B ÉR ÉNICE GREC Laurent Boudin

Bérénice Grec
email: berenice.grec@u-paris.fr

Milana Pavić-Čolić
email: milana.pavic@dmi.uns.ac.rs

Srboljub Simić
email: srboljub.simic@dmi.uns.ac.rs

AND Milana Pavi Ć-Čoli Ć

Srboljub Simi

Energy method for the Boltzmann equation of monatomic gaseous mixtures

In this paper, we present an energy method for the system of Boltzmann equations in the multicomponent mixture case, based on a micro-macro decomposition. More precisely, the perturbation of a solution to the Boltzmann equation around a global equilibrium is decomposed into the sum of a macroscopic and a microscopic part, for which we obtain a priori estimates at both lower and higher orders. These estimates are obtained under a suitable smallness assumption.

The assumption can be justified a posteriori in the higher-order case, leading to the closure of the corresponding estimate.

Introduction

In the last decade or so, the Boltzmann equation for mixtures, which was already mentioned in [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF], attracted the attention of many works. The modelling issue, for both monatomic and polyatomic gases, was for instance discussed in [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF][START_REF] Baranger | On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases[END_REF][START_REF] Simić | Non-equilibrium mixtures of gases: modelling and computation[END_REF] (see also the references therein). Many works focused on the analysis of the monatomic case, like [START_REF] Boudin | Diffusion asymptotics of a kinetic model for gaseous mixtures[END_REF][START_REF] Daus | Hypocoercivity for a linearized multispecies Boltzmann system[END_REF][START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF][START_REF] Bondesan | Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium Maxwell distributions[END_REF], which were dedicated to compactness, hypocoercivity-related and stability results. Well-posedness and regularity were investigated in [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF][START_REF] Briant | Stability of global equilibrium for the multi-species Boltzmann equation in L ∞ settings[END_REF][START_REF] Gamba | On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures[END_REF][START_REF] Alonso | The Cauchy problem for Boltzmann bi-linear systems: the mixing of monatomic and polyatomic gases[END_REF][START_REF] De La Canal | On existence, uniqueness and Banach space regularity for solutions of Boltzmann equations systems for monatomic gas mixtures[END_REF], and asymptotics questions were tackled in [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF][START_REF] Hutridurga | Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections[END_REF][START_REF] Bondesan | Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation[END_REF].

Some of the previous papers, for instance [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF], rely on the so-called micro-macro decomposition. In the present work, we aim to provide a more detailed insight on that decomposition in the mixture case. Indeed, the micro-macro decomposition of a solution of the linearized Boltzmann equation has a key role in the study of both mathematical and numerical properties of that solution. It was introduced for the monospecies Boltzmann equation in [START_REF] Guo | The Vlasov-Maxwell-Boltzmann system near Maxwellians[END_REF][START_REF] Guo | The Boltzmann equation in the whole space[END_REF] on the one hand, and in [START_REF] Liu | Energy method for Boltzmann equation[END_REF][START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF][START_REF] Yang | A new energy method for the Boltzmann equation[END_REF] on the other hand. The method consists in considering the equilibrium perturbation as the sum of a macroscopic part and a microscopic one. The macroscopic part can be decomposed on a finite-dimensional subspace, where the associate coordinates solve some conservation laws of fluid type, whereas the microscopic one still solves a kinetic equation. Nevertheless, the microscopic part is incorporated in macroscopic conservation laws and fills the gap between the usual Navier-Stokes approximation and the complete kinetic equation [START_REF] Liu | Energy method for Boltzmann equation[END_REF]. In fact, it brings information which is essential to provide proper estimates of the perturbed solutions of the kinetic equation.

In the monospecies case, the micro-macro decomposition and the underlying energy method were used for hypocoercivity estimates, see [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF], for large-time behaviour studies [START_REF] Liu | Nonlinear stability of rarefaction waves for the Boltzmann equation[END_REF][START_REF] Lee | Large-time behavior of solutions for the Boltzmann equation with hard potentials[END_REF][START_REF] Mouhot | Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus[END_REF] (see also [START_REF] Liu | Optimal time decay of the Boltzmann system for gas mixtures[END_REF] for a binary mixture), for propagation of one-dimensional waves [START_REF] Liu | Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation[END_REF], or to obtain Green's function for the Boltzmann equation [START_REF] Liu | Solving Boltzmann equation, Part I: Green's function[END_REF]. In numerical analysis, this decomposition is a major tool to build asymptotic-preserving (AP) schemes, allowing to pass, for instance, from the Boltzmann equation to the Navier-Stokes equations [START_REF] Bennoune | Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics[END_REF], to exactly conserve some physical quantities [START_REF] Gamba | Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations[END_REF], or to quantify uncertainty in kinetic equations [START_REF] Dimarco | Multi-scale control variate methods for uncertainty quantification in kinetic equations[END_REF]. As far as the mixture case is concerned, note that several attempts relying on a micro-macro decomposition were already performed, with a BGK approximation [START_REF] Jin | A micro-macro decomposition-based asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF][START_REF] Jin | A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF], in the two-species case [START_REF] Crestetto | Kinetic/fluid micro-macro numerical scheme for a two component gas mixture[END_REF], and in the general case with partial results [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF].

In this paper, we study the micro-macro decomposition and the corresponding energy method in the multicomponent mixture case, by following the strategy of [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF]. More precisely, we start from an equilibrium not depending on the time and space variables. The perturbation of this equilibrium is then decomposed into the sum of microscopic and macroscopic parts, for which we obtain lower and higher order estimates, first using relevant smallness assumptions, and exhibiting closure in the one-dimensional (in space) setting.

It is peculiar for the mixture, in contrast to the monospecies case, that the microscopic part contributes to macroscopic equations not only in the momentum and energy conservation laws, but also in the mass conservation law. This effect is crossed with the perturbation of the energy variable, which altogether makes the procedure of finding proper estimates much more involved. This problem is solved by means of introducing a suitable fluid quantity.

The paper is structured as follows. In the next section, we give a preliminary overview describing the framework for the subsequent analysis. Then, in Section 3, we discuss the methodology and main ideas relying on the micro-macro decomposition, and state our main results. They are a priori estimates on the perturbation based on the decomposition, whose proofs are exposed in Sections 4 and 5. In particular, we provide very detailed explanations for the lower-order estimate, knowing that, for the higher-order one, the same kind of computations and ideas are developed.

Preliminaries

We consider an ideal gas mixture constituted with I ≥ 2 monatomic species. Each species, indexed by 1 ≤ i ≤ I, is described thanks to a distribution function F i , which is nonnegative, and depends on time t ∈ [0, T ], T > 0, space position x ∈ R and microscopic velocity v ∈ R 3 . We denote by m i the atomic mass of species i. We emphasize that we choose to work here in a one-dimensional setting for the space variable x, not only for the sake of simplicity. Indeed, if most computations and results remain true in dimensions 2 and 3, the estimates are closed in this work by introducing the antiderivative of the macroscopic part of the decomposition, which can only be performed in a one-dimensional setting.

To consider the species altogether, we introduce the vector distribution function of the mixture, denoted by F = (F i ) 1≤i≤I . It satisfies the system of Boltzmann equations, also written in a vector form, (1)

∂ t F + v 1 ∂ x F = Q(F , F ),
where v 1 is the coordinate of velocity v in a direction of the space variable x, and Q is the vector collision operator, which only acts on the velocity variable v. The vector collision operator Q can be defined component-wise. To this end, we first need to recall the microscopic context of the collisions.

We assume that the mixture only involves elastic collisions, without chemical reactions. Consider two colliding molecules, one of species i and another one of species j, with respective pre-collisional velocities v and v * . Those velocities change after collision into post-collisional velocities v and v * , with both momentum and kinetic energy conserved, i.e.

(2)

m i v + m j v * = m i v + m j v * , 1 2 m i v 2 + 1 2 m j v * 2 = 1 2 m i v 2 + 1 2 m j v * 2 .
The previous equalities allow to introduce a parameter ω ∈ S 2 , enabling to write v and v * in terms of v and v * as

(3) v = m i v + m j v * m i + m j + m j m i + m j T ω (v -v * ), v * = m i v + m j v * m i + m j - m i m i + m j T ω (v -v * ), denoting T ω z = z -2(ω • z)ω for any z ∈ R 3 .
Then, for any i, j, we can define the operator Q ij describing the atomic interactions of species i with species j. It only acts on the velocity variable and is given by

Q ij (f i , g j )(v) = R 3 ×S 2 f i (v )g j (v * ) -f i (v)g j (v * ) B ij (v, v * , ω) dω dv * ,
for any species-related real-valued functions f i , g j of the velocity variable. The cross-section B ij allows to classify the way species i and j interact and must satisfy the micro-reversibility property

B ij (v , v * , ω) = B ji (v * , v, ω) = B ij (v, v * , ω) ≥ 0.
Moreover, in this work, we make the hard-sphere assumption, for any i, j,

(4) B ij (v, v * , ω) = β ij |(v -v * ) • ω| ,
where β ij > 0 is given. The assumption is required to ensure needed properties of the collision frequency and to deal with the nonlinearity. Eventually, we can define the i-th component of Q, with f = (f j ) 1≤j≤I , g = (g j ) 1≤j≤I , by

Q i (f , g) = I j=1 Q ij (f i , g j ).
Before recalling the main properties of the solutions to (1), let us introduce some very convenient notations. First, we define a component-wise product of two vectors A = (A i ) 1≤i≤I , B = (B i ) 1≤i≤I and a vector-valued function of A, for Φ : R → R, by

AB =      A 1 B 1 A 2 B 2 . . . A I B I      , Φ(A) =      Φ(A 1 ) Φ(A 2 ) . . . Φ(A I )     
.

This way, we can write, for instance, A 1/2 = (A i 1/2 ) 1≤i≤I , when A i ≥ 0. Finally, L 2 (R 3 ) I is endowed with its natural scalar product and norm, i.e. we set, for any vector functions f

= (f i ) 1≤i≤I , g = (g i ) 1≤i≤I ∈ L 2 (R 3 ) I , f , g I = I i=1 R 3 f i g i dv, f I = f , f I 1/2 .
Conservative properties of the Boltzmann equations are obtained thanks to the weak form of the collision operator that uses some symmetries built in the model. In the mixture setting, the weak form is carefully described, for example, in [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections[END_REF]. We only mention here the final formula. For any functions G and ψ for which it makes sense, we have

Q(G, G), ψ I = - 1 4 I i,j=1 R 3 ×R 3 ×S 2 G i (v )G j (v * ) -G i (v)G j (v * ) × ψ i (v ) + ψ j (v * ) -ψ i (v) -ψ j (v * ) B ij (v, v * , ω) dω dv * dv.
In this paper, we work in a perturbative setting, around a global equilibrium distribution function. Its notion is introduced in the so-called H-theorem, see [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF] for instance. Let us first define the entropy production functional

D(G) = Q(G, G), log G I . The H-theorem reads Proposition 1.
Assume that all the cross sections are positive almost everywhere and that G is such that both Q(G, G) and D(G) are well defined. Then (a) The entropy production is non-positive, i.e. D(G) ≤ 0.

