Nicolas Fournier
email: nicolas.fournier@univ-paris12.fr

C L Ément Mouhot
email: clement.mouhot@ceremade.dauphine.fr

ON THE WELL-POSEDNESS OF THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH A MODERATE ANGULAR SINGULARITY

Keywords: Mathematics Subject Classification (2000): 76P05 Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05] Boltzmann equation without cutoff, long-range interaction, uniqueness, Wasserstein distance, Kantorovich-Rubinstein distance

. We deduce some well-posedness and stability results in the physically relevant cases of hard and moderately soft potentials. In the case of hard potentials, we relax the regularity assumption of [6], but we need stronger assumptions on the tail of the distribution (namely some exponential decay). We thus obtain the first uniqueness result for measure initial data. In the case of moderately soft potentials, we prove existence and uniqueness assuming only that the initial datum has finite energy and entropy (for very moderately soft potentials), plus sometimes an additionnal moment condition. We thus improve significantly on all previous results, where weighted Sobolev spaces were involved.

∂ t f t (v) = R d dv * S d-1 dσB(|v -v * |, θ) f t (v ′ )f t (v ′ * ) -f t (v)f t (v * ) , (1.1)
where

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ
and θ is the so-called deviation angle defined by cos θ

= (v-v * ) |v-v * | • σ. The collision kernel B = B(|v -v * |, θ) = B(|v ′ -v ′ * |, θ
) is given by physics and is related to the microscopic interaction between particles. In dimension d = 3 it is related to the probabilistic cross-section B of the distribution of possible outgoing velocities v ′ and v ′ * arising from a collision with two particles with velocities v and v * , by the formula B = |v -v * | B. We refer to the review papers of Desvillettes [START_REF] Desvillettes | Boltzmann's Kernel and the Spatially Homogeneous Boltzmann Equation[END_REF] and Villani [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] for more details.

Conservation of mass, momentum and kinetic energy hold at least formally for solutions to (1.1), that is for all t ≥ 0,

R d f t (v) ϕ(v) dv = R d f 0 (v) ϕ(v) dv, ϕ = 1, v, |v| 2
and we classically may assume without loss of generality that R d f 0 (v) dv = 1.

1.2. Assumptions on the collision kernel. We shall assume that the collision kernel takes the form

(A1) B(|v -v * |, θ) sin d-2 θ = Φ(|v -v * |) β(dθ)
for some function Φ : R + → R + and some nonnegative measure β on (0, π].

In the case of an interaction potential V (s) = 1/r s in dimension d = 3, with s ∈ (2, ∞), one has

(1.3) Φ(z) = cst z γ , β(θ) ∼ cst θ -1-ν , with γ = s -5 s -1 , ν = 2 s -1 .
On classically names hard potentials the case when γ ∈ (0, 1) (i.e., s > 5 in dimension d = 3), Maxwellian molecules the case when γ = 0 (i.e., s = 5 in dimension d = 3), and soft potentials the case when γ ∈ (-d, 0) (i.e., s ∈ (2, 5) in dimension d = 3).

Let us emphasize that 0+ β(dθ) = +∞, which expresses the affluence of grazing collisions, but in any case, (1.4) π 0 θ 2 β(dθ) < +∞.

In this paper we shall deal with a moderate angular singularity, that is we shall assume that the collision kernel satisfies (A2)

κ 1 = π 0 θ β(dθ) < +∞, which corresponds to s ∈ (3, ∞) in (1.3)).
We will also assume that Φ behaves as a power function, namely that for some γ ∈ (-d, 1], there exists some constant C such that for all z, z ∈ R + ,

(A3(γ)) Φ(z) ≤ C z γ ; |Φ(z) -Φ(z)| ≤ C |z γ -zγ |.
Sometimes, we will need a lowerbound: there exists c > 0 such that for all z ∈ R + , (A4(γ)) Φ(z) ≥ c z γ .

In the case of hard potentials, we will also sometimes use an additionnal technical assumption in order to obtain the propagation of some exponential moments:

(A5) β(dθ) = b(cos θ) dθ, where b is nondecreasing, convex and C 1 on [-1, 1).

In the case of moderately soft potentials, we will sometimes use (A6(ν)) β(dθ) = β(θ) dθ with β(θ) ∼ θ→0 cst θ -1-ν for some positive constant.

In practise, all these assumptions are met when one deals with interaction potential V (s) = 1/r s in dimension d = 3, with s ∈ (3, ∞).

1.3. Goals, existing results and difficulties. We study in this paper the wellposedness of the spatially homogeneous Boltzmann equation for singular collision kernel as introduced above. In particular we focus on the questions of uniqueness and stability with respect to the initial condition which were open, for collision kernel with angular cutoff, until the two recent papers [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF][START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF] (except in the special case of Maxwell molecules, see below).

In the case of a collision kernel with angular cutoff, that is when π 0 β(dθ) < +∞, there are some optimal existence and uniqueness results: Mischler-Wennberg [START_REF] Mischler | On the spatially homogeneous Boltzmann equation[END_REF] in the space of L 1 non-negative functions with finite non-increasing kinetic energy (for counter-examples of spurious solutions with increasing kinetic energy, see [START_REF] Wennberg | An example of nonuniqueness for solutions to the homogeneous Boltzmann equation[END_REF] in the hard spheres case, and [START_REF] Lu | Solutions with increasing energy for the spatially homogeneous Boltzmann equation[END_REF] in the case of hard potentials with or without angular cutoff), Lu-Mouhot [START_REF] Lu | About measures solutions of the spatially homogeneous Boltzmann equation[END_REF] in the space of non-negative measures with finite non-increasing kinetic energy.

However, the case of collision kernels without cutoff is much more difficult. At the same time it is crucial from the physical viewpoint since it corresponds to the fundamental class of the interactions deriving from inverse power-law between particles. This difficulty is not surprising, since there is a difference of nature in the collision process between the two cases: on each compact time interval, each particle collides with infinitely (resp. finitely) many others in the case without (resp. with) cutoff.

Until recently, the only uniqueness result obtained for non cutoff collision kernel was concerning Maxwellian molecules, studied successively by Tanaka [START_REF] Tanaka | Probabilistic treatment of the Boltzmann equation of Maxwellian molecules[END_REF], Horowitz-Karandikar [START_REF] Horowitz | Martingale problems associated with the Boltzmann equation[END_REF], Toscani-Villani [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas[END_REF]: it was proved in [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas[END_REF] that uniqueness holds for the Boltzmann equation as soon as Φ is constant and (1.4) is met, for any initial (measure) datum with finite mass and energy, that is

R d (1 + |v| 2 ) f 0 (dv) < +∞.
There has been recently two papers in the case where β is non cutoff and Φ is not constant. The case where Φ is bounded (together with additionnal regularity assumptions) was treated in Fournier [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF], for essentially any initial (measure) datum such that R d (1 + |v|)f 0 (dv) < ∞. More realistic collision kernels have been treated by Desvillettes-Mouhot [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF] (including the physical important cases of hard and moderately soft potentials without cutoff), for initial data in some weighted W 1,1 spaces.

