Yoshinori Morimoto
email: morimoto.yoshinori.74r@st.kyoto-u.ac.jp

C.-J Xu

Chao-Jiang Xu
email: xuchaojiang@nuaa.edu.cn

ANALYTIC SMOOTHING EFFECT FOR THE NONLINEAR LANDAU EQUATION OF MAXWELLIAN MOLECULES

Keywords: Mathematics Subject Classification. Primary 35B65; Secondary 35Q82, 35S05 Landau equation, analytic smoothing effect, microlocal analysis

We consider the Cauchy problem of the nonlinear Landau equation of Maxwellian molecules, under the perturbation frame work to global equilibrium. We show that if H r x (L 2 v ), r > 3/2 norm of the initial perturbation is small enough, then the Cauchy problem of the nonlinear Landau equation admits a unique global solution which becomes analytic with respect to both position x and velocity v variables for any time t > 0. This is the first result of analytic smoothing effect for the spatially inhomogeneous nonlinear kinetic equation. The method used here is microlocal analysis and energy estimates. The key point is adopting a time integral weight associated with the kinetic transport operator.

Introduction

We consider the Cauchy problem for the spatially inhomogeneous Landau equation, a kinetic model from plasma physics that describes the evolution of a particle density f (t, x, v) ≥ 0 with position x ∈ R 3 and velocity v ∈ R 3 at time t. It reads (1.1)

∂ t f + v • ∇ x f = Q L (f, f ), f | t=0 = f 0 ,
where the term Q L (f, f ) corresponds to the Landau collision operator associated to the bilinear operator

Q L (g, f ) = ∇ v • R 3 a(v -v * ) g(v * )(∇ v f )(v) -(∇ v g)(v * )f (v) dv * .
Here a = (a i,j ) 1≤i,j≤3 stands for the non-negative symmetric matrix

(1.2) a(v) = |v| γ |v| 2 I -v ⊗ v = |v| γ+2 P v ⊥ ∈ M 3 (R), -3 ≤ γ < +∞,
where P v ⊥ is the orthogonal projection on v ⊥ . The Landau equation with γ = -3 was first proposed in 1936 [START_REF] Landau | Kinetic equation for the case of Coulomb interaction[END_REF] by the Russian theoretical physicist Lev Davidovitch Landau, as a transport equation for a system of charged particles, where the long range of the Coulomb interactions makes it impossible to use the normal Boltzmann equation. It soon became (in combination with the Vlasov equation) the most important mathematical kinetic model in the theory of collisional plasma.

The "generalized" Landau equation with γ > -3 was independently introduced by several authors ( see e.g. [START_REF] Bobylev | Expansion of the Boltzmann collision integral in a Landau series[END_REF][START_REF] Desvillettes | On asymptotics of the Boltzmann equation when collisions became grazing[END_REF]). It plays a role of a model of the Boltzmann equation for various interactions including inverse power law potential ρ -n+1 , n > 2. This equation can be obtained as a limit of the Boltzmann equation when grazing collisions prevails (see [START_REF] Villani | On a new class of week solutions to the homogeneous Boltzmann and Landau equations[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF] for a detailed study of the limiting process and further references on the subjects). Though the Landau equation has no relation to physics in the non-Coulomb case, from various mathematical points of view, it has been intensively studied by mathematicians in last two decades, because it is a simple approximation of the non-cutoff Boltzmann equation (see, e.g. [START_REF] Alexandre | Entropy dissipation and longrange interactions[END_REF][START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF]). We refer the reader to the surveys [START_REF] Lions | On Boltzmann and Landau equations[END_REF][START_REF] Villani | A review of mathematical topics in collisional kinetic theory[END_REF][START_REF] Bouchut | Kinetic equations and asymtotic theory[END_REF] and recent papers [START_REF] Bobylev | From practical system to the Landau equation: a consistency result[END_REF][START_REF] Bobylev | On some properties of the Lnadau kinetic equation[END_REF][START_REF] Carrapatoso | Landau equation for very soft and Coulomb potentials near Maxwellians[END_REF], as well as to the references therein for matters related to the derivation and basic results for that equation.

In this article, we focus our attention on the analytic smoothing effect of a solution for the Cauchy problem (1.1). More specifically, we study the Landau equation with Maxwellian molecules in a close to equilibrium framework. The Maxwellian molecules corresponds to the case when the parameter γ = 0 in the cross section (1.2). We consider the fluctuation

f = µ + √ µg,
around the Maxwellian equilibrium distribution 2 . This distribution is a stationary solution for the Landau equation since it only depends on the velocity variable v and the fact that Q L (µ, µ) = 0. We consider the linearized Landau operator around this equilibrium distribution given by L g = -µ -1/2 Q L (µ, µ 1/2 g) -µ -1/2 Q L (µ 1/2 g, µ).

(1.3) µ(v) = (2π) -3 2 e -|v| 2
The Cauchy problem for the Landau equation (1.1) is then reduced to the one for the fluctuation (1.4) ∂ t g + v • ∇ x g + L g = Γ(g, g), g| t=0 = g 0 , with Γ(g, f ) = µ -1/2 Q L ( √ µg, √ µf ). To state our main result, we define the Sobolev space H r x (L 2 v ) for r ≥ 0 by

H r x (L 2 v ) = u ∈ L 2 (R 6 x,v ); D x r u ∈ L 2 (R 6
x,v ) , with D x = √ 1 -△ x and D x = -i∂ x . Let A(R n ) denote the analytic function space on R n . Theorem 1.1. Let r > 3/2. There exists a small constant ǫ 0 > 0 such that for all

g 0 ∈ H r x (L 2 v ) satisfying g 0 H r x (L 2
v ) ≤ ǫ 0 , the Cauchy problem (1.4) admits a unique global solution such that g(t) ∈ A(R 6

x,v ), ∀t > 0 . Furthermore, there exists a c 0 > 0 such that, (1.5) e c0{ t2 (-∆x) 1/2 + t(-∆v) 1/2 } g(t) ∈ L ∞ [0, +∞), H r x (L 2 v ) , where t = min{1, t} for t ≥ 0. This article concerns the existence of a solution to the Cauchy problem for the Landau equation, as well as the smoothing properties of that solution, which is a topic studied in many previous works, as stated above. Among them we refer [START_REF] Villani | On a new class of week solutions to the homogeneous Boltzmann and Landau equations[END_REF][START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness[END_REF][START_REF] Desvillettes | On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications[END_REF][START_REF] Desvillettes | Entropy dissipation estimates for the Landau equation in the Coulomb case and applications[END_REF] concerning the existence result, [START_REF] Morimoto | Ultra-analytic effect of Cauchy problem for a class of kinetic equations[END_REF][START_REF] Morimoto | A remark on the ultra-analytic smoothing properties of the spatially homogeneous Landau equations[END_REF] about higher regularity such as (ultra-)analyticity, in the spatially homogeneous case, while in the spatially inhomogeneous case, [START_REF] Alexandre | On the Landau approximation in plasma physics[END_REF] concerning renormalized solution with defect measure, [START_REF] Guo | The Landau equation in a periodic box[END_REF][START_REF] Strain | Exponential decay for soft potentials near Maxwellian[END_REF][START_REF] Carrapatoso | Cauchy problem and exponential stability for the inhomogeneous Landau equation[END_REF][START_REF] Carrapatoso | Landau equation for very soft and Coulomb potentials near Maxwellians[END_REF][START_REF] Cao | Well-posedness of Cauchy problem for Landau equation in critical Besov space Kinetic and Related[END_REF][START_REF] Duan | Global mild solutions of the Landau and non-cutoff Boltzmann equations[END_REF] in a close to equilibrium setting, and recent regularity results by [START_REF] Cameron | Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials[END_REF][START_REF] Golse | Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation[END_REF][START_REF] Ch | C ∞ smoothing for weak solutions of the inhomogeneous Landau equations[END_REF] under boundedness conditions on the mass, energy, entropy densities. Also we want to mention the related works on the non cut-off Boltzmann equation, e.g., the papers by [START_REF] Alexandre | Entropy dissipation and longrange interactions[END_REF][START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF][START_REF] Morimoto | Gevrey smoothing effect of solutions for spatially homogeneous nonlinear Boltzmann equation without angular cutoff[END_REF][START_REF] Glangetas | Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation[END_REF][START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF][START_REF] Alexandre | The Boltzmann equation without angular cutoff in the whole space: qualitative properties of solutions[END_REF][START_REF] Alexandre | Global existence and full regularity of the Boltzmann equation without angular cutoff[END_REF][START_REF] Lerner | Gelfand-Shilov smoothing effect for the spatially inhomogeneous non-cutoff Kac equation[END_REF][START_REF] Morimoto | Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF][START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equation without cutoff for Maxwellian molecules[END_REF].

The rest of paper is arranged as follows: In Section 2 we recall the exact expression of the linear operator given in [START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF], introduce a version of the exponential weight used in the previous work [START_REF] Morimoto | Ultra-analytic effect of Cauchy problem for a class of kinetic equations[END_REF], and show the estimate for the linear term with the exponential weight. In Section 3 we give an explicit form of the nonlinear term by means of creation and annihilation operators and spherical derivatives. In Section 4, we use this to show a trilinear estimate for the inner product of the nonlinear term and test function with exponential weight. Section 5 is devoted to complete the proof of the main theorem by constructing the time local solution with analytic smoothing property and by combining this and the known global existence result.

Fourier analysis of linear Landau operator

With µ defined in (1.3), the linearized Landau operator L is defined by

L g = -µ -1/2 Q L (µ, µ 1/2 g) -µ -1/2 Q L (µ 1/2 g, µ),
is an unbounded symmetric operator on L 2 (R 3 v ). In the case of Maxwellian molecules, the linearized Landau operator may be computed explicitly (see e.g. [START_REF] Lerner | Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators[END_REF], Proposition 1), and we have

L = L 1 + L 2 ,
with

L 1 = 2 -∆ v + |v| 2 4 - 3 2 -∆ S 2 , L 2 = ∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 P 1 + -∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 P 2
where

∆ S 2 = 1 2 1≤j,k≤3 j =k L 2 k,j , L k,j = v j ∂ v k -v k ∂ vj ,
stands for the Laplace-Beltrami operator on the unit sphere S 2 , and P k denotes the orthogonal projection onto the Hermite basis

E k = Span{Φ α } α∈N 3 ,|α|=k . Here Φ 0 (v) = µ 1/2 (v), Φ α = 1 √ α! a α1 +,1 a α2 +,2 a α3 +,3 Φ 0 , α = (α 1 , α 2 , α 3 ) ∈ N 3 , α! = α 1 !α 2 !α 3 !, with a ±,j = v j 2 ∓ ∂ ∂v j , 1 ≤ j ≤ 3.
We have then,

(L 1 g, g) L 2 (R 3 v ) = 2 3 j=1 ∂ vj g 2 L 2 (R 3 v ) + 1 4 v j g 2 L 2 (R 3 v ) + 1 2 1≤j,k≤3 j =k L k,j g 2 L 2 (R 3 v ) -3 g 2 L 2 (R 3 v ) .
The operators

L 2 is bounded in L 2 (R 3 v ).
Putting

|||g||| 2 v = 2 3 j=1 ∂ vj g 2 L 2 (R 3 v ) + 1 4 v j g 2 L 2 (R 3 v ) + 1 2 1≤j,k≤3 j =k L k,j g 2 L 2 (R 3 v ) |||g||| 2 r,0 = 2 3 j=1 ∂ vj g 2 H r x (L 2 v ) + 1 4 v j g 2 H r x (L 2 v ) + 1 2 1≤j,k≤3 j =k L k,j g 2 H r x (L 2 v ) ,
we have the following coercive estimates; there exists C > 0 such that for all g ∈ S (R 3 ),

|||g||| 2 v ≤ (L g, g) L 2 (R 3 v ) + C g 2 L 2 (R 3
v ) . Ultra-analytic smoothing effect of Kolmogorov equation. We recall the Cauchy problem of Kolmogorov equation

∂ t f + v • ∇ x f -∆ v f = 0 , f (0, x, v) = f 0 (x, v) ∈ L 2 (R 2n
x,v ). Using the Fourier transform ( (x, v) ↔ (η, ξ) ) we have

f (t, η, ξ) = e -t 0 |ξ+ρη| 2 dρ f0 (η, ξ + tη) because (∂ t -η • ∇ ξ + |ξ| 2 ) f = 0. Using the following Ukai inequality (2.1) with α = 2, ∃c > 0 , c (t|ξ| 2 + t 3 |η| 2 ) ≤ t 0 |ξ + ρη| 2 dρ. Then we have e c(-t∆v-t 3 ∆x) f (t, •, • ) ∈ L 2 (R 2n x,v
) , which shows the ultra-analytic smoothing effect of (x, v) variables for any t > 0.

