Jean-Marie Barbaroux
email: barbarou@univ-tln.fr

Dirk Hundertmark
email: dirk.hundertmark@kit.edu

AND Tobias Ried
email: tobias.ried@kit.edu

Semjon Vugalter
email: semjon.wugalter@kit.edu

STRONG SMOOTHING FOR THE NON-CUTOFF HOMOGENEOUS BOLTZMANN EQUATION FOR MAXWELLIAN MOLECULES WITH DEBYE-YUKAWA TYPE INTERACTION

Keywords: 2010 Mathematics Subject Classification: 35D10 (Primary); 35B65, 35Q20, 82B40 (Secondary) Smoothing of weak solutions, Non-cutoff homogeneous Boltzmann equation, Debye-Yukawa potential, Maxwellian molecules

We study weak solutions of the homogeneous Boltzmann equation for Maxwellian molecules with a logarithmic singularity of the collision kernel for grazing collisions. Even though in this situation the Boltzmann operator enjoys only a very weak coercivity estimate, it still leads to strong smoothing of weak solutions in accordance to the smoothing expected by an analogy with a logarithmic heat equation.

Introduction and main results

We study the regularity of weak solutions of the Cauchy problem

       ∂ t f = Q( f, f ) f | t=0 = f 0 (1)
for the fully nonlinear homogeneous Boltzmann equation in d ≥ 2 dimensions with initial datum f 0 ≥ 0 having finite mass, energy and entropy,

f 0 ∈ L 1 2 (R d ) ∩ L log L(R d
). The bilinear Boltzmann collision operator Q is given by

Q(g, f ) = R d S d-1 B |v -v * |, v -v * |v -v * | • σ g(v ′ * ) f (v ′ ) -g(v * ) f (v) dσdv * . (2)
Here we use the σ-representation of the collision process, in which

v ′ = v + v * 2 + |v -v * | 2 σ, v ′ * = v + v * 2 - |v -v * | 2 σ, for σ ∈ S d-1 .
The collision kernel B takes into account the detailed scattering process by which the particles change their velocities, which, in a dilute gas, can be assumed to involve only two particles at a time (binary collisions). In the important case of inverse-power-law interactions Φ(r) = r 1-n , n > 2, it is of the form B(|v -v * |, cos θ) = |v -v * | γ b(cos θ), γ = n-(2d- 1) n-1 , where cos θ = v-v * |v-v * | • σ and b is the so called angular collision kernel. Even though an explicit formula for b is not known, one can show [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF] that b is smooth away from the singularity, non-negative, and has the non-integrable singularity

sin d-2 θ b(cos θ) θ→0 ∼ K θ 1+2ν (3)
for some K > 0 and 0 < ν < 1.

It has been noted for some time now, see [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] and the references therein, that the divergence (3) leads to a coercivity in the Boltzmann collision kernel of the form

-Q(g, f ) ≈ (-∆) ν f + lower order terms, (4)
that is, it behaves similar to a singular integral operator with leading term proportional to a fractional Laplacian. If the interaction is instead of Debye-Yukawa type

Φ(r) = r -1 e -r s , 0 < s < 2. ( 5
)
the angular collision cross-section b(cos θ) has a much weaker non-integrable singularity of logarithmic type

sin d-2 θ b(cos θ) ∼ κθ -1 log θ -1 µ (6)
for grazing collisions θ → 0, with some κ, µ > 0. For example in dimension d = 3 one has µ = 2 s -1 [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF]. Going through the calculations of [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] one can check that µ = d-2 s -1 in arbitrary dimension d ≥ 2. In this case, the coercive effects are much weaker and of the form -Q(g, f ) ≈ (log(1 -∆)) µ+1 f + lower order terms, [START_REF] Desvillettes | Boltzmann's kernel and the spatially homogeneous Boltzmann equation[END_REF] as was noticed in [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF], see also appendix A. In this work, as in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF][START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF], we consider only the socalled Maxwellian molecules approximation, where the collision kernel does not depend on vv * , but only on the collision angle θ.

Even though b has a singularity, the quantity

π 2 0 sin d θ b(cos θ) dθ < ∞, (8)
which is related to the momentum transfer in the scattering process, is finite in both cases (3) and [START_REF] Cercignani | The Boltzmann Equation and Its Applications[END_REF]. We will also assume that b(cos θ) is supported on angles θ ∈ [0, π 2 ], which is always possible due to symmetry properties of the Boltzmann collision operator.

In this article, we use the following notations and conventions: Given a vector v ∈ R d and α ≥ 0, let v α := (α + |v| 2 ) 1/2 , and v := v 1 . For p ≥ 1 and s ∈ R the weighted L p spaces are given by

L p s (R d ) := f ∈ L p (R d ) : • s f ∈ L p (R d ) , equipped with the norm f L p s (R d ) = R d | f (v)| p v sp dv 1/p .
We will also make use of the weighted (L 2 -based) Sobolev spaces

H k ℓ (R d ) = f ∈ S ′ (R d ) : • ℓ f ∈ H k (R d ) , k, ℓ ∈ R,
where H k (R d ) are the usual Sobolev spaces given by

H k (R d ) = f ∈ S ′ (R d ) : • k f ∈ L 2 (R d ) , for k ∈ R. We also use H ∞ (R d ) = k≥0 H k (R d ).
The inner product on L 2 (R d ) is given by f, g = R d f (v)g(v) dv. Further, closely related to the functions with finite (negative) entropy H( f ) := R d f log f dv is the space

L log L(R d ) = f : R d → R measurable : f L log L = R d | f (v)| log (1 + | f (v)|) dv < ∞ .
We use the following convention regarding the Fourier transform of a function f in this article,

(Ff )(η) = f (η) = R d f (v) e -2πiv•η dv.
We denote D v = -i 2π ∇ and for a suitable function G : R d → C we define the operator G(D v ) as a Fourier multiplier, that is,

G(D v ) f := F -1 [G f ].
The precise notion of a solution of the Cauchy problem (1) is given by Definition 1.1 (Weak Solutions of the Cauchy Problem (1) [START_REF] Arkeryd | Intermolecular forces of infinite range and the Boltzmann equation[END_REF][START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF]). Assume that the initial datum

f 0 is in L 1 2 (R d ) ∩ L log L(R d ). f : R + × R d → R is called a weak solution to the Cauchy problem (1), if it satisfies the following conditions 1 : (i) f ≥ 0, f ∈ C(R + ; D ′ (R d )) ∩ L ∞ (R + ; L 1 2 (R d ) ∩ L log L(R d )) (ii) f (0, •) = f 0 (iii) For all t ≥ 0, mass is conserved, R d f (t, v) dv = R d f 0 (v) dv, kinetic energy is conserved, R d f (t, v) v 2 dv = R d f 0 (v) v 2 dv, and the entropy is increasing, that is, H( f ) is decreasing, H( f (t, •)) ≤ H( f 0 ). (iv) For all ϕ ∈ C 1 (R + ; C ∞ 0 (R d )) one has f (t, •), ϕ(t, v) -f 0 , ϕ(0, •) - t 0 f (τ, •)∂ τ ϕ(τ, •) dτ = t 0 Q( f, f )(τ, •), ϕ(τ, •) dτ, for all t ≥ 0, (9)
where the latter expression involving Q is defined for test functions ϕ ∈ W 2,∞ (R d ) by

Q( f, f ), ϕ = 1 2 R 2d S d-1 b v -v * |v -v * | • σ f (v * ) f (v) ϕ(v ′ ) + ϕ(v ′ * ) -ϕ(v) -ϕ(v * ) dσdvdv * .
Weak solutions of the above type of the Cauchy problem (1) for the homogeneous Boltzmann equation are known to exist due to results by Arkeryd [START_REF] Arkeryd | On the Boltzmann equation[END_REF][START_REF] Arkeryd | Intermolecular forces of infinite range and the Boltzmann equation[END_REF], which were later extended by Villani [START_REF] Villani | On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations[END_REF]. They are known to be unique [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas[END_REF], see also the review articles [START_REF] Desvillettes | Boltzmann's kernel and the spatially homogeneous Boltzmann equation[END_REF][START_REF] Mischler | On the spatially homogeneous Boltzmann equation, Annales de l'Institut Henri Poincare[END_REF].

In [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] it has been shown that weak solutions to the Cauchy problem (1) with Debye-Yukawa type interactions enjoy an H ∞ smoothing property, i.e. starting with arbitrary initial datum f 0 ≥ 0,

f 0 ∈ L 1 2 ∩ L log L, one has f (t, •) ∈ H ∞ for any positive time t > 0.
Based upon our recent proof [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF] of Gevrey smoothing for the homogeneous Boltzmann equation with Maxwellian molecules and angular singularity of the inverse-power law type (3), we can show a stronger than H ∞ regularisation property of weak solutions in the Debye-Yukawa case.