In the present paper, we extend and improve the method of [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF]:

• it can deal with the physical collision kernels corresponding to hard and moderately soft potentials, as in [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF]: in dimension d = 3 we obtain wellposedeness for interaction potentials 1/r s with s ∈ (3, ∞), • the proof is simplified as compared to [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF]: it is shorter, allows measure initial conditions (for technical reasons, we had to consider only functions in [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF]), and it does not refer anymore to probabilistic arguments.

Finally let us compare our results with those in [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF], when applied to the case of an interaction potential V (s) = 1/r s in dimension d = 3.

• Our result is much better in the case of moderately soft potentials (s ∈ (3, 5)). Indeed, we assume only that the initial condition f 0 has finite mass, energy and entropy (plus, if s ∈ (3, 3.48), a moment condition R d |v| q f 0 (v)dv < ∞ for q large enough). All these conditions, together with

f 0 ∈ L p (R d ) ∩ W 1,1 (R d , (1 + |v|) 2 dv
) (for some p > 1 depending on the collision rate) were assumed in [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF]. • Our result is different in the case of hard potentials (s ∈ (5, ∞)). We allow any measure initial f 0 condition such that for some ε > 0, R d e ε|v| γ f 0 (dv) < ∞, where γ = (s -5)/(s -1). In [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF], the case where f 0 ∈ W 1,1 (R d , (1 + |v|) 2 dv) was treated. We thus assume much less regularity, but much more localization.

Let us remark that our result is quasi-optimal when s ∈ (3.48, 5), since the finiteness of entropy and energy is physically very reasonnable. It might be possible to relax the entropy condition, but it is not clear: one reasonnably has to assume a few regularity on f 0 to get the uniqueness, since the collision rate involves |v -v * | γ with γ < 0, and we remark that |v -v * | γ f 0 (dv)f 0 (dv * ) is infinite when f 0 contains, e.g., Dirac measures.

Let us emphasize that, as in [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF][START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF], we are only able to prove well-posedness in the case of a moderate angular singularity (assumption (A2)).

To our knowledge, there is no uniqueness result under the general assumption (1.4), except for Maxwellian molecules (see [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas[END_REF]).

ϕ Lip(R d ) = sup v =ṽ |ϕ(v) -ϕ(ṽ)| |v -ṽ| ≤ 1.
Let also L p (R d ) denote the Lebesgue space of measurable functions f such that

f L p (R d ) := R d f p dv 1/p < +∞.
Let P(R d ) be the set of probability measures on R d , and

P 1 (R d ) = f ∈ P(R d ), m 1 (f ) < ∞ with m 1 (f ) := R d |v| f (dv).
We denote by L ∞ ([0, T ], P 1 (R d )) the set of measurable families (f t ) t∈[0,T ] of probability measures on R d such that sup

[0,T ] m 1 (f t ) < +∞,
and by L ∞ ([0, T ],

P 1 (R d ))∩L 1 ([0, T ], L p (R d )) the set of measurable families (f t ) t∈[0,T ] of probability measures on R d such that sup [0,T ] m 1 (f t ) < +∞, T 0 f t L p (R d ) dt < +∞.
For v, v * ∈ R d , and σ ∈ S d-1 , we write

v ′ = v ′ (v, v * , σ) = v + v * 2 + |v -v * | 2 σ,
and we write σ = (cos θ, sin θ ξ)

with ξ ∈ S d-2 , θ ∈ [0, π],
in some orthonormal basis of R d with first vector (v

-v * )/|v -v * |.
Finally we denote x ∧ y = min{x, y} and x + = max{x, 0}, and for some set E we write 1 1 E the usual indicator function of E.

Main results

Let us define the notion of weak (measure) solutions we shall use.

Definition 2.1. Let B be a collision kernel which satisfies (A1-A2). A family

f = (f t ) t∈[0,T ] ∈ L ∞ ([0, T ], P 1 (R d )) is a weak solution to (1.1) if (2.1) T 0 dt R d f t (dv) R d f t (dv * ) Φ(|v -v * |) |v -v * | < +∞,
and if for any ϕ ∈ Lip(R d ), and any t ∈ [0, T ],

(2.2)

d dt R d ϕ(v) f t (dv) = R d f t (dv) R d f t (dv * ) A[ϕ](v, v * ),
where

(2.3) A[ϕ](v, v * ) = Φ(|v -v * |) π 0 β(dθ) ξ∈S d-2 [ϕ(v ′ ) -ϕ(v)] dξ.
Note that for any σ ∈ S d-1 , (2.4)

|v ′ -v| = |v -v * | 1 -cos θ 2 ≤ θ 2 |v -v * |,
so that thanks to assumption (A2), (2.1) ensures that all the terms in (2.2) are well-defined.

Let us now introduce the distance on P 1 (R d ) we shall use. For g, g ∈ P 1 (R d ), let H(g, g) be the set of probability measures on R d × R d with first marginal g and second marginal g. We then set

d 1 (g, g) = inf R d ×R d |v -ṽ| G(dv, dṽ), G ∈ H(g, g) = min R d ×R d |v -ṽ| G(dv, dṽ), G ∈ H(g, g) = sup R d ϕ(v) g(dv) -g(dv) , ϕ ∈ Lip 1 (R d ) . (2.5)
This distance is the Kantorovitch-Rubinstein distance, and can be viewed as a particular Wasserstein distance. We refer to Villani [START_REF] Villani | Topics in optimal transportation[END_REF]Section 7] for more details on this distance, and for proofs that the equalities in (2.5) hold.

Our main result is the following inequality, which will be applied in the sequel to hard and soft potentials separately.

Theorem 2.2. Let B be a collision kernel which satisfies (A1-A2). Let us consider two weak solutions f, f to (1.1) lying in L ∞ ([0, T ], P 1 (R d )), and satisfying

(2.6) T 0 dt R d ×R d f t (dv) f t (dv * ) + ft (dv) ft (dv * ) (1 + |v|) Φ(|v -v * |) < +∞. For s ∈ [0, T ], let R s ∈ H(f s , fs ) be such that d 1 (f s , fs ) = R d ×R d |v -ṽ| R s (dv, dṽ).
Then for all t ∈ [0, T ],

d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + κ 1 |S d-2 | 2 t 0 ds R d ×R d R s (dv, dṽ) R d ×R d R s (dv * , dṽ * ) × 8 Φ(|v -v * |) ∧ Φ(ṽ -ṽ * ) |v -ṽ| + Φ(|v -v * |) -Φ(ṽ -ṽ * ) + |v -v * | + Φ(|ṽ -ṽ * |) -Φ(v -v * ) + |ṽ -ṽ * | . (2.7)
The meaning of this inequality can be understood by means of probabilistic arguments, see [START_REF] Fournier | Uniqueness for a class of spatially homogeneous Boltzmann equations without angular cutoff[END_REF] for details. Consider however two infinite particle systems, whose velocity distributions are f and f respectively. The main ideas are that the first term on the right hand side expresses an increase of the optimal coupling due to simultaneous collisions (in both systems), whose rate is (optimally) the minimum between the two rates. Next, the second and third terms explain that the optimal coupling also increases due to a difference between the rates of collision in the two systems. Note that these two last terms equal zero in case of Maxwellian molecules.