(b) Moreover, the three following properties are equivalent: i. for any

1 ≤ i, j ≤ I, Q ij (G i , G j ) = 0;
ii. the entropy production vanishes, that is D(G) = 0; iii. there exist T > 0 and u ∈ R 3 such that, for any i, there exists n i ≥ 0 such that

G i (v) = n i m i 2π kT 3/2 e -m i 2kT |v-u| 2 .
Choosing kT = 1, u = 0, n as a nonnegative constant vector, we obtain the normalized centered Maxwell vector function M as

M i (v) = m i 2π 3/2 e -m i 2 |v| 2 , 1 ≤ i ≤ I.
Let us then recall the collision invariants in the gas mixtures setting which can be found in [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF], for instance. The collision invariants are the velocity-depending functions which make the previous weak form of Q vanish. They are moreover chosen one-to-one orthogonal and normalized with respect to a L 2 scalar product weighted in terms of nM (remembering the component-wise multiplication defined above). More precisely, we set (5)

                                                   χ 1 = 1 √ n 1      1 0 . . . 0      , χ 2 = 1 √ n 2      0 1 . . . 0      , . . . , χ I = 1 √ n I      0 0 . . . 1      , χ I+1 = 1 I j=1 n j m j      m 1 v 1 m 2 v 1 . . . m I v 1      , . . . , χ I+3 = 1 I j=1 n j m j      m 1 v 3 m 2 v 3 . . . m I v 3      , χ I+4 = 1 6 I j=1 n j      m 1 |v| 2 -3 m 2 |v| 2 -3 . . . m I |v| 2 -3      .
Then the family (χ k ) 1≤k≤I+4 satisfies, for any G,

Q(G, G), χ k I = 0, 1 ≤ k ≤ I + 4, (nM ) 1/2 χ k , (nM ) 1/2 χ I = δ k , 1 ≤ k, ≤ I + 4. (6)
In this paper, we focus on a perturbation of the global equilibrium distribution function nM . More precisely, we consider a perturbation carried by a vector-valued function f , which implies that F takes the form [START_REF] Boudin | Diffusion asymptotics of a kinetic model for gaseous mixtures[END_REF] F = nM + (nM ) 1/2 f .

Since nM does not depend on t and x, we shall carefully study the macroscopic part of the perturbation, which contains the time and space variations of f and subsequently of F , as in [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF], bringing at the same time some more details about the estimates to handle the mixture case. Note that another possibility would have been to follow [START_REF] Liu | Energy method for Boltzmann equation[END_REF], in which F is decomposed into the sum of a local Maxwellian, containing the whole macroscopic part of the distribution function, and the microscopic part. This decomposition induces other difficulties, such as the dependence of ker L on x and t.

Main ideas and results

Let us now focus our attention on the micro-macro decomposition of the perturbation f . Straightforwardly, (1) implies that f satisfies (8)

∂ t f + v 1 ∂ x f -Lf = N (f ),
where L and N are respectively the linearized Boltzmann operator and a quadratic operator defined by

Lf = (nM ) -1/2 Q(nM , (nM ) 1/2 f ) + Q((nM ) 1/2 f , nM ) , N (f ) = (nM ) -1/2 Q((nM ) 1/2 f , (nM ) 1/2 f ).
In the remainder of the paper, we shall denote by D the domain of L in L 2 (R 3 ) I . It is easy to see [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF][START_REF] Boudin | Diffusion asymptotics of a kinetic model for gaseous mixtures[END_REF] that the operator L is a non-positive self-adjoint operator, i.e. for any f , g ∈ D,

Lf , g I = f , Lg I , Lf , f I ≤ 0. The collisions invariants allow to characterize the elements of P 0 = ker L, i.e.

(9)

P 0 = ker L = Span (nM ) 1/2 χ k | 1 ≤ k ≤ I + 4 ,
which is a finite-dimensional subspace of D with dim P 0 = I + 4. Let us denote by P 1 its orthogonal complement in D with respect to the •, • I scalar product, i.e. P 1 = (ker L) ⊥ = (P 0 ) ⊥ . If we naively proceed by multiplying (8) by f and integrate with respect to t, x and v, the only term we can hope to upper-bound comes from a spectral gap estimate for L: it is a square norm of the projection of f onto P 1 , which we denote f 1 . That implies that we fail to control the norm of the projection f 0 of f onto P 0 . To control f 0 , we use the so-called micro-macro decomposition of f , as f is uniquely written as f = f 0 + f 1 . We shall get back to it later, but f 0 is called the macroscopic part of f , and f 1 its microscopic part.

The statements below will be accurately justified in the upcomings sections. The projections of f satisfy, as in [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF],

∂ t f 0 + P 0 (v 1 ∂ x f 0 ) + P 0 (v 1 ∂ x f 1 ) = 0, ∂ t f 1 + P 1 (v 1 ∂ x f 0 ) + P 1 (v 1 ∂ x f 1 ) -Lf 1 = N (f ).
As we shall see, L = L |P 1 is invertible, thus we get, from the previous equation on f 1 ,

f 1 = L-1 ∂ t f 1 + P 1 (v 1 ∂ x f 0 ) + P 1 (v 1 ∂ x f 1 ) -N (f ) .
We now plug this expression of f 1 in the previous equation on f 0 , which ensures

∂ t f 0 + P 0 (v 1 ∂ x f 0 ) + P 0 v 1 ∂ x L-1 ∂ t f 1 + P 1 (v 1 ∂ x f 0 ) + P 1 (v 1 ∂ x f 1 ) -N (f ) = 0.
We immediately observe that, when scalarily multiplying by f 0 , there is no way to exhibit an estimate on a norm of f 0 itself. There may only be hope to find an estimate of the norm ∂ x f 0 2 I . Indeed, we recall that P 0 is a finite-dimensional space, and P 1 (v 1 ∂ x f 0 ) can be expressed in terms of the space derivatives of the coordinates of f 0 in the orthonormal basis of P 0 . Then a subtle combination of arguments, some new and specific to the mixture case, some others coming from [START_REF] Liu | Energy method for Boltzmann equation[END_REF][START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF], leads to an estimate involving ∂ x f 0 2 I on the left-hand side. We still do not have any control on f 0 2

I , but such a control is needed, since the nonlinear term involves f , and not only f 1 (this is due to the fact that N (f 0 ) = 0), and derivatives of the quadratic term N (f ) involve the derivatives of f and f itself. We can see that it is possible to do so by introducing the antiderivative W 0 of f 0 with respect to x, which justifies the fact that we are working in a one-dimensional setting. Without loss of generality, as we shall explain in Subsection 4.9, we assume that

(10) R f 0 (0, x, v) dx = 0, v ∈ R 3 , so that W 0 : (t, x, v) → x -∞
f 0 (t, y, v) dy can be treated within a L 2 -framework with respect to both variables x and v. This allows to derive an estimate on ∂ x W 0 = f 0 , roughly as we did for ∂ x f 0 . Eventually, our global estimate requires the control of norms of the time and space derivatives of f 1 , which can be obtained after differentiation of the equation on f 1 with respect to t or x, thanks to the spectral gap of L. This whole process also requires a mandatory assumption to deal with the nonlinear term N (f ), which implies that W 0 and f must remain small in the • I norm, pointwise in time and space.

Let us first make the following smallness assumption, where ε > 0 will be chosen afterwards.

Assumption. The perturbation f must satisfy

(11) sup t≥0 x∈R W 0 I + f 0 I + (1 + |v|) 1/2 f 1 I ≤ ε.
This assumption is stronger than the one used in [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF], where the term f 0 2 I was not involved, but also allows to obtain an estimate on the full macroscopic part, which was not the case previously. More precisely, our first main result is an a priori estimate on norms of f , its partial derivatives, and the antiderivative of f 0 . The generic term denoted by I(0) which appears below must be understood as a linear combination of square norms of initial data of the pointwise in time integrals of the left-hand side of the estimates, with coefficients only depending on the problem data.

Proposition 2. There exists ε 0 > 0 such that, for any ε ∈ (0, ε 0 ], we can find positive constants α 0 , α 1 , Ĉ1 , Ĉ2 , Ĉ3 such that, for any solution f to (8) satisfying the smallness assumption [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF], the following lower-order a priori estimate holds

(12) 1 4 R W 0 2 I dx t=T + 1 4 R f 0 2 I dx t=T + α 0 R f 2 I dx t=T + α 1 2 R ∂ x f 2 I dx t=T + α 1 2 R ∂ t f 2 I dx t=T + Ĉ1 T 0 R f 0 2 I dx dt + Ĉ1 T 0 R ∂ x f 0 2 I dx dt + Ĉ2 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt + 2 Ĉ3 T 0 R (1 + |v|) 1/2 ∂ x f 1 2 I dx dt + 2 Ĉ3 T 0 R (1 + |v|) 1/2 ∂ t f 1 2 I dx dt ≤ I(0).
Unfortunately, estimate ( 12) is not closed. For the latter property to hold, choosing the initial data such that I(0) is of order ε 2 would have to imply [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]. However, (1+|v|) 1/2 f 1 I would only lie in H 1 t,x , which does not continuously inject in L ∞ t,x . We thus need a higher-order estimate involving more derivatives with respect to t and x. In order to obtain it, we differentiate the equations satisfied by f 0 and f as many times as necessary. To this aim, we introduce the following notation. For any p = (p 1 , p 2 ) ∈ N 2 , we set |p| = p 1 + p 2 , and, for any function g of t, x and v,

∂ p g = ∂ p 1 t ∂ p 2
x g. We emphasize that the multi-index notation does not imply any derivative with respect to v.

At this point, we make the following smallness assumption, where ε > 0 will be chosen afterwards.

Assumption. The perturbation f must satisfy

(13) sup t≥0 x∈R W 0 I + max |p|≤2 ∂ p f 0 I + (1 + |v|) 1/2 ∂ p f 1 I ≤ ε.
Under Assumption [START_REF] Daus | Hypocoercivity for a linearized multispecies Boltzmann system[END_REF], we obtain the next theorem. This time, as we explain below, the smallness assumption can be dropped, provided that we assume instead that I(0) is small. Theorem 3. There exists ε 1 > 0 such that, for any ε ∈ (0, ε 1 ], and any solution f to (8) such that the corresponding I(0) is at most of order ε 2 , the following higher-order a priori estimate holds

(14) R W 0 2 I dx t=T + R f 0 2 I dx t=T + 1≤|r|≤4 R ∂ r f 0 2 I dx t=T + R f 2 I dx t=T + 1≤|p|≤5 R ∂ p f 2 I dx t=T + T 0 R f 0 2 I dx dt + T 0 R ∂ x f 0 2 I dx dt + 1≤|r|≤4 T 0 R ∂ x ∂ r f 0 2 I dx dt + T 0 R (1 + |v|) 1/2 f 1 2 I dx dt + 1≤|p|≤5 T 0 R (1 + |v|) 1/2 ∂ p f 1 2 I dx dt ≤ I(0).
Estimate ( 14) is closed because, this time, [START_REF] De La Canal | On existence, uniqueness and Banach space regularity for solutions of Boltzmann equations systems for monatomic gas mixtures[END_REF] implies that Assumption ( 13) is automatically satisfied when I(0) is of order ε 2 . The latter closed estimate ( 14) appears as a very useful tool. For instance, it allows to obtain the stability of the global Maxwellian function (nM ) in large time, provided that the perturbation at initial time is chosen small enough in the H s (L 2 v ) norm, for s ≥ 5.

(1 + |v|) 1/2 f 1 I lies in H 5 t,x → W 2,∞ t,x , and f 0 I lies in H 4 t,x → W 2,∞ t,x . Estimate
Let us now focus on the proofs of Proposition 2 and Theorem 3, starting with the lower-order estimate.