Lemma 2.1. For any α > 0, there exists c α > 0 such that

(2.1) t 0 (1 + |ξ + ρη| 2 ) α/2 dρ ≥ c α t 1 + |ξ| 2 + t 2 |η| 2 α/2 , for all t > 0.
Remark 2.2. The following simple proof is due to Seiji Ukai. On the other hand, there exists C α > 0 such that

(2.2) t 0 (1 + |ξ + ρη| 2 ) α/2 dρ ≤ C α t 1 + |ξ| 2 + t 2 |η| 2 α/2 , ∀t > 0.
Proof. Put ρ = tτ , η = -tη, then the estimate (2.1) is equivalent to

1 0 ξ -τ η α dτ ≥ c α 1 + |ξ| 2 + |η| 2 α/2 , with the notation • = 1 + | • | 2 .
Since the case η = 0 is trivial, we assume η = 0. Notice

2 1 0 ξ -τ η α dτ ≥ 1 + 1 0 |ξ -τ η| α dτ. If |ξ| ≥ |η| 1 0 |ξ -τ η| α dτ ≥ |ξ| α 1 0 1 -τ |η| |ξ| α dτ ≥ |ξ| α 1 0 1 -τ α dτ = |ξ| α α + 1 ≥ 1 2 α (α + 1) (|ξ| 2 + |η| 2 ) α/2 . If |ξ| < |η| 1 0 |ξ -τ η| α dτ ≥ |η| α 1 0 τ - |ξ| |η| α dτ = |η| α |ξ|/|η| 0 |ξ| |η| -τ α dτ + 1 |ξ|/|η| τ - |ξ| |η| α dτ ≥ |η| α α + 1 min 0≤θ≤1 (θ α+1 + (1 -θ) α+1 ) = |η| α 2 α (α + 1) ≥ 1 2 2α (α + 1) (|ξ| 2 + |η| 2 ) α/2 .
We finally get (2.1).

We set

Ψ(t, η, ξ) = c 0 t 0 ξ + ρη dρ = c 0 t 0 ξ + (t -ρ)η dρ,
for a sufficiently small c 0 > 0 which will be chosen later on. We have that

(∂ t -η • ∇ ξ )Ψ = c 0 ξ .
Then we can use (2.1) with α = 1 to estimate Ψ.

To study the Gevery (and analytic) regularity of kinetic equations, exponential type weights were used in [START_REF] Morimoto | Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff[END_REF][START_REF] Morimoto | Ultra-analytic effect of Cauchy problem for a class of kinetic equations[END_REF] (see also [START_REF] Barbaroux | Gevrey smoothing for weak solutions of the fully nonlinear homogeneous Boltzmann and Kac equation without cutoff for Maxwellian molecules[END_REF]). Now we set

F δ,δ ′ (t, η, ξ) = e Ψ (1 + δe Ψ )(1 + δ ′ Ψ) r for 0 < δ ≤ 1, r > 3/2, 0 < rδ ′ ≤ 1.
Without the confusion, we use the same notation F δ,δ ′ for the symbol F δ,δ ′ (t, η, ξ) and also the the pseudo-differential operator F δ,δ ′ (t, D x , D v ). If A is a first order differential operator of (t, η, ξ) variables, then we have

(2.3) AF δ,δ ′ = 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ (AΨ)F δ,δ ′ , and 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ ≤ 1, since 0 ≤ a, b ≤ 1 implies |a -b| ≤ 1.
We study now the apriori estimate of solution g ∈ L

∞ (]0, T [, H r x (L 2 v )) of the equa- tion ∂ t g + v • ∇ x g + L g = Γ(g, g) .
Since in this case,

v • ∇ x g ∈ L ∞ (]0, T [, H r-1 x (L 2 -1 (R 3 v ))), L g, Γ(g, g) ∈ L ∞ (]0, T [, H r x (H -2 -2 (R 3 v ))
), we will take

g = F δ,δ ′ (t, D x , D v ) δ ′ v -4 F δ,δ ′ (t, D x , D v )g ∈ L ∞ (]0, T [, H 2r 4 (R 6 x,v ))
as test function, where H m ℓ (R 6

x,v ) is the weighted Sobolev space of order ℓ with respect to v variable. Taking the H r x (L 2 v ) inner product for r > 3 2 , we get,

(∂ t + v • ∇ x )g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) + L g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = Γ(g, g), F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v )
.

For the first term, we have

Proposition 2.3. There exist C 1 , C 2 > 0 independent of δ, δ ′ such that, for 0 < t ≤ 1, (∂ t + v • ∇ x )g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) ≥ 1 2 d dt δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -c 0 C 1 D v δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -C 2 δ ′ v -2 F δ,δ ′ g H r x (L 2
v ) . Proof. Using the Plancherel formula, we have

(∂ t + v • ∇ x )g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = 1 (2π) 6 δ ′ D ξ -2 F δ,δ ′ (∂ t -η • ∇ ξ )ĝ, η 2r δ ′ D ξ -2 F δ,δ ′ ĝ(t, η, ξ) L 2 η,ξ = 1 (2π) 6 (∂ t -η • ∇ ξ ) δ ′ D ξ -2 F δ,δ ′ ĝ, η 2r δ ′ D ξ -2 F δ,δ ′ ĝ(t, η, ξ) L 2 η,ξ + 1 (2π) 6 [ δ ′ D ξ -2 F δ,δ ′ , (∂ t -η • ∇ ξ )]ĝ, η 2r δ ′ D ξ -2 F δ,δ ′ ĝ(t, η, ξ) L 2 η,ξ
, and

(∂ t -η • ∇ ξ ) δ ′ D ξ -2 F δ,δ ′ ĝ, η 2r δ ′ D ξ -2 F δ,δ ′ ĝ(t, η, ξ) L 2 η,ξ = 1 2 d dt R 6 | δ ′ D ξ -2 F δ,δ ′ ĝ(t, η, ξ)| 2 η 2r dηdξ (2π) 6 .
We study now the commutators term. Since (∂ t -η • ∇ ξ )Ψ = c 0 ξ , we have

-[F δ,δ ′ , (∂ t -η • ∇ ξ )] = (∂ t -η • ∇ ξ )F δ,δ ′ = c 0 ξ 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ F δ,δ ′ ,
and

[ δ ′ D ξ -2 F δ,δ ′ , (∂ t -η • ∇ ξ )] = δ ′ D ξ -2 [F δ,δ ′ , (∂ t -η • ∇ ξ )] = -c 0 δ ′ D ξ -2 ξ 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ δ ′ D ξ 2 δ ′ D ξ -2 F δ,δ ′ .
Moreover, we have

δ ′ D ξ -2 ξ 1 + δe Ψ - rδ ′ ξ 1 + δ ′ Ψ δ ′ D ξ 2 = ξ 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ + δ ′ D ξ -2 ξ 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ , δ ′ D ξ 2 ,
and

δ ′ D ξ -2 ξ 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ , δ ′ D ξ 2 = 2 (δ ′ ) 2 D ξ δ ′ D ξ 2 • D ξ ξ 1 + δe Ψ - rδ ′ ξ 1 + δ ′ Ψ -δ ′ D ξ -2 (δ ′ ) 2 D 2 ξ ξ 1 + δe Ψ - rδ ′ ξ 1 + δ ′ Ψ . Now using (2.4) |∂ ξj Ψ| = c 0 t 0 ξ j -ρη j ξ -ρη dρ ≤ c 0 t, |∂ α ξ Ψ| ≤ C α c 0 t for |α| ≥ 2,
we can complete the proof of Proposition 2.3.

We study now terms concerning linear operators.

Proposition 2.4. If 0 < c 0 ≪ 1, then there exists a C > 0 independent of δ, δ ′ > 0 such that for any 0 < t ≤ 1 we have

L 1 g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) ≥ 1 2 ||| δ ′ v -2 F δ,δ ′ g||| 2 r,0 -C δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) . Proof. Note that L 1 = 2 3 j=1 (D 2 vj + v 2 j 4 ) -3 - 1 2 1≤j,k≤3 j =k L 2 k,j .
Then we have firstly

D 2 vj g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = D 2 vj F δ,δ ′ g, δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = δ ′ v -2 D 2 vj δ ′ v 2 δ ′ v -2 F δ,δ ′ g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) = D 2 vj δ ′ v -2 F δ,δ ′ g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + δ ′ v -2 [D 2 vj , δ ′ v 2 ] δ ′ v -2 F δ,δ ′ g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v )
.

Since [D 2 vj , δ ′ v 2 ] = -2δ ′2 -i4δ ′2 v j D vj ,
we have

δ ′ v -2 [D 2 vj , δ ′ v 2 ] δ ′ v -2 F δ,δ ′ g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) ≤ 2δ ′2 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + 4δ ′ D vj δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) ≤ 10δ ′2 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + 1 2 D vj δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) , which gives D 2 vj g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) ≥ 1 2 D vj δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
(2.5)

-10δ ′2 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
. Now, for the second terms, we write

v 2 j g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = v 2 j δ ′ v -2 F δ,δ ′ g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + δ ′ v -2 [F δ,δ ′ , v 2 j ]g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v )
, and note that

[F δ,δ ′ , v 2 j ] = 2(D ξj F δ,δ ′ )(t, D x , D v )v j -(D 2 ξj F δ,δ ′ )(t, D x , D v ) (2.6) = 2v j (D ξj F δ,δ ′ )(t, D x , D v ) + (D 2 ξj F δ,δ ′ )(t, D x , D v )
. By using (2.3), we have

(D ξj F δ,δ ′ )(t, D x , D v ) = 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ (D ξj Ψ)F δ,δ ′ := B j,δ,δ ′ (t, D x , D v )F δ,δ ′ (2.7)
and moreover

(D 2 ξj F δ,δ ′ )(t, D x , D v ) = (D ξj B j,δ,δ ′ )(t, D x , D v ) + B j,δ,δ ′ (t, D x , D v ) 2 F δ,δ ′ := Bj,δ,δ ′ (t, D x , D v )F δ,δ ′ . (2.8) Consequently δ ′ v -2 [F δ,δ ′ , v 2 j ]g, δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) ≤ 2 δ ′ v -2 B j,δ,δ ′ δ ′ v 2 δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) v j δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + δ ′ v -2 Bj,δ,δ ′ δ ′ v 2 δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) . Since it follows from (2.4) that δ ′ v -2 B δ,δ ′ δ ′ v 2 = B δ,δ ′ -δ ′2 δ ′ v -2 [B δ,δ ′ , |v| 2 ] is an L 2 (R 6
x,v ) bounded operator with a constant factor c 0 t and the same fact holds for Bδ,δ ′ , we have that for 0 < t ≤ 1,

v 2 j g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v )
(2.9)

≥ 1 2 v j δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -C 1 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) .
Finally, for the last terms, let Lk,j = -

ξ k ∂ ξj + ξ j ∂ ξ k . Then we have Lk,j F δ,δ ′ (t, η, ξ) = 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ F δ,δ ′ -ξ k ∂ ξj Ψ + ξ j ∂ ξ k Ψ , (2.10) and hence, in view of rδ ′ ≤ 1, -L 2 k,j g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) = R 6
Lk,j -

1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ -ξ k ∂ ξj Ψ + ξ j ∂ ξ k Ψ F δ,δ ′ ĝ η 2r × Lk,j + 1 1 + δe Ψ - rδ ′ 1 + δ ′ Ψ -ξ k ∂ ξj Ψ + ξ j ∂ ξ k Ψ δ ′ D ξ -4 F δ,δ ′ ĝ dηdξ (2π) 6 ≥ L k,j δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -c 2 0 t 2 C 2 ∂ vj δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + ∂ v k δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -c 0 tC 3 L k,j δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) × ∂ vj δ ′ v -2 F δ,δ ′ g H r x L 2 v + ∂ v k δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) -c 2 0 t 2 C 4 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) ,
where we have used Lk,j , δ ′ D ξ = 0, because δ ′ D ξ is radial. Therefore, for 0 < t ≤ 1 and 0 < c 0 ≪ 1 we get

-L 2 k,j g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) ≥ 1 2 L k,j δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
(2.11)

- 1 8 ∂ vj δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + ∂ v k δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) -C 5 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
. Combing the estimates (2.5),(2.9) and (2.11), we finish the proof of Proposition 2.4.

We recall the notations: Φ

0 (v) = µ 1/2 (v), Φ α = 1 √ α! a α1 +,1 a α2 +,2 a α3 +,3 Φ 0 , α = (α 1 , α 2 , α 3 ) ∈ N 3 , α! = α 1 !α 2 !α 3 !, with a ±,j = v j 2 ∓ ∂ ∂v j , 1 ≤ j ≤ 3. Φ e k = v k Φ 0 , Φ 2e k = 1 √ 2 (v 2 k -1)Φ 0 , Φ ej +e k = v j v k Φ 0 (j = k),
where (e 1 , e 2 , e 3 ) stands for the canonical basis of R 3 .