We now give the application of our inequality to the study of hard potentials.

Corollary 2.3. Let B be a collision kernel which satisfies (A1-A2), and (A3)(γ) for some γ ∈ (0, 1].

(i) Let ε > 0 be fixed. There exists a constant K ε > 0 such that for any pair of weak solutions

(f t ) t∈[0,T ] , ( ft ) t∈[0,T ] to (1.1), lying in L ∞ ([0, T ], P 1 (R d ))
and satisfying

(2.8) C T, f + f , ε := sup [0,T ] R d e ε|v| γ f t + ft (dv) < +∞,
there holds for all t ∈ [0, T ]:

d 1 (f t , ft ) ≤ d 1 (f 0 , g 0 ) + K ε C T, f + f , ε t 0 d 1 (f s , fs ) 1 + log d 1 (f s , fs ) ds.
(ii) As a consequence for any f 0 ∈ P 1 (R d ), there exists at most one weak solution

f ∈ L ∞ ([0, T ], P 1 (R d )) to (1.1) starting from f 0 and such that C(T, f, ε) < +∞. (iii)
Let us now give an existence and uniqueness result, assuming (here only) additionnally (A4)(γ) and (A5). Consider f 0 ∈ P 1 (R d ) such that, for some ε 0 > 0, K > 0, we have

(2.9) R d e ε 0 |v| γ f 0 (dv) ≤ K < +∞.
Then there exists a unique weak solution

(f t ) t∈[0,∞) ∈ L 1 loc ([0, ∞), P 1 (R d ))
starting from f 0 . Furthermore, there exist ε 1 > 0 and K > 0, depending only on ε 0 , K, B, such that for all T > 0, C(T, f, ε 1 ) ≤ K. (iv) Finally let us give a result on the dependence according to the initial datum.

Consider a family (f n ) n≥1 , f ∞ of weak solutions to (1.1) such that, for some ε > 0, T > 0, we have

sup n≥1 C(T, f ∞ + f n , ε) < +∞.
Then

lim n→∞ d 1 (f n 0 , f ∞ 0 ) = 0 =⇒ lim n→∞ sup [0,T ] d 1 (f n t , f ∞ t ) = 0.
Let us recall that this result applies in particular to hard potentials in dimension d = 3 (that is inverse power-law potentials with s > 5). In [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF], under very similar conditions on the collision kernel, a well-posedness and stability result was obtained in the space L ∞ ([0, T ], W 1,1 (R d , (1 + |v| 2 ) dv)). We thus relax the regularity assumption, but we require more moments.

We finally apply our inequality to the study of soft potentials.

Corollary 2.4. Let B be a collision kernel which satisfies (A1-A2), and (A3)(γ) for some γ ∈ (-d, 0). (i) Let p ∈ (d/(d + γ), ∞] be fixed. There exists a constant K p > 0 such that for any pair of weak solutions

(f t ) t∈[0,T ] , ( ft ) t∈[0,T ] to (1.1) on [0, T ], lying in L ∞ ([0, T ], P 1 (R d )) ∩ L 1 ([0, T ], L p (R d ))
, there holds

∀ t ∈ [0, T ], d 1 (f t , ft ) ≤ d 1 (f 0 , g 0 ) e Kp C(t,f,p)+C(t, f ,p)+t ,
where

∀ t ∈ [0, T ], C(t, f, p) = t 0 f s L p (R d ) ds.
Uniqueness and stability thus hold in

L ∞ ([0, T ], P 1 (R d )) ∩ L 1 ([0, T ], L p (R d )). (ii) Let p ∈ (d/(d + γ), ∞]. For any initial condition f 0 ∈ P 1 (R d ) ∩ L p (R d ), local
existence and uniqueness hold, that is there exists

T * = T * f 0 L p (R d ) , B > 0
such that there exists a unique weak solution (f t ) t∈[0,T * ) to (1.1) which furthermore belongs to

L ∞ loc [0, T * ), P 1 (R d ) ∩ L p (R d
) . (iii) Assume now furthermore that γ ∈ (-1, 0), (A4)(γ), and (A6)(ν) for some ν ∈ (-γ, 1). Consider an initial datum f 0 ∈ P 1 (R d ) with finite energy and entropy, that is

(2.10) R d f 0 (v)(|v| 2 + | log f 0 (v)|)dv < ∞.
Assume also that for some q > γ 2 /(ν + γ), f 0 ∈ L 1 (R d , |v| q dv). Then there exists a unique weak solution (f t ) t∈[0,∞) to (1.1), which furthermore belongs to

L ∞ loc [0, ∞), P 1 (R d ) ∩ L 1 (R d , (|v| q + |v| 2 ) dv) ∩ L 1 loc [0, ∞), L p (R d ) for some (explicit) p ∈ (d/(d + γ), d/(d -ν)).
Let us recall that point (iii) applies, in dimension d = 3, to the case of moderately soft potentials, that is inverse power-law potentials with s ∈ [START_REF] Bobylev | Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions[END_REF][START_REF] Desvillettes | Boltzmann's Kernel and the Spatially Homogeneous Boltzmann Equation[END_REF]. In such a case, one has γ = (s -5)/(s -1) and ν = 2/(s -1) ∈ (-γ, 1). We observe that for s ∈ (s 0 , 5), with s 0 = 2 √ 5 -1 ≃ 3.472, the choice q = 2 is possible, so that our conditions reduce to the finiteness of entropy and energy. On the contrary, for s > 3 close to 3, q has to be chosen very large, e.g., for s = 3.01, we have to take q ≃ 200.

A similar result was obtained in [6, Theorem 1.3], assuming that

f 0 ∈ L p (R d ) ∩ L 1 (R d , |v| q dv) ∩ W 1,1 (R d , (1 + |v| 2 ) dv), with p > d/(d + γ) and q > γ 2 /(ν + γ).
We thus relax a large part of these conditions.