To this aim we define the function spaces

Definition 1.2. Let µ > 0. A function f ∈ H ∞ (R d
) is defined to be in the space A µ (R d ) if there exist constants C > 0 and b > 0 such that

∂ α f L 2 ≤ C |α|+1 e b|α| 1+1/µ for all α ∈ N d 0 . ( 10
)
1 Throughout the text, whenever not explicitly mentioned, we will drop the dependence on t of a function, i.e.

f (v) := f (t, v) etc
For µ > 0 we define the family of function spaces, parametrised by τ > 0,

D e τ(log D ) µ+1 : L 2 (R d ) := f ∈ L 2 (R d ) : e τ(log D ) µ+1 f ∈ L 2 (R d ) Remark 1.3. Let µ > 0. Then A µ (R d ) = τ>0 D e τ(log D ) µ+1 : L 2 (R d ) .
The proof is rather technical and is deferred to Appendix B.

In view of the coercivity property [START_REF] Desvillettes | Boltzmann's kernel and the spatially homogeneous Boltzmann equation[END_REF] and the regularisation properties of the logarithmic heat equation

∂ t f = (log(1 -∆)) µ+1 f, ( 11
)
the spaces A µ , through their Fourier characterisation in Remark 1.

≥ 0, f 0 ∈ L 1 2 (R d ) ∩ L log L(R d ).
Then for any T 0 > 0 there exist β, M > 0 such that

e βt(log D v ) µ+1 f (t, •) ∈ L 2 (R d )
and

sup η∈R d e βt(log D v ) µ+1 | f (t, η)| ≤ M for all t ∈ (0, T 0 ]. In particular, f (t, •) ∈ A µ for all t > 0.
Remark 1.5. This regularity is much weaker than the Gevrey regularity we proved in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF] for singular kernels of the form (3), but it is much stronger than the H ∞ smoothing shown in [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF]. Moreover, it is exactly the right type of regularity one would expect for a coercive term of the form (7) from the analogy with the heat equation [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF].

For our proof we have to choose β small if T 0 is large and our bounds on β deteriorate to zero in the limit T 0 → ∞, so our Theorem 1.4 does not give a uniform result for all t > 0. Nevertheless, by propagation results due to Desvillettes, Furiolo and Terraneo [START_REF] Desvillettes | Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules[END_REF] we even have the uniform bound Corollary 1.6. Under the same assumptions as in Theorem 1.4, for any weak solution f of the Cauchy problem [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]

with initial datum f 0 ≥ 0 and f 0 ∈ L 1 2 (R d ) ∩ L log L(R d ), there exist constants 0 < K, C < ∞ such that sup 0≤t<∞ sup η∈R d e K min(t,1) (log η ) µ+1 | f (t, η)| ≤ C. ( 12
)
The strategy of the proofs of our main result Theorem 1.4 is as follows: We start with the additional assumption f 0 ∈ L 2 on the initial datum (Theorem 4.1). We use the known H ∞ smoothing [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] of the non-cutoff Boltzmann equation to allow for this. Within an L 2 framework, a reformulation of the weak formulation of the Boltzmann equation is possible which includes suitable growing Fourier multipliers. As in [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] the inclusion of Fourier multipliers leads to a nonlocal and nonlinear commutator with the Boltzmann kernel. For non-power-type Fourier multipliers this commutator is considerably more complicated than the one encountered in the H ∞ smoothing case. To overcome this, we follow the strategy we developed in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF], where an inductive procedure was invented to control the commutation error, in order to prove the Gevrey smoothing conjecture in the Maxwellian molecules case.

The main differences compared with [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF] are:

(1) For the weights needed in the proof of Theorem 1.4 we have a much stronger enhanced subadditivity bound, see Lemma 2.1. The proof is more involved than the one in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF], though.

(2) Because of the stronger form of the subadditivity bound, we can allow for a bigger loss in the induction step. We can therefore work with a more straightforward version of the 'impossible' L 2 -to-L ∞ bound, see Lemma 3.1. (3) Due to the special form of the weights we use in this paper, which are in some sense in between the power type weights used in [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] and the sub-gaussian weight used in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF], we don't have to do much of the additional songs and dances from [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF].

2. Enhanced subadditivity and properties of the Fourier weights

Lemma 2.1. Let µ > 0 and h : [0, ∞) → [0, ∞), s → h(s) = log(α + s) µ+1 for some α ≥ e µ .
Then h is increasing, concave and for any 0

≤ s -≤ s + , h(s -+ s + ) ≤ µ + 1 1 + log α h(s -) + h(s + ). ( 13
)
Remark 2.2. For α ≥ e µ , one has h(0) = µ µ+1 > 0, and from the concavity of h one concludes the subadditivity estimate

h(s -) + h(s + ) ≥ h(s -+ s + ) + h(0) > h(s -+ s + )
for all s -, s + ≥ 0. Note that this is the best possible bound for general s -, s + ≥ 0. For 0 ≤ s -≤ s + Lemma 2.1 shows that the subadditivity bound can be improved to gain the small factor µ+1 1+log α , which is strictly less than one for α > e µ , in front of h(s -). So this is indeed an enhanced subadditivity property of the function h. Lemma 2.1 plays a similar role in the proof of Theorem 1.4, as Lemma 2.6 in our previous paper [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF]. Here the situation is a bit simpler than in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF], since by choosing α large enough, we can make the term µ+1 1+log α as small as we like. Proof. Since

h ′ (s) = µ + 1 α + s log(α + s) µ ≥ 0 if α ≥ 1,
the function h is increasing. Further,

h ′′ (s) = µ + 1 (α + s) 2 log(α + s) µ-1 µ -log(α + s) ≤ µ + 1 (α + s) 2 log(α + s) µ-1 µ -log(α) ≤ 0 for α ≥ e µ , so h is concave.
For all s -, s + ≥ 0,

h(s -+ s + ) = h(s -) h(s -+ s + ) -h(s + ) h(s -) + h(s + ),
and by concavity,

s + → h(s -+ s + ) -h(s + ) is decreasing, so using 0 ≤ s -≤ s + one has h(s -+ s + ) ≤ h(s -) h(2s -) -h(s -) h(s -) + h(s + ). Since h ′ is decreasing, h(2s -) -h(s -) = 2s - s - h ′ (r) dr ≤ h ′ (s -)s -
and we get

h(s -+ s + ) ≤ h(s -) h ′ (s -)s - h(s -) + h(s + ) = h(s -) (µ + 1)s - (α + s -) log(α + s -) + h(s + ). For α ≥ 1 the function F α : [0, ∞) → R, F α (s) := (α + s) log(α + s)
, is strictly convex and thus

F α (s) ≥ F α (0) + F ′ α (0)s = α log α + (1 + log α)s ≥ (1 + log α)s.

It follows that

s - (α+s -) log(α+s -) ≤ 1 1+log α and therefore

h(s -+ s + ) ≤ h(s -) µ + 1 1 + log α + h(s + ). Proposition 2.3. Let β, t, µ > 0, α ≥ e µ and define the function G : [0, ∞) → R by G(r) := e βt2 -µ-1 (log(α+r)) µ+1 .
Then for all 0 ≤ s -≤ s + with s -+ s + = s one has

G(s) -G(s + ) ≤ 2 -µ βt(µ + 1) 1 - s + s log(α + s) µ G(s -) µ+1 1+log α G(s + ). Proof. Using G ′ (s) = 2 -µ-1 βt(µ + 1) 1 α + s log(α + s) µ G(s)
one has

G(s) -G(s + ) = s s + G ′ (r) dr ≤ 2 -µ-1 βt(µ + 1) s -s + α + s + log(α + s) µ G(s),
where we used that s + ≤ s and the fact that G is increasing. Since s -+ s + = s and 0 ≤ s -≤ s + , in particular s + ≥ s 2 , we can further estimate

s -s + α + s + = 1 - s + s s α + s + ≤ 1 - s + s 2s + α + s + ≤ 2 1 - s + s , to obtain G(s) -G(s + ) ≤ 2 -µ βt(µ + 1) 1 - s + s log(α + s) µ G(s).
The rest now follows from the enhanced subadditivity property [START_REF] Toscani | Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas[END_REF], namely

G(s) = G(s -+ s + ) ≤ G(s -) µ+1 1+log α G(s + ). 3. Extracting L ∞ bounds from L 2 : a simple proof
Following is a simple bound which controls the size of a function h in terms of its local L 2 norm and some global a priori bounds on h and its derivative.

Lemma 3.1. Let h ∈ C 1 b (R d ), i.e.

h is a bounded continuously differentiable function with bounded derivative. Then there exists a constant L < ∞ (depending only on d, h L

∞ (R d ) and, ∇h L ∞ (R d ) ) such that for any x ∈ R d , |h(x)| ≤ L Q x |h(y)| 2 dy 1 d+2 , ( 14
)
where Q x is a unit cube in R d with x being one of the corners, oriented away from the origin in the sense that x • (yx) ≥ 0 for all y ∈ Q x .

Remarks 3.2. (1) We use the norm ∇h

L ∞ (R d ) = sup η∈R d |∇h(η)|, where | • | is the Euclidean norm on R d . (2)
The exponent 1 d+2 can be improved if higher derivatives of the function h are bounded, see Section 2.3 in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF]. This was important for the results of [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF], but we don't need it here because of the stronger form of the enhanced subadditivity Lemma for the weight we consider in this paper.