Proposition 2.5. For 0 < t ≤ 1, there exists C 6 > 0 independent of 0 < δ ≤ 1, 0 < δ ′ < r -1 and 0 < c 0 ≪ 1 such that we have

L 2 g, F δ,δ ′ δ ′ v -4 F δ,δ ′ h H r x (L 2 v ) ≤ C 6 δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) δ ′ v -2 F δ,δ ′ h H r x (L 2 v ) . Proof. First we recall L 2 g = ∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 P 1 g + -∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 P 2 g.
Notice

P 1 g = 3 k=1 (g, Φ e k ) L 2 (R 3 v ) Φ e k , P 2 g = |α|=2 (g, Φ α ) L 2 (R 3 v ) Φ α .
Then

L 2 g = 3 k=1 (g, Φ e k ) L 2 (R 3 v ) Φe k + |α|=2 (g, Φ α ) L 2 (R 3 v ) Φα with Φe k = ∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 Φ e k = p e k (v)e -|v| 2 4 , Φ2 = -∆ S 2 -2 -∆ v + |v| 2 4 - 3 2 Φ α = p α (v)e -|v| 2 4 ,
where p * (v) are the polynomial of v-variables of 3 or 4 degrees. We then study one of terms, where Φ denotes Φ e k , Φ α ,

(g, Φ) L 2 (R 3 v ) Φ, F δ,δ ′ δ ′ v -4 F δ,δ ′ h H r x (L 2 v ) = η,ξ (ĝ(t, η, •), F v * (Φ)) L 2 (R 3 ξ * ) F v ( Φ)(ξ)F δ,δ ′ δ ′ D ξ -4 F δ,δ ′ ĥ(t, η, ξ) η 2r dηdξ (2π) 6 = η,ξ,ξ * ĝ(t, η, ξ * )F v * (Φ)(ξ * )F v ( Φ)(ξ)F δ,δ ′ δ ′ D ξ -4 F δ,δ ′ ĥ(t, η, ξ) η 2r dηdξdξ * (2π) 9 = η,ξ,ξ * δ ′ D ξ * -2 F δ,δ ′ (t, η, ξ * )ĝ(t, η, ξ * ) δ ′ D ξ * 2 F v * (Φ)(ξ * ) F v ( Φ)(ξ) × F δ,δ ′ (t, η, ξ) F δ,δ ′ (t, η, ξ * ) δ ′ D ξ -4 F δ,δ ′ (t, η, ξ) ĥ(t, η, ξ) η 2r dηdξdξ * (2π) 9 + 2δ ′2 η,ξ,ξ * δ ′ D ξ * -2 F δ,δ ′ (t, η, ξ * )ĝ(t, η, ξ * ) D ξ * F v * (Φ)(ξ * ) F v ( Φ)(ξ) × D ξ * F δ,δ ′ (t, η, ξ) F δ,δ ′ (t, η, ξ * ) δ ′ D ξ -4 F δ,δ ′ (t, η, ξ) ĥ(t, η, ξ) η 2r dηdξdξ * (2π) 9 + δ ′2 η,ξ,ξ * δ ′ D ξ * -2 F δ,δ ′ (t, η, ξ * )ĝ(t, η, ξ * ) F v * (Φ)(ξ * ) F v ( Φ)(ξ) × D 2 ξ * F δ,δ ′ (t, η, ξ) F δ,δ ′ (t, η, ξ * ) δ ′ D ξ -4 F δ,δ ′ (t, η, ξ) ĥ(t, η, ξ) η 2r dηdξdξ * (2π) 9 .
On the other hand, we have

F δ,δ ′ (t, η, ξ) F δ,δ ′ (t, η, ξ * ) =e c0 t 0 ( ξ+ρη -ξ * +ρη )dρ 1 + δe c0 t 0 ξ * +ρη dρ 1 + δe c0 t 0 ξ+ρη dρ × 1 + δ ′ c 0 t 0 ξ * + ρη dρ 1 + δ ′ c 0 t 0 ξ + ρη dρ r . Since ξ + ρη = |(1, ξ + ρη)| ≤ |(1, ξ * + ρη)| + |(0, ξ -ξ * )| (2.12) ≤ ξ * + ρη + |ξ| + |ξ * |, ∀ξ, ξ * ∈ R 3 ,
we have, by using (2.3) and (2.4), for 0

< t ≤ 1, 1 ≤ p ≤ 2, D p ξ * F δ,δ ′ (t, η, ξ) F δ,δ ′ (t, η, ξ * ) ≤ Ce 2c0t(|ξ|+|ξ * |) (1 + |ξ| r + |ξ * | r ). Because, for 0 ≤ p ≤ 2 |ξ| r e 2c0t|ξ| D p ξ F v ( Φ)(ξ) ∈ L 2 (R 3 ξ ), we proved that, for 0 < t ≤ 1, (g, Φ) L 2 (R 3 v ) Φ, F δ,δ ′ δ ′ v -4 F δ,δ ′ h H r x (L 2 v ) ≤ C η δ ′ D ξ * -2 F δ,δ ′ ĝ(t, η, •) L 2 ξ * δ ′ D ξ -4 F δ,δ ′ ĥ(t, η, •) L 2 ξ η 2r dη (2π) 9 ≤ C δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) δ ′ v -2 F δ,δ ′ h H r x (L 2 v ) .
Remark 2.6. Compared to (2.12), there is no constant C > 0 such that

ξ + ρη α -ξ * + ρη α ≤ C(|ξ| α + |ξ * | α ), ∀ξ, ξ * ∈ R 3 , if α > 1.
By this reason we do not seek for the ultra-analytic smoothing effect in the present paper.

In conclusion, for the linear operators, we get that there exists

C 7 > 1 indepen- dent of 0 < δ ≤ 1, 0 < δ ′ < r -1 and 0 < c 0 ≪ 1 such that, for 0 < t ≤ 1 (∂ t + v • ∇ x + L )g, F δ,δ ′ δ ′ v -4 F δ,δ ′ g H r x (L 2 v ) ≥ 1 2 d dt δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + 1 4 ||| δ ′ v -2 F δ,δ ′ g||| 2 r,0 (2.13)
-C 7 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) .

Decomposition of nonlinear operators

We compute now the nonlinear term Γ(f, g), Proposition 3.1.

(3.1) (Γ(f, g), h) L 2 (R 3 v ) = D 1 + D 2 + D 3 + D 4 + D 5 + D 6 + D 7 ,
with

D 1 = √ 2 1≤i,j≤3 i =j (f, Φ 2ej ) L 2 (R 3 v ) (a +,i g, a -,i h) L 2 (R 3 v ) , D 2 = - 1≤i,j≤3 i =j (f, Φ 0 ) L 2 (R 3 v ) (a -,i g, a -,i h) L 2 (R 3 v ) , D 3 = - 1≤i,j≤3 i =j (f, Φ ei+ej ) L 2 (R 3 v ) (a +,j g, a -,i h) L 2 (R 3 v ) , D 4 = 1≤i,j≤3 i =j (f, Φ ei ) L 2 (R 3 v ) (g, a -,i h) L 2 (R 3 v ) , D 5 = - 1 2 1≤i,j≤3 i =j (f, Φ 0 ) L 2 (R 3 v ) L i,j g, L i,j h L 2 (R 3 v ) , D 6 = 1≤i,j≤3 i =j (f, Φ ej ) L 2 (R 3 v ) (L i,j g, a -,i h) L 2 (R 3 v ) , D 7 = - 1≤i,j≤3 i =j (f, Φ ej ) L 2 (R 3 v ) (a +,i g, L i,j h) L 2 (R 3 v ) .
where the creation (resp. annihilation) operator a +,j (resp. a -,j ) is given by

a ±,j = v j 2 ∓ ∂ ∂v j and L k,j = v j ∂ v k -v k ∂ vj .
Proof. We begin by computing explicitly the bilinear term Γ(g, f ). Notice that for all f, g ∈

S (R 3 v ), Γ(f, g) = µ -1/2 (v) 1≤k,j≤3 ∂ v k R 3 a k,j (v-v * )µ 1/2 (v * )f (v * )∂ vj µ 1/2 (v)g(v) dv * -µ -1/2 (v) 1≤k,j≤3 ∂ v k R 3 a i,j (v -v * )∂ v * j µ 1/2 (v * )f (v * ) µ 1/2 (v)g(v)dv * , that is, Γ(f, g) = 1≤k,j≤3 ∂ v k - v k 2 R 3 a k,j (v-v * )µ 1/2 (v * )f (v * ) ∂ j g(v)- v j 2 g(v) dv * - 1≤k,j≤3 ∂ v k - v k 2 R 3 a k,j (v -v * )µ 1/2 (v * ) ∂ j f (v * ) - v * j 2 f (v * ) g(v)dv * . It follows that for all f, g, h ∈ S (R 3 v ), Γ(f, g),h L 2 (R 3 v ) = 1≤k,j≤3 R 6 a k,j (v -v * )µ 1/2 (v * )f (v * ) ∂ j g(v) - v j 2 g(v) × -∂ k h(v) - v k 2 h(v) dv * dv - 1≤k,j≤3 R 6 a k,j (v -v * )µ 1/2 (v * ) ∂ j f (v * ) - v * j 2 f (v * ) g(v) × -∂ k h(v) - v k 2 h(v) dv * dv.
Integrating by parts, we obtain that

R 3 a k,j (v -v * )µ 1/2 (v * ) ∂ j f (v * ) - v * j 2 f (v * ) dv * = - R 3 ∂ v * j a k,j (v -v * ) µ 1/2 (v * )f (v * )dv * ,
and this implies that

Γ(f, g), h L 2 (R 3 v ) = 1≤k,j≤3 R 3 µ(v * ) 1/2 f (v * ) × a k,j (v-v * ) ∂ j g(v)- v j 2 g(v) +∂ v * j a k,j (v-v * ) g(v), -∂ k h(v)- v k 2 h(v) L 2 (R 3 v ) dv * .
Since for the Maxweillian case,

a k,j (z) = δ kj |z| 2 -z k z j , 1 ≤ k, j ≤ 3 .
We have that

Γ(f, g), h L 2 (R 3 v ) = 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) × (v 2 j -2v j v * j + (v * j ) 2 ) ∂ k g(v) - v k 2 g(v) , -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * + 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) × -(v k -v * k )(v j -v * j ) ∂ j g(v) - v j 2 g(v) + (v k -v * k )g(v), -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * .
We obtain that

Γ(g, f ), h L 2 (R 3 v ) = A 0 + A 1 + A 2 + A 3 + A 4 + A 5 + A 6 + A 7 ,
with

A 0 = - 1≤k,j≤3 k =j R 3 v * k v * j µ(v * ) 1/2 f (v * ) × ∂ j g(v) - v j 2 g(v), -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * , A 1 = 1≤k,j≤3 k =j R 3 (v * j ) 2 µ(v * ) 1/2 f (v * ) × ∂ k g(v) - v k 2 g(v), -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * , A 2 = 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v k g(v), -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * , A 3 = - 1≤k,j≤3 k =j R 3 v * k µ(v * ) 1/2 f (v * ) g(v), -∂ k h(v) - v k 2 h(v) L 2 (R 3 v ) dv * , A 4 = 1 4 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) × -v 2 k v 2 j + v 2 k v j v * j + v k v 2 j v * k + v 2 k v 2 j -2v 2 k v j v * j g(v), h(v) L 2 (R 3 v ) dv * = 0, since 1≤k,j≤3 k =j v k v 2 j v * k -v 2 k v j v * j = 0. We have also (3.2) A 5 = 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) × (v 2 j -2v j v * j )∂ k g(v) + -v k v j + v k v * j + v j v * k ∂ j g(v), -∂ k h(v) L 2 (R 3 v )
dv * , and

A 6 = - 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v k ∂ j g(v), (-v k v j +v k v * j +v j v * k )h(v) L 2 (R 3 v ) dv * + 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) (-v k v j + v k v * j + v j v * k )g(v), v j ∂ k h(v) L 2 (R 3 v ) dv * , A 7 = - 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v j ∂ k g(v), (v k v j -2v k v * j )h(v) L 2 (R 3 v ) dv * + 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) (v k v j -2v k v * j )g(v), v j ∂ k h(v) L 2 (R 3 v ) dv * .
It follows from (3.2) that

A 5 = 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v j ∂ k g(v) -v k ∂ j g(v), -v j ∂ k h(v) L 2 (R 3 v ) dv * + 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) -2v j v * j ∂ k g(v)+ v k v * j +v j v * k ∂ j g(v), -∂ k h(v) L 2 (R 3 v ) dv * .
In the sequel, we shall use several times the (obvious) formula 1≤k,j≤3

k =j α k,j = 1≤k,j≤3 k =j α j,k . We notice that 1≤k,j≤3 k =j v j ∂ k g(v) -v k ∂ j g(v), -v j ∂ k h(v) L 2 (R 3 v ) = 1 2 1≤k,j≤3 k =j v j ∂ k g(v) -v k ∂ j g(v), -v j ∂ k h(v) L 2 (R 3 v ) + 1 2 1≤k,j≤3 k =j v k ∂ j g(v) -v j ∂ k g(v), -v k ∂ j h(v) L 2 (R 3 v ) = - 1 2 1≤k,j≤3 k =j v j ∂ k g(v) -v k ∂ j g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) , and
1≤k,j≤3 k =j -2v j v * j ∂ k g(v) + v k v * j + v j v * k ∂ j g(v), -∂ k h(v) L 2 (R 3 v ) = 1≤k,j≤3 k =j v j ∂ k g(v), v * j ∂ k h(v) L 2 (R 3 v ) + v * j ∂ k g(v), v j ∂ k h(v) L 2 (R 3 v ) - 1≤k,j≤3 k =j v k ∂ j g(v), v * j ∂ k h(v) L 2 (R 3 v ) - 1≤k,j≤3 k =j v * j ∂ k g(v), v k ∂ j h(v) L 2 (R 3 v ) = 1≤k,j≤3 k =j v j ∂ k g(v) -v k ∂ j g(v), v * j ∂ k h(v) L 2 (R 3 v ) + v * j ∂ k g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) .
This implies that