The rest of the paper is dedicated to the proof of these results: we establish Theorem 2.2 in Section 3. Applications to hard and soft potentials are studied in Sections 4 and 5 respectively.

The general inequality

As a preliminary step, we shall parameterize precisely the post-collisional velocities. We follow here the approach of [START_REF] Fournier | A stochastic particle numerical method for 3D Boltzmann equations without cutoff[END_REF], which was strongly inspired by Tanaka [START_REF] Tanaka | Probabilistic treatment of the Boltzmann equation of Maxwellian molecules[END_REF], and we extend it to any dimension d ≥ 2.

The first step is to define a parameterization of the sphere orthogonal to some given vector X ∈ R d . This parameterization shall not be smooth of course. We identify in the sequel S 0 = {-1, +1}. For X ∈ R d \{0}, we set S X to be the symmetry with respect to the hyperplane

H X = e d - X |X| ⊥
(where e d = (0, . . . , 0, 1)) if e d = X/|X|, and S X = Id else. We set

C X = U ∈ R d ; |U| = |X| and U, X = 0 .
Then we parameterize C X by S d-2 as follows: we set

∀ ξ = (ξ 1 , ..., ξ d-1 ) ∈ S d-2 , Π(ξ) = (ξ 1 , . . . , ξ d-1 , 0) ∈ S d-1 ⊂ R d and Γ(X, ξ) = |X| S X Π(ξ) .
It is easy to check that for a given X, the map ξ ∈ S d-2 → Γ(X, ξ) is a bijection onto C X and is a unitary parameterization. Therefore, for ξ ∈ S d-2 , θ ∈ [0, π], and X, v, v * ∈ R d , one may write

v ′ = v ′ (v, v * , θ, ξ) = v + cos θ -1 2 (v -v * ) + sin θ 2 Γ(v -v * , ξ)
and for all ϕ ∈ Lip(R d ), recalling (2.3)

A[ϕ](v, v * ) = Φ(|v -v * |) π 0 β(dθ) S d-2 dξ ϕ v ′ (v, v * , θ, ξ) -ϕ(v) .
A problem of this parameterization is its lack of smoothness. To overcome this difficulty, we shall prove the following fine version of a Lemma due to Tanaka [START_REF] Tanaka | Probabilistic treatment of the Boltzmann equation of Maxwellian molecules[END_REF], whose proof may be found in [8, Lemma 2.6] in dimension 3.

Lemma 3.1. There exists a measurable map ξ 0 :

R d × R d × S d-2 → S d-2 such that for any X, Y ∈ R d \ {0}, the map ξ → ξ 0 (X, Y, ξ
) is a bijection with jacobian 1 from S d-2 into itself (when d ≥ 3), and

(3.1) ∀ ξ ∈ S d-2 , Γ(X, ξ) -Γ Y, ξ 0 (X, Y, ξ) ≤ 3 |X -Y |.
This implies that for all v, v * , ṽ, ṽ * ∈ R d , all θ ∈ [0, π], all ξ ∈ S d-2 , we have

v ′ (v, v * , θ, ξ) -v ′ (ṽ, ṽ * , θ, ξ 0 (v -v * , ṽ -ṽ * , ξ) ≤ |v -ṽ| + 2 θ |v -ṽ| + |v * -ṽ * | . (3.2) Proof of Lemma 3.1. The case d = 2 is trivial, therefore we assume d ≥ 3. Let us consider X, Y ∈ R d \ {0}. If X/|X| = Y /|Y |, it is enough to choose ξ 0 (X, Y, ξ) = ξ. Indeed in this case S X = S Y so that Γ(X, ξ) -Γ(Y, ξ) = |X| -|Y | ≤ |X -Y |.
Γ Y, ξ 0 (X, Y, ξ) = |Y | |X| R X,Y Γ(X, ξ) ∈ C Y .
For any X, Y ∈ R d \ {0}, the application ξ → ξ 0 (X, Y, ξ) is the restriction to S d-2 of the following orthogonal linear transformation on R d-1

∀ Z ∈ R d-1 , O X,Y (Z) = Π -1 • S Y • R X,Y • S X • Π(Z).
Therefore it has unit jacobian. Finally let us check the control (3.1):

Γ(X, ξ) -Γ Y, ξ 0 (X, Y, ξ) = Γ(X, ξ) - |Y | |X| R X,Y Γ(X, ξ) ≤ Γ(X, ξ) 1 - |Y | |X| + |Y | |X| Γ(X, ξ) -R X,Y Γ(X, ξ) ≤ |X -Y | + |Y | Y |Y | - X |X| ≤ 3 |X -Y |.
Since the transformaction ξ 0 (X, Y, •) has unit jacobian, one may finally rewrite (2.3), for all ϕ ∈ Lip(R d ), all X, Y ∈ R d (which may depend on v, v * , θ), as

(3.3) A[ϕ](v, v * ) = Φ(|v-v * |) π 0 β(dθ) S d-2 dξ ϕ v ′ v, v * , θ, ξ 0 (X, Y, ξ) -ϕ(v) .
We may finally give the Proof of Theorem 2.2. We denote

h ϕ t := R d ϕ(v) f t -ft (dv) for ϕ ∈ Lip 1 (R d ), t ∈ [0, T ].
We also set h t = d 1 (f t , ft ), and we recall that

h t = R d ×R d |v -ṽ| R t (dv, dṽ) = sup ϕ ∈ Lip 1 (R d ) h ϕ t .
Step 1. Let us thus consider ϕ ∈ Lip 1 (R d ). Using (2.2), that R t ∈ H(f t , ft ) and (3.3), we immediately obtain, using the map ξ 0 built in Lemma 3.1,

d dt h ϕ t = R d ×R d f t (dv) f t (dv * ) A[ϕ](v, v * ) - R d ×R d ft (dṽ) f t (dṽ * ) A[ϕ](ṽ, ṽ * ) = R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) A[ϕ](v, v * ) -A[ϕ](ṽ, ṽ * ) = R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) π 0 β(dθ) S d-2 dξ Φ(|v -v * |) ϕ v ′ (v, v * , θ, ξ) -ϕ(v) (3.4)
-Φ(|ṽ -ṽ * |) ϕ v ′ (ṽ, ṽ * , θ, ξ 0 v -v * , ṽ -ṽ * , ξ -ϕ(ṽ) .

We now use the shortened notation v ′ = v ′ (v, v * , θ, ξ) and ṽ′ = v ′ ṽ, ṽ * , θ, ξ 0 v -v * , ṽ -ṽ * , ξ .