Remark 3.3. If f ∈ L 1 1 (R d ), its Fourier transform satisfies f ∈ C 1 b (R d ) by the Riemann-Lebesgue lemma. Since ∇ η f (η) = 2πiv f (η) one has the a priori bound ∇ f L ∞ (R d ) ≤ 2π f L 1 1 (R d ) . If f is a weak solution of the homogeneous Boltzmann equation, we can also bound ∇ f L ∞ (R d ) ≤ 2π f 0 L 1 2 (R d )
uniformly in time due to conservation of energy. Proof. We first consider the one-dimensional case and prove the d-dimensional result by iteration in each coordinate direction.

Let u ∈ C 1 b (R) and q ≥ 1. Then for any r ∈ R we have

|u(r)| q ≤ max q u ′ L ∞ (R) , u L ∞ (R) I r |u(s)| q-1 ds, ( 15
)
where

I r = [r, r + 1] if r ≥ 0 and I r = [r -1, r] if r < 0.
Indeed, assuming for the moment r ≥ 0,

|u(r)| q - I r |u(s)| q ds ≤ I r |u q (r) -u q (s)| ds,
and by the fundamental theorem of calculus,

|u q (r) -u q (s)| ≤ q I r |u(t)| q-1 |u ′ (t)| dt ≤ q u ′ L ∞ (R) I r |u(t)| q-1 dt.
Combined with the trivial estimate

I r |u(s)| q ds ≤ u L ∞ (R) I r |u(s)| q-1
ds one arrives at inequality (15) for r ≥ 0. The case r < 0 is analogous.

For the case d > 1 we remark that for any

y ∈ R d , h(y 1 , . . . , y j-1 , • , y j+1 , . . . , y d ) L ∞ (R) ≤ h L ∞ (R d )
, and

∂ j h(y 1 , . . . , y j-1 , • , y j+1 , . . . , y d ) L ∞ (R) ≤ ∇h L ∞ (R d )
and setting q = d + 2 iterative application of (15) in each coordinate direction yields for

x ∈ R d |h(x)| d+2 ≤ max (d + 2) ∇h L ∞ (R d ) , h L ∞ (R d ) d I x 1 ו••×I x d |h(y)| d+2-d dy, hence |h(x)| ≤ max (d + 2) ∇h L ∞ (R d ) , h L ∞ (R d ) d d+2 Q x |h(y)| 2 dy 1 d+2 =: L h 2 d+2 L 2 (Q x )
where

Q x = I x 1 × • • • × I x d
is a unit cube directed away from the origin with x ∈ R d at one of its corners.

Smoothing property of the Boltzmann operator

A central step in the proof of Theorem 1.4 is to prove a version for L 2 initial data first. This is the content of Theorem 4.1 below. In the remainder of this article we will always assume that the collision kernel satisfies assumptions ( 6) and ( 8).

Theorem 4.1. Let f be a weak solution of the Cauchy problem (1) with initial datum f

0 ≥ 0, f 0 ∈ L 1 2 (R d ) ∩ L log L(R d ) and in addition f 0 ∈ L 2 (R d ).
Then for all T 0 > 0 there exist β, M > 0 such that for all t ∈ [0, T 0 ] sup

η∈R d e βt(log η α) µ+1 | f (t, η)| ≤ M and e βt(log D v α) µ+1 f (t, •) ∈ L 2 (R d ) where α = e d 2 + d+2 2 µ .
We give the proof of Theorem 4.1 in section 5. To prepare for its proof, let α ≥ e µ and β > 0 and define the Fourier multiplier

G : R + × R d → R + by G(t, η) := e βt(log η α) µ+1 , η α := α + |η| 2 1 2
and for Λ > 0 the cut-off multiplier

G Λ : R + × R d → [0, ∞) by G Λ (t, η) := G(t, η)1 Λ (|η|) wehre 1 Λ is the characteristic function of the interval [0, Λ]. The associated Fourier multiplication operator is denoted by G Λ (t, D v ), G Λ (t, D v ) f := F -1 G Λ (t, •) f (t, •)
By Bobylev's identity, the Fourier transform of the Boltzmann operator for Maxwellian molecules is

Q(g, f )(η) = S d-1 b η |η| • σ ĝ(η -) f (η + ) -ĝ(0) f (η) dσ, η ± = η ± |η|σ 2 , (16)
Note that, due to the cut-off in Fourier space,

G Λ f, G 2 Λ f ∈ L ∞ ([0, T 0 ]; H ∞ (R d )) for any finite T 0 > 0 and Λ > 0, if f ∈ L ∞ ([0, T 0 ]; L 1 (R d ))
, and even analytic in a strip containing

R d v .
In particular, by Sobolev embedding,

G Λ f, G 2 Λ f ∈ L ∞ ([0, T 0 ]; W 2,∞ (R d )), so Q( f, f )(t, •), G 2 Λ f (t, •) is well-defined.
≤ f 0 ∈ L 1 2 (R d ) ∩ L log L(R d )
, and let T 0 > 0. Then for all t ∈ (0, T 0 ], β > 0, α ∈ (0, 1), and

Λ > 0 we have G Λ f ∈ C [0, T 0 ]; L 2 (R d ) and 1 2 G Λ (t, D v ) f (t, •) 2 L 2 - 1 2 t 0 f (τ, •), ∂ τ G 2 Λ (τ, D v ) f (τ, •) dτ = 1 2 1 Λ (D v ) f 0 2 L 2 + t 0 Q( f, f )(τ, •), G 2 Λ (τ, D v ) f (τ, •) dτ. ( 17
)
Informally, equation (17) follows from using ϕ(t,

•) := G 2 Λ (t, D v ) f (t, •) in the weak formulation of the homogenous Boltzmann equation. Recall that G 2 Λ f ∈ L ∞ ([0, T 0 ]; W 2,∞ (R d ))
for any finite T 0 > 0, so it still misses the required regularity in time needed to be used as a test function. The proof of Proposition 4.2 is analogous to Morimoto et al. [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF], see also Appendix A in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF].

For weak solutions of the homogeneous Boltzmann equation we have (see also Corollary A.5):

Proposition 4.3. Let g be a weak solution of the Cauchy problem (1) with initial datum g

0 ∈ L 1 2 (R d ) ∩ L log L(R d ).
Then there exist constants C g 0 , C g 0 > 0 depending only on the dimension d, the angular collision kernel b,

g 0 L 1 , g 0 L 1 2 and g 0 L log L such that for all f ∈ H 1 (R d ) one has -Q(g, f ), f ≥ C g 0 log(α + e) µ+1 log D v α µ+1 2 f 2 L 2 -C g 0 f 2 L 2 , ( 18
)
uniformly in t ≥ 0.

Remark 4.4. The above estimate makes the intuition [START_REF] Desvillettes | Boltzmann's kernel and the spatially homogeneous Boltzmann equation[END_REF] on the coercivity of the Boltzmann collision operator precise. It was already used in Motimoto, Ukai, Xu and Yang [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF] to show H ∞ smoothing and goes back to Alexandre, Desvillettes, Villani and Wennberg [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF], where they proved the corresponding sub-elliptic estimate for Boltzmann collision operators with the singularity arising from power-law interaction potentials and more general singularities.

Since we need to carefully fine-tune some of the constants in our inductive procedure, we need a precise information about the dependence of the constants on α in this inequality. Therefore we will give the proof of the coercivity estimate in the form stated above in Appendix A.

Together with Proposition 4.2 the coercivity estimate from Proposition 4.3 implies Corollary 4.5 (A priori bound for weak solutions). Let f be a weak solution of the Cauchy problem (1) with initial datum f 0 ≥ 0 satisfying f 0 ∈ L 1 2 ∩ L log L, and let T 0 > 0. Then there exist constants C f 0 , C f 0 > 0 (depending only on the dimension d, the collision kernel b, f 0 L 1 2 and f 0 L log L ) such that for all t ∈ (0, T 0 ], β, µ > 0, α ≥ 0, and Λ > 0 we have

G Λ f 2 L 2 ≤ 1 Λ (D v ) f 0 2 L 2 + 2 C f 0 t 0 G Λ f 2 L 2 dτ + 2 t 0 β - C f 0 (log(e + α)) µ+1 log D v α µ+1 2 G Λ f 2 L 2 dτ + 2 t 0 G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f dτ. ( 19
)
Proof. In order to make use of the coercivity property of the Boltzmann collision operator, we write

Q( f, f ), G 2 Λ f = G Λ Q( f, f ), G Λ f = Q( f, G Λ f ), G Λ f + G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f
and estimate the first term with (18).

Since ∂ τ G 2 Λ (τ, η) = 2β log η α µ+1 G 2 Λ (t, η), we further have f, ∂ τ G 2 Λ f = 2β log D v α µ+1 2 G Λ f 2 L 2
, and inserting those two results into (17), one obtains the claimed inequality (19).

≥ 0, f 0 ∈ L 1 2 (R d ) ∩ L log L(R d ).
Then for all t, β, µ, Λ > 0 and α ≥ e µ one has the bound

| G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f | ≤ βt(µ + 1) R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G(η -) µ+1 1+log α | f (η -)| × G Λ (η + )| f (η + )| G Λ (η)| f (η)| dσdη. ( 20
)
Remark 4.7. The bound (20) is very similar to the one we derived in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF]. In particular, it is a trilinear expression in the weak solution f . The f (η -) term is multiplied by a faster-thanpolynomially growing function. If the Fourier multiplier G were only growing polynomially, the factor G(η -) µ+1 1+log α would be replaced by 1, making the analysis much easier. We will therefore rely on the inductive procedure we developed in [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equations without cutoff for Maxwellian molecules[END_REF] to treat exactly this type of situation.