A 5 = - 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v j ∂ k g(v) -v k ∂ j g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) dv * + 1≤k,j≤3 k =j R 3 v * j µ(v * ) 1/2 f (v * ) v j ∂ k g(v) -v k ∂ j g(v), ∂ k h(v) L 2 (R 3 v ) + ∂ k g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) dv * .
On the other hand, we may write that

A 6 + A 7 = - 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v k ∂ j g(v), (-v k v j + v k v * j + v j v * k + v j -2v j v * k )h(v) L 2 (R 3 v ) dv * + 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) (-v k v j + v k v * j + v j v * k + v k v j -2v k v * j )g(v), v j ∂ k h(v) L 2 (R 3 v ) dv * .
It follows that

A 6 + A 7 = - 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v k ∂ j g(v), (v k v * j -v j v * k )h(v) L 2 (R 3 v ) dv * + 1 2 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) (v j v * k -v k v * j )g(v), v j ∂ k h(v) L 2 (R 3 v ) dv * .
This implies that

A 6 + A 7 = - 1 4 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) v k ∂ j g(v) -v j ∂ k g(v), (v k v * j -v j v * k )h(v) L 2 (R 3 v ) dv * + 1 4 1≤k,j≤3 k =j R 3 µ(v * ) 1/2 f (v * ) (v j v * k -v k v * j )g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) dv * ,
that is,

A 6 + A 7 = 1 2 1≤k,j≤3 k =j R 3 v * j µ(v * ) 1/2 f (v * ) v j ∂ k g(v) -v k ∂ j g(v), v k h(v) L 2 (R 3 v ) dv * + 1 2 1≤k,j≤3 k =j R 3 v * k µ(v * ) 1/2 f (v * ) v j g(v), v j ∂ k h(v) -v k ∂ j h(v) L 2 (R 3 v ) dv * .
We obtain that

Γ(f, g), h L 2 (R 3 v ) = E 1 + E 2 + E 3 + E 4 + E 5 + E 6 + E 7 , with E 1 = 1≤k,j≤3 k =j f, v 2 j µ 1/2 L 2 (R 3 v ) ∂ k g - v k 2 g, -∂ k h - v k 2 h L 2 (R 3 v ) , E 2 = - 1≤k,j≤3 k =j f, v k v j µ 1/2 L 2 (R 3 v ) ∂ j g - v j 2 g, -∂ k h - v k 2 h L 2 (R 3 v ) , E 3 = 1≤k,j≤3 k =j f, µ 1/2 L 2 (R 3 v ) v k g, -∂ k h - v k 2 h L 2 (R 3 v ) , E 4 = - 1≤k,j≤3 k =j f, v k µ 1/2 L 2 (R 3 v ) g, -∂ k h - v k 2 h L 2 (R 3 v ) , E 5 = - 1 2 1≤k,j≤3 k =j f, µ 1/2 L 2 (R 3 v ) v j ∂ k g -v k ∂ j g, v j ∂ k h -v k ∂ j h L 2 (R 3 v ) , E 6 = 1≤k,j≤3 k =j f, v j µ 1/2 L 2 (R 3 v ) v j ∂ k g -v k ∂ j g, ∂ k h + v k 2 h L 2 (R 3 v ) , E 7 = 1≤k,j≤3 k =j f, v j µ 1/2 L 2 (R 3 v ) ∂ k g - v k 2 g, v j ∂ k h -v k ∂ j h L 2 (R 3 v )
. This is exactly (3.1).

Trilinear estimates with exponential weights

Proposition 4.1. Let 0 < t ≤ 1 and 0 < c 0 ≪ 1. If r > 3 2 , then there exists C 0 > 0 independent of δ, δ ′ , c 0 such that we have for suitable functions f, g, h,

(Γ(f, g), F δ,δ ′ δ ′ v -4 F δ,δ ′ h) H r x (L 2 v ) ≤ C 0 F δ,0 f H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ g||| r,0 + δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) (4.1) × ||| δ ′ v -2 F δ,δ ′ h||| r,0 + δ ′ v -2 F δ,δ ′ h H r x (L 2 v ) .
We consider the nonlinear term (Γ(f, g),

F δ,δ ′ δ ′ v -4 F δ,δ ′ h) H r x (L 2 v )
. For instance we estimate a term of D 5 , that is,

R 3 x (f, Φ 0 ) L 2 (R 3 v ) L k,j g, D x 2r L k,j F δ,δ ′ δ ′ v -4 F δ,δ ′ h L 2 (R 3 v ) dx = R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j ĝ(η, ξ)dη × η 2r Lk,j F δ,δ ′ δ ′ D ξ -4 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 = R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j ĝ(η, ξ)dη × η 2r F δ,δ ′ Lk,j δ ′ D ξ -4 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 + R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j ĝ(η, ξ)dη × η 2r [ Lk,j , F δ,δ ′ ] δ ′ D ξ -4 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 , := Γ 1 + Γ 2 .
We consider

F δ,δ ′ (t, η, ξ) η r = e Ψ (1 + δe Ψ ) η (1 + δ ′ Ψ) r := F δ,0 G δ ′ . By means of W + V ≤ W + V , we have t 0 ξ -ρη dρ ≤ t 0 ξ + ξ * -ρ(η -η) -ρη dρ + t 0 ξ * dρ ≤ t 0 ξ * -ρ(η -η) dρ + t 0 ξ -ρη dρ + t ξ * .
Noting that Fδ (X) = e X /(1 + δe X ) is an increasing function and that Fδ (X + Y ) ≤ 3 Fδ (X) Fδ (Y ) , we have

F δ,0 (t, η, ξ) ≤ 9F δ,0 (t, η -η, ξ * )F δ,0 (t, η, ξ)e c0t ξ * , Since Ψ(t, η, ξ) ∼ c 0 t(1 + |ξ| 2 + t 2 |η| 2 ) 1/2 and (1 + Y )/(a + bY ) for any constants a ≥ b > 0 is increasing in Y , there exists a constant C > 0 independent of δ ′ > 0 and (t, ξ), η, η such that G δ ′ (t, η, ξ) ≤ C η -η r + η r (1 + δ ′ Ψ(t, η, ξ)) r .
Consequently, for another constant C 7 > 0 independent of δ ′ > 0 and (t, ξ), η, η we have

F δ,δ ′ (t, η, ξ) η r (4.2) ≤ C 7 ( η -η r + η r ) F δ,0 (t, η -η, ξ * )F δ,δ ′ (t, η, ξ)e c0t ξ * . Γ 1 = R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j δ ′ D ξ -2 ĝ(η, ξ)dη × η 2r δ ′ D ξ 2 F δ,δ ′ Lk,j δ ′ D ξ -4 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 = R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j δ ′ D ξ -2 ĝ(η, ξ)dη × η 2r F δ,δ ′ Lk,j δ ′ D ξ -2 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 + R 6 R 3 f (η -η, ξ * ) Φ0 (ξ * ) dξ * (2π) 3 Lk,j δ ′ D ξ -2 ĝ(η, ξ)dη × η 2r [ δ ′ D ξ 2 , F δ,δ ′ ] δ ′ D ξ -2 Lk,j δ ′ D ξ -2 F δ,δ ′ ĥ(η, ξ) dηdξ (2π) 6 := Γ 1,1 + Γ 1,2 ,
where we have used again Lk,j , δ ′ D ξ = 0. By means of (4.2) and Young inequality, we get

|Γ 1,1 | • r F δ,0 (•, ξ * ) f (•, ξ * ) L 2 (R 6 ) e c0t ξ * -|ξ * | 2 /4 L 2 (R 3 ξ * ) × F δ,δ ′ (•, ξ) Lk,j δ ′ D ξ -2 ĝ(•, ξ) L 2 ξ L 1 L k,j δ ′ v -2 F δ,δ ′ h H 3/2+ǫ x L 2 v + F δ,0 (•, ξ * ) f (•, ξ * ) L 2 ξ * L 1 e c0t ξ * -|ξ * | 2 /4 L 2 (R 3 ξ * ) × • r F δ,δ ′ (•, ξ) Lk,j δ ′ D ξ -2 ĝ(•, ξ) L 2 (R 6 ) L k,j δ ′ v -2 F δ,δ ′ h H r x (L 2 v ) F δ,0 f H r x (L 2 v ) F δ,δ ′ δ ′ v -2 L k,j g H r x (L 2 v ) L k,j δ ′ v -2 F δ,δ ′ h H r x (L 2 v )
. By the same calculus as in (2.6), we note that

F δ,δ ′ (t, η, D v ) = δ ′ v -2 F δ,δ ′ (t, η, D v ) δ ′ v 2 + 2δ ′2 3 j=1 v j δ ′ v -2 (D ξj F δ,δ ′ )(t, η, D v ) -δ ′2 δ ′ v -2 3 j=1 (D 2 ξj F δ,δ ′ )(t, η, D v ).
By means of (2.3) and (2.4), there exists a constant

C 8 > 0 independent of t, η, δ, δ ′ > 0 such that for H(v) ∈ L 2 2 (R 3 ) we have F δ,δ ′ (t, η, D v )H L 2 v ≤ δ ′ v -2 F δ,δ ′ (t, η, D v ) δ ′ v 2 H L 2 v + C 8 δ ′ c 0 F δ,δ ′ (t, η, D v )H L 2 v , and hence F δ,δ ′ (t, η, D v )H L 2 v ≤ 2 δ ′ v -2 F δ,δ ′ δ ′ v 2 H L 2 v if c 0 C 8 < 1/2. Consequently, we obtain F δ,δ ′ δ ′ v -2 L k,j g H r x (L 2 v ) ≤ 2 δ ′ v -2 F δ,δ ′ L k,j g H r x (L 2 v ) ≤ 2 L k,j δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + 2 δ ′ v -2 [L k,j , F δ,δ ′ ]g H r x (L 2 v ) L k,j δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + D v k δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + D vj δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) + δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) .
Here we have used (2.10) at the third inequality. As a consequence, we obtain

|Γ 1,1 | F δ,0 f H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ g||| r,0 + δ ′ v -2 F δ,δ ′ g H r x (L 2 v ) × ||| δ ′ v -2 F δ,δ ′ h||| r,0 + δ ′ v -2 F δ,δ ′ h H r x (L 2
v ) . Since it follows from the almost same calculation as in (2.6), (2.7) and (2.8) that

δ ′ D ξ 2 , F δ,δ ′ δ ′ D ξ -2 = 3 j=1 (D ξj F δ,δ ′ ) 2δ ′ 2 D ξj δ ′ D ξ -2 + (D 2 ξj F δ,δ ′ )δ ′ 2 δ ′ D ξ -2 = F δ,δ ′ (t, η, ξ) 3 j=1 B j,δ,δ ′ (t, η, ξ) 2δ ′ 2 D ξj δ ′ D ξ -2 + Bj,δ,δ ′ (t, η, ξ)δ ′ 2 δ ′ D ξ -2 ,
and since the last factor is a bounded operator, the estimation for Γ 1,2 is similar to the one for Γ 1,1 , As for the estimation of Γ 2 , we recall (2.10). Then

[ Lk,j , F δ,δ ′ ] δ ′ D ξ -2 = iF δ,δ ′ B k,δ,δ ′ ξ j -B j,δ,δ ′ ξ k δ ′ D ξ -2 = iF δ,δ ′ B k,δ,δ ′ δ ′ D ξ -2 ξ j -2i δ ′ 2 D ξj δ ′ D ξ 2 -B j,δ,δ ′ δ ′ D ξ -2 ξ k -2i δ ′ 2 D ξ k δ ′ D ξ 2 . Writing F δ,δ ′ B k,δ,δ ′ δ ′ D ξ -2 = F δ,δ ′ δ ′ D ξ -2 B k,δ,δ ′ + [ δ ′ D ξ 2 , B k,δ,δ ′ ] δ ′ D ξ -2 ,
we see that the estimation for Γ 2 is quite similar to the one for Γ 1 , because there exist bounded operators R k , R j , R j,k such that

[ Lk,j , F δ,δ ′ ] δ ′ D ξ -2 = F δ,δ ′ δ ′ D ξ -2 R k ξ j + R j ξ k + R j,k .
Thus we have a desired estimate for D 5 . For the other terms D j , we remark that [â ±j , δ ′ D ξ ] = 0, but it is bounded, whence we also can estimate them by the same procedure. We omit the detail.

End of proof of main theorem

In the first subsection we show the local existence of solution in [0, 1] and its analytic smoothing effect. We remark the stability and uniqueness of this local solution in the next subsection. In the last subsection we complete the proof of the main theorem by using the global existence theorem given by Guo [START_REF] Guo | The Landau equation in a periodic box[END_REF].

Existence of analytic time-local solution.