Noting that for all x, y ∈ R, x = x ∧ y + (x -y) + , we easily deduce from (3.4) that

d dt h ϕ t = R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) π 0 β(dθ) S d-2 dξ Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) × ϕ(v ′ ) -ϕ(ṽ ′ ) -ϕ(v) + ϕ(ṽ) + Φ(|v -v * |) -Φ(|ṽ -ṽ * |) + × ϕ(v ′ ) -ϕ(v) + Φ(|ṽ -ṽ * |) -Φ(|v -v * |) + × ϕ(ṽ) -ϕ(ṽ ′ ) =: I ϕ 1 (t) + I ϕ 2 (t) + I ϕ 3 (t),
where the last equality stands for a definition. Using that ϕ ∈ Lip 1 (R d ), (2.4), and (A2), we get

I ϕ 2 (t) + I ϕ 3 (t) ≤ κ 1 |S d-2 | 2 R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) Φ(|v -v * |) -Φ(|ṽ -ṽ * |) + |v -v * | + Φ(|ṽ -ṽ * |) -Φ(|v -v * |) + |ṽ -ṽ * | .
Next, using again that ϕ ∈ Lip 1 (R d ), we get that for all ε ∈ (0, π),

I ϕ 1 (t) ≤ R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) ε 0 β(dθ) S d-2 dξ Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) × |v ′ -v| + |ṽ ′ -ṽ| + R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) π ε β(dθ) S d-2 dξ Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) × |v ′ -ṽ′ | -|v -ṽ| + R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) π ε β(dθ) S d-2 dξ Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) × |v -ṽ| -ϕ(v) -ϕ(ṽ) =: J ϕ 1 (t, ε) + J ϕ 2 (t, ε) + J ϕ 3 (t, ε),
where the last equality stands for a definition. First for J ϕ 2 (t, ε), using (3.2) and (A2), we immediately get, by symmetry, that

J ϕ 2 (t) ≤ 2 π ε θ β(dθ) |S d-2 | R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) |v -ṽ| + |v * -ṽ * | ≤ 4 κ 1 |S d-2 | R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) |v -ṽ|.
Next, setting

α ε = |S d-2 | ε 0 θ β(dθ),
it is not hard to obtain, using (2.4), the fact that R t ∈ H(f t , ft ) and a symmetry argument, that

J ϕ 1 (t, ε) ≤ α ε 2 R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) × Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) |v| + |ṽ| + |v * | + |ṽ * | ≤ α ε R d f t (dv) R d f t (dv * ) Φ(|v -v * |) |v| + α ε R d ft (dṽ) R d ft (dṽ * ) Φ(|ṽ -ṽ * |) |ṽ| ≤ C t, f, f α ε ,
where the constant C t, f, f belongs to L 1 ([0, T ]) due to (2.6).

Finally for J ϕ 3 (t, ε) we notice that the integrand is nonnegative (since ϕ ∈ Lip 1 (R d )) and does not depend on θ, ϕ. Hence, denoting

S ε := |S d-2 | π ε β(dθ) < +∞,
we have, for any A > 0,

J ϕ 3 (t, ε) ≤ K ϕ 1 (t, ε, A) + K ϕ 2 (t, ε, A), where K ϕ 1 (t, ε, A) = A S ε R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) |v -ṽ| -ϕ(v) -ϕ(ṽ) K ϕ 2 (t, ε, A) = S ε R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) 1 1 {Φ(|v-v * |)∧Φ(|ṽ-ṽ * |)>A} |v -ṽ|.
Using that R t ∈ H(f t , ft ), and that it achieves the Wasserstein distance, we get

K ϕ 1 (t, ε, A) = A S ε d 1 (f t , ft ) -h ϕ t .
Next, we obtain

K ϕ 2 (t, ε, A) ≤ S ε R d f t (dv) R d f t (dv * ) |v| Φ(|v -v * |) 1 1 {Φ(|v-v * |)>A} +S ε R d ft (dṽ) R d ft (dṽ * ) |ṽ| Φ(|ṽ -ṽ * |) 1 1 {Φ(|ṽ-ṽ * |)>A} ≤ S ε C A t, f, f .
Due to (2.6), we observe that lim

A→∞ T 0 C A t, f, f dt = 0.
Step 2. Gathering all the previous estimates, we observe that for any ϕ ∈ Lip

1 (R d ), t ∈ [0, T ], ε > 0, A > 0, we have d dt h ϕ t ≤ H t + Γ ε,A (t) + A S ε h t -h ϕ t , (3.5) where Γ ε,A (t) := α ε C t, f, f + S ε C A t, f, f , and
H t := κ 1 |S d-2 | 2 R d ×R d R t (dv, dṽ) R d ×R d R t (dv * , dṽ * ) 8 Φ(|v -v * |) ∧ Φ(ṽ -ṽ * ) |v -ṽ| + Φ(|v -v * |) -Φ(ṽ -ṽ * ) + |v -v * | + Φ(|ṽ -ṽ * |) -Φ(v -v * ) + |ṽ -ṽ * | . Recall that h t = sup ϕ∈Lip 1 (R d ) h ϕ t ,
H s + Γ ε,A (s) ds,
so that for all t ∈ [0, T ],

h t ≤ h 0 + t 0 H s ds + T 0 Γ ε,A (t) dt.
This inequality holding for any ε > 0, A > 0, we easily conclude that (3.6) holds, since

lim A→∞ T 0 Γ ε,A (t) dt = α ε T 0 C t, f, f dt with T 0 C t, f, f dt < +∞ and α ε = |S d-2 | ε 0 θ β(dθ) --→ ε→0 0 due to (A2).

Application to hard potentials

4.1. Propagation of exponential moments. We first prove a lemma on the propagation (and appearance) of exponential moment, which is a variant of results first obtained in [START_REF] Bobylev | Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems[END_REF][START_REF] Bobylev | Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions[END_REF] (and also developed in [START_REF] Mouhot | Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials[END_REF][START_REF] Mischler | Cooling process for inelastic Boltzmann equations for hard spheres. I. The Cauchy problem[END_REF]).