Proof. Bobylev's identity and a small computation show that

| G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f | = F G Λ Q( f, f ) -Q( f, G Λ f ) , F G Λ f L 2 ≤ R d S d-1 b η |η| • σ G Λ (η)| f (η)| | f (η -)| | f (η + )||G(η) -G(η + )| dσ dη
since G Λ is supported on the ball {|η| ≤ Λ} and |η + | ≤ |η|. We further have

|η ± | 2 = |η| 2 2 1 ± η • σ |η| , |η -| 2 + |η + | 2 = |η| 2 ,
in particular by the support assumption on the collision kernel b, η•σ |η| ∈ [0, 1], and therefore

0 ≤ |η -| 2 ≤ |η| 2 2 ≤ |η + | 2 ≤ |η| 2 .
From Proposition 2.3 it now follows that

G(η) -G(η + ) = G(|η| 2 ) -G(|η + | 2 ) ≤ βt(µ + 1) 1 - |η + | 2 |η| 2 log η α µ G(η -) µ+1 1+log α G(η + ),
which completes the proof.

Lemma 4.8.

R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η + )| f (η + )| G Λ (η)| f (η)| dσdη ≤ c b,d G Λ f 2 L 2 + log D v α µ 2 G Λ f 2 L 2 , ( 21
)
where c b,d = 1 2 max{1, 2 µ-1 } max{2 d-1-µ (log 2) µ , 1 + 2 d-1 } |S d-2 | π 2 0 sin d θ b(cos θ) dθ.
Proof. Using Cauchy-Schwartz, in the form ab ≤ a 2 2 + b 2 2 , one can split the integral into

R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η + )| f (η + )| G Λ (η)| f (η)| dσdη ≤ 1 2 R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η) 2 | f (η)| 2 dσdη + 1 2 R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η + ) 2 | f (η + )| 2 dσdη
and we will treat the two terms separately. To estimate the first integral, one introduces polar coordinates such that η |η| • σ = cos θ and thus, since

|η + | 2 = |η| 2 1 + η |η| • σ = |η| 2 cos 2 θ 2 ,
obtains

I := 1 2 R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η) 2 | f (η)| 2 dσdη = 1 2 |S d-2 | π 2 0 sin d-2 θ b(cos θ) sin 2 θ 2 dθ R d log η α µ G Λ (η) 2 | f (η)| 2 dη ≤ 1 2 |S d-2 | π 2 0 sin d θ b(cos θ) dθ log D v α µ 2 G Λ f 2 L 2 .
Notice that the θ integral is finite due to the assumptions on the angular collision kernel. This is another instance where cancellation effects play an important role in controlling the singularity for grazing collisions.

It remains to bound the second integral, and we will do this after a change of variables η → η + . This change of variables is well-known to the experts, see, for example, [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF][START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF]. We give some details for the convenience of the reader.

Observe that

η + •σ |η + | = |η + | |η| and η•σ |η| = 2 η + •σ |η + | 2
-1, and by Sylvester's determinant theorem, one has

∂η + ∂η = 1 2 1 + η |η| ⊗ σ = 1 2 d 1 + η |η| • σ = 1 2 d-1 η + • σ |η + | 2 = 1 2 d-1 |η + | 2 |η| 2 . Since 0 ≤ |η -| ≤ |η + | and |η| 2 = |η -| 2 + |η + | 2 , in particular |η| 2 ≤ 2|η + | 2 , it follows that ∂η + ∂η ≥ 2 -d and log η α = 1 2 log(α + |η| 2 ) ≤ 1 2 log 2 + 1 2 log(α + |η + | 2 ) = 1 2 log 2 + log η + α . For all x, y ≥ 0 one has        (x + y) µ ≤ 2 µ-1 (x µ + y µ
) for µ ≥ 1 by convexity, and

(x + y) µ ≤ x µ + y µ for µ < 1,
where the second statement is a consequence of the fact that for 0 < µ < 1 the function 0 ≤ s → h(s) = (1 + s) µs µ is monotone decreasing for all s > 0 with h(0) = 1. Therefore,

log η α µ ≤ max{1, 2 µ-1 } 2 -µ (log 2) µ + log η + α µ .
After those preparatory remarks, we can estimate

I + := 1 2 R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ G Λ (η + ) 2 | f (η + )| 2 dσdη = 1 2 S d-1 R d b        2 η + • σ |η + | 2 -1               1 - η + • σ |η + | 2        log η α µ G Λ (η + ) 2 | f (η + )| 2 ∂η + ∂η -1 dη + dσ ≤ 2 d-1 max{1, 2 µ-1 } × 2 -µ (log 2) µ R d S d-1 b        2 η + • σ |η + | 2 -1               1 - η + • σ |η + | 2        G Λ (η + ) 2 | f (η + )| 2 dσdη + + R d S d-1 b        2 η + • σ |η + | 2 -1               1 - η + • σ |η + | 2        log η + α µ G Λ (η + ) 2 | f (η + )| 2 dσdη + .

Introducing new polar coordinates with pole

η + |η + | , such that cos ϑ = η + •σ |η + | ≥ 1 √ 2 , i.e. ϑ ∈ [0, π 4
], one then gets

I + ≤ 2 d-1 max{1, 2 µ-1 } |S d-2 | π 4 0 sin d ϑb(cos 2ϑ) dϑ × 2 -µ (log 2) µ G Λ f 2 L 2 + log D v α µ 2 G Λ f 2 L 2 .
Estimating We now have all the necessary pieces together to start the inductive proof of Theorem 4.1 for initial data that are in addition square integrable.

The proof is based on gradually removing the cut-off Λ in Fourier space, in such a way that the commutation error can be controlled, even though it contains fast growing terms. For fixed T 0 , µ > 0 and α ≥ e µ we define Definition 5.1 (Induction Hypothesis Hyp Λ (M).). Let M ≥ 0 and Λ > 0. Then for all 0 ≤ t ≤ T 0 ,

sup |ξ|≤Λ G(t, ξ) µ+1 1+log α | f (t, ξ)| ≤ M.
Remark 5.2. Recall that the Fourier multiplier G also depends on β > 0 and α ≥ e µ and we suppress this dependence here.

The induction step itself will be divided into two separate steps:

Step 1 Hyp Λ (M) =⇒ G √ 2Λ f L 2 ≤ C via a Gronwall argument. Step 2 L 2 → L ∞ bound: G √ 2Λ f L 2 ≤ C =⇒ Hyp Λ (M) for intermediate Λ = 1+ √ 2 2 Λ.
Here it is essential that M does not increase during the induction procedure. This can be accomplished by choosing β small enough at very beginning.

Lemma 5.3 (Step 1)

. Fix T 0 , µ > 0 and α ≥ e µ and let M ≥ 0 and Λ > 0. Let further C f 0 , C f 0 and c b,d be the constants from Corollary 4.5 and Lemma 4.8, respectively. If

0 < β ≤ β 0 (α) := C f 0 (log(e + α)) µ+1 log α log α + 2T 0 (µ + 1)c b,d M , ( 22
)
then for any weak solution of the Cauchy problem [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]

with initial datum f 0 ≥ 0, f 0 ∈ L 1 2 ∩ L log L, Hyp Λ (M) =⇒ G √ 2Λ f L 2 (R d ) ≤ 1 √ 2Λ (D v ) f 0 L 2 (R d ) e T 0 A f 0 (α) ,
where A f 0 (α)

:= C f 0 + C f 0 log α 2(log(e+α)) µ+1 depends on f 0 only through f 0 L 1 , f 0 L 1 2 and f 0 L log L . Proof. Assume Hyp Λ (M) is true. Since |η -| = |η| sin θ 2 ≤ |η| √ 2
by the assumption on the angular cross-section, the hypothesis implies

sup |η|≤ √ 2Λ G(η -) µ+1 1+log α | f (η -)| ≤ M.
With this uniform estimate at hand, we can bound the commutation error by 20). By Lemma 4.8, this can be further bounded by

| G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f | ≤ 2βt(µ + 1)M R d S d-1 b η |η| • σ 1 - |η + | 2 |η| 2 log η α µ × G √ 2Λ (η + )| f (η + )| G √ 2Λ (η)| f (η)| dσdη, see equation (
| G Λ Q( f, f ) -Q( f, G Λ f ), G Λ f | ≤ βT 0 (µ + 1)c b,d M G √ 2Λ f 2 L 2 + log D v α µ 2 G √ 2Λ f 2 L 2
for all 0 ≤ t ≤ T 0 . Thus, the a priori bound from Corollary 4.5 yields

G √ 2Λ f 2 L 2 ≤ 1 √ 2Λ (D v ) f 0 2 L 2 + 2 C f 0 + βT 0 (µ + 1)c b,d M t 0 G √ 2Λ f 2 L 2 dτ + 2 t 0 β log D v α µ+1 2 G √ 2Λ f 2 L 2 + βT 0 (µ + 1)c b,d M log D v α µ 2 G √ 2Λ f 2 L 2 - C f 0 (log(e + α)) µ+1 log D v α µ+1 2 G √ 2Λ f 2 L 2 dτ ( 23
)
Choosing β ≤ β 0 (α) as defined in ( 22) ensures that the integrand in the last term on the right hand side of ( 23) is negative. Indeed, setting