Lemma 5.1 (local existence for a linear equation). Let r > 3/2 and 0 < c 0 ≪ 1. Assume that 0 < δ ≤ 1. Then there exist ǫ 0 > 0 and C 9 > 1 independent of δ such that for any 0

< T ≤ 1, g 0 ∈ H r x (L 2 v ), f ∈ L ∞ ([0, T ]; H r x (L 2 v )) satisfying F δ,0 f L ∞ ([0,T ];H r x (L 2 v )) ≤ ǫ 0 , (5.1)
the Cauchy problem

∂ t g + v • ∇ x g + Lg = Γ(f, g), g| t=0 = g 0 , (5.2) admits a weak solution g ∈ L ∞ ([0, T ]; H r x (L 2 v )) satisfying F δ,0 g 2 L ∞ ([0,T ];H r x (L 2 v )) + T 0 |||F δ,0 g(s)||| 2 r,0 ds ≤ C 9 g 0 2 H r x (L 2 v ) . (5.3) Proof. Consider Q = -∂ t + (v • ∇ x + L -Γ(f, •)) * ,
where the adjoint operator (•) * is taken with respect to the scalar product in H r x (L 2 v ). Then, by using (2.13) and (4.1) with δ, δ ′ , c 0 → 0 we see that for all h ∈ C ∞ ([0, T ], S(R 6

x,v )), with h(T ) = 0 and 0

≤ t ≤ T , Re h(t), Qh(t) H r x (L 2 v ) = - 1 2 d dt ( h 2 H r x (L 2 v ) ) + Re(v • ∇ x h, h) H r x (L 2 v ) + Re(Lh, h) H r x (L 2 v ) -Re(Γ(f, h), h) H r x (L 2 v ) ≥ - 1 2 d dt h(t) 2 H r x (L 2 v ) + 1 4 |||h(t)||| 2 r,0 -C 8 h(t) 2 H r x (L 2 v ) -C 0 f (t) H r x (L 2 v ) |||h(t)||| 2 r,0 , because L is a selfadjoint operator and Re(v •∇ x h, h) H r x (L 2 v ) = 0. Since (5.1) implies f L ∞ ([0,T ];H r x (L 2 v )) ≤ 2ǫ 0 , we have - d dt e 2C8t h(t) 2 H r x (L 2 v ) + 1 4 e 2C8t |||h(t)||| 2 r,0 ≤ 2e 2C8t h(t) H r x (L 2 v ) Qh(t) H r x (L 2 v ) , if 16ε 0 C 0 < 1. Since h(T ) = 0, for all t ∈ [0, T ] we have h(t) 2 H r x (L 2 v ) + 1 4 T t |||h(τ )||| 2 r,0 dτ ≤ 2 T t e 2C8(τ -t) h(τ ) H r x (L 2 v ) Qh(τ ) H r x (L 2 v ) dτ ≤ 2e 2C8T h L ∞ ([0,T ];H r x (L 2 v )) Qh L 1 ([0,T ],H r x (L 2 v )) , so that (5.4) h L ∞ ([0,T ];H r x (L 2 v )) ≤ 2e 2C8T Qh L 1 ([0,T ],H r x (L 2 v )
) . We consider the vector subspace

W = {w = Qh : h ∈ C ∞ ([0, T ], S(R 6 x,v )), h(T ) = 0} ⊂ L 1 ([0, T ], H r x (L 2 v )
). This inclusion holds because it follows from Proposition 3.1 that for g

∈ H r x (L 2 v ) |(Γ(f, •) * h, g) H r x (L 2 v ) | = |(h, Γ(f, g)) H r x (L 2 v ) | f H r x (L 2 v ) g H r x (L 2 v ) v 2 h H r x (H 2 v )
, and hence, for all t ∈ [0, T ],

Γ(f, •) * h H r x (L 2 v ) f H r x (L 2 v ) v 2 h H r x (H 2 v ) . Since g 0 ∈ H r x (L 2 v ), we define the linear functional G : W -→ C w =Qh → (g 0 , h(0)) H r x (L 2 v )
where h ∈ C ∞ ([0, T ], S(R 6 x,v )), with h(T ) = 0. According to (5.4), the operator Q is injective. The linear functional G is therefore well-defined. It follows from (5.4) that G is a continuous linear form on (W,

• L 1 ([0,T ];H r x (L 2 v )) ), |G(w)| ≤ g 0 H r x (L 2 v ) h(0) H r x (L 2 v ) ≤ 2e 2C8T g 0 H r x (L 2 v ) Qh L 1 ([0,T ];H r x (L 2 v )) = 2e 2C8T g 0 H r x (L 2 v ) w L 1 ([0,T ];H r x (L 2 v ))
. By using the Hahn-Banach theorem, G may be extended as a continuous linear form on

L 1 ([0, T ]; H r x (L 2 v )), with a norm smaller than 2e 2C8T g 0 H r x (L 2 v ) . Hence there exists g ∈ L ∞ ([0, T ]; H r x (L 2 v )) satisfying g L ∞ ([0,T ],H r x (L 2 v )) ≤ 2e 2C8T g 0 H r x (L 2 v ) , such that ∀w ∈ L 1 ([0, T ]; H r x (L 2 v )), G(w) = T 0 (g(t), w(t)) H r x (L 2 v ) dt. This implies that for all h ∈ C ∞ 0 ((-∞, T ), S(R 6 x,v )), G(Qh) = T 0 (g(t), Qh(t)) H r x (L 2 v ) dt = (g 0 , h(0)) H r x (L 2 v ) . This shows that g ∈ L ∞ ([0, T ]; H r x (L 2 v )
) is a weak solution of the Cauchy problem (5.2).

It remains to show (5.3). Noting that g ∈ L ∞ ([0, T ]; H r x (L 2 v )) implies, for any δ > 0, δ ′ > 0,

v t r D v r + t 2r D x r δ ′ v -2 F δ,δ ′ g ∈ L ∞ ([0, T ]; H r x (L 2 v
)), we multiply the first equation of (5.2) by F δ,δ ′ δ ′ v -4 F δ,δ ′ g and take its H r x (L 2 v ) inner product. Then it follows from (2.13) and (4.1) that for 0

< t ≤ T ≤ 1 1 2 d dt δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v ) + 1 4 ||| δ ′ v -2 F δ,δ ′ g||| 2 r,0 -C 8 δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
(5.5)

≤ 2C 0 F δ,0 f H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ g||| 2 r,0 + δ ′ v -2 F δ,δ ′ g 2 H r x (L 2 v )
. Since ǫ 0 is chosen small enough that 16ǫ 0 C 0 < 1, we get

δ ′ v -2 F δ,δ ′ g 2 L ∞ ([0,T ];H r x (L 2 v )) + 1 4 T 0 ||| δ ′ v -2 F δ,δ ′ g(s)||| 2 r,0 ds ≤ 2e 3C8T δ ′ v -2 g 0 2 H r x (L 2
v ) , which yields (5.3) with C 9 = 8e 3C8 , by letting δ ′ → 0. Theorem 5.2 (analytic time-local solution). Let r > 3/2. There exists an ǫ 1 > 0 such that for all g

0 ∈ H r x (L 2 v ) satisfying g 0 H r x (L 2
v ) ≤ ǫ 1 , the Cauchy problem (1.4) admits a solution such that g(t) ∈ A(R 6

x,v ), 0 < ∀t ≤ 1 .

Furthermore, there exists a 0 < c 1 < 1 such that,

e c1{t 2 (-∆x) 1/2 +t(-∆v) 1/2 } g(t) ∈ L ∞ [0, 1], H r x (L 2 v ) , more precisely for any 0 < t ≤ 1 e c1{t 2 (-∆x) 1/2 +t(-∆v) 1/2 } g(t)|| H r x (L 2 v ) ≤ C 9 g 0 H r x (L 2 v ) . (5.6)
Proof. Consider the sequence of approximate solution defined by (5.7)

∂ t g n+1 + v • ∇ x g n+1 + L g n+1 = Γ(g n , g n+1 ). g n+1 | t=0 = g 0 ,
with g 0 = e -Ψ(t,Dx,Dv ) g 0 . We apply Lemma 5.1 with f = g n and g = g n+1 by assuming √ C 9 ǫ 1 ≤ ǫ 0 . Then it follows from (5.3) with T = 1 that for n ≥ 1

(5.8) F δ,0 g n 2 L ∞ ([0,1];H r x (L 2 v )) + 1 0 |||F δ,0 g n (s)||| 2 r,0 ds ≤ C 9 g 0 2 H r x (L 2 v ) ≤ ǫ 2 0 holds inductively because F δ,0 g 0 H r x (L 2 v ) ≤ g 0 H r x (L 2 v ) ≤ ǫ 1 ≤ ǫ 0 . Setting w n = g n+1 -g n , from (5.7) we have ∂ t w n + v • ∇ x w n + Lw n = Γ(g n , w n ) + Γ(w n-1 , g n ) ,
with w n | t=0 = 0. Similar to the computation for (5.5), we obtain 1 2

d dt δ ′ v -2 F δ,δ ′ w n 2 H r x (L 2 v ) + 1 4 ||| δ ′ v -2 F δ,δ ′ w n ||| 2 r,0 -C 8 δ ′ v -2 F δ,δ ′ w n 2 H r x (L 2 v ) ≤ 2C 0 F δ,0 g n H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ w n ||| 2 r,0 + δ ′ v -2 F δ,δ ′ w n 2 H r x (L 2 v ) + 2C 0 F δ,0 w n-1 H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ g n ||| 2 r,0 + δ ′ v -2 F δ,δ ′ g n 2 H r x (L 2 v ) 1/2 × ||| δ ′ v -2 F δ,δ ′ w n ||| 2 r,0 + δ ′ v -2 F δ,δ ′ w n 2 H r x (L 2 v ) 1/2 ≤ 2C 0 F δ,0 g n H r x (L 2 v ) + 1 16 ||| δ ′ v -2 F δ,δ ′ w n ||| 2 r,0 + δ ′ v -2 F δ,δ ′ w n 2 H r x (L 2 v ) + 32C 2 0 F δ,0 w n-1 2 L ∞ ([0,1];H r x (L 2 v )) × ||| δ ′ v -2 F δ,δ ′ g n ||| 2 r,0 + δ ′ v -2 F δ,δ ′ g n 2 H r x (L 2 v ) , which implies δ ′ v -2 F δ,δ ′ w n 2 L ∞ ([0,1];H r x (L 2 v )) + 1 8 1 0 ||| δ ′ v -2 F δ,δ ′ w n (τ )||| 2 r,0 dτ ≤ 64C 2 0 e 3C8 1 0 |||F δ,0 g n (τ )||| 2 r,0 dτ + F δ,0 g n 2 L ∞ ([0,1];H r x (L 2 v )) × F δ,0 w n-1 2 L ∞ ([0,1];H r x (L 2 v )) ≤ 64C 2 0 e 3C8 × 2ǫ 2 0 F δ,0 w n-1 2 L ∞ ([0,1];H r x (L 2
v )) , because of (5.8) and 16ε 0 C 0 < 1. Letting δ ′ → 0, we see that there exists a 0 < λ < 1 such that

F δ,0 w n 2 L ∞ ([0,T ];H r x (L 2 v )) ≤ λ F δ,0 w n-1 2 L ∞ ([0,T ];H r x (L 2 v ))
if ǫ 0 > 0 is small enough so that 128C 2 0 e 3C8 ǫ 2 0 ≤ λ. By taking δ → 0 we see that there exists a local solution g

∈ L ∞ ([0, T ]; H r x (L 2 v )) of the Cauchy problem (1.4) such that e Ψ (g n -g) L ∞ ([0,1];H r x (L 2 v )) → 0 as n → ∞, and
e Ψ g 2 L ∞ ([0,1];H r x (L 2 v )) + 1 0 |||e Ψ g(s)||| 2 r,0 ds ≤ C 9 g 0 2 H r x (L 2 v ) ≤ ǫ 2 0 .
By means of Lemma 2.1, we get the desired estimate (5.6).

5.2. The stability and uniqueness of time-local solution.

Proposition 5.3 (Stability and uniqueness). Let r > 3/2, 0 < c 0 ≪ 1 and let

ǫ 0 > 0 satisfy 16C 0 ǫ 0 < 1 for the constant C 0 in Proposition 4.1. Let 0 < T ≤ 1 and let g j (t) ∈ L ∞ ([0, T ]; H r x (L 2 v )), j = 1, 2 be two solutions of the Cauchy problem (1.4) with initial data g 1,0 , g 2,0 ∈ H r x (L 2 v ), respectively. Assume that e Ψ g 1 L ∞ ([0,T ];H r x (L 2 v )) ≤ ǫ 0 ,
and there exists M > 0, such that

e Ψ g 2 2 L ∞ ([0,T ];H r x (L 2 v )) + T 0 |||e Ψ g 2 (s)||| 2 r,0 ds ≤ M.
Then there exists a C M > 1 independent of c 0 such that

e Ψ (g 1 -g 2 ) 2 L ∞ ([0,T ];H r x (L 2 v )) (5.9)
+ T 0 |||e Ψ (g 1 -g 2 )(s)||| 2 r,0 ds ≤ C M g 0,1 -g 0,2 2 H r x (L 2 v ) .
Remark 5.4. It is easy to see that this proposition holds in the case where c 0 = 0, that is, e Ψ is replaced by 1. Therefore the uniqueness of solutions belonging to L ∞ ([0, T ]; H r x (L 2 v )) holds under above conditions with e Ψ = 1. Proof. Putting w = g 1 -g 2 , we have ∂ t w + v • ∇ x w + Lw = Γ(g 1 , w) + Γ(w, g 2 ) , with an initial datum w| t=0 = g 0,1 -g 0,2 ∈ H r x (L 2 v ). Similar to the proof of Theorem 5.2, we obtain 1 2

d dt δ ′ v -2 F δ,δ ′ w 2 H r x (L 2 v ) + 1 4 ||| δ ′ v -2 F δ,δ ′ w||| 2 r,0 -C 8 δ ′ v -2 F δ,δ ′ w 2 H r x (L 2 v ) ≤ 2C 0 F δ,0 g 1 H r x (L 2 v ) + 1 16 ||| δ ′ v -2 F δ,δ ′ w||| 2 r,0 + δ ′ v -2 F δ,δ ′ w 2 H r x (L 2 v ) + 32C 2 0 F δ,0 w 2 H r x (L 2 v ) × ||| δ ′ v -2 F δ,δ ′ g 2 ||| 2 r,0 + δ ′ v -2 F δ,δ ′ g 2 2 H r x (L 2 v ) , which implies d dt δ ′ v -2 F δ,δ ′ w 2 H r x (L 2 v ) + 1 8 ||| δ ′ v -2 F δ,δ ′ w||| 2 r,0 -3C 8 δ ′ v -2 F δ,δ ′ w 2 H r x (L 2 v ) ≤ 64C 2 0 F δ,0 w 2 H r x (L 2 v ) ||| δ ′ v -2 F δ,δ ′ g 2 ||| 2 r,0 + M .
By integrating from 0 to t ∈ (0, T ], we get Finally, letting δ → 0 we obtain the desired estimate (5.9).

