in velocity) and spreading this bound, thanks to the spreading property of the Q + operator, in order to include larger and larger velocities and finally compare it to an exponential. The case of the spatially non-homogeneous equation [10][1] was dealt by finding a spreading method that was invariant along the flow of characteristics.
The creation of "upheaval points" (localised in space because of boundary effects) is essantially the method developed in [1] for general continuous initial datum or the one of [11][10] for constructive purposes. There is a new technical treatment of the time of appearance of such lower bounds but does not present any new difficulties.
The main issue is the use of a spreading method that would be invariant along the flow of characteristics [10] [1]. In the case of Maxwellian diffusion boundary conditions characteristic trajectories are no longer well defined. Indeed, once a trajectory touches the boundary it is absorbed an re-emitted in all the directions. Characteristic trajectories are therefore only defined in between two consecutive rebounds and one cannot hope to use the invariant arguments developed in [10] [1].
The case of the torus, studied in [10], indicates that without boundaries the exponential lower bound is created after time t = 0 as quickly as one wants. In the case of a bounded domain with specular reflection boundary conditions [1], this minoration also occurs immediately. Together, it roughly means that one can expect to obtain an exponential lower bound on straight lines uniformly on how close the particle is from the boundary. Therefore we expect the same kind of uniform bounds to arise on each characteristic trajectory in between two consecutive rebounds as long as the Maxwellian diffusion emitted by the boundary is uniformly non-negative.
Our strategy is therefore to first prove that the boundary condition produces a strictly positive quantity uniformly towards the interior of Ω and then to find a method to spread either this diffusion or the localised "upheaval points". More precisely, if there is no contact during a time τ > 0 we expect to be able to use the spreading method developed in [10] from the initial lower bounds. Else there is a contact during the interval [0, τ ] we cannot hope to use the latter spreading method, nor its more general characteristics invariant version derived in [1], since the Maxwellian diffusion boundary condition acts like an absorption for particles arriving on the boundary. But this boundary condition also diffuses towards the interior of the domain Ω what it has absorbed. This diffusion follows an exponential law and therefore of the shape of a Maxwellian lower bound that we manage to keep along the incoming characteristic trajectory.
Collision kernels satisfying a cutoff property as well as collision kernels with a non-cutoff property will be treated following the strategy described above. The only difference is the decomposition of the Boltzmann bilinear operator Q we consider in each case. In the case of a non-cutoff collision kernel, we shall divide it into a cutoff collision kernel and a remainder. The cutoff part will already be dealt with and a careful control of the L ∞ -norm of the remainder will give us the expected exponential lower bound, which decreases faster than a Maxwellian.
1.3. Organisation of the paper. Section 2 is dedicated to the statement and the description of the main results proven in this article. It contains three different parts Section 2.1 defines all the notations which will be used throughout the article. The last subsections, 2.2 and 2.3, are dedicated to a mathematical formulation of the results related to the lower bound in, respectively, the cutoff case and the non-cutoff case, described above. It also defines the concept of mild solutions to the Boltzmann equation in each case. Section 3 deals with the case of the immediate Maxwellian lower bound for collision kernels with angular cutoff. As described in our strategy, it follows three steps. Section 3.1 generates the localised "upheaval points" for general initial datum. A constructive approach to that problem is given in Section 3.4.
The uniform positivity of the Maxwellian diffusion is proven in Section 3.2. Finally, Section 3.3 combines the standard spreading methods with the diffusion process to prove the expected instantaneous Maxwellian lower bound.
To conclude, Section 4 proves the immediate appearance of an exponential lower bound in the case of collision kernels without angular cutoff. We emphasize here that this Section actually explains the adaptations required compared to the case of collision kernels with the cutoff property.

Main results
We begin with the notations we shall use all along this article.
Functional spaces. This study will hold in specific functional spaces regarding the v variable that we describe here and use throughout the sequel. For all p in [1, ∞], we use the shorthand notation for Lebesgue spaces L p v = L p R d . For p ∈ [1, ∞] and k ∈ N we use the Sobolev spaces W k,p v by the norm Physical observables and hypotheses. In the sequel of this study, we are going to need bounds on some physical observables of solution to the Boltzmann equation (1.1).