Lemma 4.1. Let B be a collision kernel satisfying assumptions (A1-A2-A5) and

(A3)-(A4)(γ) for some γ ∈ (0, 1]. Let f 0 ∈ P 1 (R d ).
(i) Assume that for some ε 0 > 0, some s ∈ (0, 2),

R d e ε 0 |v| s f 0 (dv) ≤ C ε 0 ,s < +∞.
Then there exists ε 1 > 0 and a constant C > 0, depending only on s, ε 0 , C ε 0 ,s , such that for any T > 0, any weak solution

(f t ) t∈[0,T ] to (1.1) satisfies sup [0,T ] R d e ε 1 |v| s f t (dv) ≤ C < +∞.
(ii) Assume now only that e 0 = R d |v| 2 f 0 (dv) < ∞. For any s ∈ (0, γ/2), any τ > 0, there exists ε > 0 and C > 0, depending only on s, τ , and an upperbound of e 0 such that for any T > 0, any weak solution

(f t ) t∈[0,T ] to (1.1) satisfies sup t∈[τ,T ] R d e ε|v| s f t (dv) ≤ C < +∞.
Proof of Lemma 4.1. We first recall that for any t ∈ [0, T ], (4.1)

R d |v| 2 f t (dv) = R d |v| 2 f 0 (dv) =: e 0 ,
and we observe that for all v ∈ R d , all t ≥ 0, since γ ∈ (0, 1] and since

f t ∈ P 1 (R d ), (4.2) R d |v -v * | γ f t (dv * ) ≥ |v| γ - R d |v * | γ f t (dv * ) ≥ |v| γ -e γ/2 0 .
Let us fix 0 < s < 2. We define for any

p ∈ R + m p (t) := R d |v| sp f t (dv).
Step 1. The evolution equation (2.2) yields

(4.3) dm p dt = R d ×R d Φ(|v -v * |) K p (v, v * ) f t (dv) f t (dv * ),
where, using (A5) and a symmetry argument,

K p (v, v * ) := 1 2 π 0 S d-2 |v ′ | sp + |v ′ * | sp -|v| sp -|v * | sp b(cos θ) dθ dξ. Let us split b = b c η + b r η for some η ∈ (0, π) with b c η (cos θ) = b(cos θ) 1 1 θ≥η + b(cos η) + b ′ (cos η) (cos θ -cos η) 1 1 0≤θ≤η for θ ∈ (0, π]. Due to (A5), we know that b c
η ≤ b, so that b r η ≥ 0. We can split correspondingly K p = K c,η p + K r,η p . We also easily check that for each η ∈ (0, π), b c η is convex, non-decreasing, and bounded on [-1, 1). We are thus in a position to apply [3, Corollary 1], which yields that for p > 2/s, (4.4)

K c,η p (v, v * ) ≤ α p (η) |v| 2 + |v * | 2 sp/2 -K(η) |v| sp + |v * | sp where (α p (η)) p is strictly decreasing and satifies (4.5) ∀ p > 2/s, 0 < α p < C(η) sp + 1
for some constant C(η) depending on an upper bound of b c η , and some constant K(η) depending on a lower bound of the mass of β c η . Therefore K can be made uniform according to η as η → 0. For the other part of the collision kernel we use for instance [6, Lemma 2.1] and assumption (A2) to deduce that (as soon as sp ≥ 2)

(4.6) K r,η p (v, v * ) ≤ δ(η) |v| sp + |v * | sp with δ(η) ≤ cst π 0 θ b r η (cos θ) dθ → 0 as η → 0, due to (A2).
Combining (4.4,4.5,4.6) and fixing carefully η we thus find for all p > 2/s

K p (v, v * ) ≤ ᾱp |v| 2 + |v * | 2 sp/2 -K |v| sp + |v * | sp
for some constant K > 0 and where ( ᾱp ) p is strictly decreasing and satisfies, for some constant C > 0,

∀ p > 2/s, 0 < ᾱp < C sp + 1 .
We of course deduce that for p large enough, say p ≥ p 0 > 2/s, (4.7)

K p (v, v * ) ≤ ᾱp |v| 2 + |v * | 2 sp/2 -|v| sp -|v * | sp -K |v| sp + |v * | sp ,
changing if necessary the value of K > 0.

We now insert (4.7) in (4.3). Using (A4)(γ) and (4.2), we get, for p ≥ p 0 , (

dm p dt ≤ ᾱp Q p -K ′ (m p+γ/s -e γ/2 0 m p ) 4.8)
for some new constant K ′ > 0 and with

Q p := R d ×R d |v| 2 + |v * | 2 sp/2 -|v| sp -|v * | sp Φ(|v -v * |) f t (dv) f t (dv * ).
Step 2. Using (A3)(γ) and following line by line the proof of [14, Lemma 4.7] from [14, eq. (4.13)] which is the same as (4.8) here to [14, eq. (4.19)] (this proof is itself essentially based on [START_REF] Bobylev | Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions[END_REF]), we obtain the following conclusion. Set k p = [sp/4 + 1/2] (here [•] stands for the integer part). Set also, with the usual Gamma function,

z p := m p Γ(p + 1/2)
and Z p := max k=1,..,kp z (2k+γ)/s z p-2k/s , z 2k/s z p-2k/s+γ/s .

Then for some constants A ′ > 0, A ′′ > 0, A ′′′ > 0, for all p ≥ p 0 , (4.9)

dz p dt ≤ A ′ p γ/s-1/2 Z p -A ′′ p γ/s z 1+ γ sp p + A ′′′ z p .
Step 3. Next, point (i) can be checked following the ideas of [12, Proposition 3.2] (for γ = 1) and using that classically, for any p ≥ 0, sup t∈[0,∞) m p ≤ C p , for some constant depending only on B and m p (0), see e.g., [START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF]

Then (i) if a = 0, then ρ(t) = 0 for all t ∈ [0, T ]; (ii) if a > 0, then ∀ t ∈ [0, T ], m(a) -m(ρ(t)) ≤ t
where m(x) = We may now give the Proof of Corollary 2.3. We thus consider γ ∈ (0, 1], and we assume (A1)-(A2)-(A3)(γ). We also consider some ε > 0 fixed.

Step 1. Let us first prove point (i). Let us consider two weak solutions (f t ) t∈[0,T ] , ( ft ) t∈[0,T ] to (1.1), lying in L ∞ ([0, T ], P 1 (R d )) and satisfying (2.8). We are in position to apply Theorem 2, since (A3)(γ) and (2.8) clearly guarantee that (2.6) holds. We thus know that (2.7) holds. Using (A3)(γ), simple computations show that

Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) |v -ṽ| ≤ C |v| γ + |v * | γ |v -ṽ|, while Φ(|v -v * |) -Φ(|ṽ -ṽ * |) + |v -v * | ≤ C |(|v -v * | γ -|ṽ -ṽ * | γ )| |v -v * | ∧ |ṽ -ṽ * | + |(|v -v * | -|ṽ -ṽ * |)| ≤ Cγ (|v -v * | ∧ |ṽ -ṽ * |) γ-1 |v -v * | -|ṽ -ṽ * | (|v -v * | ∧ |ṽ -ṽ * |) +C |v -v * | γ + |ṽ -ṽ * | γ |v -ṽ| + |v * -ṽ * | ≤ C(1 + γ) |v| γ + |v * | γ + |ṽ| γ + |ṽ * | γ |v -ṽ| + |v * -ṽ * | .
We hence obtain by inserting these inequalities in (2.7) and using symmetry properties, that for some constant D > 0,

d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + D t 0 ds R d ×R d R s (dv, dṽ) R d ×R d R s (dv * , dṽ * ) × |v| γ + |v * | γ + |ṽ| γ + |ṽ * | γ |v -ṽ|.
Recall now that R s ∈ H(f s , fs ) achieves the Wasserstein distance. It is thus clear (recall that C(T, f + f , ε) was defined in (2.8)) that sup