4. Proof of the lower-order estimate 4.1. Estimates on L and N . The linearized operator L can be written as L = Kν, where K is compact [START_REF] Boudin | Diffusion asymptotics of a kinetic model for gaseous mixtures[END_REF] and ν is a multiplicative operator, called the collision frequency, given by

ν i (v) = I j=1 n j m j 2π 3/2 R 3 ×S 2 e - m j 2 |v * | 2 B ij (v, v * , ω) dω dv * , v ∈ R 3 , 1 ≤ i ≤ I.
Note that, thanks to the hard-sphere assumption (4) on the cross sections, as in [START_REF] Liu | Energy method for Boltzmann equation[END_REF], ν satisfies a growth estimate. More precisely, there exist positive constants ν 0 and ν0 , such that

(15) 0 < ν 0 ≤ ν 0 (1 + |v|) ≤ ν i (v) ≤ ν0 (1 + |v|), v ∈ R 3 , 1 ≤ i ≤ I.
Besides, in [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF], a constructive spectral gap estimate on L is proved. In our notation, it means that there exists λ > 0 such that, for any h ∈ (ker L) ⊥ , (16) Lh, h I ≤ -λ ν 1/2 h 2 I . The spectral gap estimate [START_REF] Dimarco | Multi-scale control variate methods for uncertainty quantification in kinetic equations[END_REF] on L and the lower bound ν 0 on ν from [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF], together with the Cauchy-Schwarz inequality, yield

ν 0 g 2 I ≤ ν 1/2 g 2 I ≤ - 1 λ Lg, g I ≤ 1 λ g I Lg I .
Since P 1 = (ker L) ⊥ and L is self-adjoint, it is clear that L(P 1 ) = P 1 , hence L = L |P 1 is an invertible operator on P 1 . Now, for any h ∈ P 1 , writing g = L-1 h and setting C inv = (ν 0 λ) -1 > 0, the previous inequality implies that [START_REF] Dolbeault | Hypocoercivity for linear kinetic equations conserving mass[END_REF] L-1 h I ≤ C inv h I , which ensures the boundedness of L-1 on P 1 . Eventually, we write an estimate on the nonlinear operator N . Its i-th component, 1 ≤ i ≤ I, is given by

N i (f ) = M -1/2 i I j=1 n 1/2 j Q ij (M 1/2 i f i , M 1/2 j f j ).
Thanks to Lemma 8 stated in Appendix A, we immediately have

(1 + |v|) -1/2 N (f ) I ≤ C β (1 + |v|) 1/2 f 2 I , ( 18
) (1 + |v|) -1/2 ∂ N (f ) I ≤ 2 C β (1 + |v|) 1/2 f I (1 + |v|) 1/2 ∂ f I , (19)
where ∂ denotes either time or space partial differentiation. 4.2. Micro-macro decomposition. The micro-macro decomposition method lies on the orthogonal decomposition of D onto P 0 = ker L and P 1 = (ker L) ⊥ = im L. In order to perform it, let P 0 and P 1 respectively denote the orthogonal projections on P 0 and P 1 . It is clear that

LP 0 = P 0 L = 0, LP 1 = P 1 L.
Moreover, thanks to (9), we can write, for any g ∈ D,

P 0 g = I+4 k=1 (nM ) 1/2 χ k , g I (nM ) 1/2 χ k . (20)
We decompose the perturbation f following the direct orthogonal sum P 0 ⊕ P 1 . Then, f 0 = P 0 f and f 1 = P 1 f satisfy

f = P 0 f + P 1 f = f 0 + f 1 ,
f 0 , f 1 I = 0. Functions f 0 and f 1 are respectively referred to as the macroscopic (or fluid) and microscopic (or non-fluid) components of f . The coordinates of f 0 in the orthonormal basis (nM ) 1/2 χ k 1≤k≤I+4 of P 0 , also known as the fluid quantities, are given by

ρ i (t, x) = (nM ) 1/2 χ i , f I , 1 ≤ i ≤ I, q k (t, x) = (nM ) 1/2 χ I+k , f I , k = 1, 2, 3, e(t, x) = (nM ) 1/2 χ I+4 , f I , so that (21) f 0 = I i=1 ρ i (nM ) 1/2 χ i + 3 k=1 q k (nM ) 1/2 χ I+k + e(nM ) 1/2 χ I+4 .
Note that ( 21) is a very convenient form of f 0 , as it is written as a sum of tensorized functions with respect to t and x on the one hand, and v on the other hand.

If we project the equation ( 8) on the perturbation f onto P 0 and P 1 , respectively, we obtain the following equations respectively satisfied by f 0 and f 1 , i.e.

∂ t f 0 + P 0 (v 1 ∂ x f 0 ) + P 0 (v 1 ∂ x f 1 ) = 0, ( 22
)
∂ t f 1 + P 1 (v 1 ∂ x f 0 ) + P 1 (v 1 ∂ x f 1 ) -Lf 1 = N (f ). (23)
The nonlinear term P 0 N (f ) vanished in [START_REF] Guo | The Boltzmann equation in the whole space[END_REF] by combining the representation (20) of elements of P 0 , the definition of N , and the one (6) of the collision invariants. This means that N (f ) lies in P 1 , which we take into account, together with the commutation of P 1 and L to rewrite [START_REF] Hutridurga | Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting[END_REF] as an equality regarding f 1 , that is ( 24)

f 1 = L-1 ∂ t f 1 + P 1 (v 1 ∂ x f 0 ) + P 1 (v 1 ∂ x f 1 ) -N (f ) .
Of course, [START_REF] Jin | A BGK-penalization-based asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF] does not provide a direct expression of f 1 , since its right-hand side still depends on the nonlinear term N (f ) = N (f 0 + f 1 ) and first-order derivatives of f 0 and f 1 . Nevertheless, this equality is crucial for the rest of the paper. Let us also emphasize here that time or space differentiations commute with L, P 0 and P 1 , since these three operators only act on the velocity variable.

Let us now state a result on the fluid quantities, which is important for the further analysis. It comes from the equation ( 22) satisfied by f 0 and provides conservation laws for (ρ i ) 1≤i≤I , (q k ) k∈{1,2,3} and e. Proposition 4. The fluid quantities of f 0 satisfy the following conservation laws

∂ t ρ i + n i n j m j ∂ x q 1 + P 0 (v 1 ∂ x f 1 ), χ i (nM ) 1/2 I = 0, 1 ≤ i ≤ I, (25)
∂ t q 1 + 1 n j m j ∂ x i √ n i ρ i + 6 n j 3 e + P 0 (v 1 ∂ x f 1 ), χ I+1 (nM ) 1/2 I = 0, (26)
∂ t q k + P 0 (v 1 ∂ x f 1 ), χ I+k (nM ) 1/2 I = 0, k = 2, 3, (27)
∂ t e + 1 3 6 n j n j m j ∂ x q 1 + P 0 (v 1 ∂ x f 1 ), χ I+4 (nM ) 1/2 I = 0. ( 28
)
The principle of the proof is very simple (checking the equations), but the computations inside are tedious. The proof is provided in Appendix B for the sake of completeness. Note that the term with f 1 in ( 25) is peculiar to the mixture and does not appear in the monospecies case.

In the following, we shall also need to introduce the fluid quantity appearing in [START_REF] Lee | Large-time behavior of solutions for the Boltzmann equation with hard potentials[END_REF] with its space derivative, that is

(29) = 1 n j m j i √ n i ρ i + 2 n j √ 6 e .
Using ( 25) and ( 28), clearly satisfies

(30) ∂ t + n j 3/2 + 2 3 n j n j m j ∂ x q 1 + 1 n j m j P 0 (v 1 ∂ x f 1 ), i √ n i χ i + 1 3 6 n j χ I+4 (nM ) 1/2 I = 0,
so that q 1 and can easily be linked through [START_REF] Lee | Large-time behavior of solutions for the Boltzmann equation with hard potentials[END_REF] and [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF]. Note that, thanks to [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF], the perturbation f has zero total macroscopic quantities at initial time, i.e.

R ρ i dx t=0 = 0, 1 ≤ i ≤ I, R q k dx t=0 = 0, 1 ≤ k ≤ 3, R e dx t=0 = 0.
Integrating the conservation laws ( 25)-( 28) with respect to x ∈ R obviously ensures that

R ρ i (t, x) dx = R ρ i (0, x) dx, 0 ≤ t ≤ T, 1 ≤ i ≤ I, (31) R q k (t, x) dx = R q k (0, x) dx, 0 ≤ t ≤ T, 1 ≤ k ≤ 3, (32) R e(t, x) dx = R e(0, x) dx, 0 ≤ t ≤ T. ( 33
)
That allows to define, for any t, the antiderivatives (R i ), (Q k ) and E of (ρ i ), (q k ) and e with respect to x, which are also the coordinates of W 0 in the basis (χ k (nM ) 1/2 ) of P 0 . Denote by L the antiderivative of , then we have the straightforward corollary of Proposition 4.

Corollary 5. The fluid quantities in W 0 satisfy

∂ t Q 1 + + P 0 (v 1 f 1 ), χ I+1 (nM ) 1/2 I = 0, ( 34
)
∂ t L + n j 3/2 + 2 3 n j n j m j q 1 (35) + 1 n j m j P 0 (v 1 f 1 ), i √ n i χ i + 1 3 6 n j χ I+4 (nM ) 1/2 I = 0.
We conclude this subsection by the following lemma, which is useful in the proofs of the upcoming a priori estimates. It relies on the fact that P 0 is finite-dimensional. Lemma 6. The norms • I , v 1 • I and (1 + |v|) 1/2 • I are equivalent on P 0 . Hence, there exists a constant C eq > 0, depending on n and m, such that for any g ∈ P 0 , (36) v 1 g I ≤ C eq g I and

(1 + |v|) 1/2 g I ≤ C eq g I .

Moreover, from (20), we can deduce that there exists a constant C χ > 0 only depending on n and m, such that for any f ∈ D, (37)

P 0 (v 1 f ) 2 I = I+4 k=1 v 1 (nM ) 1/2 χ k , f I 2 ≤ C χ f 2 I .

4.3.

Handling the lower-order estimate on f 1 through f . Since we deal with f 0 estimates separately, it is equivalent to treat f or f 1 to obtain an estimate on the microscopic part. We choose to proceed with f . Let us scalarily multiply (8) by f , integrate with respect to t ∈ [0, T ] and x ∈ R to obtain

(38) 1 2 R f 2 I dx T 0 + T 0 R v 1 ∂ x f , f I dx dt - T 0 R Lf , f I dx dt ≤ T 0 R N (f ), f I dx dt.
The second term on the left-hand side of this equation vanishes by conservativity with respect to the x variable, since

T 0 R v 1 ∂ x f , f I dx dt = 1 2 T 0 R ∂ ∂x I i=1 R 3 v 1 f 2 i dv dx dt.
The term with L can be lower-bounded thanks to the L spectral gap estimate [START_REF] Dimarco | Multi-scale control variate methods for uncertainty quantification in kinetic equations[END_REF] and to [START_REF] Desvillettes | A kinetic model allowing to obtain the energy law of polytropic gases in the presence of chemical reactions[END_REF] in the following way

λν 0 (1 + |v|) 1/2 f 1 2 I ≤ λ ν 1/2 f 1 2 I ≤ -Lf 1 , f 1 I = -Lf , f I .
Eventually, we deal with the term involving N . We first notice that

N (f ), f I = N (f ), f 1 I = (1 + |v|) -1/2 N (f ), (1 + |v|) 1/2 f 1 I ,
since P 0 N (f ) = 0. Then, using [START_REF] Gamba | Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations[END_REF], we have

(39) | N (f ), f I | ≤ C β (1 + |v|) 1/2 f 2 I (1 + |v|) 1/2 f 1 I ≤ 2C β (1 + |v|) 1/2 f 0 2 I + (1 + |v|) 1/2 f 1 2 I (1 + |v|) 1/2 f 1 I .
Using the equivalence of norms (36) on P 0 , we can find a constant K eq > 0, depending on m, n, the cross sections and L through its null space, such that (38) becomes

(40) 1 2 R f 2 I dx T 0 + λν 0 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt ≤ K eq T 0 R f 0 2 I + (1 + |v|) 1/2 f 1 2 I (1 + |v|) 1/2 f 1 I dx dt.
Of course, we can see that, if we choose ε small enough in assumption ( 11), there will only remain a nonnegative contribution of a norm of f 0 on the right-hand side of (40). This is why we have to focus now on an estimate on f 0 .

4.4.