We consider here a function f (t, x, v) 0 defined on [0, T ) × Ω × R d and we recall the definitions of its local hydrodynamical quantities.
• its local energy • its local weighted energy Our results depend on uniform bounds on those quantities and therefore, to shorten calculations we will use the following In our theorems we are giving a priori lower bound results for solutions to (1.1) satisfying some properties about their local hydrodynamical quantities. Those properties will differ depending on which case of collision kernel we are considering. We will take them as assumptions in our proofs and they are the following.
• In the case of hard or Maxwellian potentials with cutoff (γ 0 and ν < 0): • In the case of a singularity of the kinetic collision kernel (γ ∈ (−d, 0)) we shall make the additional assumption • In the case of a singularity of the angular collision kernel (ν ∈ [0, 2)) we shall make the additional assumption 2) implies the boundedness of the local entropy and if γ 0 we have E ′ f E f and so in some cases several assumptions might be redundant. Moreover, in the case of the torus with periodic conditions or the case of bounded domain with specular boundary reflections [1], solutions to (1.1) also satisfied the conservation of the total mass and the total energy. The case with Maxwellian diffusion boundary conditions only preserves, in general (see [12] for instance), the total mass: Characteristic trajectories. The characteristic trajectories of the equation are only defined between two consecutive rebounds against the boundary and they are given by straight lines that we will denote by Because Ω is a closed set, we can define the first time of contact between a backward trajectory and ∂Ω: and the contact point between such a trajectory and the boundary: Maxwellian lower bound for cutoff collision kernels. The final theorem we prove in the case of cutoff collision kernel is the immediate appearance of a uniform Maxwellian lower bound. We use, in that case, the Grad's splitting for the bilinear operator Q such that the Boltzmann equation reads where we used the following definitions As already mentionned, the characteristics of our problem can only be defined in between two consecutive rebounds against ∂Ω. We can therefore define a mild solution of the Boltzmann equation in the cutoff case, which is expressed by a Duhamel formula along the characteristics. This weaker form of solutions is actually the key point for our result and also gives a more general statement.
T ) the solution f is bounded from below, almost everywhere, by a global Maxwellian distribution with density ρ and temperature θ, i.e.
If we add the assumptions of uniform boundedness of f 0 and of the local mass and entropy of the solution f we can use the arguments originated in [11] to construct explicitely the initial "upheaval point", without any compactness argument. We refer the reader to Section 3.4 which gives the following corollary.
and that f has a bounded local mass and entropy Then conclusion of Theorem 2.2 holds true with the constants ρ and θ being explicitely constructed in terms of τ , E f , H f and L pγ f .

2.3.
Exponential lower bound for non-cutoff collision kernels. In the case of non-cutoff collision kernels (0 ν < 2 in (1.6)), Grad's splitting does not make sense anymore and so we have to find a new way to define mild solutions to the Boltzmann equation (1.1). The splitting we are going to use is a standard one and it reads where we used the following definitions We would like to use the properties we derived in the study of collision kernels with cutoff. Therefore we will consider additional splitting of Q.
For ε in (0, π/4) we define a cutoff angular collision kernel . This yields the splitting we shall use to deal with non-cutoff collision kernels, Thanks to the splitting (2.13), we are able to define mild solutions to the Boltzmann equation with non-cutoff collision kernels. This is obtained by considering the Duhamel formula associated to the splitting (2.13) along the characteristics (as in the cutoff case).
is a mild solution of the Boltzmann equation with non-cutoff angular collision kernel associated to the initial datum f 0 (x, v) if there exists 0 < ε 0 < π/4 such that for all 0 < ε < ε 0 : Then for all τ ∈ (0, T ) and for any exponent K such that Moreover, in the case ν = 0, one can take K = 2 (Maxwellian lower bound).
As in the angular cutoff case, if we further assume that f 0 presents no vacuum area and that f has uniformly bounded local mass and entropy, our results are entirely constructive. Corollary 2.6. As for Corollary 2.3, if f 0 is bounded uniformly from below as well as the local mass of f , the local entropy of f is uniformly bounded from above then the conclusion of Theorem 2.5 holds true with constants being explicitely constructed in terms of τ ,

The cutoff case: a Maxwellian lower bound
In this section we are going to prove a Maxwellian lower bound for a solution to the Boltzmann equation (1.1) in the case where the collision kernel satisfies a cutoff property.