B = T 0 (µ + 1)c b,d M and C = C f 0 (log(e+α)) µ+1 , so that β ≤ C log α log α+2B
, one sees that

β log η α + βB -C log η α ≤ - 2CB log α + 2B log η α + CB log α log α + 2B = CB log α -log(α + |η| 2 ) log α + 2B ≤ 0,
and further, since log α ≥ µ > 0,

βB ≤ CB log α log α + 2B = C log α 2 2B log α + 2B ≤ C log α 2 .
It follows that

G √ 2Λ f 2 L 2 ≤ 1 √ 2Λ (D v ) f 0 2 L 2 + 2 C f 0 + βT 0 (µ + 1)c b,d M t 0 G √ 2Λ f 2 L 2 dτ ≤ 1 √ 2Λ (D v ) f 0 2 L 2 + 2A f 0 (α) t 0 G √ 2Λ f 2 L 2 dτ. Now Gronwall's lemma implies G √ 2Λ f 2 L 2 ≤ 1 √ 2Λ (D v ) f 0 2 L 2 e 2A f 0 (α)t ≤ 1 √ 2Λ (D v ) f 0 2 L 2 e 2A f 0 (α)T 0 .
Lemma 5.4 (Step 2). Let β, µ > 0, T 0 > 0, and

Λ ≥ Λ 0 := 2 √ d √ 2 -1 . If there exist finite constants B 1 , B 2 ≥ 0 such that for all 0 ≤ t ≤ T 0 f (t, •) L 1 1 (R d ) ≤ B 1 , and (G √ 2Λ f )(t, •) L 2 (R d ) ≤ B 2 ,

then there exists a constant K depending only on the dimension d and the bounds B

1 , B 2 , such that for all |η| ≤ Λ := 1+ √ 2 2 Λ and t ∈ [0, T 0 ] | f (t, η)| ≤ K G(t, η) -2 d+2 .
Proof. By Remark 3.3 f satisfies the conditions of Lemma 3.

1 with ∇ f L ∞ (R d ) ≤ 2πB 1 , uniformly in t ∈ [0, T 0 ]. Obviously, also f L ∞ (R d ) ≤ f L 1 (R d ) ≤ B 1 . It follows that for any η ∈ R d | f (η)| ≤ (2π(d + 2)B 1 ) d d+2       Q η | f | 2 dη       1 d+2
.

where Q η is a unit cube with one corner at η, such that η • (ζη) ≥ 0 for all ζ ∈ Q η . Since its diameter is √ d, the condition Λ ≥ Λ 0 and the choice of Λ guarantee that for |η| ≤ Λ the cube Q η always stays inside a ball around the origin with radius √ 2Λ. By the orientation of Q η and since the Fourier weight G is a radial and increasing function in η, we can further estimate

| f (η)| ≤ (2π(d + 2)B 1 ) d d+2 G(η) -2 d+2       Q η G(η) 2 | f | 2 dη       1 d+2 ≤ (2π(d + 2)B 1 ) d d+2 G(η) -2 d+2 G √ 2Λ f 2 d+2 L 2 (R d ) ≤ 2π(d + 2)B 1 B 2 d 2 d d+2 G(η) -2 d+2 which is the claimed inequality with K = 2π(d + 2)B 1 B 2 d 2 d d+2 . Proof of Theorem 1.4. Let µ > 0 and T 0 > 0 be fixed. Set α * = e d 2 + d+2 2 µ ≥ e µ , which is chosen in such a way that µ+1 1+log α * = 2 d+2 and the function s → log(α * + s) µ+1 is concave. Choosing Λ 0 = 2 √ d √ 2-1
as in Lemma 5.4, we define the length scales for our induction by

Λ N := Λ N-1 + √ 2Λ N-1 2 = 1 + √ 2 2 Λ N-1 =       1 + √ 2 2       N Λ 0 , N ∈ N.
By conservation of energy, we have Lemma 5.4. By Lemma 5.3 a good (in particular uniform in N ∈ N) choice for B 2 is

f (t, •) L 1 1 ≤ f (t, •) L 1 2 = f 0 L 1 2 =: B 1 in view of
B 2 := f 0 L 2 (R d ) e T 0 A f 0 (α * ) .

Define further

M := max        2B 1 + 1, 2π(d + 2)B 1 B 2 d 2 d d+2        ,
where the second expression is just the constant K from Lemma 5.4.

For the start of the induction, we need Hyp Λ 0 (M) to hold. Since

sup t∈[0,T 0 ] sup |η|≤Λ 0 G(η) µ+1 1+log α * | f (η)| ≤ e µ+1 1+log α * βT 0( 1 2 log(α * +Λ 2 0 )) µ+1 B 1 ,
there exists β > 0 small enough, such that for the the above choice of M, Hyp Λ 0 (M) is true for all 0 < β ≤ β.

For the induction step, assume that Hyp Λ N (M) is true. Setting

β = min{β 0 (α * ), β}
with β 0 (α) from Lemma 5.3, all the assumptions of Lemma 5.3 are fulfilled and it follows that

G √ 2Λ N f L 2 (R d ) ≤ 1 √ 2Λ N (D v ) f L 2 (R d ) e T 0 A f 0 (α * ) ≤ B 2 .
Notice that the right hand side of this inequality does not depend on M. Lemma 5.4 now implies that for all

|η| ≤ Λ N = Λ N+1 G(t, η) 2 d+2 | f (t, η)| ≤ K ≤ M for all t ∈ [0, T 0 ]. By the choice of α * this means that Hyp Λ N+1 (M) is true.
By induction, it follows that Hyp Λ N (M) holds for all N ∈ N, in particular

sup t∈[0,T 0 ] sup η∈R d e βt(log η α * ) µ+1 | f (η)| ≤ M.
Another application of Lemma 5.3 implies

G √ 2Λ N f L 2 (R d ) ≤ f 0 L 2 (R d ) e T 0 A f 0 (α * ) for all N ∈ N. Passing to the limit N → ∞, it follows that G f L 2 (R d ) ≤ B, that is, e βt(log D v α * ) µ+1 f (t, •) ∈ L 2 (R d ).

Smoothing effect for arbitrary physical initial data

Proof of Theorem 1.4. Let T > 0 be arbitrary (but finite). By the already known H ∞ smoothing property of the homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction, see [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF], for any 0 < t 0 < T one has

f ∈ L ∞ ([t 0 , T ]; H ∞ (R d )), in particular f (t, •) ∈ L 2 (R d ) for all t 0 ≤ t ≤ T . Using f (t 0 , •) ∈ L 1 2 ∩ L log L ∩ L 2 as new initial datum, Theorem 4.1 implies that there exist β, M > 0 such that e βt(log D v α * ) µ+1 f (t, •) ∈ L 2 (R d ) and e βt(log • α * ) µ+1 f (t, •) L ∞ (R d ) ≤ M for all t ∈ [t 0 , T ]
. By the characterisation of the spaces A µ (see Appendix B), and since t 0 and T are arbitrary, it follows that f (t, •) ∈ A µ (R d ) for all t > 0.

Sketch of the Proof of Corollary 1.6. In the notation of [START_REF] Desvillettes | Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules[END_REF], basically, the only thing that needs to be checked is that the function ψ α : [0, ∞) → [0, ∞), r → ψ α (r) := (log √ α + r) µ+1 satisfies (i) ψ α (r) → ∞ for r → ∞ (ii) ψ α (r) ≤ r for r large enough (iii) there exists R ≥ 1 such that for all 0 ≤ λ ≤ 1

ψ α (λ 2 |η| 2 ) ≥ λ 2 ψ α (|η| 2 ) whenever λ|η| ≥ R.
Property (iii) is fulfilled by any concave function ψ with ψ(0) ≥ 0. This clearly is the case for ψ α if α ≥ e µ , see Lemma 2.1. So we take the α from Theorem 4.1 and conclude propagation with Theorem 1.2 from [START_REF] Desvillettes | Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules[END_REF].

Appendix A. Coercivity of the Boltzmann collision operator with Debye-Yukawa Potential

Since we need to take care of the dependence of the constants within our inductive approach, we present a slightly modified version of the coercivity estimate first proved by Morimoto et al. [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF], based upon the ideas of Alexandre et al. [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF].