δ ′ v -2 F δ,δ ′ w(t)

5.3.

Time global solution and its analytic smoothing. To show the existence of a global solution and its analyticity smoothing, we refer the following theorem that was first proved by Guo [START_REF] Guo | The Landau equation in a periodic box[END_REF] in the torus T 3

x case and was extended to the whole space R 3

x by Yang-Yu [START_REF] Yang | Optimal convergence rates of the Landau equation with external forcing in the whole space[END_REF]; Theorem 5.5 ([22, 34]). There exist some positive constants ǫ 2 > 0, C 10 > 1 such that if g 0 ∈ H 8 (R 6

x,v ) satisfies g 0 H 8

x,v ≤ ǫ 2 then the Cauchy problem (1.4) admits a unique global solution g(t) ∈ L ∞ ([0, ∞); H 8

x.v (R 6 )) fulfilling sup 0≤t<∞ g(t) H 8

x,v ≤ C 10 g 0 H 8 x,v .

Assume that 0 < ǫ 3 ≤ (c 1 ) 8 /(8!C 9 ) min{ǫ 1 /C 10 , ǫ 2 } and g 0 H r

x (L 2 v ) ≤ ǫ 3 . Let 1 ≤ τ ≤ 2 and apply Theorem 5.2 with the initial time t 0 = τ -1, in view of √ C 9 ǫ 3 < ǫ 1 . Then for any τ ∈ [START_REF] Alexandre | Entropy dissipation and longrange interactions[END_REF][START_REF] Alexandre | Regularizing effect and local existence for the non-cutoff Boltzmann equation[END_REF] we have e c1((-∆x) 1/2 +(-∆v) 1/2 ) g(τ

) H r x (L 2 v )
= e c1(((τ -t0) 2 (-∆x) 1/2 +(τ -t0)(-∆v) For τ > 2, apply again Theorem 5.2 with the initial time t 0 = τ -1. Then we obtain sup τ ≥2 e c1((-∆x) 1/2 +(-∆v) 1/2 ) g(τ

) H r x (L 2 v ) ≤ C 9 sup τ ≥2 g(τ -1) H r x (L 2 v ) ≤ C 9 ǫ 1 .
This together with (5.10) and Theorem 5.2 complete the proof of (1.5).

5.4.

Global stability and uniqueness. It follows from Proposition 5.3 and the stability of global solutions in Theorem 5.5 that the stability of analytic global solutions holds. In fact, for any fixed T > 2, if t ∈ [2, T ] then e c1((-∆x) 1/2 +(-∆v) 1/2 ) (g 1 (t) -g 2 (t)

) H r x (L 2 v ) ≤ C M g 1 (t -1) -g 2 (t -1) H r x (L 2 v )
≤ C M C T g 1 (1) -g 2 (1) H 8

x,v ≤ C M 8!C T (c 1 ) 8 e c1((-∆x) 1/2 +(-∆v) 1/2 ) (g 1 (1) -g 2 ( 1)

) H r x (L 2 v ) ≤ (C M ) 8!C T (c 1 ) 8 g 0,1 -g 0,2 H r x (L 2 v ) .
The case 1 ≤ t ≤ 2 is an easy consequence of Proposition 5.3. Thus the uniqueness of global solutions holds. Now the proof of Theorem 1.1 is complete.

  ) M + |||e Ψ g 2 (τ )||| 2 r,0 dτLet δ ′ → 0 and denote the right hand side by Y δ (t). ThenY ′ δ (t) ≤ 64e 3C8T0 C 2 0 M + e Ψ g 2 (t)||| 2 r,0 Y δ (t), a.e. t ∈ [0, T ],

	2 H r x (L 2 v ) +	1 8	0	t	||| δ ′ v -2 F δ,δ ′ w(τ )||| 2 r,0 dτ
	≤ w(0) 2 H r x (L 2 v ) + 64e 3C8T0 C 2 0	0	t	F δ,0 w(τ ) 2 H r x (L 2
	so that we obtain
	Y δ (t) ≤ w(0) 2 H r x (L 2 v ) e 64e 3C 8 T 0 C 2 0 (MT + T 0 |||e Ψ g2(s)||| 2 r,0 ds) .

v

  1/2 ) g(τ ) H r ) ≤ C 9 ǫ 3 , which implies the existence of local solution g(t) ∈ L ∞ ([0, 2]; H r ) 8 C 9 ǫ 3 ≤ ǫ 2 .By using Theorem 5.5 with the initial time t 1 = 1, we obtain a global solutiong(t) ∈ L ∞ ([1, ∞[; H 8 x,v (R 6 )) satisfying ) 8 C 9 C 10 ǫ 3 ≤ ǫ 1 .

	(5.10)	x (L 2 v )
	≤ C 9 g(t 0 ) H r x (L 2
		x (L 2 v )) satisfying
	sup [1,2] (c 1 sup g(t) H 8 8! x,v ≤ [1,∞[ g(t) H r x (L 2 v ) ≤ sup [1,∞[ g(t) H 8 x,v ≤	8! (c 1

v

Acknowledgements. The research of the first author is supported by JSPS Kakenhi Grant No.17K05318. The research of the second author is supported partially by "The Fundamental Research Funds for Central Universities". Both authors would like to thank Nicolas Lerner and Karel Pravda-Starov for stimulating discussions on the topic of smoothing effect for kinetic equations.


est destinée au dépôt

The solubility of PuO 2 (am) was determined in the aqueous K + -HCO 3 --CO 3 2--OH --H 2 O system extending to high concentrations of carbonate and bicarbonate. X-ray absorption spectroscopy (XAS) and solvent extraction were used to identify species and oxidation states in the aqueous phase. The dominant aqueous species in relatively concentrated CO 3 2-and HCO 3 -solutions were determined by XAS to be Pu(CO 3 ) 5 6-. The solubility of PuO 2 (am) increased dramatically with increasing total carbonate concentrations, indicating that carbonate makes strong complexes with Pu(IV). The dominant Pu(IV)-carbonate species that reasonably described all of the experimental data were Pu(CO 3 ) 5 6-in high concentrations of carbonate and bicarbonate and Pu(OH) 2 (CO 3 ) 2 2-in low concentrations of bicarbonate. Data suggest the possible presence of another species in the low carbonate and high pH region. However, because of uncertainty in Pu(IV) concentrations in this region, no attempt was made to fit the data in this region. The logarithm of the thermodynamic equilibrium constants for the PuO 2 (am) dissolution reactions involvingPu(CO 3 ) 5 6-and Pu(OH) 2 (CO 3 ) 2 2-[(PuO 2 (am) + 5 CO 3 2-+ 4 H + ← → Pu(CO 3 ) 5 6-+ 2 H 2 O) and (PuO 2 (am)

+ 2 HCO 3 -← → Pu(OH) 2 (CO 3 ) 2 2-
)] were found to be 33.32 and -4.78, respectively. These values, when combined with the solubility product of PuO 2 (am) (log K sp = -56.85 [1]), provided logarithm of the equilibrium constants of 34.18 and 44.76, respectively, for (Pu 4+ + 5 CO 3 2-← → Pu(CO 3 ) 5 6-

) and (Pu 4+ + 2 CO 3 2-+ 2 OH -← → Pu(OH) 2 (CO 3 ) 2 2-.

A

  Thermodynamic model for the solubility of PuO2(am) in the aqueous Na+-H+-OH-Cl-H2O-ethylenediaminetetraacetate system Dhanpat Rai, Heather Bolton, Da. Moore, Nancy Hess, G. Choppin, Pierre Vitorge To cite this version: Dhanpat Rai, Heather Bolton, Da. Moore, Nancy Hess, G. Choppin, et al.. Thermodynamic model for the solubility of PuO2(am) in the aqueous Na+-H+-OH-Cl-H2O-ethylenediaminetetraacetate system. Radiochimica Acta, 1999, 86 (3-4), pp.89-99. 10.1524/ract.2001.89.2.067. cea-02359297 , N.J.Hess, A.R.Felmy, D.A.Moore, M.Yui, P.Vitorge

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Roda Bounaceur

Pierre-Alexandre Glaude

Baptiste Sirjean

René Fournet

Pierre Montagne

Matthieu Vierling

Michel Molière

Development of a model for auto-ignition delays and its use for the prediction of premix combustion reliability

Except in diesel engine applications, auto-ignition is an unwanted event from a general safety and reliability standpoint. It is especially undesirable in the premixing process involved in most low NOx combustion technologies. Therefore, in addition to auto-ignition temperature, autoignition delay (AID) is a key data for the design of modern combustors including gas turbine ones. The authors have investigated the detailed kinetic mechanisms leading to autoignition and established practical AID correlations involving the fuel composition, its temperature, pressure and equivalence ratio. The correlations brought about during this program offer a good reconciliation between calculated and experimental AID through a wide range of fuel composition, initial temperature and pressure. Validations were mainly done against data acquired with experimental setups consisting in shock tubes and rapid compression machines. The auto-ignition delay times of methane, pure light alkanes and various blends representative of several natural gas and process-derived fuels have been reviewed. For each fuel mixture, this study procures a simple equation linking the auto-ignition delay time to the temperature, pressure and equivalence ratio. As a direct application of this work, the authors have evaluated the risk of auto ignition in the premixing zone of a combustor characterized by a residence time and an associated probability density function. The results of this simulation stress the key role of larger hydrocarbon in the risk of flash-back events.

INTRODUCTION

Auto-ignition is a complex, highly non-linear phenomenon that has been the matter of extensive experimental and theoretical research for decades, but still strongly resists attempts of modeling through combustion kinetics and fluid dynamics. Nonetheless, auto-ignition data represent crucial inputs in process safety and combustion reliability and are often sorely lacking when the fuel composition and the pressure depart from usual values. Therefore, the prediction of auto-ignition in terms of both AIT (auto-ignition temperature) and AID (auto-ignition delay) over extended pressure and fuel composition ranges would represent a great step forward for the engineering community. In a previous paper [START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF] the LRGP laboratory and GE Power have set out a method that aimed at evaluating the Auto-Ignition Temperature of a series of industrial gas fuels and was successfully validated against a number of experimental results compiled from the abundant literature found in this field. Using the same methodology the present paper addresses the prediction of AID and continues this overall effort to improve the knowledge of auto-ignition phenomena. The first part of this paper will set out the modeling of autoignition delay times based on the same methodology as the one developed in the previous paper, using detailed kinetic modeling. For different gas and blend of interest, auto-ignition delay times are calculated for temperatures ranging from 600 to 1500 K, pressures from 1 to 25 bar and with equivalence ratio ranging from 0.2 to 4.0. A second part will be devoted to stress the relevance of AID in modern combustion systems taking as example the occurrence of "spontaneous ignition" in premix combustion, a risk that materializes when the residence time of the fuel in the premixing zone exceeds the local auto-ignition delay, which depends on the fuel composition, equivalence ratio, pressure and temperature.