[0,T ] R d ×R d R s (dv * , dṽ * ) |v * | γ + |ṽ * | γ ≤ A ε C(T, f + f , ε)
for some constant A ε . We thus get

d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + D A ε C T, f + f , ε t 0 d 1 (f s , fs ) ds +D t 0 ds R d ×R d R s (dv, dṽ) |v| γ + |ṽ| γ |v -ṽ|.
Next, for any s ∈ [0, T ] and a > 0, we have

R d ×R d R s (dv, dṽ) |v| γ + |ṽ| γ |v -ṽ| ≤ 2 a γ d 1 (f s , fs ) + R d ×R d R s (dv, dṽ) |v| γ + |ṽ| γ |v| + |ṽ| 1 1 {|v|>a} + 1 1 {|ṽ|>a} ≤ 2 a γ d 1 (f s , fs ) + L ε R d ×R d
R s (dv, dṽ) e -εa γ /2 e ε|v| γ + e -εa γ /2 e ε|ṽ| γ ≤ 2 a γ d 1 (f s , fs ) + L ε e -εa γ /2 C T, f + f, ε , for some constant L ε such that

|v| γ + |ṽ| γ |v| + |ṽ| e ε|v| γ /2 + e ε|ṽ| γ /2 ≤ L ε e ε|v| γ + e ε|ṽ| γ .
Choosing a such that a γ = 2 log d 1 (f s , fs )/ε , we finally get

R d ×R d R s (dv, dṽ) |v| γ + |ṽ| γ |v -ṽ| ≤ 4 ε d 1 (f s , fs ) log d 1 (f s , fs ) +L ε C T, f + f , ε d 1 (f s , fs ).
We finally obtain, setting

K ε = D (A ε + L ε + 4/ε), that d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + K ε C T, f + f, ε t 0 d 1 (f s , fs ) 1 + log d 1 (f s , fs ) ds.
Step 2. Points (ii) and (iv) are immediate consequences of point (i) and Lemma 4.2 applied with µ(x) = x 1 + | log x| .

Step 3. Finally, we check point (iii). We thus assume (A1)-(A2)-(A3)(γ)-(A4)(γ)-(A5), and consider an initial condition f 0 ∈ P 1 (R d ) satisfying (2.9) for some ε 0 > 0.

Then we know from Lemma 4.1-(i) that any weak solution starting from f 0 satisfies (2.8) for some ε 1 > 0. We thus deduce the uniqueness part from point (ii). Next, we approximate f 0 by a sequence of initial conditions f n 0 with finite entropy satisfying (2.9) uniformly (in n), and such that d 1 (f 0 , f n 0 ) tends to 0. Then, using for example the existence result of Villani [17, Theorem 1], we know that for each n, there exists a weak solution (f n t ) t≥0 to (1.1) starting from f n 0 . Due to Lemma 4.1-(i), we deduce that there exists ε 1 > 0 such that for all T > 0, sup n C(T, f n t , ε 1 ) < ∞. It is then not hard to deduce from point (i) and Lemma 4.2 that there exists (f t ) t≥0 such that for all T > 0, C(T, f, ε 1 ) < ∞ and lim n sup [0,T ] d 1 (f n t , f t ) = 0. An easy consequence is that (f t ) t≥0 is a weak solution to (1.1) starting from f 0 .

Application to soft potentials

The application to soft potentials is easier, since we shall apply the standard Gronwall Lemma instead of that of Yudovitch.

Proof of Corollary 2.4. We consider γ ∈ (-d, 0), and assume that (A1)-(A2)-(A3)(γ).

We observe at once that for α ∈ (-d, 0), and for q ∈ (d/(d + α), ∞], there exists a constant C α,q such that for any g

∈ P 1 (R d ) ∩ L q (R d ), any v ∈ R d , R d g(v * ) |v -v * | α dv * ≤ |v * -v|<1 g(v * ) |v -v * | α dv * + |v * -v|≥1 g(v * ) dv * ≤ C α,q g L q (R d ) + 1. (5.1)
Step 1. We first prove point (i). Let thus p ∈ (d/(d + γ), ∞]. We consider two solutions f, f as in the statement. In order to apply Theorem 2.2, we have to check that (2.6) holds. But using (5.1), since p > d/(d + γ), we get for t ∈ [0, T ]

R d f t (dv) R d f t (dv * ) (1 + |v|) |v -v * | γ ≤ R d f t (dv) (1 + |v|) sup v∈R d R d f t (dv * ) |v -v * | γ ≤ R d f t (dv) (1 + |v|) C γ,p f t L p (R d ) + 1 .
The same estimate holds for f , and therefore we conclude that the estimate (2.6) holds using that f and f belong to

L ∞ ([0, T ], P 1 (R d )) ∩ L 1 ([0, T ], L p (R d )).
Hence we deduce that (2.7) holds. Simple computations using (A3)(γ) show that

Φ(|v -v * |) ∧ Φ(|ṽ -ṽ * |) |v -ṽ| ≤ C|v -v * | γ |v -ṽ|, while Φ(|v -v * |) -Φ(|ṽ -ṽ * |) + |v -v * | ≤ C |v -v * | γ -|ṽ -ṽ * | γ |v -v * | ∧ |ṽ -ṽ * | + |v -v * | -|ṽ -ṽ * | ≤ C|γ| (|v -v * | ∧ |ṽ -ṽ * |) γ-1 (|v -v * | -|ṽ -ṽ * |) (|v -v * | ∧ |ṽ -ṽ * |) +C |v -v * | γ ∨ |ṽ -ṽ * | γ (|v -v * | -|ṽ -ṽ * |) ≤ C(1 + |γ|) (|v -v * | γ + |ṽ -ṽ * | γ ) (|v -ṽ| + |v * -ṽ * |) .
Inserting these inequalities in (2.7) and using a symmetry argument, we obtain that for some constant D > 0,

d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + D t 0 ds R d ×R d R s (dv, dṽ) R d ×R d R s (dv * , dṽ * ) |v -v * | γ + |ṽ -ṽ * | γ |v -ṽ|.
Recall now that R s ∈ H(f s , fs ) and achieves the Wasserstein distance. Hence,

sup v,ṽ R d ×R d R s (dv * , dṽ * ) |v -v * | γ + |ṽ -ṽ * | γ ≤ sup v R d f t (dv * ) |v -v * | γ + sup ṽ R d ft (dṽ * ) |ṽ -ṽ * | γ ≤ C γ,p f t L p (R d ) + C γ,p ft L p (R d ) + 2 ≤ C γ,p f t L p (R d ) + ft L p (R d ) + 2,
where we used (5.1). Since

R d ×R d R s (dv, dṽ) |v -ṽ| = d 1 (f s , fs )
we obtain finally, choosing K p := D (C γ,p + 2),

d 1 (f t , ft ) ≤ d 1 (f 0 , f0 ) + K p f t L p (R d ) + ft L p (R d ) + 1 t 0 d 1 (f s , fs ) ds.
The Gronwall Lemma then allows us to conclude the proof.