The strategy to tackle this result follows the main idea used in [11][10][1] which relies on finding an "upheaval point" (a first minoration uniform in time and space but localised in velocity) and spreading this bound, thanks to a spreading property of the Q + operator, in order to include larger and larger velocities.
As described in the introduction (Section 1.2), we need a method that translates the usual spreading argument in the case of our problem and combine it with a strict positivity of the diffusion process. Roughly speaking, either the characteristic we are looking at comes from the diffusion of the boundary or the spreading will be generated on a straight line as in [10].
Thus our study will be split into three parts, which are the next three subsections The first step (Section 3.1) is to partition the position and velocity spaces so that we have an immediate appearance of an "upheaval point" in each of those partitions.
As discussed in the introduction, the standard spreading method fails in the case of characteristics trajectories bouncing against ∂Ω. We therefore study of the effects of the Maxwellian diffusion (Section 3.2).
The last one (Section 3.3) is to obtain uniform lower bounds less and less localised (and comparable to an exponential bound in the limit) in velocity. The strategy we use relies on the spreading property of the gain operator that we already used and follows the method in [10] combined with the treatment of localised bounds in [1], together with the previous focus on the Maxwellian diffusion process.
A separated part, Section 3.4, is dedicated to a constructive approach to the Maxwellian lower bound.
3.1. Initial localised lower bounds. In Section 3.1.2 we use the continuity of f together with the conservation of total mass (2.4) and the uniform boundedness of the local energy to obtain a point in the phase space where f is strictly positive. Then, thanks to the continuity of f , its Duhamel representation (2.11) − (2.10) and the spreading property of the Q + operator (Lemma 3.2) we extend this positivity to high velocities at that particular point (Lemma 3.3).
Finally, in Section 3.1.3, the free transport part of the solution f will imply the immediate appearance of the localised lower bounds (Proposition 3.5).
Moreover we define constants that we will use in the next two subsections in order to have a uniform lower bound.
3.1.1. Controls on the gain and the loss operators. We first introduce two lemmas, proven in [10], that control the gain and loss terms in the Grad's splitting (2.8) we are using under the cutoff assumption. One has an L ∞ bound on the loss term (Corollary 2.2 in [10]). Lemma 3.1. Let g be a measurable function on R d . Then The spreading property of Q + is given by the following lemma (Lemma 2.4 in [10]), where we define As a consequence in the particular quadratic case δ = r = R, we obtain for any v ∈ R d and ξ ∈ (0, 1).

3.1.2.
First "upheaval" point. We start by the strict positivity of our function at one point for all velocities. Then there exist ∆ > 0, (x 1 , v 1 ) in Ω × R d such that for all n ∈ N there exist r n > 0, depending only on n, and t n (t), α n (t) > 0 such that , with α 0 > 0 independent of t and the induction formula Remark 3.4. It is essentially the same method used to generate "upheaval points" in [1]. The main difference being that we need to control the characteristics before they bounce against the boundary, whence the existence of the bound t n (t).
Proof of Lemma 3.3. The proof is an induction on n and mainly follows the method in [1] Lemma 3.3.
Step 1: Initialization. We recall the assumptions that are made on the solution f ((2.4) and assumption (2.1)): Since Ω is bounded, and so is included in, say, B(0, R X ), we also have that Therefore, exactly the same computations as in [1] (step 1 of the proof of Lemma 3.3) are applicable and lead to the existence of (x 1 , v 1 ) such that f (0, x 1 , v 1 ) > 0 and, by uniform continuity of f , to Lemma 3.3 in the case n = 0.
Step 2: Proof of the induction. We assume the conjecture is valid for n.
We thus use the fact that f is a mild solution to write f (t, x, v) under its Duhamel form starting at t n (t) without contact with the boundary (2.10). The control we have on the L operator, Lemma 3.1, allows us to bound from above the second integral term (the first term is positive). Moreover, this bound on L is independent on t, x and v since it only depends on an upper bound on the energy e f (t,x,·) (and its local L p norm l p f (t,x,·) ) which is uniformly bounded by E f (and by L p f ). This yields , see Lemma 3.1, and we used v 2r n + v 1 .