KINETIC MODELING

Several comprehensive kinetic models for the oxidation and combustion of natural gas and light hydrocarbons have been published in the literature in the last years. The University of Leeds modeled successfully the combustion of methane at typical flame temperatures [START_REF] Hughes | Development and testing of a comprehensive chemical mechanism for the oxidation of methane[END_REF]. Another popular detailed mechanism is GRI-mech 3.0 [START_REF] Smith | GRI-Mech 3.0[END_REF], which was optimized for the combustion of methane at high temperature. Mixtures of light hydrocarbons representative of natural gas can be modeled by GDF-Kin [START_REF] De Ferrières | Experimental and numerical investigation of low-pressure laminar premixed synthetic natural gas/O2/N2 and natural gas/H2/O2/N2 flames[END_REF] and Konnov's models [START_REF] Konnov | The pseudo-catalytic promotion of nitric oxide oxidation by ethane at low temperatures[END_REF], which have been regularly updated, and contain also the reactions for the formation of nitrogen oxides. Models for the combustion of light alkanes have been developed at the Lawrence Livermore National Laboratory (LLNL) and include the formation of peroxy radicals that are responsible for the reactivity at low temperature [START_REF] Mehl | Kinetic modeling of gasoline surrogate components and mixtures under engine conditions[END_REF]. AramcoMech 1.3 has been written hierarchically with systematic validations for each reactant over a wide range of pressure and equivalence ratio at National University of Ireland at Galway (NUIG) [START_REF] Metcalfe | A Hierarchical and Comparative Kinetic Modeling Study of C1 -C2 Hydrocarbon and Oxygenated Fuels[END_REF]. All these models have been validated against experimental ignition delay times, but at temperatures much higher than AIT. As shown previously [START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF], most of these mechanisms aim at the simulation of flames in burner or engine conditions and their validity is assumed only in high temperature conditions, i.e. above 1000K. Some mechanisms, such as LLNL or AramcoMech 1.3, contain the low temperature alkyl radical species in order to account for the formation of peroxy radicals but they do not involve all the alkanes contained in natural gas fuels. Peroxy radicals are responsible for cool flames and the for NTC (Negative Temperature Coefficient) behavior that is specific of alkanes. Auto-ignition, a key consideration in process safety, is characterized by much lower temperatures, slower reactions and much longer timescales than flames. Some specific reactions, like those of peroxy radicals, can be neglected at high temperature but become critical reaction paths at low temperature because they yield branching agents and fragile molecules, the fast decomposition of which dramatically promotes the global reactivity and finally the ignition. The following scheme can be used to summarize the oxidation of hydrocarbons at low temperature:

A new model has been developed in our group for the prediction of AIT of gas mixtures [START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF]. It contains reaction bases for the lighter species, i.e. the CO/H2 sub-mechanism, methane, and ethane [START_REF] Glaude | Experimental and Modeling Study of Premixed Laminar Flames of Ethanol and Methane[END_REF], and uses the software EXGAS [START_REF] Buda | Progress toward a unified detailed kinetic model for the autoignition of alkanes from C4 to C10 between 600 and 1200 K[END_REF] for the low temperature chemistry of C3-C5 alkanes. This software has been patiently developed at the LRGP for years to enable the automatic generation of mechanisms for the gasphase oxidation of hydrocarbons. This mechanism contains 209 species involved in 1473 reactions; it can be obtained upon request. The model has proven its ability to predict rather well AITs, which correspond at the lowest temperatures of reaction, with typical timescales amounting in minutes. In the present work, the model has been refined for the prediction of the ignition delay (AID) times of gas mixtures at low and high temperatures in order to account for the widest conditions encountered in industrial processes. Its validation has been made against experimental results from the abundant literature found on this subject. The corresponding simulations were performed using CHEMKIN II [START_REF] Kee | CHEMKIN-II: A FORTRAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics[END_REF]: this software solves the mass and energy equations that are involved in the main laboratory reactors. Figure 1 compares the results of such simulation performed for methane with some experimental auto-ignition delay times obtained in shock tube. The model reproduces very well the AIDs measured at moderate pressure and in stoichiometric conditions [START_REF] Crossley | The effect of higher alkanes on the ignition of methane-oxygen-argon mixtures in shock waves[END_REF] (fig. 1a) as well as those measured at elevated pressures and in rich conditions [START_REF] Petersen | Kinetics modeling of shock-induced ignition in low-dilution CH4/O2 mixtures at high pressures and intermediate temperatures[END_REF] (fig. 1b). Simulation are within the range of reproducibility of the experiments and the apparent activation energy is well simulated. The capability of the model to deal with hydrogen and hydrogen/methane blends has be demonstrated previously [START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF]. Ethane and propane, two significant alkane components of natural gas, are known to increase its reactivity. Figure 2 displays the comparison between simulation and experimental results data for lean and rich mixtures [START_REF] De Vries | Ethane ignition and oxidation behind reflected shock waves[END_REF]. Propane is the first of the alkane series that exhibits the specific low temperature behavior consisting in cool flame and NTC. The model can reproduce these properties: Figure 3a compares the AID simulation results with those obtained with a rapid compression machine [START_REF] Gallagher | A rapid compression machine study of the oxidation of propane in the negative temperature coefficient regime[END_REF]. Figure 3b moreover shows the suitability of the model for reproducing high temperature shock tube experiments [15]. Similar conclusions apply to nbutane (Figure 4a) and n-pentane (Figure 5) which exhibit still stronger NTC behaviors at low temperature: the experimental results come from experiments performed in Lille with a rapid compression machine [START_REF] Minetti | Experimental and modeling study of oxidation and autoignition of butane at high pressure[END_REF][START_REF] Minetti | The chemistry of pre-ignition of npentane and 1-pentene[END_REF]. The high temperature ignition of n-butane is correctly modeled in shock tube both at low and high pressure (Figure 4b) [START_REF] Ogura | Chemical Kinetic Mechanism for High Temperature Oxidation of Butane Isomers[END_REF][START_REF] Burcat | Shock-tube investigation of comparative ignition delay times for C1-C5 alkanes[END_REF]. Note that discrepancies are larger in the case of simulations of Rapid Compression Machine results in the NTC region (Fig 4a,5a). The complexity of the device, especially the nonuniform temperature and the gas motion, is not captured by the assumption of an adiabatic reacting core in the 0-D calculation. Eventually, the ability of the model to simulate mixtures or natural gas has also been checked, as displayed: Figure 6 shows that the model correctly simulates the results given by a high pressure shock tube experiments in the cases of: (i) a mixture of 70% methane/30% propane/ [START_REF] Petersen | Methane/propane oxidation at high pressures: Experimental and detailed chemical kinetic modeling[END_REF] and (ii) an Algerian natural gas [START_REF] Lamoureux | Natural gas ignition delay times behind reflected shock waves: Application to modelling and safety[END_REF]. Ignition delay times of alkanes vary according to the temperature, pressure and equivalence ratio of the mixture. Above 1000 K, a linear variation according to the temperature is found using an Arrhenius plot but at lower temperature, a nonlinear behavior is observed. Additions of C2+ alkanes to methane decrease the ignition delay times but the intensity of this "promoting effect" significantly depends on temperature and pressure. In this respect we have performed numerous comparisons between experimental data and the results of simulations using the present model; these comparisons involve large ranges of mixtures, equivalence ratios, pressures and temperatures. The outcome of this comparison work is that the AID values and the variation trends agree well, even at low temperature, where experimental uncertainty are the highest and the chemical mechanisms the most complicated.

PREDICTIVE CALCULATIONS FOR PROCESS GASES

The model being validated, we have used it to predict the AIDs of several gas fuels of interest in gas turbine applications, in order to assess auto-ignition risks, in function of pressure and equivalence ratio. Table 1 shows the compositions of the selected gas mixtures. Mixture B1 is methane, taken as a reference. Mixtures B2 and B3 that contain propane and butane respectively, are representative of natural gas fuels. B4 represents a classical natural gas burned in gas turbines while B5 stands for methane enriched in hydrogen. B6 is a typical Coke Oven Gas; B7 a low BTU Blast Furnace Gas, and B8 a typical blend of the B6 and B7. For all blends, the equivalence ratio ranged from 0.2 to 4.0 and pressure from 1 to 25 bar. The simulations assumed a closed vessel without thermal transfer. For instance, figure 7 illustrates the results obtained with the natural gas B4. In the high temperature range, above 1000K, one observes a linear Arrhenius behavior, whereas the curve exhibits a deviation at low temperature, due to the specific behaviors developed by the C3 to C5 components. This effect is more important with very rich mixtures (e.g. with an equivalence ratio of 3.7) and at lower pressure (results not shown in figure 7). Using the present model, we have evaluated the AID at different initial temperatures, pressures, and equivalence ratios, for the 8 blends listed in table 1. Then we have used these results as samples to build up a generic correlation based on an Arrhenius-like equation which is a widely used form in auto-ignition studies:

𝐴𝐼𝐷 ∝ 𝜑 𝑥 𝑃 𝑦 𝑒𝑥𝑝 ( 𝐸 𝑅𝑇 ) (1)
While AIDs monotonically decrease with the initial temperature for a given equivalence ratio (Figure 7), the apparent activation energy is found lower at high temperature than at low temperature. Furthermore, AIDs vary nonmonotonically with the equivalence ratio for a given initial temperature. Therefore, a single set of parameters cannot represent the entire range of possible conditions. Therefore, for the purpose of process safety studies linked to gas turbine (GT) operation, we defined "reduced" parameters ranges: temperature between 600 and 900 K; pressure between 1 and 25 bar; equivalence ratio between 0.2 and 4.0. Then we tested the adequacy of the generic correlation (1) with the blend B4. On this range, correlation coefficients are above 0.999 except 0.992 and 0.986 for fuels B2 and B3, respectively. Figure 8 compares the AIDs calculated with the full kinetic mechanism to those obtained with the fitted correlation: the correlation proves effective on the reduced temperature range. Although the agreement deteriorates at low pressure and high equivalence ratio because of the enhancement of the NTC behavior of the C2+ alkanes, combustion in gas turbines is not ultra-rich (>2) but lean and occurs at elevated pressure; moreover C2+ species remain minor components in many blends of interest. Consequently, practical correlations for the evaluation of the order of magnitude of AIDs can be derived and enable a fast and easy estimation as compared with a detailed modeling, which should be preferred, however, for a deeper investigation or for better accuracy. In conclusion the generic correlation (2) represents an interesting tool for a fast evaluation of the auto-ignition delays of industrial gas fuels used in variable temperature, pressure and fuel/air ratio conditions.

IMPORTANCE OF AID IN PREMIX COMBUSTION

Premix combustion is a good example of the relevance of auto-ignition delay in industrial processes. Indeed autoignition sets a feasibility barrier that imposes on combustion engineers in lean premix combustion. The principle of lean premix combustion is to get fuel and air thoroughly mixed inside a premixing zone (PrZ) located upstream of the actual combustion zone (CbZ). Because the emission of NOx is virtually an exponential function of the temperature which itself increases with the equivalence ratio, one needs to obtain an air/fuel mixture as homogenous as possible since any richer spot would generate a locally high NOx contribution that might defeat the low NOx objective. In this regard, figure 9 [START_REF] Molière | Stationary gas turbines and primary energies: A review of fuel influence on energy and combustion performances[END_REF] shows the effect of "unmixedness" (expressed as the standard deviation S of the equivalence ratio  calculated throughout the PrZ, S=[ (i -mean) 2 /N] 1/2 where i represents a point of the premix zone and N the number of points) on the overall NOx emission: the combustor is fed with a generic CH4/air premixing device operating at PPrZ = 16 bar, TPrZ = 650 K (377°C). overall fuel/air flow during mixing and (ii) the residence time of the fuel molecules in this zone. However since the premix is performed above the AIT in modern engines, this residence time must be designed to be substantially lower than the AID associated with the specific fuel considered and the values of TPrZ and PPrZ. Otherwise an unwanted "spontaneous ignition" ("SI") would occur in the PrZ and the flame would go backward from the CbZ into the PrZ which, as a result, is equivalent to a "flashback", the latter term being normally reserved for cases where the flame retro-propagates in the absence any upstream auto-ignition. This point can be accounted for through a simple lagrangian approach in which one molecule of fuel is tracked during its transit through the PrZ and one compares its residence time to the AID value. If Q represents the volume flow of the fuel/air mixture and V the volume of the PrZ, the mean value of "t", i.e. the average residence time "tm" equals V/Q. However, the residence time (t) of any group of molecules crossing the volume "V" of the PrZ is a random variable that typically obeys a log-normal law, assuming the distribution is monomodal. A log-normal process is typically used for a phenomenon which results from many independent random variables. Figure 10 shows the corresponding probability density function Pdf(t) and the cumulated distribution function Cdf(t) which is the Pdf's integral that tends towards 1 when t becomes infinite.

Pdf(t) = 1 tσ 𝑙𝑜𝑔 √2π 𝑒𝑥𝑝 [ -(ln(t)-t 𝑚,𝑙𝑜𝑔 ) 2 2σ 𝑙𝑜𝑔 ] (3)
Cdf(t) = 1 2 erf [ ln(t)-t 𝑚,𝑙𝑜𝑔 √2σ 𝑙𝑜𝑔 ] (4)
where "erf" stands for the error function.

where:

-S is defined as: S = {𝑙𝑛[1 + (𝜎/𝑡 𝑚 ) 2 ]} 0.5 , -L is defined as: L= erfc -1 (R) and:

-R = 5.56 10 -4 tm/H The data (𝜎/𝑡 𝑚 ), which is the ratio between the standard deviation and the average residence time, is called in statistics the "coefficient of variation" of the distribution Pdf(t). When one considers very low SI risks, e.g. one SI event over 45,000 hours, L can be taken in the order of 4.5. Inequality ( 5) is interesting as it can be used in two ways:

A) Adapting the design of PrZ in function of the fuel: An equivalent form of inequality ( 5) is:

𝜎/𝑡 𝑚 < √exp {2[L -√L 2 -ln (AID/t 𝑚 ] 2 } -1 (6)
The corresponding curve (𝜎/𝑡 𝑚 ) = f(AID/tm) is drawn in figure 11. In order to have less than one SI in e.g. 45,000 operation hours (about 5 years), one must stay in the region located below this curve which imposes an upper limit for the value of , the standard deviation of the residence time.