Proposition A.1 (Coercivity Estimate). Let g ≥ 0, g ∈ L 1 1 (R d ) ∩ L log L(R d ).
Then there exists a positive constant C g depending only on the dimension d, the collision kernel b, g L 1 1 and g L log L and constants C > 0, R ≥ √ e, depending only on the dimension d and on the collision kernel b, such that for all α ≥ 0 and all 0 L log L , see the proof of Lemma 3 in [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF]. In particular, if g is a weak solution of the Cauchy problem (1) with initial datum

≤ f ∈ H 1 (R d ) one has -Q(g, f ), f ≥ C g log(α + e) µ+1 log D v α µ+1 2 f 2 L 2 -C g log R µ+1 + C g L 1 f 2 L 2 .
g 0 ∈ L 1 2 (R d ) ∩ L log L(R d ), we have g L 1 = g 0 L 1 , g L 1 1 ≤ g L 1 2 ≤ g 0 L 1 2 and g L log L ≤ log 2 g 0 L 1 + H(g 0 ) + C δ,d g 0 1-δ L 1 2
, for small enough δ > 0. This implies

C g ≥ C g 0 .
Applying the remark to the constant C ′ g in Lemma A.3, we arrive at Corollary A.5. Let g be a weak solution of the Cauchy problem (1) with initial datum g 0 ∈ L 1 2 (R d ) ∩ L log L(R d ) and angular collision kernel b satisfying [START_REF] Cohen | A simple proof of the Denjoy-Carleman theorem[END_REF] and [START_REF] Desvillettes | Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules[END_REF]. Then the conclusion of Proposition A.1 holds with C g and g L 1 replaced by C g 0 and g 0 L 1 , i.e.

-Q(g, f ), f ≥ C g 0 log(α + e) µ+1 log D v α µ+1 2 f 2 L 2 -C g 0 log R µ+1 + C g 0 L 1 f 2 L 2 ,
uniformly in t ≥ 0.

Proof of Proposition A.1. We have Q(g, f ), f = Re Q(g, f ), f and by Bobylev's identity,

-Re Q(g, f ), f = Re R d ×S d-1 b η |η| • σ ĝ(0) f (η) -ĝ(η -) f (η + ) f (η) dσdη = 1 2 R d ×S d-1 b η |η| • σ f (η) f (η + ) , 2ĝ(0) -ĝ(η -) -ĝ(η -) 0 f (η) f (η + ) C 2 dσdη = 1 2 R d ×S d-1 b η |η| • σ f (η) f (η + ) , ĝ(0) -ĝ(η -) -ĝ(η -) ĝ(0) f (η) f (η + ) C 2 dσdη - 1 2 R d ×S d-1 b η |η| • σ f (η) f (η + ) , -ĝ(0) 0 0 ĝ(0) f (η) f (η + ) C 2 dσdη =: I 1 -I 2 .
To estimate

I 2 = 1 2 R d ×S d-1 b η |η| • σ ĝ(0) | f (η + )| 2 -| f (η)| 2 dσdη
, we do a change of variables η + → η as in [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] in the first part, treating b as if it were integrable, and using a limiting argument to make the calculation rigorous (this is a version of the cancellation lemma of [START_REF] Alexandre | Entropy dissipation and long-range interactions[END_REF] on the Fourier side). We then obtain with ĝ(0) = g L 1

I 2 = |S d-2 | π/2 0 sin d-2 θ b(cos θ)        1 cos d θ 2 -1        dθ g L 1 (R d ) f 2 L 2 (R d ) .
In particular, since 1

cos d θ 2 -1 = d 8 θ 2 + O(θ 3
), the θ-integral is finite and it follows that

|I 2 | ≤ C g L 1 (R d ) f 2 L 2 (R d ) .
For the integral I 1 , we note that since g ≥ 0, the matrix in I 1 is positive definite by Bochner's theorem and has the lowest eigenvalue ĝ(0) -|ĝ(η -)|. Therefore,

I 1 ≥ 1 2 R d ×S d-1 b η |η| • σ ĝ(0) -|ĝ(η -)| | f (η)| 2 + | f (η + )| 2 dσdη ≥ 1 2 R d | f (η)| 2 S d-1 b η |η| • σ ĝ(0) -|ĝ(η -)| dσdη
and by Lemma A.3,

I 1 ≥ C ′ g 2 {|η|≥R} | f (η)| 2 log η α log(α + e) µ+1 dη ≥ C ′ g 2(log(α + e)) µ+1 log D v α µ+1 2 f 2 L 2 - C ′ g 2 log(α + R 2 ) 2 log(α + e) µ+1 f L 2 ≥ C ′ g 2(log(α + e)) µ+1 log D v α µ+1 2 f 2 L 2 - C ′ g 2 log R µ+1 f L 2 .
In the last inequality we used the fact that for R ≥ √ e the function α → log(α+R 2 ) 2 log(α+e) is decreasing. Combining the estimates of I 1 and I 2 and setting C g = C ′ g /2, we arrive at the claimed sub-elliptic estimate for the Boltzmann operator with Debye-Yukawa singularity.

It remains to give the

Proof of Lemma A.3. Since g ≥ 0, g ∈ L 1 1 ∩ L log L, there exists a constant C g > 0 such that for all

η ∈ R d ĝ(0) -|ĝ(η)| ≥ C g |η| 2 ∧ 1 . It is therefore enough to bound S d-1 b( η |η| • σ)(|η -| 2 ∧ 1) dσ. Recall that |η -| 2 = |η| 2 2 1 -η |η| • σ ,

and, choosing spherical coordinates with pole η

|η| such that η |η| • σ = cos θ, we obtain

S d-1 b η |η| • σ |η -| 2 ∧ 1 dσ = |S d-2 | π 2 0 sin d-2 θ b(cos θ) |η| 2 sin 2 θ 2 ∧ 1 dθ ≥ |S d-2 | 4π 2 π 2 0 sin d-2 θ b(cos θ) |η| 2 θ 2 ∧ 1 dθ.
By the assumption (6) on the singularity for grazing collisions on b, there exists a θ 0 > 0 small enough such that We prove a precise correspondence between the decay in Fourier space and the growth rate of derivatives of functions in A µ .

|S d-2 | 4π 2 π 2 0 sin d-2 θ b(cos θ) |η| 2 θ 2 ∧ 1 dθ ≥ κ 2 |S d-2 | 4π 2 θ 0 0 θ -1 log θ -1 µ |η| 2 θ 2 ∧ 1 dθ. Let R > 0 be large enough, such that 1 R < θ 0 . Then for |η| ≥ R we have κ 2 |S d-2 | 4π 2 θ 0 0 θ -1 log θ -1 µ |η| 2 θ 2 ∧ 1 dθ ≥ κ 2 |S d-2 | 4π 2 θ 0 1 |η| θ -1 log θ -1 µ dθ = κ 2 |S d-2 | 4π 2 1 µ + 1        log |η| µ+1 -log 1 θ 0 µ+1        ≥ C log |η| µ+1
Theorem B.1. Let µ > 0. Then A µ (R d ) = τ>0 D e τ(log D ) µ+1 : L 2 (R d ) .
Invoking a classic theorem by Denjoy and Carleman (see, for instance, [START_REF] Cohen | A simple proof of the Denjoy-Carleman theorem[END_REF][START_REF] Krantz | A Primer of Real Analytic Functions[END_REF][START_REF] Rudin | Real and Complex Analysis[END_REF]) one can show that the classes A µ for µ > 0 are not quasi-analytic, that is, they contain non-vanishing C ∞ functions of arbitrarily small support.

Proof. Let µ > 0 be fixed and assume first that e τ(log

D ) µ+1 f L 2 < ∞ for some τ > 0. Let α ∈ N d 0 with |α| = n for some n ∈ N 0 . Then ∂ α f 2 L 2 = R d |(2πiη) α f (η)| 2 dη ≤ (2π) 2n R d η 2n | f | 2 dξ = 2n(2π) 2n ∞ 0 t 2n-1 ν f ({ η > t}) dt
where we introduced the (finite) measure ν f (dη

) := | f (η)| 2 dη. Since η ≥ 1 for all η ∈ R d , one has ν f ({ η > t}) = ν f (R d ) = f 2 L 2 (R d ) for t < 1.
For t ≥ 1 we estimate

ν f ({ η > t}) ≤ e -2τ(log t) µ+1 R d e 2τ(log η ) µ+1 ν f (dη) = e -2τ(log t) µ+1 e τ(log D ) µ+1 f 2 L 2 (R d ) , since 1 ≤ t → e 2τ(log t) µ+1 is increasing. It follows that ∂ α f 2 L 2 ≤ (2π) 2n f 2 L 2 (R d ) + 2n(2π) 2n e τ(log D ) µ+1 f 2 L 2 (R d ) ∞ 1 t 2n-1 e -2τ(log t) µ+1 dt. ( 24
)
To extract the required growth in n from the latter integral, we essentially apply Laplace's method. Indeed, substituting the logarithm and rescaling suitably yields The function h : (0, ∞) → R, h(t) : = tt µ+1 is strictly concave and attains its maximum at t * = (µ + 1) -

≤ n τ 1/µ         (µ + 1) -1/µ + √ π 2 √ µ τ µ + 1 1 2µ n -µ+1 2µ         e 2τ -1/µ µ(µ+1) -(1+1/µ) n 1+1/µ . (27)
Therefore, inserting the obtained bound into (24), there exist constants C > 0 and b > 0, depending on τ and µ, such that ∂ α f L 2 ≤ C |α|+1 e b|α| 1+1/µ for all α ∈ N d 0 .

(28)

For µ ∈ (0, 1) the global bound (25) does not hold, but, as in the proof of Laplace's method for the asymptotics of integrals, one can find a suitable δ > 0 such that the bound (25) holds on [t * -δ, t * + δ] and the contribution to the integral outside of this interval is of much smaller order. So the right hand side of (26) still provides an upper bound modulo lower order terms and we conclude (28) also in this case.