We already saw that (X s,t (x, v)) s∈[tn(t),t] stays in B(x 1 , ∆/2 n ). Therefore, by calling v * the integration parametre in the operator Q + we can apply the induction property to f (s, X s,t (x, v), v * ) which implies, in (3.4), Applying the spreading property of Q + , Lemma 3.2, with ξ = 1/4 gives us the expected result for the step n + 1 since B(v 1 , r n+1 ) ⊂ B(0, v 1 + 2r n ).

3.1.3.
Partition of the phase space and first localised lower bounds. We are now able to prove the immediate appearance of localised "upheaval points". We emphasize here that the following proposition is proven with exactly the same arguments as in [1]. Then there exists ∆ > 0 such that for all 0 < τ 0 ∆, there exists δ T (τ 0 ), δ X (τ 0 ), δ V (τ 0 ), R min (τ 0 ), a 0 (τ 0 ) > 0 such that for all N in N there exists N X in N * and x 1 , . . . , x N X in Ω and v 1 , . . . , v N X in B(0, R min (τ 0 )) and • Ω ⊂ Proof of Proposition 3.5. We are going to use the free transport part of the Duhamel form of f (2.10), to create localised lower bounds out of the one around (x 1 , v 1 ) in Lemma 3.3.
Ω is bounded so let us denote its diameter by d Ω . Take τ 0 in (0, ∆]. Let n be large enough such that r n 2d Ω /τ 0 + v 1 , where r n is defined by (3.2) in Lemma 3.3. It is possible since (r n ) increases to infinity. Now, define R min (τ 0 ) = 2d Ω /τ 0 . Thanks to Lemma 3.3 applied to this particular n we have that where we used the fact that α n (t) is an increasing function.
Definition of the constants. We notice that for all x in ∂Ω we have that n(x) · (x − x 1 ) > 0, because Ω has nowhere null normal vector by hypothesis. But the function x −→ n(x) · x − x 1 x − x 1 is continuous (since Ω is C 2 ) on the compact ∂Ω and therefore has a minimum that is atteined at a certain X(x 1 ) on ∂Ω. Hence, To shorten following notations, we define on Ω × R d − {0} the function where we defined t(x, v) = min{t 0 : x + tv ∈ ∂Ω}, the first time of contact against the boundary of the forward characteristic (x + sv) s 0 defined for v = 0 and continuous on Ω × R d − {0} (see [1] Lemma 5.2 for instance).
We denote d 1 to be half of the distance from x 1 to ∂Ω. We define two sets included in [0, ∆] × Ω × R d : By continuity of t(x, v) and of n (on ∂Ω), we have that is compact and does not intersect the grazing set [0, ∆] × Λ 0 defined by (1.7). Therefore, f is continuous in Λ and thus is uniformly continuous on Λ. Hence, there exist The map Φ (defined by (3.7)) is uniformly continuous on the compact [0, ∆] × Ω × S d−1 and therefore there exist δ ′′ We conclude our definitions by taking Proof of the lower bounds. We take N ∈ N and notice that Ω is compact and therefore there exists x 1 , . . . , x N X in Ω such that Ω ⊂ Moreover, we construct them such that x 1 is the one defined in Lemma 3.3 and we then take v 1 to be the one defined in Lemma 3.3. We define Because Ω is convex we have that The latter equalities imply that there is no contact with ∂Ω between times τ 0 /2 and τ 0 when starting from x 1 to go to x i with velocity v i . Using the fact that f is a mild solution of the Boltzmann equation, we write it under its Duhamel form without contact (2.10), but starting at τ 0 /2. We drop the last term which is positive. As in the proof of Lemma 3.3 we can control the L operator appearing in the first term in the right-hand side of (2.10) (corresponding to the free transport).
where we used (3.5) for the second inequality. We see here that v i belongs to B(0, R min (τ 0 )) and that B(0, R min (τ 0 )) ⊂ B(v 1 , r n ) and therefore We first notice that (τ 0 , x i , v i ) belongs to Λ since either x i belongs to B(x 1 , d 1 ) or x 1 − x i d 1 but by definition of v i and λ(x 1 ) (see (3.6)), ) and we will prove that (t, x, v) also belongs to Λ. If and (t, x, v) thus belongs to Λ (1) ⊂ Λ.