Lower-order estimate on f 0 : first steps. In order to deal with the estimate on f 0 , we plug the expression (24) of f 1 into the equation ( 22) satisfied by f 0 , then scalarily multiply the new equation by f 0 , and integrate with respect to t and x, to obtain

(41) 1 2 R f 0 2 I dx T 0 + L 1 + L 2 + L 3 + L 4 = 0,
where the term with P 0 (v 1 ∂ x f 0 ), f 0 I vanishes, again by conservativity in x, and where we set

L 1 = T 0 R v 1 ∂ x L-1 ∂ t f 1 , f 0 I dx dt = - T 0 R L-1 ∂ t f 1 , P 1 (v 1 ∂ x f 0 ) I dx dt, L 2 = T 0 R v 1 ∂ x L-1 P 1 (v 1 ∂ x f 0 ), f 0 I dx dt = - T 0 R L-1 P 1 (v 1 ∂ x f 0 ), P 1 (v 1 ∂ x f 0 ) I dx dt, L 3 = T 0 R v 1 ∂ x L-1 P 1 (v 1 ∂ x f 1 ), f 0 I dx dt = - T 0 R ∂ x f 1 , v 1 L-1 P 1 (v 1 ∂ x f 0 ) I dx dt, L 4 = - T 0 R v 1 ∂ x L-1 N (f ), f 0 I dx dt = T 0 R N (f ), L-1 P 1 (v 1 ∂ x f 0 ) I dx dt.
Let us show how to handle these terms by performing the following preliminary computations on

L 1 , L 2 , L 3 , L 4 . 4.4.1. Term L 1 .
Combining the equivalence of norms (36) on P 0 with the boundedness of L-1 , and setting K 1 = C inv C eq > 0, we obtain

(42) |L 1 | ≤ K 1 T 0 R ∂ t f 1 I ∂ x f 0 I dx dt. 4.4.2. Term L 2 . It is clear that V = P 1 (v 1 P 0 ) is a finite-dimensional subspace of P 1 . Consequently, L-1
|V is a bounded invertible operator, as well as its inverse. Hence, there exists a constant C V > 0, only depending on L, such that, for any g ∈ P 0 ,

P 1 (v 1 g) I ≤ C V L-1 P 1 (v 1 g) I .
Thus, thanks to the previous inequality and to the spectral gap estimate ( 16) on L, we get

L 2 = - T 0 R L-1 P 1 (v 1 ∂ x f 0 ), L L-1 P 1 (v 1 ∂ x f 0 ) I dx dt ≥ λ T 0 R ν 1/2 L-1 P 1 (v 1 ∂ x f 0 ) 2 I dx dt ≥ λ 2 T 0 R P 1 (v 1 ∂ x f 0 ) 2 I dx dt, ( 43
)
where we chose λ 2 = λν 0 /C V > 0. 4.4.3. Term L 3 . We apply the Cauchy-Schwarz inequality to yield

|L 3 | ≤ T 0 R ∂ x f 1 I v 1 L-1 P 1 (v 1 ∂ x f 0 ) I dx dt.
Then, introducing C K > 0 as the boundedness constant of the compact operator K = L + ν, we notice that

v 1 L-1 P 1 (v 1 ∂ x f 0 ) I ≤ 1 ν 0 ν L-1 P 1 (v 1 ∂ x f 0 ) I ≤ 1 ν 0 K L-1 P 1 (v 1 ∂ x f 0 ) I + L L-1 P 1 (v 1 ∂ x f 0 ) I ≤ C eq ν 0 (C K C inv + 1) ∂ x f 0 I ,
where we also used the boundedness of L-1 and P 1 (as a projector), and the norm equivalence argument on P 0 , involving the constant C eq . Setting K 3 = C eq (C K C inv + 1)/ν 0 > 0, we obtain (44)

|L 3 | ≤ K 3 T 0 R ∂ x f 0 I ∂ x f 1 I dx dt, 4.4.4. Term L 4 .
We first have to estimate

N (f ), L-1 P 1 (v 1 ∂ x f 0 ) I = (1 + |v|) -1/2 N (f ), (1 + |v|) 1/2 L-1 P 1 (v 1 ∂ x f 0 ) I ≤ (1 + |v|) -1/2 N (f ) I (1 + |v|) 1/2 L-1 P 1 (v 1 ∂ x f 0 ) I .
We treat the norm with N using ( 18) and the same kind of argument as in (39), including norm equivalence on P 0 . It ensures that

(1 + |v|) -1/2 N (f ) I ≤ C eq f 0 2 I + (1 + |v|) 1/2 f 1 2 I .
Besides, the norm with L-1 is treated in the same way as for L 3 , i.e.

(1 + |v|) 1/2 L-1 P 1 (v 1 ∂ x f 0 ) I ≤ (1 + |v|) L-1 P 1 (v 1 ∂ x f 0 ) I ≤ K 3 ∂ x f 0 I .
Therefore, setting K 4 = C eq K 3 > 0, we can write

(45) |L 4 | ≤ K 4 T 0 R f 0 2 I + (1 + |v|) 1/2 f 1 2 I ∂ x f 0 I dx dt.
4.4.5. Before we proceed. Let us explain what the current situation on f 0 is. Applying Young's inequality in (42) and (44) with a parameter δ 0 > 0 to be chosen later, and plugging (42)-( 45) in (41), we get

(46) 1 2 R f 0 2 I dx T 0 + λ 2 T 0 R P 1 (v 1 ∂ x f 0 ) 2 I dx dt ≤ K 1 T 0 R 1 2δ 0 ∂ t f 1 2 I + δ 0 2 ∂ x f 0 2 I dx dt + K 3 T 0 R δ 0 2 ∂ x f 0 2 I + 1 2δ 0 ∂ x f 1 2 I dx dt + K 4 T 0 R ∂ x f 0 I f 0 2 I + (1 + |v|) 1/2 f 1 2 I dx dt.
We intend afterwards to combine (46) with the estimate (40) for f 1 . All the terms are product of exactly two norms, apart from the ones coming from the nonlinear operator. This is why, in order to proceed, we shall need the same kind of smallness assumption as in [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF], namely [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF].

The term (1 + |v|) 1/2 f 1 2 I appears in the left-hand side of (40), and in the right-hand side of both estimates in the terms coming from N , with a multiplication by the arbitrary small parameter ε when using the smallness assumption. So they can be put on the left-hand side to obtain a still positive coefficient for (1 + |v|) 1/2 f 1 2 I . On the contrary, the time and space derivatives of f 1 only appear in the right-hand sides of our estimates, and so does f 0 2 I . We have no way to control them for the time being. Last, the space derivative of f 0 appears on the right-hand side of (46) as its I-norm, and on the left-hand side in P 1 (v 1 ∂ x f 0 ) I , which will provide a helpful contribution.

Consequently, our next two steps are natural: deal with the norms of the time and space derivatives of f 1 , and P 1 (v 1 ∂ x f 0 ) I .

4.5.

Handling the derivatives of f 1 . We proceed in the same way as in Subsection 4.3. We denote by ∂ any time or space partial differentiation. We differentiate (8) accordingly, scalarily multiply it by ∂ f and integrate with respect to t and x. Then, using the spectral gap (16) of L, the growth property (15) of ν and the estimate [START_REF] Gamba | On existence and uniqueness to homogeneous Boltzmann flows of monatomic gas mixtures[END_REF] on ∂ N (f ), we obtain, similarly to (40), the existence of C > 0, depending on m, n and β and L through its null space, such that

(47) 1 2 R ∂ f 2 I dx T 0 + λν 0 T 0 R (1 + |v|) 1/2 ∂ f 1 2 I dx dt ≤ C T 0 R (1 + |v|) 1/2 f I (1 + |v|) 1/2 ∂ f 1 I ∂ f 0 I + (1 + |v|) 1/2 ∂ f 1 I dx dt.
Let us deal with the time derivative, since no other term involving ∂ t f 0 will appear in our estimates. In fact, using ( 22) and (37), we get, using again the Cauchy-Schwarz inequality,

∂ t f 0 2 I = P 0 (v 1 ∂ x f ) 2 I ≤ C χ ∂ x f 2 I .
Consequently, (47) becomes, for the time derivative,

(48) 1 2 R ∂ t f 2 I dx T 0 + λν 0 T 0 R (1 + |v|) 1/2 ∂ t f 1 2 I dx dt ≤ C T 0 R (1 + |v|) 1/2 f I (1 + |v|) 1/2 ∂ t f 1 I ∂ x f 0 I + (1 + |v|) 1/2 ∂ t f 1 I dx dt,
where C = C max(C χ , 1). We can thus observe that (47) and (48) have exactly the same structure and can be rewritten, for some constant K 5 > 0 only depending on n, m and β, as

(49) 1 2 R ∂ f 2 I dx T 0 + C 0 T 0 R (1 + |v|) 1/2 ∂ f 1 2 I dx dt ≤ K 5 T 0 R (1 + |v|) 1/2 f I (1 + |v|) 1/2 ∂ f 1 I ∂ x f 0 I + (1 + |v|) 1/2 ∂ f 1 I dx dt,
where we set C 0 = λν 0 > 0.

4.6. Lower bound for P 1 (v 1 ∂ x f 0 ) I . Let us now focus on the term with P 1 (v 1 ∂ x f 0 ) I . The following lemma allows to estimate the L 2 -norm of P 1 (v 1 ∂ x f 0 ) in terms of ∂ x f 0 , up to a contribution in ∂ x f 1 which can be as small as desired. We must emphasize that this result is the main improvement in the mixture case, compared to its monospecies counterparts from [28, estimate (2.22) p.185] and [30, Lemma 3.1 p.139]. Indeed, in the monospecies case, the lower bound of the term P 1 (v 1 ∂ x f 0 ) I did not give any ∂ x ρ contribution, only ∂ x q and ∂ x e parts of ∂ x f 0 I . Lemma 7. There exists θ 0 ∈ (0, 1] such that, for any f ∈ D and any 0 < θ < θ 0 , (50)

T 0 R P 1 (v 1 ∂ x f 0 ) 2 I dx dt ≥ γ θ T 0 R ∂ x f 0 2 I dx dt -θC 2 T 0 R ∂ x f 1 2 I dx dt + 2θ R q ∂ x dx T 0
where γ θ > 0 only depends on n, m and θ, C 2 > 0 on n and m.

Proof. By orthogonality, we first have

P 1 (v 1 ∂ x f 0 ) 2 I = v 1 ∂ x f 0 2 I -P 0 (v 1 ∂ x f 0 ) 2 I .
The first term of the right-hand side writes

v 1 ∂ x f 0 2 I = i 1 m i (∂ x ρ i ) 2 + 4 6 n j i √ n i m i (∂ x ρ i )(∂ x e) + 7 3
1 n j i n i m i (∂ x e) 2 + n j n j m j 3 ∂ x q 1 2 + ∂ x q 2 2 + ∂ x q 3 2 .
It can become a sum of square quantities, as

(51) v 1 ∂ x f 0 2 I = i 1 m i ∂ x ρ i + 2 √ n i 6 n j ∂ x e 2 + 5 3
1 n j i n i m i (∂ x e) 2 + n j n j m j 3 ∂ x q 1 2 + ∂ x q 2 2 + ∂ x q 3 2 .
The term with P 0 decomposes on our basis of P 0 into

P 0 (v 1 ∂ x f 0 ) = ∂ x q 1 i √ n i n j m j χ i (nM ) 1/2 + ∂ x q 1 2 n j 6 n j m j χ I+4 (nM ) 1/2 + 1 n j m j i √ n i ∂ x ρ i + 2 n j √ 6 ∂ x e χ I+1 (nM ) 1/2 .
It is thus easy to check that (52)

P 0 (v 1 ∂ x f 0 ) 2 I = (∂ x ) 2 + 5 3 n j n j m j (∂ x q 1 ) 2 .
Therefore, we immediately get

(53) P 1 (v 1 ∂ x f 0 ) 2 I = i 1 m i ∂ x ρ i + 2 √ n i 6 n j ∂ x e 2 + 5 3 1 n j n j m j (∂ x e) 2 + n j n j m j 4 3 ∂ x q 1 2 + ∂ x q 2 2 + ∂ x q 3 2 -(∂ x ) 2 .
Thanks to the Cauchy-Schwarz inequality, we have

i √ n i ∂ x ρ i + 2 n j √ 6 ∂ x e 2 = i √ n i ∂ x ρ i + 2n i 6 n j ∂ x e 2 ≤   j n j m j   i 1 m i ∂ x ρ i + 2 √ n i 6 n j ∂ x e 2 ,
which implies that