For the converse assume that (28) holds. We want to show that there exists a τ > 0 such that e τ(log D ) µ+1 f ∈ L 2 (R d ). Using that e 2τ(log η ) µ+1 = 1 + η 1 2τ(µ + 1)t -1 (log t) µ e 2τ(log t) µ+1 dt one obtains e τ(log

D ) µ+1 f 2 L 2 (R d ) = f 2 L 2 (R d ) + ∞ 1 2τ(µ + 1)t -1 (log t) µ e 2τ(log t) µ+1 ν f ({ η > t}) dt. ( 29
)
Next we estimate for t > 1 and any n ∈ N 0 , since |η| 2 ≥ t 2 -1 on { η > t},

ν f ({ η > t} ≤ 1 (2π) 2n (t 2 -1) n R d (2π) 2n |η| 2n | f | 2 dη.
By the multinomial theorem, we have (in the standard multi-index notation) by assumption. Since this holds for any n ∈ N 0 , we even have

|η| 2n =        
ν f ({ η > t} ≤ exp inf n∈N 0 2n log A -n log(t 2 -1) + 2bn 1+1/µ = exp        2 inf n∈N 0        bn 1+1/µ -n log √ t 2 -1 A              
where for notational convenience we set A = C n+1 √ d 2π . If √ t 2 -1 < A, then the infimum in the above exponent is just zero, so ν({ η > t}) ≤ 1 in this case. If, however, √ t 2 -1 ≥ A, we get inf

n∈N 0        bn 1+1/µ -n log √ t 2 -1 A        ≤ bn 1+1/µ * -n * log √ t 2 -1 A where n * = 1 b µ µ+1 log √ t 2 -1 A µ
, and ⌊a⌋ denotes the greatest integer smaller or equal to a ∈ R. Obviously,

n * ≤        1 b µ µ + 1 log √ t 2 -1 A        µ < n * + 1,
so we get the bound inf

n∈N 0        bn 1+1/µ -n log √ t 2 -1 A        ≤ - µ β µ        1 µ + 1 log √ t 2 -1 A        µ+1 + log √ t 2 -1 A .
In particular, there exists T * > 1 and β > 0 such that for t > T * , one has inf

n∈N 0        bn 1+1/µ -n log √ t 2 -1 A        ≤ -β(log t) µ+1 .
This shows,

ν f ({ η > t}) ≤ e -β(log t) µ+1
for large enough t, and choosing τ < β/2 in (29) we get the finiteness of e τ(log D ) µ+1 f L 2 (R d ) .

4. 1 .Proposition 4 . 2 .

 142 L 2 reformulation and coercivity. Let f be a weak solution of the Cauchy problem (1) with initial datum f 0 satisfying 0

4. 2 .Proposition 4 . 6 (

 246 Controlling the commutation error. Bound on the Commutation Error). Let f be a weak solution of the Cauchy problem (1) with initial datum f 0

π 4 0 2 0

 42 sin d ϑb(cos 2ϑ) dϑ ≤ π sin d θ b(cos θ) dθ and combining the bounds on I and I + proves inequality (21). 5. Smoothing effect for L 2 initial data: Proof of Theorem 4.1

Remark A. 2 . 1 Lemma A. 3 . 1 ∩ 1 1,

 21311 Of course, the above lower bound holds for a much larger class of functions, essentially, log D v α µ+1 2 f L 2 should be finite.As a first step in the proof of Proposition A.Assume that the angular collision kernel b satisfies[START_REF] Cohen | A simple proof of the Denjoy-Carleman theorem[END_REF] and (8) and let g ≥ 0, g ∈ L 1 L log L. Then there exists a constant C ′ g > 0, depending only on b, the dimension d, and g L 1 , g L and g L log L , as well as a constant R ≥ √ e depending only on d and b, such thatS d-2 b η |η| • σ ĝ(0) -|ĝ(η -)| dσ ≥ C ′ g log η α log(α + e)µ+1 1 {|η|≥R} . Remark A.4. The constant C g (respectively C ′ g ) is an increasing function of g L 1 , g -1

e one has log |η| = 1 2 log |η| 2 ≥

 2 for some constant C > 0 depending only on the dimension and the collision kernel b. We conclude by noting that for all |η| ≥ √ log η α log(e + α) , since for any α ≥ 0 the function [e, ∞) ∋ s → H(s) : = log slog(α+s) log(α+e) is non-decreasing with H(e) = 0. Appendix B. Properties of the function spaces A µ

∞ 1 t

 1 2n-1 e -2τ(log t) µ+1 dt = n τ 1/µ n 1+1/µ (t-t µ+1 ) dt.(25)

  -1) n e 2bn 1+1/µ

  3, capture exactly the gain of regularity that is to be expected for the Boltzmann equation with Debye-Yukawa type angular singularity. Indeed, our main result is

Theorem 1.4. Let f be a weak solution of the Cauchy problem (1) for the homogeneous Bolzmann equation for Maxwellian molecules with angular collision kernel satisfying

[START_REF] Cohen | A simple proof of the Denjoy-Carleman theorem[END_REF]

and

[START_REF] Desvillettes | Propagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules[END_REF]

, and initial datum f 0

  1/µ . If µ ≥ 1, h ′′ is negative and decreasing, so by Taylor's theorem we can bound

	n τ	1/µ	0	∞	e 2τ -1/µ n 1+1/µ (t-t µ+1 ) dt	(26)

h(t) ≤ h(t * ) + h ′′ (t * ) 2 (tt * ) 2 1 {t>t * }

and obtain with h

(t * ) = µ(µ + 1) -(1+1/µ) , h ′′ (t * ) = -µ(µ + 1) 1/µ ,

Acknowledgements. It is a pleasure to thank the REB program of CIRM for giving us the opportunity to start this research. J.-M. B. was partially supported by the project SQFT ANR-12-JS01-0008-01. D. H., T. R., and S. V. gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. D. H. also thanks the Alfried Krupp von Bohlen und Halbach Foundation for financial support. Furthermore, we thank the University of Toulon and the Karlsruhe Institute of Technology for their hospitality.


François Cheviré

Frédéric Clabau

Olivier Larcher

Emmanuelle Orhan

Franck Tessier
email: franck.tessier@univ-rennes1.fr

Roger Marchand

Cheviré François

Clabau Frédéric

Larcher Olivier

Orhan Emmanuelle

Tessier Franck

Marchand Roger

Tunability of the optical properties in the Y6(W,Mo)(O,N)12 system

Keywords: Rare earth tungstates, solid solution, optical properties, UV absorbers

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Over the past years, the interest in ternary metal oxynitrides as pigments has been considerable because they exhibit bright colors and are free of toxic heavy metals [1][2][3][4][5]. For instance, starting from the defect fluorite-type tungstates R 14 WO 33 and R 6 WO 12 (R = La  Lu and Y) -that could be also respectively formulated A 4 X 7.33  0.67 and A 4 X 6.86  1.14 (A for cations and X for anions) -Diot et al. have evidenced new oxynitride solid solution domains between the starting oxide and the limit formulation A 4 X 6  2 [6]. This work has shown the possibility to tune selectively the absorption of the pigments by controlling the nitrogen content in the oxynitrides which exhibit a wide range of color from pale yellow to brown and kaki respectively for the nitrogen-rich products. The following compositions -Y 6 WO 9.1 N 1.9  3.0 , Sm 6 WO 8.3 N 2.5  3.2 and Sm 14 W 4 O 23.4 N 6.4  6.2 -were of interest as new yellow pigment able to compete with BiVO 4 . Using the same concept of tunability of the optical absorption with the chemical composition in order to answer specific needs, we have recently reported the optical properties of fluorite-type compositions within the Y 6 WO 12 -CeO 2 solid solution and have studied their potential as inorganic UV absorber materials [7]. Keeping starting from the tungstate Y 6 WO 12 , here we have investigated the influence of both cationic and anionic substitutions on the optical properties through the study of two fluorite-type solid solutions: Y 6 (W 1-x Mo x )O 12 and Y 6 W(O 12-3/2x N x ).

Experimental

Preparation

Oxide precursors were synthesized by the amorphous citrate route as described in previous works [7,8]. Y 2 O 3 (99.99%), H 2 WO 4 (99%) and MoO 3 (99%) were used as the starting materials and C 6 H 8 O 7 (> 99%) as the chelating agent. Oxynitride powders were prepared by annealing the corresponding oxide precursors under flowing ammonia using the nitridation setup detailed elsewhere [4,8].

Characterization

Powders were analyzed by X-ray diffraction (Philips PW3710) at room temperature using CuK  radiation. Nitrogen and oxygen contents were determined with a LECO TC-436 analyzer using the inert gas fusion method. Diffuse reflectance spectra were collected using a Varian Cary 100 Scan spectrometer equipped with the integrating sphere Labsphere (DRC-CA-30I) and the optical gap was then determined using the theory of Kubelka-Munk [9].