In the other case where The latter inequality combined with (3.9) and that |t − τ 0 | δ ′′ T (τ 0 ) and which in turn implies so that (t, x, v) belongs to Λ (2) .
We can now conclude the proof. We proved that (τ 0 , x i , v i ) belongs to Λ and that for all Remark 3.6. In order to lighten our presentation and because τ 0 can be chosen as small as one wants, we will only study the case of solutions to the Boltzmann equation which satisfies Proposition 3.5 at τ 0 = 0. Then we will immediatly create the exponential lower bound after at τ 1 for all τ 1 > 0. Then we apply the latter result to F (t, x, v) = f (t + τ 0 , x, v) to obtain the exponential lower bound for f at time τ 0 + τ 1 which can be chosen as small as one wants.

3.2.
Global strict positivity of the Maxwellian diffusion. In this subsection, we focus on the positivity of the Maxwellian diffusion boundary condition. More precisely, we prove that the boundary of the domain Ω diffuses in all directions a minimal strictly positive quantity.
Proof of Proposition 3.7. Let τ 0 be in (0, ∆], take t in [τ 0 , ∆] and x ∂ on ∂Ω. We consider (x 1 , v 1 ) constructed in Lemma 3.3 and will use the same notations as in the proof of Proposition 3.5.
In the spirit of the proof of Proposition 3.5 we define which gives, because ∂Ω is convex, that X t/2,s t/2 s t is a straight line not intersecting ∂Ω apart at time s = t. We can thus write f under its Duhamel form without contact (2.10) starting at t/2. We keep only the first term on the right hand-side and control the operator L by Lemma 3.1.
where we used (3.5) for the second inequality since t/2 belongs to [τ 0 /2, ∆]. Note that we choose n exactly as in the proof of Proposition 3.5 and, for the same reasons, v ∂ thus belongs to B(v 1 , r n ) which implies Here again, the continuity of f away from the grazing set implies the existence of δ(τ 0 ) independent of t, x and v ∂ such that We now deal with the scalar product v * · n(x ∂ ) appearing in the Maxwellian diffusion.
We notice that for all x in ∂Ω we have that n(x) · (x − x 1 ) > 0, because Ω has nowhere null normal vector by hypothesis. But the function x −→ n(x) · (x − x 1 ) is continuous (since Ω is C 2 ) on the compact ∂Ω and therefore has a minimum that is atteined at a certain X(x 1 ) on ∂Ω. Hence, We define δ ′ (x 1 ) = B(x 1 ) and a mere Cauchy-Schwarz inequality implies that which in turns implies To conlude we combine (3.11) and (3.12) at point x ∂ inside the integrale to get, with δ ′′ (τ 0 ) = min (δ(τ 0 ), 2δ This yields the expected result with Remark 3.8. For now on, τ 0 used in Proposition 3.7 will be the same as the one used in Proposition 3.5. Moreover, as already mentioned and explained in Remark 3.6, we will consider in the sequel that τ 0 = 0.

Spreading of the initial localised bounds and global Maxwellian lower bound.
In Section 3.1 we proved the immediate appearance of initial lower bounds that are localised in space and in velocity. The present subsection aims at increasing these lower bounds in a way that larger and larger velocities are taken into account, and compare it to an exponential bound in the limit.
3.3.1. Spreading of the initial "upheaval points". First, we pick N in N * , construct δ V , R min and cover Ω with 1 i N X B(x i , δ X /2 N ) as in Proposition 3.5 where we dropped the dependencies in τ 0 and N. Then for any sequence (ξ n ) in (0, 1) and for all τ > 0 we define three sequences in R + by induction. First, Second, with the notation r n = R min + r n , where b ∂ defined in Proposition 3.7 and C L in Lemma 3.3. And finally, with C Q being defined in Lemma 3.3, a n+1 (τ ) = min (a n (τ ), b n (τ )) 2 C Q r d+γ n ξ d/2−1 n+1 We express the spreading of the lower bound in the following proposition.
Proof of Proposition 3.9. We are interested in immediate appearance of a lower bound so we can always choose δ t δ X and also R min 1. This Proposition will be proved by induction on n, the initialisation is simply Proposition 3.5 so we consider the case where the proposition is true for n < N.
There are two possible cases depending on t ∂ (x, v) t or t ∂ (x, v) > t. We remind here that t ∂ (see (2.6)) is the first time of contact of the backward trajectory starting at x with velocity v.