(∂ x ) 2 ≤ i 1 m i ∂ x ρ i + 2 √ n i 6 n j ∂ x e 2 .
Let θ ∈ (0, 1). Then we split (∂ x ) 2 in (53) into the sums of itself respectively multiplied by θ and (1 -θ). Applying the previous Cauchy-Schwarz inequality, it follows from (53) that

(54) P 1 (v 1 ∂ x f 0 ) 2 I ≥ θ i 1 m i ∂ x ρ i + 2 √ n i 6 n j ∂ x e 2 + 5 3
1 n j n j m j (∂ x e) 2 + n j n j m j 4 3 ∂ x q 1 2 + ∂ x q 2 2 + ∂ x q 3 2 -θ(∂ x ) 2 .
The term -θ|∂ x | 2 appearing in (54) is absorbed thanks to the conservation laws. More precisely, multiplying [START_REF] Lee | Large-time behavior of solutions for the Boltzmann equation with hard potentials[END_REF] by ∂ x and integrating with respect to t and x gives

T 0 R |∂ x | 2 dx dt = - T 0 R ∂ t q 1 ∂ x dx dt - T 0 R ∂ x f 1 , v 1 χ I+1 (nM ) 1/2 I ∂ x .
Then, using integrations by parts, one with respect to t, one for x, for the first term on the right-hand side, and Cauchy-Schwarz and Young's inequalities, we obtain with (37) that

T 0 R |∂ x | 2 dx dt ≤ - T 0 R ∂ t ∂ x q 1 dx dt - R q 1 ∂ x dx T 0 + 1 2 T 0 R |∂ x | 2 dx dt + C χ 2 T 0 R ∂ x f 1 2 I dx dt, ensuring that (55) 1 2 T 0 R |∂ x | 2 dx dt + R q 1 ∂ x dx T 0 ≤ T 0 R ∂ t ∂ x q 1 dx dt + C χ 2 T 0 R ∂ x f 1 2 I dx dt.
Further, we multiply (30) by ∂ x q 1 , integrate with respect to x and t and use again Cauchy-Schwarz and Young's inequalities as well as (37), to get that there exists a some constant K χ > 0 depending only on n and m such that

T 0 R ∂ t ∂ x q 1 dx dt = 1 n j m j n 3/2 j + 2 3 n j T 0 R |∂ x q 1 | 2 dx dt + 1 2 T 0 R |∂ x q 1 | 2 dx dt + K χ 2 T 0 R ∂ x f 1 2 I dx dt.
This estimate, combined with (55), gives, after multiplication by 2θ,

(56) θ T 0 R |∂ x | 2 dx dt + 2θ R q 1 ∂ x dx T 0 ≤ 2θ n 3/2 j + 2 3 n j n j m j + 1 2 T 0 R |∂ x q 1 | 2 dx dt + θ(C χ + K χ ) T 0 R ∂ x f 1 2 I dx dt. Setting (57) θ 0 = min 1, n j 3 n j m j + 6 n 3/2 j + 4 n j ∈ (0, 1],
we can come back to (54) with a fixed 0 < θ < θ 0 , so that the term in ∂ x q 1 2 in (56) can be absorbed by the following term, taken in (54),

n j n j m j T 0 R 1 3 (∂ x q 1 ) 2 dx dt.
Let us now introduce G :

P 0 → R + , f 0 → θ i 1 m i ρ i + 2 √ n i 6 n j e 2 + 5 3
1 n j n j m j e 2 + n j n j m j q 1 2 + q 2 2 + q 3 2 ,
which defines a positive definite quadratic form on the finite-dimensional subspace P 0 , thus equivalent to • 2 I on P 0 . Therefore, there exists a constant γ θ > 0, only depending on n, m and θ, such that, for any f 0 ∈ P 0 , G(f 0 ) ≥ γ θ f 0 2 I . All the previous considerations lead to the following estimate (58)

T 0 R P 1 (v 1 ∂ x f 0 ) 2 I dx dt ≥ γ θ T 0 R ∂ x f 0 2 I dx dt -θ(C χ + K χ ) T 0 R ∂ x f 1 2 I dx dt + 2θ R q 1 ∂ x T 0 .
We obtain the required estimate (50) by setting C 2 = (C χ + K χ ).

Remark 1. Let us emphasize that the previous lemma holds in higher dimensions. We explain in Appendix C how to handle the proof of Lemma 7 in a three-dimensional setting.

4.7.

Estimate using an antiderivative of f 0 . As we already pointed out at the end of Subsection 4.4, we still fail to control f 0 I . Since we work in a one-dimensional space setting, following the strategy of [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF], we write an estimate on the antiderivative W 0 of f 0 .

We first integrate [START_REF] Guo | The Boltzmann equation in the whole space[END_REF] with respect to the space variable between -∞ and x ∈ R, so that (59)

∂ t W 0 + P 0 v 1 ∂ x W 0 + P 0 v 1 f 1 = 0.
We then scalarily multiply (59) by W 0 and integrate with respect to t and x. Since the second term has a conservative form in x and using the expression (24) of f 1 , we get

(60) 1 2 R W 0 2 I dx T 0 + L1 + L2 + L3 + L4 = 0,
where we set

L1 = T 0 R P 0 (v 1 L-1 ∂ t f 1 ), W 0 I dx dt L2 = - T 0 R L-1 P 1 (v 1 f 0 ), P 1 (v 1 f 0 ) I dx dt L3 = - T 0 R P 1 (v 1 f 1 ), L-1 P 1 (v 1 f 0 ) I dx dt L4 = T 0 R N (f ), L-1 P 1 (v 1 W 0 ) I dx dt.
The term L2 in (60) is treated in the same way as the corresponding term L 2 in Subsections 4.4.2 and 4.6. Indeed, we can prove a result of the same kind as Lemma 7 involving f 0 instead of ∂ x f 0 , by using, among others properties, ( 34)- [START_REF] Yang | A new energy method for the Boltzmann equation[END_REF]. Hence, we can write, for any θ ∈ (0, θ 0 ),

L2 ≥ λ 2 γ θ T 0 R f 0 2 I dx dt -λ 2 θ C 2 T 0 R f 1 2 I dx dt + 2λ 2 θ R Q 1 dx T 0 .
The term L3 has the same structure as L 3 in 4.4.3. Consequently, involving the same constant K 3 as in (44), we have

| L3 | ≤ K 3 T 0 R f 0 I f 1 I dx dt.
The term L4 with the nonlinear operator is also treated as L 4 in 4.4.4 to yield

| L4 | ≤ K 4 T 0 R W 0 I f 0 2 I + (1 + |v|) 1/2 f 1 2 I dx dt.
The main difference with Subsection 4.4 comes from the treatment of L1 . Integrating by parts with respect to t, and then replacing ∂ t W 0 by its expression in (59), yields

| L1 | ≤ T 0 R f 1 I L-1 P 1 v 1 P 0 (v 1 f ) I dx dt + R L-1 f 1 , v 1 W 0 I dx T 0 .
Thanks to the boundedness of L-1 and the norm equivalence argument on P 0 (36), and computing directly P 0 (v 1 f ) as in (37), we obtain

L-1 P 1 v 1 P 0 (v 1 f ) I ≤ C inv C eq C χ f I . Setting K1 = C inv C eq C χ > 0, we get | L1 | ≤ K1 T 0 R f 1 I f I dx dt + R L-1 f 1 , v 1 W 0 I dx T 0 .
Let us sum up the situation on W 0 . Taking into account the estimates on L1 , L2 , L3 , L4 in (60), and applying Young's inequality in the estimates on L1 and L3 with a parameter δ 1 > 0 to be chosen later, we get

(61) 1 2 R W 0 2 I dx T 0 + λ 2 γ θ T 0 R f 0 2 I dx dt -θ λ 2 C 2 T 0 R f 1 2 I dx dt + 2λ 2 θ R Q 1 dx T 0 ≤ K1 T 0 R ( 1 2δ 1 f 1 2 I + δ 1 2 f 2 I ) dx dt + K 3 T 0 R ( δ 1 2 f 0 2 I + 1 2δ 1 f 1 2 I ) dx dt + K 4 T 0 R W 0 I f 0 2 I + (1 + |v|) 1/2 f 1 2 I dx dt + R L-1 f 1 , v 1 W 0 I dx T 0 .
4.8. Proof of the global lower-order estimate. We now carefully mix all the estimates we got so far, starting with the use of the smallness assumption [START_REF] Cercignani | The Boltzmann equation and its applications[END_REF]. We also have to treat pointwise in time integrals. Recall that all terms at initial time are put in a generic term denoted by I(0). Terms at time T must be handled more shrewdly. First, imposing ε ≤ C 0 /(2K eq ), (40) becomes

(62) 1 2 R f 2 I dx t=T + C 0 2 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt ≤ εK eq T 0 R
f 0 2 I dx dt + I(0).

In the same way, if ε ≤ C 0 /(3K 5 ), (49) can be rewritten as

(63) 1 2 R ∂ f 2 I dx t=T + C 0 2 T 0 R (1 + |v|) 1/2 ∂ f 1 2 I dx dt ≤ εK 5 2 T 0 R ∂ x f 0 2 I dx dt + I(0).
Let us now deal with the estimates on elements of P 0 . In (61), we can find three terms at time T , which are

1 2 R W 0 2 I dx + 2λ 2 θ R Q 1 dx - R L-1 f 1 , v 1 W 0 I dx.
We then notice that, if we set

δ 2 = (C inv C eq ) 2 , R L-1 f 1 , v 1 W 0 I dx ≤ C inv C eq R f 1 I W 0 I dx ≤ 1 4 R W 0 2 I dx + δ 2 R f 1 2 I dx.
Moreover, we can write

-2 R Q 1 dx ≤ R (Q 1 ) 2 dx + R 2 dx ≤ R W 0 2 I dx + R 2 dx.
Hence, for θ ≤ (8λ 2 ) -1 , we get

-2θλ 2 R Q 1 dx ≤ 1 8 R W 0 2 I dx + θλ 2 R 2 dx.
All in all, the terms at time T in (61) satisfy, for any 0

< θ ≤ (8λ 2 ) -1 , 1 2 R W 0 2 I dx + 2λ 2 θ R Q 1 dx - R L-1 f 1 , v 1 W 0 I dx ≥ 1 8 R W 0 2 I dx -δ 2 R f 1 2 I dx -θλ 2 R | | 2 dx. Let us set Ĉ1 = λ 2 γ θ , δ 1 = Ĉ1 ( K1 + K 3 ) -1 /2, Ĉ2 = θ 0 λ 2 C 2 + K1 + K1 +K 3 2δ 1 + K 4 . Using that f 2 I = f 0 2 I + f 1 2
I and the previous inequality in (61), we obtain

(64) 1 8 R W 0 2 I dx t=T + Ĉ1 2 T 0 R f 0 2 I dx dt ≤ Ĉ2 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt + θλ 2 R | | 2 dx t=T + δ 2 R f 1 2 I dx t=T + I(0),
where we imposed ε ≤ min( Ĉ1 /(4K 4 ), 1). Eventually, we similarly tackle the estimate (46). We apply (50), and choose δ 0 = Ĉ1 (K

1 + K 3 ) -1 /2, Ĉ3 = max(θ 0 C 2 + K 3 /(2δ 0 ), K 1 /(2δ 0 )), to obtain (65) 1 4 R f 0 2 I dx t=T + Ĉ1 2 T 0 R ∂ x f 0 2 I dx dt ≤ Ĉ3 T 0 R (1 + |v|) 1/2 ∂ x f 1 2 I dx dt + Ĉ3 T 0 R (1 + |v|) 1/2 ∂ t f 1 2 I dx dt + εK 4 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt + θλ 2 R |∂ x | 2 dx t=T + I(0),
with ε ≤ Ĉ1 /4. Now, let us set