B) Given a design of the premix zone, evaluate the acceptability of a candidate fuel:

An alternate, equivalent form of equation ( 5) is:

AID/t m > [1 + (𝜎/t m ) 2 ] 0.5 exp{1.41 L × ln[1 + (𝜎/t m ) 2 ] 0.5 } (7)
Figure 12 that illustrates this condition shows the minimum (AID/tm) ratio that is required from a candidate fuel in order to have an SI risk smaller than the specified value (here again: 1 over 45,000 hrs). For instance, if the average of the residence time in the PrZ is 2 ms and its standard deviation does not exceed 0.2 ms (/tm = 0,1), then one finds a minimum (AID/tm) ratio of 1.9, hence AID could be theoretically as low as 3.8 ms. These simple considerations can be applied to the particular case of the premixing of natural gas fuels containing some amounts of heavier hydrocarbons. This can be illustrated with three fuels studied in the previous section, the AID's of which can be calculated from the corresponding correlations drawn from table 2 (TPrZ = 823°C; PPrZ = 25 bar, PrZ = 0,7): -B1 (100% CH4) : AID = 1.25 s -B2 (93% CH4-7% C3H8): AID = 0.47 s -B3 (95% CH4-5% C4H10): AID = 0.14 s Considering an average residence time of 60 ms and a standard deviation of 18 ms with thus (/tm) = 0.3, one sees, after placing the three representative points in figure 12, that: B3 (rich in C4) would trigger at least one spontaneous ignition; pure methane (B1) would not and B2 (rich in C3) would lie in a "grey area". This emphasizes again the criticality of C2+ hydrocarbon components in natural gas fuels. One can draw the following general interpretation from this simple approach. To avoid auto ignition in the premixing zone the standard deviation of the residence time distribution in this zone must be reduced by appropriate aerodynamics design. This is achieved in particular by: (i) limiting the volume of the premix zone in order to minimize low speed and local separation zones that increase the residence time; (ii) favoring, on the contrary, intense swirls in the fuel/air mixing process since fast rotational motion greatly improves mixing while not increasing substantially the time of transit. These basic guidelines are actually implemented in all modern combustor designs.

Certainly this oversimplified 0-dimensional approach suffers strong limitations. It ignores factors as: operational transients, possible multimodal distribution of the residence time and non-uniform temperature and fuel/air ratio conditions during the premixing process. Therefore it has essentially an illustrative value of the key role of auto-ignition delays rather than a practical reach.

CONCLUSION

This work was primarily devoted to the elaboration of a kinetic-based model to quantify the auto-ignition delay (AID) of complex fuels in pressure and temperature conditions different from ambient ones. This work gave the opportunity to emphasize the relevance of this data not only in safety applications but also and especially in the fuel/air premixing processes that are now of common use for the design of DLE systems.

The simple AID correlations obtained, which add to the AIT correlations set out in a previous paper [START_REF] Bounaceur | Prediction of Auto-Ignition Temperatures and Delays for Gas Turbine Applications[END_REF], represent potential useful criteria for the evaluation of auto-ignition risks as well as data-driven although not absolute guidelines in the design of premixing devices.

Figure 1 :

 1 Figure 1: Auto-ignition delay times of methane in shock tube under high pressure. Points are experimental data, lines are simulations.

Figure 2 :

 2 Figure 2: Auto-ignition delay times of ethane in shock tube for different equivalence ratios. Points are experimental data, lines are simulations.

Figure 3 :

 3 Figure 3: Auto-ignition delay times of propane in (a) rapid compression machine and (b) shock tube. Points are experimental data, lines are simulations.

Figure 4 :

 4 Figure 4: Auto-ignition delay times of n-butane in (a) rapid compression machine and (b) shock tube. Points are experimental data, lines are simulations.

  bar - = 0.7 -Ogura et al., 2007 P = 10 bar - = 1 -Burcat et al., 1971

Figure 5 :

 5 Figure 5: Auto-ignition delay times of n-pentane in rapid compression machine. Points are experimental data, lines are simulations.

Figure 6 :

 6 Figure 6: Auto-ignition delay times for (a), 70% methane-30% propane (b) natural gas in shock tube. Points are experimental data, lines are simulations.

Figure 7 :

 7 Figure 7: AID simulations for the blend B5 in function of pressure and equivalence ratio: B5 is representative of a natural gas.

Figure 8 :

 8 Figure 8: AID of mixture B4 determined by kinetic simulations (dots) and correlation (lines) as a function of the temperature. The following correlation expresses the AID as a function of the operating parameters for the different blends: log 10 AID = 𝑘 0 + 𝑘 1 × 1000 𝑇 + 𝑘 2 × log 10 𝑃 + 𝑘 3 × log 10 𝜑 (2) AID represents the auto-ignition delay [ms], P the pressure [bar], T the temperature [K] and  the equivalence ratio.Table 2 summarizes the coefficients computed for the

Figure 9 :

 9 Figure 9: NOx emission in function of  and its standard deviation inside the premix zone Now the degree of premixedness at the outlet of the PrZ directly depends from: (i) the turbulence intensity of the

Figure 10 :

 10 Figure 10: Probability density function Pdf(t) and cumulated distribution function Cdf(t) of the residence time t (for the sake of legibility, AID has been taken exaggeratedly close to tm)The meanings of the distribution parameters present in these definitions are given in Appendix 1. By definition, Cdf(AID), as shown in figure10, represents the probability of a group of molecules residing less than AID seconds in the PrZ. Consequently [1-Cdf(AID)] represents the probability of an SI occurring, due to auto-ignition, while this group of molecules crosses the PrZ, which is the probability of a "one shot" SI. As shown in Appendix 1, the condition for having -

Figure 11 :

 11 Figure 11: condition to fulfill in order to have a given maximum frequency of SI (here: 1 event/45000 hours).

Figure 12 :

 12 Figure 12: minimum (AID/tm) ratio to have a spontaneous ignition risk smaller than a specified value.

Table 1 :

 1 Composition of the fuels studied (% mol.)

Table 2

 2 summarizes the coefficients computed for the different blends. These expressions are valid for pressures from 1 to 25 atm, equivalence ratios from 0.2 to 4 and temperatures from 600 to 900 K.

Table 2 :

 2 Coefficients ki contained in correlation (2) for the 8 blends considered.

		B1	B2	B3	B4	B5	B6	B7	B8
	k0 -5.87 -1.35 -0.20 -3.16 -7.62 -7.41 -7.31 -7.41
	k1	8.26 4.44 3.10 5.96 9.49 9.26 9.51 9.35
	k2 -0.83 -1.08 -1.16 -0.98 -0.59 -0.67 -0.14 -0.58
	k3 -0.58 -0.87 -1.21 -0.72 -0.51 -0.48 -0.57 -0.47

DISCLAIMER

The information contained in this technical paper represents the best and most current information owned by the authors on the subject and is believed to be correct. However the authors and their respective organizations make no representations or warranties regarding the accuracy and will not be responsible for damages resulting from use or reliance upon this information.

APPENDIX 1: EXPRESSION OF THE PROBABILITY OF SPONTANEOUS IGNITION DUE TO AUTO-IGNITION IN A PREMIXING ZONE

The Pdf and Cdf functions of the residence time 't" variable write:

where tm,log and log are respectively the average and standard deviation of the random variable "ln(t)". The average and standard deviation of the random variable "t", that are noted tm and  respectively and are expressed in seconds, are correlated to tm,log and log by the equations:

The probability of spontaneous ignition occurring "in one shot", i.e. the probability of an SI occurring while a sole group of molecules crosses the PrZ, is represented by [1-Cdf(AID)]

where the "erfc" function is the complementary of the "erf" function: erfc(x) = 1 -erf (x) Using the parameters tm and , it comes:

During a "reference operation period" H (in hours) the number of molecules groups crossing the PrZ equals 3600* H/tm; therefore the frequency of spontaneous ignition writes:

Consequently, if one wishes to have, for instance, less than one spontaneous ignition in H operation hours (Fr(SI)= 1), then he must satisfy the following equation in which  and tm are expressed in seconds and H in hours:

The meaning of the "residue" R is as follows: if the the erfc function reaches the value R, then at least one spontaneous ignition will occur over H operation hours. For very great value of the reference period H, the value of R is very low, e.g.: R = 1.24 10 -11 for H = 45,000 hours and tm in the order of 0.001 s.

The inequality [START_REF] Kee | CHEMKIN-II: A FORTRAN Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics[END_REF] involves quadratic expression in "S": 0.5 S 2 -1.41 L . S + ln( 𝐴𝐼𝐷 𝑡𝑚 ) > 0 (8) where: S is defined as: S = {𝑙𝑛[1 + (𝜎/𝑡 𝑚 ) 2 ]} 0.5 (9) -L, which is defined as: L= erfc -1 (R), can be extracted from a table of the erfc function or from a numeric approximation of the reverse of the erfc function. Since R depends on the ratio (tm/H), L depends on the same but this dependence is loose due to the asymptotic profile of the erfc -1 (R) function when R becomes very low; therefore a value of L in the order of e.g. 4.5 can be taken. The right member of inequality ( 8) is an explicit function of (AID/tm) and an implicit one of (/tm) (through eq. ( 9)). Therefore (AID/tm) and (/tm) are linked by inequality [START_REF] Glaude | Experimental and Modeling Study of Premixed Laminar Flames of Ethanol and Methane[END_REF].


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UMR 7194 CNRS web page « Siwaliks : les homininés du haut bassin de l'Indus depuis le Pliocène ».

Human Fossils of the Siwaliks: Myths and Realities

Anne Dambricourt Malassé UMR 7194 CNRS "Natural History of Prehistoric Man", National Center for Scientific Research (CNRS) -National Museum of Natural History-Perpignan University Via Domitia, France.

Two American biologists, [START_REF] Herrera | Ancestral DNA, Human Origins, and Migrations[END_REF], were quick to publish a long synthesis on Masol [START_REF] Dambricourt Malassé | The first Indo-French Prehistorical Mission in Siwaliks and the discovery of anthropic activities at 2.6 million years. In Human origins in the Indian sub-continent, (dir. A. Dambricourt Malassé), Fascicule thématique Palevol[END_REF] All the experts replied that this tooth was not human but that of a cervid. M.P. Singh repeated his misdeeds in 2003 in the peer review Human Evolution which agreed to publish four photos without scale and with totally unrealistic interpretations. The article is downloadable to verify (see references). There is absolutely no relation between the photos and the descriptions, for example figure 4 is a small rectangular fragment of mandible with fractured roots. There is no dental crown. But M.P. Singh describes 1) a parabolic mandiblei.e. with the arched shape characteristic of Homo and 2) a molar whose the crown surface shows the Dryopithecus pattern, i.e. five cuspids. It is not necessary to be a great expert to see that figure 5 is not a femur. In the acknowledgments M.P. Singh mentions he had submitted this article to Nature which rejected it. Till now no one understands how the reviewer accepted such a manuscript.

The consequences of this paper are disastrous as unqualified biologists like these two Americans present M.P. Singh as the historical discoverer of Pliocene hominin in Asia reducing the researchers of the "Siwaliks" to epigones without event mentioning their names nor the French Institutions. To give the measure of this humiliation in relation to Dr. Mukesh Singh, it is as if the discoverers of the Australopithecus Lucy had been presented as anonymous epigones of a mythomegalomane.

There are no known hominin fossils in the Siwaliks and the first discovery of human activity during Pliocene will always historically accredited to the Indo-French cooperation.

  taking up the problem of the geographical and phylogenetic origins of the hominins who let the oldest intentional cut marks collected in Pliocene deposits in association with choppers, in Asia and not in Africa. The merit of this historical discovery goes to the "Siwaliks" program fully funded by two great European research organizations, the National Center for Scientific Research (CNRS) and the National Museum of Natural History, Paris which entrusted the direction to Dr. Anne Dambricourt Malassé, paleoanthropologist. The first Indo-French fieldworks in the Siwaliks were initiated in 2002 by Dr. Mukesh Singh, President of the "Society for Archaeological and Anthropological Research" at Chandigarh (Union Territory of India). Nevertheless, it is imperative to denounce a serious untruth in this eagerness to seize the emerging paleoanthropological problematic. This precipitation transmits a very serious sham put in place by a former student of the Panjab University of Chandigarh since 1988, Mohinder Pal Singh, while no paleoanthropologist cites him. Since the official recognition of Masol in 2016 by the Prime Minister of India and the President of the French Republic in Chandigarh, he claims the international recognition for the discovery of fossils of 3.4 Ma allegedly humans. In 2018 the Archaeological Survey of India asked to confirm what Mohinder Pal Singh said, because he presented himself as the former teacher of the author of this paper "Human Fossils of the Siwaliks: Myths and Realities", graduate of the Academy of the University of Paris. Yet, Mohinder Pal Singh does not hold a doctorate but a master's degree in Anthropology from the Panjab University. So, he could never have become of Lecturer. He was laboratory assistant at the Blood Bank of PGI in Chandigarh. If M.P. Singh had collected the slightest bit of Homo at to 3.4 Ma in the Himalayan foothills, he would have been invited to the international symposium of the Pontifical Academy of Sciences organized by Yves Coppens in April 2019 and heard among the distinguished guests. Human Fossils of the Siwaliks: Myths and Realities In 1988 Mohinder Pal Singh published an incisor in the Proceedings of the Academy of Sciences of India claiming that it is human. Then, he visited paleontologists from the European Centre for Prehistoric Research in Tautavel (CERP, France) and the current director of the French mission in India, Anne Dambricourt Malassé, at the Institute of Human Paleontology in Paris.

Acknowledgements

The author thanks Dr. Mukesh Singh, President of the "Society for Archaeological and Anthroppological Research" and the Archaeological Survey of India for the efforts to restitute the historical truth.