1 st case: no contact against the boundary from 0 to t. We can therefore use the Duhamel formula without contact (2.10) and bound it by the second term on the right-hand side (every term being positive). Then Lemma 3.1 on the linear operator L and the fact that gives the following inequality (3.17) We now see that for all s in the interval considered in the integral above where we used that δ T δ X . We can therefore apply the induction hypothesis to f (t, X s,t (x, v), v * ) into (3.17), where we denoted by v * the integration parameter in Q + . This yields Applying the spreading property of Q + , Lemma 3.2, with ξ = ξ n+1 gives us the expected result for the step n + 1 with the lower bound a n+1 (τ ).
2 nd case: there is at least one contact before time t. In that case we have that t ∂ (x, v) t and we can use the Duhamel formula with contact (2.11) for f . Both terms on the right-hand side are positive so we can lower bound f (t, x, v) by the first one. Denoting x ∂ = x ∂ (x, v) (see Definition (2.7)), this yields where we used the definition of f ∂ (1.2).
Thanks to the previous subsection (Proposition 3.7), we obtain straightforwardly the expected result for step n + 1 with the lower bound b n+1 (τ ).

A Maxwellian lower bound: proof of Theorem 2.2. In this subsection we prove Theorem 2.2.
We take f being the mild solution described in Theorem 2.2 and we suppose, thanks to Remarks 3.6 and 3.8, that f satisfies Propositions 3.5 and 3.7 with τ 0 = 0.
In Proposition 3.9, we showed that we can spread the localised lower bound with larger and larger velocities taken into account, i.e. by taking ξ n = 1/4 the sequence r n is strictly increasing to infinity. We can consider r 0 > 0 and find an n 0 in N such that B(0, R min ) ∀v ∈ B(0, R min ), B(0, r 0 ) ⊂ B(v, r n 0 ). By setting N to be this specific n 0 and applying Proposition 3.9 with this N we obtain a uniform lower bound: This bound is uniform in x and the same arguments as in the proof of Proposition 3.9 allows us to spread it in the same manner in an even easier way since it is a global lower bound. Therefore, without loss of generality, we can assume n 0 = 0 and that the following holds, ∀n ∈ N, ∀t ∈ τ − τ 2 n+1 r n , τ , ∀x ∈ Ω, f (t, x, v) min (a n (τ ), b n (τ )) 1 B(0,rn) (v), with (r n ), (a n ) and (b n ) satisfying the same inductive properties (3.14)-(3.16)-(3.15) (with r 0 = r n 0 , a 0 = a n 0 and b 0 = b no ), with (ξ n ) to be chosen later.
The proof of Theorem 2.2 is then done in two steps. The first one is to establish the Maxwellian lower bound at time τ using a slightly modified version of the argument in [10] Lemma 3.3. The second is to prove that the latter bound holds for all t > τ . 1 st step: a Maxwellian lower bound at time τ . A natural choice for (ξ n ) is a geometric sequence ξ n = ξ n for a given ξ in (0, 1). With such a choice we have that (3.19) r with c r > 0 depending only on r 0 and ξ.
It has been proven in [10] Lemma 3.3 that for a function satisfying the property (3.21) with c n α 2 n , for some α > 0, there exist ρ and θ strictly positive explicit constants such that It thus only remains to show that there exists α 1 and α 2 strictly positive such that b n α 2 n 1 and a n α 2 n 2 .
We recall the inductive definition of (a n ) for n 0, with ξ n = ξ n , First, using (3.19) we have that for all n 0, which is bounded from above for all n since 0 γ + 1. Therefore, if we denote by C 3 any explicit non-negative constant independent of n, we have for all n 0 a n+1 C 3 r d+γ n ξ ((n+1))(d/2−1) 2 n+2 (R min + r n+1 ) min (a n , b n ) 2 .
Thus using (3.19) and (3.20) to bound r n we have min (a n , b n ) 2 n min (a n , b n ) 2 .
We could always have chosen b 0 and then b 1 respectively smaller than a 0 and a 1 (by always bounding from below by the minimum) and we assume that it is so. We can therefore define ∀n 1, k n = min {0 k n − 1 : a n−k b n−k } .