α 0 = max 2δ 2 , 4 Ĉ2 C 0 > 0, α 1 = 4 Ĉ3 C 0 > 0.
The remaining pointwise in time terms are handled by choosing θ small enough, say θ ≤ θ 1 , where θ 1 > 0 only depends on the problem data, so that

θλ 2 R | | 2 dx t=T ≤ 1 8 R f 0 2 I dx t=T , θλ 2 R |∂ x | 2 dx t=T ≤ α 1 4 R ∂ x f 2 I dx t=T .
We then multiply (62) by α 0 and both estimates (63) (for derivatives in t and x) by α 1 , to jointly add them to (64)-( 65). Assuming moreover that

ε ≤ min Ĉ2 2K 4 , Ĉ1 2α 1 K 5 , Ĉ1 4α 0 K eq ,
we can finally write

1 8 R W 0 2 I dx t=T + 1 8 R f 0 2 I dx t=T + α 0 4 R f 2 I dx t=T + α 1 4 R ∂ x f 2 I dx t=T + α 1 2 R ∂ t f 2 I dx t=T + Ĉ1 4 T 0 R f 0 2 I dx dt + Ĉ1 4 T 0 R ∂ x f 0 2 I dx dt + Ĉ2 2 T 0 R (1 + |v|) 1/2 f 1 2 I dx dt + Ĉ3 T 0 R (1 + |v|) 1/2 ∂ x f 1 2 I dx dt + Ĉ3 T 0 R (1 + |v|) 1/2 ∂ t f 1 2 I dx dt ≤ I(0),
which yields [START_REF] Crestetto | Kinetic/fluid micro-macro numerical scheme for a two component gas mixture[END_REF]. To summarize on ε, we set

ε 0 = min Ĉ2 2K 4 , Ĉ1 2α 1 K 5 , Ĉ1 4α 0 K eq , Ĉ1 4 , Ĉ1 4K 4 , C 0 3K 5 , C 0 2K eq , 1 .
The previous inequality holds as soon as ε ≤ ε 0 . This concludes the proof of Proposition 2.

4.9. Comments on Assumption (10) on f 0 . Let us explain how to handle the computations when Assumption [START_REF] Briant | The Boltzmann equation for a multi-species mixture close to global equilibrium[END_REF] does not hold. In this case, we know from ( 31)-( 33) that R ρ i dx, R q k dx and R e dx remain constants with respect to t, one of them, at least, then being nonzero. Consequently, one of the antiderivatives of the macroscopic perturbation would not lie in L 2 (R). To deal with that issue, we consider a nonnegative C ∞ -compactly supported function ψ in the variable x satisfying R ψ(y) dy = 1, and set

Ψ(x) = x -∞ ψ(y) dy, F 0 in (v) = R f 0 (0, y, v) dy, W 0 (t, x, v) = W 0 (t, x, v) -Ψ(x)F 0 in (v).
As a function of v, W 0 clearly belongs to P 0 . And now W 0 must replace W 0 in any computations involving its L 2 norm. We just have to track the changes implied by this replacement in Subsection 4.7. Since ∂ t W 0 = ∂ t W 0 , (60) becomes, using integration by parts, (66)

1 2 R W 0 2 I dx T 0 + L1 + L2 + L3 + L4 + T 0 R ψ L-1 P 1 (v 1 f ), P 1 (v 1 F 0 in ) I dx dt - T 0 R Ψ P 0 (v 1 F 0 in ), f 0 I dx dt + T 0 R ψΨ P 0 (v 1 F 0 in ), F 0 in I dx dt = 0,
where L1 and L4 have the same expressions as L1 and L4 with W 0 instead of W 0 , and L2 and L3 remains unchanged, and all are treated in the same way as in Subsection 4.7. The change then only lies in the terms involving F 0 in . The last one does not depend on f and its absolute value equals

T 2 P 0 (v 1 F 0 in ), F 0 in I ≤ C F 0 in 2 I ,
which can be considered as a contribution to I(0). Since ψ ∈ L 2 (R), the remaining term involving it is easily upper-bounded using Young's inequality by

κ T 0 R f 2 I dx dt + C F 0 in 2 I ,
where κ can be chosen as small as needed. The last remaining term (the one with Ψ) is handled by writing the equation satisfied by R Ψf 0 dx. It is obtained by multiplying ( 22) by Ψ and integrating with respect to x ∈ R and s ∈ [0, t]:

R Ψf 0 (t, x, v) dx = R Ψf 0 (0, x, v) dx + t 0 R ψP 0 (v 1 f ) dx ds.
It implies that, for some constant K > 0 depending on ψ and T ,

R Ψf 0 (t, x, •) dx 2 I ≤ K R Ψf 0 (0, x, •) dx 2 I + T 0 R P 0 (v 1 f ) 2 I dx dt .
The first term on the right-hand side of the previous estimate is again a contribution to I(0). Consequently, using Young's inequality, we have

T 0 R Ψ P 0 (v 1 F 0 in ), f 0 I dx dt = T 0 P 0 (v 1 F 0 in ), R Ψf 0 dx I dt ≤ I(0) + κ T 0 R f 2 I dx dt,
where κ can again be chosen as small as needed. Both contributions upper-bounded by κ-multiplied terms in (66) can then be absorbed by similar counterparts on the left-hand side of (12).

Elements of proof for the higher-order estimate

Unlike what we did in the previous section about the lower-order estimate, for the sake of simplicity, we shall now use the corresponding smallness assumption [START_REF] Daus | Hypocoercivity for a linearized multispecies Boltzmann system[END_REF] as soon as possible in our computations for the higher-order estimate.

Estimate involving ∂ p f 1 , 1 ≤ |p| ≤ 5. We take the ∂ p derivative of the linearized Boltzmann equation ( 8), scalarily multiply it by ∂ p f and integrate with respect to t and x, to obtain

(67) 1 2 R ∂ p f 2 I dx T 0 - T 0 R L∂ p f , ∂ p f I dx dt = T 0 R ∂ p N (f ), ∂ p f I dx dt.
In the second integral, we use the spectral gap property (16) of L, so that

- T 0 R L∂ p f , ∂ p f I dx dt ≥ C 0 T 0 R (1 + |v|) 1/2 ∂ p f 1 2 I dx dt.
Let us focus on the term with the nonlinear operator N . We first notice that

∂ p N (f ) = (nM ) -1/2 0≤|p |≤|p| |p| |p | Q (nM ) 1/2 ∂ p f , (nM ) 1/2 ∂ p-p f .
Thanks to Lemma 8, we get [START_REF] Daus | Hypocoercivity for a linearized multispecies Boltzmann system[END_REF]. Hence, in any case (when |p| ≤ 2, it is straightforward), we can write, for some constant C p > 0 only depending on m, n, β and p,

(68) (1 + |v|) -1/2 ∂ p N (f ) I ≤ C β 0≤|p |≤|p| |p| |p | (1 + |v|) 1/2 ∂ p f I (1 + |v|)
(1 + |v|) -1/2 ∂ p N (f ) I ≤ C p ε |p |≤|p| (1 + |v|) 1/2 ∂ p f I .
Consequently, we have, for any p, 1 ≤ |p| ≤ 5,

T 0 R | ∂ p N (f ), ∂ p f I | dx dt ≤ C p ε T 0 R |p |≤|p| (1 + |v|) 1/2 ∂ p f I (1 + |v|) 1/2 ∂ p f I dx dt.
All in all, summing (67) for all indices p, 1 ≤ |p| ≤ 5, we get, for ε small enough,

(69) 1 2 1≤|p|≤5 R ∂ p f 2 I dx t=T + C 0 2 1≤|p|≤5 T 0 R (1 + |v|) 1/2 ∂ p f 1 2 I dx dt ≤ C ε 1≤|p|≤5 T 0 R ∂ p f 0 2 I dx dt + I(0),
where C > 0 only depends on the data of the problem.

Estimate involving ∂ r f 0 , 1 ≤ |r| ≤ 4. We take the ∂ r derivative of [START_REF] Guo | The Boltzmann equation in the whole space[END_REF] and observe that ∂ r f 0 satisfies exactly the same kind of equation as f 0 itself. Consequently, (46) also holds for ∂ r f 0 instead of f 0 , with some changes only on the nonlinear term Lr 4 , (70)

1 2 R ∂ r f 0 2 I dx T 0 + λ 2 T 0 R P 1 (v 1 ∂ x ∂ r f 0 ) 2 I dx dt ≤ K 1 T 0 R ( 1 2δ 0 ∂ t ∂ r f 1 2 I + δ 0 2 ∂ x ∂ r f 0 2 I )dx dt + K 3 T 0 R ( δ 0 2 ∂ x ∂ r f 0 2 I + 1 2δ 0 ∂ x ∂ r f 1 2 I )dx dt + | Lr 4 |,
where

Lr 4 = - T 0 R v 1 ∂ x L-1 ∂ r N (f ), ∂ r f 0 I dx dt = T 0 R ∂ r N (f ), L-1 P 1 (v 1 ∂ x ∂ r f 0 ) I dx dt.
Using the smallness assumption ( 13) and (68), we get

(71) | Lr 4 | ≤ C r K 3 ε   |p |≤|r| (1 + |v|) 1/2 ∂ p f I   ∂ x ∂ r f 0 I .
The second term in (70) is handled thanks to Lemma 7, leading to (72)

T 0 R P 1 (v 1 ∂ x ∂ r f 0 ) 2 I dx dt ≥ γ θ T 0 R ∂ x ∂ r f 0 2 I dx dt -θC 2 T 0 R ∂ x ∂ r f 1 2 I dx dt + 2θ R ∂ r q ∂ x ∂ r dx T 0 .
Proof of Theorem 3. Following the same reasoning as in the previous section, we combine the lower-order estimate [START_REF] Crestetto | Kinetic/fluid micro-macro numerical scheme for a two component gas mixture[END_REF], the estimate (69) on the derivatives of f 1 , and the estimate on the derivatives of ∂ x f 0 obtained from (70)-(72). In the right-hand side of estimate (69), the treatment of the pure time derivatives ∂ p t f 0 is done as in Subsection 4.5. They are thus controlled by terms which involve at least one space derivative ∂ x ∂ p-1 t f , still multiplied by ε. In the end, choosing θ and ε small enough, the estimate ( 14) is proved.

Appendix A. Estimate on Q In order to deal with the terms involving N (f ), we need the following result. It requires the hard-sphere assumption (4) and its proof is provided here, despite its similarity to the one in [20, Lemma A.1] (see also [START_REF] Liu | Boltzmann equation: micro-macro decompositions and positivity of shock profiles[END_REF]Lemma B.1]), for the sake of completeness.

Lemma 8. Assuming that all the cross sections satisfy the hard-sphere assumption (4), there exists C β > 0, only depending on m, n and the cross sections, such that, for any f , g ∈ D, (73)

(1 + |v|) -1/2 (nM ) -1/2 Q (nM ) 1/2 f , (nM ) 1/2 g I ≤ C β (1 + |v|) 1/2 f I (1 + |v|) 1/2 g I .
Proof. Denote by A the left-hand side of (73). We can write, thanks to the Cauchy-Schwarz inequality and the hard-sphere assumption (4) on the cross sections,

A 2 = i R 3 (1 + |v|) -1 (n i M i ) -1 Q i (nM ) 1/2 f , (nM ) 1/2 g 2 dv ≤ I i,j n j R 3 (1 + |v|) -1 R 3 ×S 2 M j (v * ) 1/2 f i (v )g j (v * ) -f i (v)g j (v * ) B ij dω dv * 2 dv ≤ 2I i,j n j R 3 β ij 1 + |v| R 3 ×S 2 M j (v * ) 1/2 f i (v )g j (v * )|(v -v * ) • ω| dω dv * 2 + R 3 ×S 2 M j (v * ) 1/2 f i (v)g j (v * )|(v -v * ) • ω| dω dv * 2 dv. Denote β = max β ij > 0. Noticing that |(v -v * ) • ω| ≤ (|v| + |v * |),
and that v * → M j (v * ) 1/2 |v * | is a bounded function on R 3 , there exists a constant C n,β > 0, only depending on n and β, such that A is upper-bounded by The previous inequality allows to upper-bound the first addend in the same way as the second one thanks to the Fubini theorem, which concludes the proof.