Step 2. We now check point (ii). We only have to prove the existence of solutions, since uniqueness follows from point (i). Using some results of Villani [17, Theorems 1 and 3], we know that for γ ∈ (-d, 0), for any f 0 ∈ P 1 (R d ) such that

f 0 (v) |v| 2 + | log f 0 (v)| dv < +∞,
there exists a weak solution f ∈ L ∞ ([0, ∞), (1 + |v| 2 ) dv) to (1.1) starting from f 0 . Then the existence result of point (ii) follows immediately from point (i) together with the following a priori estimates, which guarantee that if f 0 ∈ P Step 3. We now assume additionnally that γ ∈ (-1, 0), (A4)(γ), and (A6)(ν) for some ν ∈ (-γ, 1). We consider an initial datum f 0 with finite energy and entropy (2.10), and such that for some q > q 0 = γ 2 /(ν + γ), f 0 ∈ L 1 (R d , |v| q dv). Applying the result of Villani [17, Theorem 1], we know that there exists a weak solution (f t ) t∈[0,∞) to (1.1).

To conclude the proof, it suffices to apply point (i), and to check that for any weak solution (f t ) t∈[0,∞) to (1.1) starting from f 0 , (a) f ∈ L ∞ loc ([0, ∞), L 1 (R d , (|v| 2 + |v| q )dv)), (b) there exists p > p 0 := d/(d + γ) such that f ∈ L 1 loc ([0, ∞), L p (R d )). Point (a) follows from a straightforward application of (2.2), using (A1)-(A2)-(A3)(γ) and that γ ∈ (-1, 0), and concluding with the Gronwall Lemma.

To check point (b), we follow the line of [6, Proposition 3.3] (see (3.2), (3.3) and (3.4) in [START_REF] Desvillettes | Regularity, stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions[END_REF]), which was relying on exploiting the entropy production and its regularization property obtained by Alexandre-Desvillettes-Villani-Wennberg [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]. Exactly as in [6, (3.2)], we get that for any α > 0, (5.3)

T 0 (1 + |v|) γ-α f t (v) L d/(d-ν) (R d ) dt ≤ C (1 + T )
for any α > 0 any T > 0 and some constant C > 0 (depending on α). Using point (a), we also now that for all T > 0, (5.4) C T := sup

[0,T ] (1 + |v|) q f t (v) L 1 (R d ) < ∞.
By interpolation between estimates (5.3) and (5.4), we see that for all T > 0, for another constant C T depending on T ,

1 . 1 .

 11 The Boltzmann equation. We consider a spatially homogeneous gas in dimension d ≥ 2 modeled by the Boltzmann equation. Therefore the time-dependent density f = f t (v) of particles with velocity v ∈ R d solves

1. 4 .

 4 Notation. Let us denote by Lip(R d ) the set of globally Lipschitz functions ϕ : R d → R, and by Lip 1 (R d ) the set of functions ϕ ∈ Lip(R d ) such that

  Now assume that X/|X| = Y /|Y |. Then let us define R X,Y to be the axial rotation of R d transforming X/|X| into Y /|Y | around a line perpendicular to the plane determined by X and Y . Let us then define ξ 0 by the identity

f

  L p (R d ) dt ≤ C T for any 1 < p < d/(d -ν) as soon as, for instance(5.6) q > (α -γ) p -1 1 -p(d -ν)/d . Since p 0 = d/(d + γ) < d/(d -ν) (because γ > -ν) and since by assumption,q > q 0 = (-γ) p 0 -1 1 -p 0 (d -ν)/d = γ 2 γ + ν ,we clearly have (5.6) when choosing with α > 0 small enough and p > p 0 close enough. This concludes the proof of point (b).

  Proof of Corollary 2.3. We first recall the following variant of a classical lemma used by Yudovitch[START_REF] Yudovich | Non stationary flow of an ideal incompressible liquid[END_REF] in his Cauchy theorem for bidimensional incompressible non-viscious flow. See [4, Lemme 5.2.1, p. 89] for a proof.

	4.2. Lemma 4.2. Consider a nonnegative bounded function ρ on [0, T ], a real number
	a ∈ [0, ∞), and a strictly positive, continuous and non-decreasing function µ = µ(x) on (0, ∞). Assume furthermore that 1 0 dx µ(x) = +∞,
	and that for all t ∈ [0, T ],	t
	ρ(t) ≤ a +	0	µ(ρ(s)) ds.
			, Theorem 1-(ii)] and [6,
	Lemma 2.1].
	Step 4. Finally, point (ii) can be proved following line by line the proof of [14,
	Lemma 4.7].

  -arctan ||f 0 || L p ), we have (5.2)f t L p ≤ tan (arctan f 0 L p + C t) . If 1 + γ ≥ 0, we immediately conclude, since |v -v * | 1+γ ≤ 1 + |v| + |v * |, thatIf 1 + γ ≤ 0, we use (5.1) (with α = 1 + γ and q = p, which is valid since p > d/(d + γ) > d/(d + α)), and we deduce thatd dt R d |v| f t (dv) ≤ κ 1 |S d-2 | 2 C 1+γ,p f t L p (R d ) + 1 =: A f t L p (R d ) + A ′ ,so that for 0 ≤ t < T * , we have, recalling (5.2), L p + C s) ds + A ′ t.

							t
	R d	|v| f t (dv) ≤	R d	|v| f 0 (dv) + A	0	tan (arctan f 0
	Next, we easily check, using (2.2), (2.4) and (A2) that
		d dt R d |v| f t (dv) ≤ κ 1	|S d-2 | 2	R d f t (dv)
		d dt R d	|v| f t (dv) ≤ κ 1	|S d-2 | 2	1 + 2	R d	|v| f t (dv) ,
	so that for t ≥ 0, we have
		R d	|v|f t (dv) ≤ e κ 1 |S d-2 |t	R d	|v|f 0 (dv) + 1 .

1 (R d ) ∩ L p (R d ), then this bound propagates locally (in time): first there exists C = C(B) such that (see [6, Proposition 3.2] and its proof) for any p ∈ (d/(d + γ), ∞], any weak solution to (1.1) satisfies

d dt f t L p ≤ C 1 + f t 2 L p (R d ) , so that for 0 ≤ t < T * := 1 C (π/2 R d f t (dv * ) |v -v * | 1+γ .