We therefore proved the existence of α > 0 such that for all n in N, min(a n , b n ) α 2 n which implies that there exist ρ and θ strictly positive explicit constants such that 2θ .
2 nd step: a Maxwellian lower bound for all T > t τ . To complete the proof of Theorem 2.2, it remains to prove that (3.23) actually holds for all t τ . All the results and constants we obtained so far do not depend on an explicit form of f 0 but just on uniform bounds and continuity that are satisfied at all times, positions and velocities (by assumption). Therefore, we can do the same arguments starting at any time and not t = 0. So if we take τ > 0 and consider τ t < T we just have to make the proof start at t − τ to obtain Theorem 2.2.

3.4.
A constructive approach to the initial lower bound and positivity of the diffussion. In previous subsections, we can see that explicit and constructive constants are obtained from given initial lower bounds and uniform positivity of the Maxwellian diffusion. Therefore, a constructive approach to the latter two will lead to completely explicit constants in the Maxwellian lower bound, depending only on a priori bounds on the solution and the geometry of the domain.
Localised "upheaval points". A few more assumptions on f 0 and f suffice to obtain a completely constructive approach for the "upheaval points". This method is based on a property of the iterated Q + operator discovered by Pulvirenty and Wennberg [11] and reformulated by Mouhot ([10] Lemma 2.3) as follows. Let g(v) be a nonnegative function on R d with bounded energy e g and entropy h g and a mass ρ g such that 0 < ρ g < +∞.
Then there exist R 0 , r 0 , η 0 > 0 and v ∈ B(0, R 0 ) such that with R 0 , r 0 , η 0 being constructive in terms on ρ g , e g and h g .
We now suppose that 0 < ρ f 0 < +∞, h f 0 < +∞ and that and we consider R 0 , r 0 , η 0 and v from Lemma 3.10 associated to the function ϕ.
We can do this iteration one more time since R 0 ∆ 1 d 1 /3 and this yields Applying Lemma 3.10 and remembering that ∆ 1 1 we obtain that f is uniformly continuous on the compact [0, T /2] × Ω × B(0, R 0 ) so there exists δ T , δ X , δ V > 0 such that From (3.27) and (3.28), we find (3.29) To conclude we construct x 2 , . . . , x N X such that Ω ⊂ 1 i N X B (x i , δ X ). We can use exactly the same arguments as for x 1 on each x i on the time interval [τ 0 , ∆ i ] and we reach the conclusion (3.29) on each B(x i , δ X ). This gives, with ∆ = min(∆ i ), Remark 3.11. We emphasize here that even though we used compactness arguments, they appeared to be solely a technical artifice. The constants a 0 (τ 0 ), v, R 0 and r 0 are entirely explicit and depends on a priori bounds on f . The only point of concern would be that ∆ is not constructive since it depends on the covering. However, in previous sections, only the constant bounding the diffusive process (Proposition 3.7 contains a dependency in ∆ (see (3.11) and (3.13)) but depend on an upper bound on ∆ which is less than 1.
Starting from this explicit bound we can use the proofs made in the previous subsections, that are constructive, to therefore have a completely constructive proof as long as the bound on the diffusion (Proposition 3.7) can be obtained without compactness arguments.
Constructive approach of positivity of diffusion. A quick look at the proof of Proposition 3.7 shows that we only need to construct δ(τ 0 ) in (3.11) and δ ′ (x 1 ) in (3.12) explicitly.
The first equality (3.11) is obtained constructively combining the arguments to obtain it pointwise (see proof of Proposition 3.7) together with the method of the proof of Lemma 3.3.
Indeed, take x ∂ on ∂Ω and fix an x 0 in ω. We can grow the initial lower bound (3.30) at x 1 for t in [τ 0 , ∆] such that it includes B(0, 2d Ω /τ 0 + v ) (as in Lemma 3.3). Then, as in the beginning of the proof of Proposition 3.5 we obtain that, We can now do that for all v in Note that this point is always well defined since Ω is convex and For any given x(v) we apply the same argument as for x 0 so that the lower bound includes v. Therefore, there is an infimum constant A min satisfying, which is a constructive version of (3.11). We emphasize here that A min exists and is indeed independent on v since it depends on the number of iteration of Lemma 3.3 (itself determined by the norm of v which is bounded) and the initial lower bound at x(v) which is uniform in space by (3.30).