C n,β i,j R 3 ×R 3 ×S 2 f i (v ) 2 g j (v * ) 2 (1 + |v|) dω dv * dv + R 3 ×R 3 f i (v) 2 g j (v * ) 2 (1 + |v|) dv * dv.

Appendix B. Conservation laws

In this section, we provide the proof of Proposition 4. We take the projected Boltzmann equation ( 22) satisfied by f 0 and scalarily multiply it by each χ i (nM ) 1/2 , 1 ≤ i ≤ I + 4.

We first choose 1 ≤ i ≤ I to obtain the conservation law for ρ i , that is

∂ t f 0 , (nM ) 1/2 χ i I + P 0 (v 1 ∂ x f 0 ), (nM ) 1/2 χ i I + P 0 (v 1 ∂ x f 1 ), (nM ) 1/2 χ i I = 0.
The first term obviously gives ∂ t ρ i . The second term can of course be rewritten as

P 0 (v 1 ∂ x f 0 ), (nM ) 1/2 χ i I = v 1 ∂ x f 0 , (nM ) 1/2 χ i I ,
which we can explicitly compute, helped by parity arguments,

v 1 ∂ x f 0 , (nM ) 1/2 χ i I = ∂ x R 3 v 1 f 0 i (n i M i ) 1/2 dv = n i ∂ x q 1 n j m j R 3 m i v 2 1 M i dv = n i ∂ x q 1 n j m j .
The third term being unchanged, we recover [START_REF] Jin | A micro-macro decomposition-based asymptotic-preserving scheme for the multispecies Boltzmann equation[END_REF].

To obtain the conservation law on q 1 , we can write

∂ t q 1 + P 0 (v 1 ∂ x f 0 ), (nM ) 1/2 χ I+1 I + P 0 (v 1 ∂ x f 1 ), (nM ) 1/2 χ I+1 I = 0.
The second term also simplifies thanks to parity arguments

v 1 ∂ x f 0 , (nM ) 1/2 χ I+1 I = i ∂ x R 3 v 1 f 0 i (n i M i ) 1/2 χ I+1 i dv = i m i n j m j ∂ x R 3 ρ i v 1 2 √ n i M i + e 6 n j (m i |v| 2 -3)v 1 2 n i M i dv = 1 n j m j i √ n i ∂ x ρ i + 6 n j 3 ∂ x e .
This finishes the derivation of the conservation law (26) for q 1 . The one for q 2 and q 3 (27), is then straightforward.

Finally, the conservation law ( 28) for e reads ∂ t e + P 0 (v 1 ∂ x f 0 ), (nM ) 1/2 χ I+4 I + P 0 (v 1 ∂ x f 1 ), (nM ) 1/2 χ I+4 I = 0.

Let us compute the second term of the above equation, which gives

v 1 ∂ x f 0 , (nM ) 1/2 χ I+4 I = I i=1 R 3 v 1 ∂ x f 0 i (n i M i ) 1/2 χ I+4 i dv = 1 6 n j i n i ∂ x R 3 m i |v| 2 -3 q 1 n j m j m i v 1 v 1 M i dv = i 2n i 6 n j ∂ x q 1 n j m j = 1 3 6 n j n j m j ∂ x q 1 .
Summarizing, the conservation law [START_REF] Liu | Energy method for Boltzmann equation[END_REF] for e is obtained.

Appendix C. Proof of Lemma 7 in three dimensions

In what follows, the one-dimensional notations are straightforwardly extended in three dimensions. The only significant difference with respect to the one-dimensional case is the treatment of ∇ x q, which requires the use of the Korn inequality to conclude. Let us rewrite the main equalities and estimates in the three-dimensional setting. We first note that the conservation laws on the fluid quantities in f 0 still hold, i.e. ∂ t ρ i + n i n j m j ∇ x • q + P 0 (v • ∇ x f 1 ), χ i (nM ) 1/2 I = 0, 1 ≤ i ≤ I, ∂ t q k + ∇ x + P 0 (v • ∇ x f 1 ), χ I+1 (nM ) 1/2 I = 0, k = 1, 2, 3,

∂ t e + 1 3
6 n j n j m j ∇ x • q + P 0 (v • ∇ x f 1 ), χ I+4 (nM ) 1/2 I = 0,

∂ t + n j 3/2 + 2 3 n j n j m j ∇ x • q + 1 n j m j P 0 (v • ∇ x f 1 ), i √ n i χ i + 1 3 6 n j χ I+4 (nM ) 1/2 I = 0,
where is still defined by [START_REF] Liu | Nonlinear stability of rarefaction waves for the Boltzmann equation[END_REF]. Of course, we follow the strategy of Section 4.6. We first compute

v • ∇ x f 0 2 I = i 1 m i ∇ x ρ i + 2 √ n i 6 n j ∇ x e 2 + 5 3
1 n j i n i m i |∇ x e| 2 + n j n j m j   (∇ x • q) 2 + l,m (∂ x l q m ) 2 + ∂ x l q m ∂ xm q l   .
In the same way, as in (52), we can write

P 0 (v • ∇ x f 0 ) 2 I = |∇ x | 2 + 5 3 n j n j m j (∇ x • q) 2 .
Then, for some θ ∈ (0, 1), (54) becomes

P 1 (v • ∇ x f 0 ) 2 I ≥ θ i 1 m i ∇ x ρ i + 2 √ n i 6 n j ∇ x e 2 + 5 3
1 n j n j m j |∇ x e| 2 + n j n j m j   - 2 3 (∇ x • q) 2 + l,m (∂ x l q m ) 2 + ∂ x l q m ∂ xm q l   -θ|∇ x | 2 .
Besides, using the three-dimensional laws on q k and , (56) can be rewritten into

θ T 0 R 3 |∇ x | 2 dx dt + 2θ R 3 q • ∇ x dx T 0 ≤ 2θ n 3/2 j + 2 3 n j n j m j + 1 2 T 0 R 3 (∇ x • q) 2 dx dt + θC χ T 0 R 3 ∇ x f 1 2 I dx dt.
Thanks to the Korn inequality, for θ small enough, there exists C θ > 0, only depending on θ, such that

- 2 3 + θ R 3 (∇ x • q) 2 dx + l,m R 3 (∂ x l q m ) 2 + ∂ x l q m ∂ xm q l dx ≥ C θ R 3 |∇ x q| 2 dx.
Eventually, in the same way as in Subsection 4.6, the norm equivalence argument on P 0 (36) allows to obtain

T 0 R 3 P 1 (v • ∇ x f 0 ) 2 I dx dt ≥ γ θ T 0 R 3 ∇ x f 0 2 I dx dt -θC 2 T 0 R 3 ∇ x f 1 2 I dx dt + 2θ R 3 q • ∇ x dx T 0 ,
for some constants γ θ > 0 depending on n, m and θ, and C 2 > 0 depending on n and m.

2 I g 2 I

 22 Let us focus on the first addend, since the second one clearly equals C n,β (1 + |v|) 1/2 f thanks to the Fubini theorem. We perform the change of variables (v, v * , ω) → (v , v * , ω) in the integral, which becomesR 3 ×R 3 ×S 2 f i (v) 2 g j (v * ) 2 (1 + |v |) dω dv * dv.The collision rules (3) then ensure that, for some constant C m > 0 only depending on m (uniform with respect to the indices i and j),|v | ≤ |v| + 2m j m i + m j |v * | ≤ C m (|v | + |v * |), ∀v, v * ∈ R 3 ,so that, up to a value change of C m > 0, still only depending on m, 1 + |v | ≤ C m (2 + |v| + |v * |), ∀v, v * ∈ R 3 .

  1/2 ∂ p-p f For 3 ≤ |p| ≤ 5, if the sub-index p of p satisfies |p | ≤ 2, the smallness assumption (13) can be used for the term (1 + |v|) 1/2 ∂ p f I , and if 3 ≤ |p | ≤ 5, then |p -p | ≤ 2, and the term (1 + |v|) 1/2 ∂ p-p f I can be handled through

I

.

This work was supported by the European COST Action CA18232 Mat-Dyn-Net, the joint Austrian-French-Serbian Danube PHC project No. 45286PC, and by the French CNRS PICS08057 and IEA00375 projects. S. S. and M. P.-Č. acknowledge support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-9/2021-14/200125). M. P.-Č.'s work is supported by the Alexander-von-Humboldt Foundation.


Marc-Antoine A Fardin

Olivier Cardoso

Benoit Lasne

Guillaume Grégoire

Jean-Paul P Decruppe

Médéric Argentina

Sandra Lerouge
email: sandra.lerouge@univ-paris-diderot.fr

Taylor-like vortices in Shear Banding Flow of Giant Micelles

Keywords: numbers: 47.50.-d, 47.20.-k, 83.60.-a, 05.45.-a, 83.85.-Ei

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

In contrast to Newtonian fluids, complex fluids are structured at the mesoscopic scale, typically from 10 nm to 100 µm, corresponding, for example, to the mesh size in entangled polymers solutions or to the bubble size in foams. Such a length scale provides internal degrees of freedom with characteristic times in the order of 10 -3 to 100 s (position, orientation, composition, ...) that can easily be excited by the flow, often resulting in a new organization of the structure of the fluid and of the flow itself. The relation between stress and shear rate σ = f ( γ) becomes strongly nonlinear and the flow evolves towards a spatially heterogeneous shear-banded state where regions of differing viscosities coexist. The shear-banding phenomenon is ubiquitous in complex fluids, from surfactants solutions [START_REF] Berret | Molecular gels[END_REF][START_REF] Salmon | [END_REF], to telechelic polymers [3,4], emulsions [5,6], granular materials [7] and foams [8]. Among these systems, surfactant giant micelles have generated an abundant literature [START_REF] Berret | Molecular gels[END_REF] and are often considered as model systems for the study of shear-banded flows. The σ = f ( γ) curve is made of two stable shear-thinning branches, characteristic of two distinct states of the system, separated by a stress plateau (σ = σ p ) that extends from γl to γh . In the plateau regime, the flow is unstable and evolves towards a banded state where the viscous and fluid phases coexist at constant stress σ p [9]. This picture is supported, among others, by investigations focused on the determination of the flow field [10,11,12,13], revealing a discontinuity of the velocity profile in the flow plane. These 1D measurements also allowed to establish the existence of large spatio-temporal fluctuations of the local flow field [14,15], suggesting the presence of three-dimensional flow [15]. Recently, using the scattering properties of the induced band, we showed that the interface between bands is not flat as usually assumed but undulates along the vorticity direction [16,17]. Such a scenario has been qualitatively reproduced by a numerical perturbation analysis of the modified Johnson-Segalman equation [18]. In addition, the model predicts the formation of velocity rolls stacked along the vorticity direction. In this letter, we report the first direct observation of Taylor-like vortices in the shear-banding flow of a semidilute micellar solution. We follow simultaneously the interface and flow dynamics and show that, as the interface between bands becomes unstable, a secondary axisymmetric vortex flow develops. The Taylor-like vortices are stacked along the vorticity axis and are mainly located in the high shear band, their size depending on the applied shear rate. The radial velocities are found to be two orders of magnitude lower than the azimuthal base flow. Using flow visualizations, we show that the interface and flow dynamics are strongly correlated : as the control parameter increases, we observe a transition between a steady vortex flow towards a state dominated by continuous creations and annihilations of cellular structures.