Quantum mechanics

Calculations have been performed on the Y 6 WO 12 unit cell in the framework of the density functional theory (DFT) with the CRYSTAL06 code [START_REF] Dovesi | CRYSTAL06 User's Manual[END_REF]. Becke's threeparameter hybrid nonlocal exchange functional [START_REF] Becke | [END_REF] combined with the Lee-Yang-Parr gradient corrected correlation functional, B3LYP [12], has been used. The standard B3LYP hybrid method has been extensively used for molecules and also provides an accurate description of crystalline structures in regard to bond lengths, binding energies, and band gap values [START_REF] Hu | Encyclopedia of Computational Chemistry[END_REF][START_REF] Muscat | [END_REF]. Pseudopotentials have been used to describe the cores of W [15] and Y [16] atomic centers while the all electrons basis set 6-31G* described the oxygen atoms. The k-points sampling was chosen to be 40 points in the irreducible part of the Brillouin zone.

Results and Discussion

Y 6 WO 12

The symmetry of the tungstates R 6 WO 12 depends on the rare earth elements: cubic or pseudo-cubic from La to Pr, pseudo-tetragonal from Nd to Gd, and rhombohedral from Tb to Lu and for Y [START_REF] Mccarthy | Proceedings of the 5 th Materials Research Symposium[END_REF]. The latter compounds show a structural analogy with the mixed-valent binary oxide Pr 7 O 12 [START_REF] Ray | [END_REF][19][20] as well as the ternary oxide Y 6 UO 12 [21]. Yttrium based-system was preferentially studied because of the absence of localized 4f-4f transitions which could induce parasitic absorption bands. The crystal structure was refined for Y 6 WO 12 by Diot et al. and described as yttrium being sevenfold coordinated in a monocapped trigonal prism while the tungsten is sixfold coordinated by oxygen atoms forming a regular octaedra [22]. As reported in the literature for Y 6 WO 12 [23,24], a phase transition from rhombohedral to cubic occurs at 1765 ± 25°C. Above this temperature, the unit cell corresponds to a fluorite and the compound can be formulated Y 3.43 W 0.57 O 6.86  1.14 . A random distribution on both the cationic and anionic sub-networks has to be considered which means that there is no more oxygen/vacancies ordering and that yttrium and tungsten atoms occupy the same crystallographic site in spite of their very different ionic radii, 0.96 Å and 0.71 Å respectively in 7-fold coordination [25]. The average coordination for the cations is then 6.86; however the oxygen/vacancies disorder and the cubic symmetry let think to an effective coordination of 8 that is relatively large for Y 3+ and even more for W 6+ .

Recently, Yoshimura et al. have shown that the cubic Y 6 WO 12 can be obtained and stabilized as a metastable phase at low temperature (600°C) using a complexationcalcination route. By heating above 1000-1200°C, the symmetry changes to the stable rhombohedral phase [26]. Before any further discussion, let us make some remarks at the level of the simple atomic bond. From this simple DOS figure, it is easy to understand that the interaction of an anion less electronegative than the oxygen with the tungsten at equivalent distances will result in levels in the valence band above the levels resulting from the W-O interaction. It is the case of nitrogen which introduction into the structure causes a reduction of the VB  CB band gap. The same holds true for the substitution of tungsten by a more electronegative cation as for example molybdenum whose interaction with oxygen will tend to close the band gap. However, the substitution of the yttrium by another trivalent cation should not influence the bandgap value.

Y 6 WO 12-3x N 2x

Here, low nitrogen contents were investigated in the way to prepared white or pale yellow compositions that display an absorption edge located around 400 nm (3.1 eV) giving them potential interest as inorganic UV absorbers. The oxide precursor used for the ammonolysis experiments is the metastable cubic phase obtained by the citrate route which has been discussed above. As mentioned in literature [3,8], the use of such precursors improves widely the reactivity of the oxide powders with ammonia. The experimental data are gathered in Table I and the diffuse reflectance spectra of the nitrided compositions shown in Figure 2A.

The nitridation results in a decrease of the diffuse reflectance maximum intensity and also in a continuous red shift of the absorption edge from 340 nm to about 525 nm.

By adjusting the amount of nitrogen introduced into the yttrium tungstate structure, it becomes possible to synthesize an oxynitride composition that exhibits an absorption edge located at 400 nm, i.e. 3.1 eV. However, the spectral selectivities of the nitrided compositions appear to be too low to make them suitable for any UV absorption applications. Figure 2B displays the evolution of the bandgap vs. nitrogen content in the nitrided compositions. The bandgap tends to decrease linearly until the nitrogen content of x = 1 then the curve deviates to follow an asymptote around 2.5 eV that corresponds to a yellow color. Actually, no orange coloration of the powders was observed in this system.

Y 6 W 1-x Mo x O 12

As observed in the case of pure yttrium tungstate Y 6 WO 12 , the XRD patterns of the Mo-substituted products obtained by the amorphous citrate method at 600°C reveal a cubic symmetry while the rhombohedral symmetry is observed for the samples heated up to 1200°C. The absorption edges of the former phases are much more spread out that the ones of the latter (Fig. 3A). The calcination temperature (crystallinity) as well as the symmetry can give rise to such straightening of the absorption edges. First, the conduction band has mainly a metallic character, i.e. 5d(W) and 4d(Mo) orbitals;

Secondly, the coordination sphere of the transition metal in the cubic phase is less regular and larger (cubic site with oxygen vacancies) than the regular octahedra in the rhombohedral phase; So as the electronic band structure is very sensitive to the interatomic distances, the VB  CB electronic transitions should be more defined for the rhombohedral phases than the cubic ones. This could explain the better absorption behavior of the rhombohedral phases. In addition, as molybdenum is more electronegative than tungsten, the energy level of the 4d(Mo) orbitals is lower than that of the 5d(W) orbitals. Consequently, a progressive substitution of Mo for W induces a red shift of the absorption edge, from 3.65 eV (Y 6 WO 12 ) to 2.80 eV (Y 6 MoO 12 ) (Fig. 3).

The non linear variation of the optical bandgap along the solid solution suggests that the 4d(Mo) and 5d(W) levels form two bands relatively distinct from each other [27].

Among the rhombohedral phases, the compositions corresponding to 0.3 < x < 0.5 appear to be attractive materials for anti-UV purposes, as they show bandgap values close to 3.1 eV with very narrow spectral selectivities (Eg ~ 0.4 -0.5 eV).

Conclusion

Using precursors prepared through the amorphous citrate route, the optical properties of new oxynitride and oxide fluorite-type solid compositions have been characterized within the system Y 6 (W,Mo)(O,N) 12 . While the oxynitride and oxide cubic-type phases show less interest as new anti-UV pigments, promising compositions have been isolated among the rhombohedral phases.

  DFT calculations carried out on the rhombohedral form of Y 6 WO 12 and illustrated on Fig. 1 by the density of states (DOS) curves indicate that the VB  CB electronic transitions are of the 2p(O)  5d(W) type. The calculated energy gap corresponding to the minimal transition is 4.04 eV in very good agreement with the one obtained by diffuse reflectance : 3.91 eV. This energy depends strongly on the W-O distances in the structure, the shorter the distance, the greater the gap energy. The gap between the 5d(W) and 4d(Y) bands is large, i.e. 3.49 eV, so the yttrium orbitals are not directly involved in the optical band gap.

Figure 1 :

 1 Figure 1: Total DOS and partial DOS projected on Y, W and O for the rhombohedral

Figure 2 :Figure 3 :

 23 Figure 2 : Diffuse reflectance spectra and evolution of the optical gap vs. nitrogen

Table I :

 I Experimental and optical data for the Y 6 (W,Mo)(O,N) 12 system.

	Thermal threatment	Chemical composition	%wt(N)	Eg (eV)	Eg (eV)	Color
		Oxynitride phases
	500°C/NH 3 /5h	Y 6 WO 11.6 N 0.3 o 2.1	0.4	3.76	-	white
	550°C/NH 3 /5h	Y 6 WO 10.9 N 0.7 o 2.4	1.2	3.22	-	yellowish
	600°C/NH 3 /60h	Y 6 WO 10.1 N 1.3 o 2.6	2.0	2.75	-	yellow
	800°C/NH 3 /15h	Y 6 WO 9.3 N 1.8 o 2.9	2.9	2.56	-	mustard
		Cubic phases
	600°C/air/2h	Y 6 WO 12	-	3.69	0.56	white
	600°C/air/2h	Y 6 W 0.9 Mo 0.1 O 12	-	3.75	1.05	cream
	600°C/air/2h	Y 6 W 0.7 Mo 0.3 O 12	-	3.28	0.82	pale yellow
	600°C/air/2h	Y 6 W 0.5 Mo 0.5 O 12	-	3.13	0.69	light yellow
	600°C/air/2h	Y 6 MoO 12	-	3.08	0.63	yellow
		Rhombohedral phases
	1200°C/air/2h	Y 6 WO 12	-	3.91	0.53	white
	1200°C/air/2h	Y 6 W 0.9 Mo 0.1 O 12	-	3.33	0.65	cream
	1200°C/air/2h	Y 6 W 0.7 Mo 0.3 O 12	-	3.20	0.51	pale yellow
	1200°C/air/2h	Y 6 W 0.5 Mo 0.5 O 12	-	3.11	0.42	light yellow
	1200°C/air/2h	Y 6 MoO 12	-	3.00	0.35	yellow