The second inequality (3.12) is purely geometric as long as we fix a x 1 satisfying the initial lower bound, which is the case with x 0 used above. We therefore obtained an entirely constructive method for the positivity of the diffusion process.

The non-cutoff case: an exponential lower bound
In this section we prove the immediate appearance of an exponential lower bound for solutions to the Boltzmann equation (1.1) in the case of a collision kernel satisfying the non-cutoff property.
The definition of being a mild solution in the case of a non-cutoff collision kernel, Definition 2.4 and equation (2.13), shows that we are in fact dealing with a cutoff kernel to which we add a non locally integrable remainder. As we shall see in Section 4.1, S ε enjoys the same kind of L ∞ control than the operator L whilst Q 1 ε , the noncutoff part of the gain operator, has an L ∞ -norm that decreases to zero as ε goes to zero.
The strategy used is therefore utterly identical to the cutoff case: creation of localised "upheaval points" and spreading of these initial lower bounds up to an exponential lower bound. The difference will be that, at each step n of the spreading process we will choose an ε n and a ξ n such that the perturbation added by the noncutoff part − Q 1 εn L ∞ v still preserves a uniform positivity in larger and larger balls in velocity.
Note that the uniform positivity of the Maxwellian diffusion still holds in the non-cutoff case since it only comes from an initial positivity and the geometry of the domain.
4.1. Controls on the operators S ε and Q 1 ε . We gather here two lemmas, proven in [10], which we shall use in this section. They control the L ∞ -norm of the linear operator S ε and of the bilinear operator Q 1 ε . We first give a property satisfied by the linear operator S, (2.13), which is Corollary 2.2 in [10], where we define  We will compare the lower bound created by the cutoff part of our kernel to the remaining part Q 1 ε . To do so we need to control its L ∞ -norm. This is achieved thanks to Lemma 2.5 in [10], which we recall here. (1) If Φ satisfies (1.4) with 2 + γ 0 or if Φ satisfies (1.5), then ∀v ∈ R d , (2) If Φ satisfies (1.4) with 2 + γ < 0, then ∀v ∈ R d , with p > d/(d + γ + 2).

4.2.
Proof of Theorem 2.5. As explained at the beginning of this section, the main idea is to compare the loss due to the non-cutoff part of the operator Q 1 ε with the spreading properties of the cutoff operator Q + ε . More precisely, due to Lemmas 3.1, 4.1 and 4.2 we find that for all 0 < ε < ε 0 , Moreover, by definitions (3.1), (2.9) and (4.1), the following behaviours happen: if ν belongs to (0, 2) and for ν = 0. This shows that the contribution of Q 1 ε decreases with ε so this operator should not affect the spreading method whereas the contribution of S ε increases, which is why we lose the Maxwellian lower bound to get a faster exponential one.
We just briefly describe the changes to make into the proof of the cutoff case to obtain Theorem 2.5.
Localised "upheaval points". The creation of localised initial lower bounds (Proposition 3.5 in the cutoff case) depends on the Boltzmann operator for two different reasons: • the creation of a first lower bound in a neighborhood of a point (x 1 , v 1 ) in the phase space (Lemma 3.3) • the creation of localised lower bounds in Ω via the free transport part. Since L ε + S ε satisfies the same bounds as L in the cutoff case, the second step can be made identically as in the proof of Proposition 3.5. It remains to prove the creation of an initial bound in a neighborhood of (x 1 , v 1 ).
Note that the step n = 0 holds here since it only depends on the continuity of the solution f and f 0 .
The key difference will be that the equivalent statement of Lemma 3.3 can only be done on time intervals of the form [τ 0 , ∆] for any τ 0 > 0. Indeed, take τ 0 > 0 and therefore ∀n ∈ N, ∀t ∈ [τ 0 , ∆], α n (t) α n (τ 0 ). By (4.4) − (4.5), as ε n tends to 0 we have that n b CO εn goes to +∞ and so the action of −(Q − ε + Q 2 ε ) seems to decrease the lower bound to 0 exponentially fast. The idea to overcome this difficulty is to find a time interval t (1) n − t (2) n = ∆ n , at each step to be sufficiently small to counterbalance the effect of n b CO εn .