In our experiments, the working fluid is a 11% wt. semidilute aqueous mixture of surfactant (cetyltrimethylammonium bromide) and salt (sodium nitrate) at 28˚C. It is contained in a transparent Perspex cylindrical Couette device (gap e=1.13 mm, inner radius R i =13.33 mm, height h=40 mm) designed for imaging of the flow gradient-vorticity plane and side views of the cell (Fig. 1.a). The device is attached to the shaft of a commercial rheometer (Physica MCR500), with the inner cylinder rotating. The experiments are performed with the shear rate as the control parameter and the response of the system is recorded from two distinct channels, in a simultaneous way. The first one is dedicated to flow vizualizations in the flow-vorticity (u θ , u z ) plane using seeding anisotropic reflective particles (anisotropic mica platelets from Merck at a volume fraction of 6.10 -5 ), as it was done for the study of the dynamic regimes of inertial Taylor-Couette flows [19]. The fluid is illuminated by ambient light and the intensity I o (z) reflected in the velocity gradient direction is collected on a digital camera (CCD1). The second channel is dedicated to the observation of a cross-section ((u r , u z ) plane), using either a He-Ne laser sheet (λ=633 nm, thickness 1.5 mm) propagating along the velocity gradient axis or white light. A digital camera (CCD2) records the scattered intensity at 90 • . This channel provides information on the interface dynamics and allows the computation of the flow field using particle image velocimetry (PIV). Figure 1.b shows the steady-state shear stress as a function of the shear rate for the sample sedded with mica flakes. This flow curve, measured in strain-controlled mode, exhibits a stress plateau with a significant positive slope at σ p = 150 ± 1 Pa, extending from γl = 7 ± 0.5s -1 to γh = 95 ± 5s -1 . Along the stress plateau, the system is organized into two bands separated by an interface that undulates along the vorticity direction with a well-defined wavelength [16,17]. The proportion of the induced band and the wavelength of the undulations are increasing functions of the applied shear rate (inset in Fig. 1.b). The flow structure associated with the interfacial undulations is illustrated in Fig. 2 for γ=40 s -1 . On the snapshot of a cross-section taken when steady-state is achieved, we overlay the corresponding instantaneous flow field computed using PIV (see Fig. 2.a). If, at first sight, no particular pattern can be extracted, axial and radial velocity components are detected, suggesting that the flow is not purely orthoradial. The existence of a secondary flow is confirmed by side views of the Couette cell under ambient illumination that reveals a succession of axisymmetric dark and bright stripes stacked along the vorticity axis (Fig. 2.b). Such a picture is reminiscent of the primary pattern of the classical Taylor instability [19] and points out that the base flow is replaced by a vortex flow. In fact, the variations of the reflected intensity results from differences in the orientation of the seeding flow-aligned platelets : dark stripes are associated with regions where the flakes do not reflect the light along the observation direction, indicating radial flow, while bright stripes correspond to flow perpendicular to the observation direction [20]. In other words, the vortices axis are centred on bright stripes, the radial flow coming from the adjacent dark stripes. The comparison between the reflected light distribution I o (z) and the interface profile noted I(z) (Fig. 2.d andc) shows that the flow is oriented radially at the crests level of the interface profile. This means that, the interfacial wavelength λ is twice the wavelength λ Io characterizing the axial structure of the flow. To differentiate the inflow and outflow and quantify the corresponding velocities, we use the flow field infered from PIV. As mentioned above, due to experimental limitations, it is not possible to distinguish cellular structures from our measured flow field. Nonetheless, to test its validity, we compute the following quantity that provides a measurement of the mean local orientation of velocity vectors with respect to the vorticity direction :

I p (z) = i |v z (z, r i )| / v 2 z (z, r i ) + v 2 r (z, r i ) .
Taking into account the relation between the intensity profile I o (z) and the radial velocity component [21], we expect the mean orientation profile I p (z) to be similar to side views of the Couette cell, indicating that, even if our computed instantaneous flow field is noisy, some relevant information can be extracted from the latter. Fig. 2.f displays the radial component v r (z) at a given position close to the interface between bands. First, we find that the outflow and inflow are aligned respectively with the outer and inner crests of the interface profile. Second, radial velocities are in the order of 0.1 mm.s -1 , namely at least two orders of magnitude lower than the mean base azimuthal velocity. The weakness of the secondary flow makes the determination of the tracers displacement difficult and explains the absence of pattern in our mapping of the local velocities in the cross-section.

I o (z).
From those experiments we can conclude that vortices develop in the shear-banding flow of giant micelles. However, they do not allow the determination of the radial structure of the vortex flow. To address this question, we take advantage of the scattering contrast between both bands, due to concentration effects. Using white light illumination of the sample, free of seeding particles, we are able to directly image the cellular structures (Fig. 2.g). The latter are detectable because of local index variations probably due to the concentration difference between the two phases. The counter-rotating vortices form concomitantly with the interface undulations and are stacked along the vorticity direction. They grow from the inflow regions and finally span the high shear rate band with a reduced extension in the low shear region when the steady-state is achieved (see Movie in Suplemental). To investigate the role of the control parameter on the flow dynamics, we record the interface profile I(z) and the reflected light distribution I o (z) as a function of time and construct in gray scale the corresponding spatiotemporal diagrams I(z, t) I o (z, t) . Note that the I o (z, t) diagrams (Fig. 3.b and d) are fully reproduced by plotting as a function of time, an intensity profile along a vertical line in the flow cross section going through cellular structures in Figure 2.e. Fig. 3.a and b show the patterns observed for γ=35 s -1 . After a transient of duration τ corresponding to the formation of a flat interface between bands, we observe the destabilization of the interface together with the development of axisymmetric cellular structures. Our experimental resolution do not allow to determine whether the Taylor-like velocity rolls are triggered by the interfacial instability or vice-versa. The growth of the interface instability and the appearance of radial and axial components in the flow field are followed by saturation in a steady-state characterized by well-defined wavelength and amplitude of the interface, and a steady vortex structure. The patterns are only slightly modulated in time with a period T . This modulation corresponds to a small variation of amplitude in the inner crests (dark regions in Figs. 3.a andc)) of the interface profile [17] and produces a rope-like structure in the spatiotemporal sequence of the reflected light I o (z). It could result from periodic oscillations of the vortices boundaries. These extremely regular patterns are representative of the system dynamics on a large part of the stress plateau, for γ typically between 15±1 s -1 and 60±5 s -1 . In this range of γ, we find that the asymptotic wavelength of the patterns increases with the shear rate (Fig. 4.a) while the period of the modulation keeps a constant value (Fig. 4.b). For comparison, the wavelength of the spatial structure I p (z) obtained from PIV measurements has been added in Fig. 4.a. For all shear rates, the height of a vortex pair and the interfacial wavelength are in good agreement. The circulation cells essentially span the high shear rate band in the radial direction, but their extension along the vorticity direction can reach about two gap widths as the control parameter is increased. For higher values of the shear rate ( γ > 65 s -1 ), the system exhibits a more complex dynamics as illustrated in Fig. 3.c and d. Once again, the dynamics of the tracers is remarkably correlated with the one of the interface. After the transient, the interface amplitude saturates but the wavelength continuously evolves with time due to nucleation and merging of inner crests. Since a pair of vortices develops between two consecutive inner crests, such a dynamics corresponds to a succession of creations and annihilations of pairs of vortices. These processes do not tend toward a stationary situation, rather leading to a chaotic dynamics. Note that at low shear rates (below 15 s -1 ), the reduced size of the induced band prevents the of the dynamics. The present results demonstrate that the shear-banding flow our semi-dilute giant micelles solution is threedimensional, far away from the classical picture usually used in this field [10,11]. Very recently, the formation of Taylor-like velocity rolls has been predicted by a nonlinear numerical study of the modified Johnson-Segalman model [18] : the 3D flow is triggered by the instability of the 1D gradient banded state with respect to interfacial undulations along the vorticity direction, with normal stress and shear rate jumps accross the interface as the underlying mechanism [22]. Some of our observations are qualitatively captured by this model. However, it does not reproduce the complex flow dynamics we report. Alternatively, another possible origin for the Taylor-like vortices is the elastic instability originally observed in dilute polymer solutions [23,24] and more recently in rod-like systems [25,26]. In working conditions where the streamlines are curved and the fluid is elastic, the base orthoradial flow can become unstable due to a centripetal body force called 'hoop stress' [23]. Such an hypothesis has been proposed to explain fast temporal fluctuations of the flow field that lead to suspicions for the presence of 3D shear-banded flow in a concentrated giant micelles system [15]. Following the analysis in Ref. [15], we compute a modified Weissenberg number Σ = (e/R i ) 1/2 W i as a function of the applied shear rate γ . Here W i is an approximate Weissenberg number obtained from the stress ratio N 1 /σ [START_REF] Larson | The structure and rheology of complex fluids[END_REF], and (e/R i ) the gap ratio. The critical value Σ c above which an elastic instability appears is around 6 [23,[START_REF] Pakdel | [END_REF]. This stability criterion has been established in the framework of the Oldroyd-B model. If we assume that each band can be modelled by an Oldroyd-B equation [29], we can compare the modified Weissenberg number Σ in each band with the critical value Σ c . Note that, if the Oldroyd-B model does not describe shear-thinning properties, the shearthinning character is supposed to reduce the instability threshold [30]. In the linear regime, Σ = 0 and it is found to increase with γ in the nonlinear regime, namely above 1 s -1 (data not shown). Along the first branch, Σ never exceeds 0.5 while for γ ≥ γh , Σ > Σ c , suggesting that the induced band is likely to undergo an elastic instability, leading to formation of cellular structures. In summary, our results provide the first direct evidence for the existence of Taylor-like velocity rolls in the shearbanding flow of wormlike micelles. Moreover, we showed that the dynamics of the Taylor-like vortices is strongly correlated with the interface dynamics, leading to fundamental questions about the physical mechanism driving the three dimensional flow. Between the interface instability and the formation of Taylor-like vortices, one challenging task is now to determine which one of these phenomena drives the other, and under which physical mechanism. More generally, our results could potentially explain some fluctuating behaviors observed recently in giant micelles systems and open new perspectives towards the understanding of shear-induced instabilities and heterogeneous flows in complex fluids.

FIG. 1 :

 1 FIG. 1: (a) Experimental setup for observations in the (u θ , uz) and (ur, uz) planes. (b) Steady-state flow curve and corresponding banding structure in the (ur, uz) plane for different shear rates along the stress plateau. The left and right sides of each picture corresponds respectively to the inner and outer cylinders.

Fig. 2 .

 2 (d-e) show that the axial structure I p (z) is consistent with the intensity profile I o (z) gathered from

FIG. 2 :

 2 FIG. 2: (a) Stationary banding structure at γ=40 s -1 and corresponding velocity field computed from the PIV algorithm in the (r, z) plane. (b) Side view of the Couette cell under ambient illumination. (c) Interface profile I(z) extracted from (a). (d) Intensity distribution Io(z) along the vertical axis inferred from (b). (e) Mean orientation Ip(z) of velocity vectors with respect to the vertical axis. (f) Radial velocity profile vr at a position r = Ri + 0.5e in the gap highlighting the inflow and outflow. (g) Direct observation in the plane (r, z) of vortices stacked along the vorticity direction. In this last case, the sample is free of tracers and illuminated with white light.

FIG. 3 :

 3 FIG. 3: Evolution in gray scale of the interface profile I(z) and the reflected intensity distribution Io(z) as a function of time during step shear rate from rest to (a-b) γ=35 s -1 and (c-d) γ=75 s -1 . The black and white dots observable in Figs. 3.c and d correspond to a bubble, trapped in a recirculation zone and that periodically crosses the observation fields.

FIG. 4 :

 4 FIG. 4: (a) Comparison between the asymptotic interfacial wavelength λ ( ) and the height of a vortex pair inferred respectively from Io(z) ( ) and Ip(z) (•). (b) Period T of the temporal modulation of the interface profile (•) and of the rope-like structure of the reflected intensity ( ). The error bars represent the standard deviation obtained over several experiments. The time and length scales are computed by direct Fourier transform.

Acknowledgements

The authors thanks A. and S. Asnacios, N. Biais, J.L. Counord, O. Dauchot and B. Ladoux for fruitful discussions, and the ANR JCJC-0020 for financial support.